Tomographic Imaging Using Non-Linear Projections
with Applications in MRI
by
Rainuka Gupta
B.S. Mechanical Engineering
Massachusetts Institute of Technology, 1999
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2001
© 2001 Rainuka Gupta. All Rights Reserved.
The
m
Author ....
*w
he
rue
ew
ct
ekcuonc copEs 01 tr t8
dr*umbnt I w~phIlor n port.
BARKER
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 16 2001
LIBRARIES
Department of Mechanical Engineering
January 19, 2001
C ertified by ...
A ccepted by .........
...............................
Derek Rowell
Professor of Mechanical Engineering
Thesis Supervisor
.............................................
Ain Sonin
Professor of Mechanical Engineering
Chairman, Graduate Thesis Committee
Tomographic Imaging Using Non-Linear Projections with
Applications in MRI
by
Rainuka Gupta
Submitted to the Department of Mechanical Engineering
on January 19, 2001, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
Magnetic resonance imaging traditionally relies on spatially encoding an object using localized magnetic gradients. In this project, an alternative method for imaging in which
radially varying gradients in the applied radiofrequency field define circular projections
of proton density was developed. The physics of RF gradient imaging was explored, and
an algorithm for image reconstruction using convolution backprojection using nonlinear
projections was developed. The optimal filter for reconstruction using nonlinear projections was found to be the same as that for linear projections. A closed form solution for
the point spread function of an object in the neighborhood of the origin was developed.
A numerical solution for the PSF of an object arbitrarily located in the object space was
created. Numerical simulations were used to verify the integrity of the reconstruction
algorithm. It was found that the model that was created for the reconstruction of objects
in the neighborhood of the origin was valid across the image space. An NMR probe was
created to experimentally determine the feasibility of imaging using RF gradients. It was
found that a 1r relationship existed between the measured nutation frequency and the
distance of the sample from the RF coil. This relationship was used to correlate projections obtained by rotating a sample around an axis offset from the RF coil. A sample
containing two tubes of water was imaged by collecting a set of twenty-four projections,
filtering the projections, and backprojecting the data. It was found that effects from the
distal wire of the coil significantly affected the spatial encoding. Mapping of the magnetic
field accounting for both conductors increased the fidelity of the image.
Thesis Supervisor: Derek Rowell
Title: Professor of Mechanical Engineering
2
Acknowledgments
There are a number of people whom I would like to thank for their assistance during the
course of this project. I would like to thank my advisor, Dr. Derek Rowell, for giving
me the opportunity to work on this project and for all the assistance and guidance along
the way. It was a pleasure to work with him. I would also like to thank Dr. David Cory
at the magnet lab for collaborating with us and for sharing his NMR expertise. Thanks
also to Fred Cote at the Edgerton Center student shop for his help in the fabrication of
the hardware for the project. Thanks to every one at the Newman Laboratory for their
assistance, particularly Lori Humphrey. Also, everyone at the Francis Bitter Magnet
Laboratory who helped out-Ross Mair, Dan Caputo, Mike Kutney, Rich Gostic, and
Evan Fortunato were repeat victims who patiently answered my many questions. An
extra special thank you to Kristin Jugenheimer and Byron Stanley who both helped out
in so many ways and whose support was greatly appreciated. Finally, thank you to my
friends and family for their love and support.
3
Contents
1
2
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1
M otivation.
1.2
Proposed Imaging Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3
Objective
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Background
15
2.1
NMR Theory ......
2.2
Current Applications of RF Gradients in NMR . . . . . . . . . . . . . . . . 20
2.3
3
12
...........................
. . . . . . . 15
2.2.1
Development of Rotating Frame Imaging . . . . . . . . . . . . . . . 20
2.2.2
Use of RF Gradients to Determine T Values . . . . . . . . . . . . . 21
2.2.3
Applications in Magic-Angle Spinning . . . . . . . . . . ..
2.2.4
Use of RF Gradients in p- and n- COSY Experiments . . . . . . . . 22
2.2.5
Quantification of Flow Velocities
2.2.6
Reduction of Acquisition Times Using Symmetrical RF Fields . . . 23
2.2.7
Three Dimensional Spatial Encoding via RF Gradients . . . . . . . 23
21
. . . . . . . . . . . . . . . . . . . 22
Proposed Image Reconstruction Using Nutation . . . . . . . . . . . . . . . 23
2.3.1
One Dimensional Imaging . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2
Two Dimensional Imaging . . . . . . . . . . . . . . . . . . . . . . . 25
Image Reconstruction from Circular Projections
3.1
...
Theoretical Analysis
28
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4
3.2
4
3.1.1
Back Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2
Convolution Back Projection . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3
Optimal Reconstruction Filter . . . . . . . . . . . . . . . . . . . . . 33
3.1.4
Filter Truncation Effects . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.5
Reconstruction of a Point Object at the Origin . . . . . . . . . . . . 37
Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1
Calculation of the Point Spread Function . . . . . . . . . . . . . . . 42
3.2.2
Reconstruction of Photographic Images . . . . . . . . . . . . . . . . 47
Experimental Work: RF Gradient Imaging
49
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2
Coil Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3
Hardware Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4
4.3.1
Base Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.2
Variable Distance Sample Position
4.3.3
Multiple Sample Holder
4.3.4
Rotating Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . 55
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Experimental Results and Analysis
68
5.1
Calibration of Frequency to Space . . . . . . . . . . . . . . . . . . . . . . . 68
5.2
Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3
R otation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.1
DANTE Pulse Program
. . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.2
Single Sample Rotation Experiment . . . . . . . . . . . . . . . . . . 75
5.3.3
Single Sample Experiment with Uniform Nutation Pulse Width
5.3.4
Two Sample Rotation Experiment . . . . . . . . . . . . . . . . . . . 99
6 Discussion
6.1
Coil Design
. . 84
106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5
7
6.2
M agnetic Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107
6.3
Spatial M arkers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108
6.4
Number of Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109
Conclusion
111
A Reconstruction Using the Hankel Transform for an Arbitrary Object 113
B MATLAB Scripts
118
B.1 Numerical Simulations ..............
. . . . . . . . . . . 118
B.1.1
Creating Circular Projections .......
. . . . . . . . . . . 118
B.1.2
Convolving the Filter Kernel . . . . . . .
. . . . . . . . . . . 120
B.1.3
Back Projecting an Image . . . . . . . .
. . . . . . . . . . . 123
B.1.4
Backprojection Over an Enlarged Image Space
. . . . . . . . . . . 125
B.1.5
Collecting Projections from an Image . .
. . . . . . . . . . . 127
B.2 Image Reconstruction . . . . . . . . . . . . . . .
. . . . . . . . . . . 129
B.2.1
Data Extraction . . . . . . . . . . . . . .
. . . . . . . . . . . 129
B.2.2
Data Processing . . . . . . . . . . . . . .
. . . . . . . . . . . 130
B.2.3
Single Sample Rotation Data . . . . . .
. . . . . . . . . . . 131
B.2.4
Single Sample with Uniform Pulse Data
. . . . . . . . . . . 134
B.2.5
Mapping the Magnetic Field . . . . . . .
. . . . . . . . . . . 137
B.2.6
Backprojection . . . . . . . . . . . . . .
. . . . . . . . . . . 138
B.2.7
Filtering of the Backprojection
. . . . .
. . . . . . . . . . . 140
B.2.8
Two Sample Data . . . . . . . . . . . . .
. . . . . . . . . . . 141
B.2.9
Windowing the Enlarged Image Space
. . . . . . . . . . . . . . . . 145
C Engineering Drawings
146
D Pulse Programs
154
D .1
zg Pulse Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
154
D.2 Nutation Pulse Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D.3 DANTE/Nutation Pulse Program . . . . . . . . . . . . . . . . . . . . . . . 156
E Experimental Data
158
7
List of Figures
1-1
Geometry of the imaging scheme. . . . . . . . . . . . . . . . . . . . . . . . 13
2-1
Displacement of the axis of M under the influence of B. . . . . . . . . . . . 16
2-2
Precession of the magnetization about B, and B1 . . . . . . . . . . . . . . 17
2-3
Frequency spectrum of a sample with and without an applied RF gradient
2-4
Schematic for one dimensional imaging. . . . . . . . . . . . . . . . . . . . . 25
2-5
Geometry of the magnetic fields and the object in two dimensions. . . . . . 26
3-1
Schematic of object space and spin density. . . . . . . . . . . . . . . . . . . 29
3-2
Result of backprojection
3-3
Effect of convolving the PSF with a kernel . . . . . . . . . . . . . . . . . . 32
3-4
Geometry for calculating the point spread function. . . . . . . . . . . . . . 34
3-5
Optimal filter for image reconstruction. . . . . . . . . . . . . . . . . . . . . 36
3-6
Cross section of the PSF reconstructed using a truncated filter . . . . . . . 37
3-7
Geometry used for point spread function simulation . . . . . . . . . . . . . 43
3-8
Effect of filtering the PSF . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3-9
Point spread function over image space . . . . . . . . . . . . . . . . . . . . 45
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3-10 Point spread function with enlarged image space . . . . . . . . . . . . . . . 46
3-11 Comparison of original and reconstructed photographic images . . . . . . . 48
4-1
Geom etry of coil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4-2
Inductance of a single turn of wire . . . . . . . . . . . . . . . . . . . . . . . 51
4-3
Drawing of disk for mounting probe.
8
. . . . . . . . . . . . . . . . . . . . . 53
4-4
Picture of the mounting frame for the RF coil. . . . . . . . . . . . . . . . . 54
4-5
Mounting fixture of variable positioning fixture
4-6
Drawing of sandwich pieces of variable positioning mechanism. . . . . . . . 56
4-7
Picture of assembled positioning mechanism. . . . . . . . . . . . . . . . . . 57
4-8
Drawing of fixture for imaging multiple samples simultaneously. . . . . . . 57
4-9
Picture of coil with multiple samples. . . . . . . . . . . . . . . . . . . . . . 58
. . . . . . . . . . . . . . . 56
4-10 Drawing of disk in rotary fixture. . . . . . . . . . . . . . . . . . . . . . . . 59
4-11 Picture of fixture for sample rotation. . . . . . . . . . . . . . . . . . . . . . 60
4-12 FID data from the zg sequence. . . . . . . . . . . . . . . . . . . . . . . . . 62
4-13 Fourier transform of FID data from the zg sequence.
. . . . . . . . . . . . 63
4-14 Schematic of nutation pulse program. . . . . . . . . . . . . . . . . . . . . . 63
4-15 FID data from the nutation sequence . . . . . . . . . . . . . . . . . . . . . 66
4-16 Fourier transform of "FID" data from the nutation sequence . . . . . . . . 67
4-17 Fourier transform of "FID" data from a dual sample. . . . . . . . . . . . . 67
5-1
Plot of nutation frequency as a function of distance to the coil. . . . . . . . 69
5-2
Fourier transform of two water samples located close to the coil. . . . . . . 71
5-3
Fourier transform of two water samples located far from the coil. . . . . . . 72
5-4
Photograph of setup of modified rotation fixture.
5-5
DANTE sequence in time and space . . . . . . . . . . . . . . . . . . . . . . 75
5-6
Geometry of rotation experiment. . . . . . . . . . . . . . . . . . . . . . . . 76
5-7
Plot of nutation frequency as the sample is rotated through a circle. . . . . 77
5-8
Image of intensity of the Fourier spectrum of each of the projections.
5-9
Spatial mapping of the intensity of the nutation frequency for each projection. 81
. . . . . . . . . . . . . . 74
. . . 79
5-10 Reconstructed image of a column of water. . . . . . . . . . . . . . . . . . . 83
5-11 Comparison of predicted and experimental values of the nutation frequency. 86
5-12 Mapping of the intensity of the nutation frequency in the frequency domain. 87
5-13 Mapping of the intensity of the nutation frequency component in the spatial
dom ain.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9
5-14 Reconstruction of object in the image space. . . . . . . . . . . . . . . . . . 90
5-15 Three dimensional view of the image reconstruction. . . . . . . . . . . . . . 91
5-16 Contour of the image reconstruction.
. . . . . . . . . . . . . . . . . . . . . 92
5-17 Simulation of reconstruction of an object using sixteen projections.
. . . . 93
5-18 Three dimensional view of simulated reconstruction. . . . . . . . . . . . . . 94
5-19 Simulation of magnetic field produced by two wires. . . . . . . . . . . . . . 95
5-20 Reconstruction of sample using the contribution of the proximal and the
distal w ires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5-21 Reconstruction of image using an expanded reconstruction space.
. . . . . 97
5-22 Three dimensional view of the filtered backprojection. . . . . . . . . . . . . 98
5-23 Schematic of geometry of two sample rotary experiment.
. . . . . . . . . . 99
5-24 Frequency spectrum of a typical "FID" of a sample containing two columns
of water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5-25 Predicted and experimental nutation frequencies as a function of rotation
an gle.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5-26 Comparison of predicted nutation frequencies and measured values. . . . . 103
5-27 Reconstruction of two tube sample
. . . . . . . . . . . . . . . . . . . . . . 104
A-1 Geometry used for reconstructing a point object located off the origin. . . . 114
A-2 Geometry for Graf's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 116
10
List of Tables
3.1
Definition of terms used in the theoretical development of the reconstruction algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
E.1 Raw data for frequency to space calibration
. . . . . . . . . . . . . . . . . 159
E.2 Raw data for initial single sample rotation experiment . . . . . . . . . . . . 160
E.3 Raw data for single sample rotation experiment
. . . . . . . . . . . . . . . 161
E.4 Raw data for dual sample rotation experiment . . . . . . . . . . . . . . . . 162
E.5 Raw data for dual sample rotation experiment . . . . . . . . . . . . . . . . 163
11
Chapter 1
Introduction
1.1
Motivation
One of the most prevalent forms of medical imaging is nuclear magnetic resonance (NMR)
imaging. Its non-invasive nature and its ability to produce high-quality images give it
greater diagnostic capability than conventional techniques and make it a preferred imaging
method [1] . Magnetic resonance imaging usually involves manipulating static magnetic
gradient fields to spatially encode an object in the core of the magnet. In this work, an
alternative method for imaging in which the object space is encoded using a spatially
varying radiofrequency (RF) field in a static, homogeneous background magnetic field
was developed. This may potentially be an alternative method for microscopic imaging.
1.2
Proposed Imaging Method
An object can be considered to be an array of magnetic spin densities associated with
the magnetic moments of the protons in the object. The proposed method encodes the
object using a spatially varying radiofrequency field. The RF field, which is radially
symmetric, produces data sets that are projections of the spin densities around circular
paths. Figure 1-1 shows a schematic of the imaging method. An object is placed in the
12
X
y
uniform static field B.
RF field
Bi
..............
-:
sample
RF field is stepped around
the sample
RF current
n
T| sinwt
Figure 1-1: Geometry of the imaging scheme.
spatially invariant background magnetic field B 0 oriented in the z direction. A radially
symmetric RF field B 1 is applied in the x direction. In the figure, this RF field emanates
from a long conductor; in practice, the coil on the NMR probe will act as the transducer.
The dependence of the magnetic field B 1 on the radial distance from the coil spatially
encodes the object. The magnitude of the magnetic field affects the nuclear spin of the
object and causes a measurable precessional tilt. The nutation frequency, the frequency
of precession of the magnetic moment about the B 1 field, depends on the magnitude of
B 1, which depends on the distance from the sample to the coil. It is therefore possible to
measure the nutation frequency and correlate it to a radial projection with respect to the
coil. By surrounding the object with a ring of transducers that are equidistantly spaced
from a central point in the object space and collecting projection data from each of the
transducers, it is possible to reconstruct an image of the object.
1.3
Objective
The objective of this thesis is to explore the feasibility of RF gradient imaging. The
basic physics of magnetic resonance are explained in Chapter 2. The development of RF
gradient imaging and its applications are discussed as well. A basic description of one and
13
two dimensional RF imaging concludes Chapter 2. An algorithm for image reconstruction
using circular projections was developed. The development of the algorithm and numerical
simulations validating the algorithm are contained in Chapter 3. Chapter 4 describes
the experimental work that was conducted to determine the feasibility of implementing
RF imaging. The results of the experimental work are analyzed in Chapter 5. A brief
discussion of some of the issues that arose during the experimentation is contained in
Chapter 6. Finally, conclusions are presented in Chapter 7.
14
Chapter 2
Background
2.1
NMR Theory
The classical theory of magnetism can be used to describe the basic physics of nuclear
magnetic resonance. Stable atomic nuclei possess angular momentum, I, quantized in
integral or half integral units of h =
h,
where h is Planck's constant.
The angular
momentum depends on whether the number of constituent nucleons is even or odd [2].
The angular momentum, I, of a proton causes its distributed charge to rotate about a
central axis. This results in a magnetic dipole moment, mp, such that
MP = -I
where -y is the gyromagnetic ratio [3].
(2.1)
For a sample containing many nuclei, the net
magnetic moment, M, is the combined sum of the individual magnetic moments.
When an arbitrary external magnetic field with flux density B0 is applied, the resulting
motion can be described as
dM= M x B.
dt
(2.2)
[3]. The precessional frequency of the magnetic moment M about the magnetic field B,,
is described by the Larmor frequency, wO, where
WO = -yBo
15
(2.3)
z
B0
Figure 2-1: Displacement of the axis of M under the influence of B.
[3]. At equilibrium, the net magnetic moment is aligned with the magnetic field. In order
to measure M, it is necessary to tilt it away from the B0 field.
Convention dictates that B, is aligned in the z direction and the moment is tilted
into the xy plane for measurement. When an additional circularly polarized magnetic
field B 1 sin wt is oriented in the xy plane at the Larmor frequency, a transformation can
be made to a rotating coordinate frame in which B1 is stationary [3]. As a result of the
applied B1 field, an additional torque is applied to the moment, rotating it about x',
moving it into the xy plane by an angle a (see Figure 2-1).
The angle a, known as the nutation angle, depends on the magnitude of B1 and the
duration of the applied RF excitation pulse such that
a = yB1 T,
(2.4)
where T is the length of the RF pulse [3]. During the application of a rotating magnetic
field of frequency wo, the magnetization simultaneously precesses about the longitudinal
polarizing field B, and about the RF field B1 at w, (see Figure 2-2). In the reference
frame rotating with B 1 around B0 , the moment simply precesses about B 1 .
16
M
Figure 2-2: Precession of the magnetization about B, and B 1 . Reproduced from [2].
When a = 2'7, M will lie along the y' axis where x'y' is the reference frame rotating
at the Larmor frequency around z. The pulse required to flip M through 900 is termed a
90* pulse. For the duration of the RF pulse, the magnetic moment precesses around the
x' axis. Since the tilt angle depends on both T and B, either of these parameters can be
adjusted to vary a. Also, since the gyromagnetic ratio affects a, different nuclei will be
tilted by different amounts by the same pulse.
After an RF pulse is applied, before the signal relaxes, the xy component of M is
a rotating RF field that can be detected with a suitable receiver coil. A typical signal
consists of a free induction decay (FID), which decays to zero as the signal loses phase
coherence due to changes in the local molecular magnetic field, inhomogeneities in the
B, field, and relaxation effects. If a spatially varying static magnetic field is applied, the
amplitude of the Larmor frequencies in the resulting FID will vary according to the spin
density. Thus, reconstruction of the spin density will be possible by observing the FID.
In a homogeneous field with flux density B,, the equation of motion is given by Equation 2.2. Bloch's equations can be used to describe the behavior of the components of
the magnetization [3]. Following the application of an RF pulse, the z component of the
magnetization M, can be described as
dMz
dt
MZ - MO(
(2.5)
=
17
where M is the equilibrium value of the magnetization and T is the longitudinal relaxation time [3]. If the magnetization has a component perpendicular to the z direction,
the transverse magnetization will decay due to local interactions as described by
cdtit
-
A li2
(2 .6 )
T2
and,
dM
=
dt
(2.7)
T2
where T 2 is the transverse relaxation time [3]. The longitudinal relaxation, T 1 , is the spinlattice relaxation due to energy exchanges between the spin system of the sample and
the surrounding thermal reservoir, or lattice, due to locally fluctuating magnetic fields
such as those produced by dipole-dipole interactions. The T is typically found by using
a saturation-recovery sequence and monitoring the amplitude of the free induction decay
(FID) as the length of the pulse is varied. The transverse relaxation, T2 , describes the
spin-spin relaxation. This is the loss of magnetization from the xy plane due to a loss of
phase coherence in the xy plane and due to the reduction of the signal magnitude through
T1 relaxation. The effective T2 relaxation, T2* accounts for the T2 relaxation and other
effects such as magnetic field inhomogeneities. T2* can be measured by monitoring the
envelope of the FID decay.
Traditionally, static gradients are applied to an object. The static gradients will cause
different spin densities to reach different levels of excitation. For example, in the sample
shown in Figure 2-3, a homogeneous background field is applied to a sample containing
two tubes of water. Since the magnetic field acting upon both tubes is uniform, each of the
tubes will have the same Larmor frequency. Thus, when the FID is Fourier transformed,
a single peak at a frequency of w, = -yB
0
results. If, however, a spatially varying gradient,
GZ, were applied such that B, = B0 + yGzz, then the two samples would be subjected
to different magnetic field intensities, which would result in different levels of excitation.
The Fourier transform of the sample will result in two frequencies: w, = -y(B
and W2 = y(B 0 + G-z 2 ) [3].
18
0
+ G~z)
6
A
(a)
UU
Bo z
(b)
F
zz~
Bz= BO+ I Gzzz,
v:1IBo+Gzz) v2= (B0+Gz Z2)
Z2
Figure 2-3: Frequency spectrum of a sample with and without an applied RF gradient.
Reproduced from [3].
This demonstrates how the Fourier transform of the free induction decay is essentially
a projection of the signal intensity in the plane in the direction of the applied gradient.
The Fourier transform of the projection is a one dimensional image of the object. Conventional imaging using magnetic resonance involves collecting a series of projections and
reconstructing the spin density of the original object.
The alternative approach to magnetic resonance imaging that is proposed here involves
manipulating the applied RF field. The precession of the magnetic moment about the
RF field, B 1 , is termed nutation. Nutation occurs only during the period of time that the
RF pulse is applied. The application of an RF pulse causes the moment to rotate by the
angle a (as defined by Equation 2.4). The initial amplitude of the FID, A, is proportional
to the tilt angle, so
A = Ksina
19
(2.8)
where K is a constant. Since a is a function of time, the amplitude of the FID is also a
function of time
A = K sin(yBi T).
(2.9)
It is possible to apply a series of RF pulses and measure the initial amplitude of the FID
in between each pulse. Since the initial amplitude of the FID is directly affected by the
tilt angle a, the changes in FID amplitude will correlate to changes in the tilt angle.
The rate at which the tilt angle changes as a function of time is the nutation frequency.
Since nutation only occurs for the duration of the applied RF pulse, the time in between
pulses is insignificant and does not affect the measure of the nutation frequency. This
method is used to experimentally measure the nutation frequency.
In this work, the
nutation frequency is used to spatially encode the spin density of the object space. A
set of projection data measuring the nutation frequency will subsequently be used to
reconstruct an image of the object space.
2.2
Current Applications of RF Gradients in NMR
2.2.1
Development of Rotating Frame Imaging
The use of nutation and the manipulation of the RF field to facilitate data acquisition
using NMR was first developed in 1979 [4].
Rotating frame imaging (RFI), as it was
termed, involved using a spatially varying radiofrequency field in a static, homogeneous
background magnetic field to spatially encode the object. The applied RF field would be
spatially varying in a single direction. The advantage of this approach is that the object
can be spatially encoded with good sensitivity without rapidly switching static gradient
fields. Theoretically, RF fields can be applied in many directions to produce resolution
in all three directions of space. It is also possible to vary the phases of various pulses
to achieve different methods of encoding the object space. Hoult described the effects of
field modulation and the effect of applying a second B 1 gradient to encode the space.
20
2.2.2
Use of RF Gradients to Determine T Values
The principle of rotating frame imaging was useful in many applications of NMR. Since
RFI is performed using a homogeneous B, field instead of employing gradient pulses,
chemical shift information is preserved. Also, RFI is valuable for imaging nuclei with
short T2 values since the acquisition time in RFI is very short. It was found that RFI
could be used in conjunction with the saturation recovery pulse sequence to determine the
T1 relaxation while retaining chemical shift information [5]. This is done by applying an
RF gradient B 1 in the x direction such that the magnetization precesses with a frequency
of
V1 =x.
(2.10)
27r
Thus, every point along the x direction has a distinct characteristic frequency.
The
spatial distribution can be detected by collecting a series of free induction decays. The
FIDs are collected using a pulse program that applies a series of pulses and measures
the initial amplitude of the FID between each pulse. The magnetization is saturated in
between pulses so that the starting orientation of the magnetizations is identical prior to
the application of each pulse. The intensities of the one dimensional images produced
using the RFI method are used to calculate the T, values for a number of samples. This
technique is applicable to nucleides with single-line spectra. It was thought that this
method would be superior to conventional methods for determining T for samples with
very short T2 values.
2.2.3
Applications in Magic-Angle Spinning
Magic-angle spinning can also incorporate RFI in its application [6]. Magic-angle spinning
is used to spatially map NMR parameters in solids and broadband spectrum systems. By
combining RFI with magic-angle spinning, the static magnetic field gradients, B", combine
with the RF gradients, B 1 cos wt, to create an effective field Boe where
JBoel =
BO 21
+ B1.
(2.11)
The resolution of a magic angle imaging experiment is improved by combining it with
RFI. This is due to the fact that a large RF field results in a high rotation rate which
removes the full dipolar linewidth. Usually, however, when magic angle imaging is done,
RFI is not used since the signal to noise ratio suffers immensely.
2.2.4
Use of RF Gradients in p- and n- COSY Experiments
Radial RF pulses have also been employed as an alternative to phase cycling in p- and ntype COSY experiments [7]. A new probe was developed that produced two RF fields of
different symmetries from the same transmitter. The two fields are therefore coherent but
not in phase. One of the fields can be used to excite and detect the spin magnetization
while the other is used to introduce spatially varying coherence pathway transformations.
Thus, the second field manipulates the magnetization with the same effect as phase cycling
by allowing quadrature detection and suppressing axial resonances. The benefits of the
radial pulse approach include short switching times, no distortions of lineshape, and no
eddy currents due to gradient switching.
2.2.5
Quantification of Flow Velocities
A spatially varying RF field has also been used to measure flow velocities without using
other magnetic field gradients [8]. A probe with an amplitude modulated RF field was used
to excite the sample. The RF field determines both the spatial variation of the excitation
pulse and the variation of the coupling to the receiver. The pair of resonances that appear
in the frequency spectrum as a result of a single excitation pulse may be used to recover
the velocity spectrum. Since each NMR resonance reports on the flow characteristics
from the compound which gave rise to the resonance, complex flows consisting of multiple
components can be analyzed in a single excitation.
22
2.2.6
Reduction of Acquisition Times Using Symmetrical RF
Fields
For objects having cylindrical symmetry, radial RF imaging can also be useful [9]. The
time required to image can be reduced by taking advantage of symmetry of the object.
Cylindrically symmetric gradients can be created by the RF field produced by an infinitely
long wire. When the wire is aligned with B, (along the z direction), the magnetic field in
the xy plane varies with radial distance to the coil. This potentially could be applied to
the study of Couette flows or for imaging flow at surfaces and boundary layers.
2.2.7
Three Dimensional Spatial Encoding via RF Gradients
Although RFI has primarily been used for one dimensional imaging, efforts have been
made to create two and three dimensional images. An RF coil in which the RF field
intensity was simultaneously orthogonal to the flux lines of B1 and parallel to the static
field B, would permit full three dimensional imaging with only RF gradients [10]. A cone
shaped saddle coil was designed that would permit the magnetization of the RF field
in three dimensions to vary in a known manner. This coil was used to create an image
with spatial resolution along the B,, axis. This has the potential for allowing for three
dimensional RF imaging; however, this has not yet been implemented.
2.3
Proposed Image Reconstruction Using Nutation
It is proposed that the nutation frequency can also be used for image reconstruction.
Different nuclei will see different tilt angles by the same RF pulse. Also, since the B1 field
is spatially varying, nuclei in different positions will see different RF field strengths and
will nutate at different frequencies. If the manner in which the B1 field varies is known,
then it is possible to reconstruct the original spin densities of the object based on the
measured nutation frequency.
23
If an object is considered to be an array of spin densities, as seen in Figure 1-1, then
all of the magnetic moments in the object will align themselves with B, when the object
is placed in a spatially invariant background field oriented in the z direction. When a
radially symmetric RF field B 1 is applied, the dependence of the magnetic field B1 on the
radial distance from the coil will spatially encode the object. Each radius, r, from the
coil has a distinct RF magnetic field strength; thus, all of the spin densities that lie along
a single arc of radius r will have the same excitation and the same measured nutation
frequency. The Fourier spectrum of the nutation frequency components can be used in
a similar manner to the spectrum resulting from applied static gradients. The measured
response has the shape of a free induction decay and can be manipulated as a FID. To
distinguish between a free induction decay and the measured nutation frequency response,
the measured FID amplitude will be subsequently referred to as a "FID." The amplitude
of the nutation frequency component is a line integral of the spin densities as a function
of r, the distance from the coil. When the object is surrounded by a ring of transducers
that are equidistantly spaced from a central point in the object space, projection data
from each of the transducers can be used to reconstruct an image of the object. A basic
description of one and two dimensional RF imaging is contained in the following two
sections.
2.3.1
One Dimensional Imaging
For the sample shown in Figure 2-4, a spatially varying B 1 field is applied at the Larmor
frequency for a time period T. To reconstruct the spin density distribution, it is necessary
to determine the contribution of a differential element 6r of the sample. Each differential
element, which can be represented by c(r)6r, contributes to the differential magnitude of
the "FID," 6A, such that
6A(-yT) = a(r) sin(yTB1 (r))Jr.
24
(2.12)
B6r
0
R
R01
2
R
r
B (r)
r
Figure 2-4: Schematic for one dimensional imaging.
The measured amplitude of the "FID" is the integral over all the elements so
ac(r) sin (7TB1(r)) dr.
A(-T) =
(2.13)
Since B 1 (r) is a single valued, monotonic function of r, a change of variables results in
A(yT) = 10 &(B,) sin (-yTB 1 ) dB 1
fo B'(BI)
(2.14)
where &(B1 ) is the transformed spin density distribution, and B'(B1 ) is the slope of B 1
evaluated at r(Bi). The spin density of the object can be obtained by taking the inverse
sine transform of Equation 2.14 so that
__(B_)
B'(B1 )
1
00
fA(yT)
sin (yTBI(r)) d(yT).
27r o
(2.15)
The spin density a(r) may be calculated from this inverse sine transform.
2.3.2
Two Dimensional Imaging
Each of the one dimensional images is actually a projection of the spin density along the
gradient. A series of such projections can be collected for reconstruction of an image in the
two dimensional case using convolution and backprojection. For the two dimensional case,
25
kz
Y
spin density
cr(r0)
z
X
x/
contour of constant
RF flux density
BN(r)
aW
(a)
(b)
Figure 2-5: Geometry of the magnetic fields and the object in two dimensions.
the spin densities are distributed in the xy plane. A spatially invariant background field
B0 is applied along the z direction (see Figure 2-5). In polar coordinates, the spin density
is c(r, 9); an element of the spin density can therefore be represented as a(r,9)rdrd9.
If a single valued, monotonic, radially symmetric RF field B 1 (r) is applied then the
contribution of an element of the spin density to the "FID" amplitude will be
6A(-yT) = o(r, 9) sin(yB 1 (r)T)r(Jr)(6).
(2.16)
Expression 2.16 can be integrated to produce the total amplitude of the "FID"
A(-yT)
=
j00*
=
jf,
-
Jo p(r) sin(-yBj(r)T)dr
2 7
r a (r,0)r
{J
2r
sin(yB 1 (r)T)d~dr
o(r, 9)rd9} sin(yB 1 (r)T)dr
(2.17)
where p(r) is the projection of the spin density a(r) around the circular path at radius r.
Since B 1 (r) is single valued and monotonic, the same change of variables can be made as
26
in section 2.3.1:
A(yT) =
/ro 13(B,
o
1
)
B (B1 )
sin('}TB1)dB1
(2.18)
where B'(B1 ) is the slope of the radial dependence of B 1 expressed in terms of the flux
B 1 , and P(B) is the projection expressed in terms of the flux density. Taking the inverse
sine transform of expression 2.18 produces P(B 1 ). The actual projection data p(r) may be
obtained once P(B 1 ) is known. This set of projections can subsequently be backprojected
to reconstruct the image of the object.
27
Chapter 3
Image Reconstruction from Circular
Projections
Since the data produced using RF imaging is encoded in radial projections, it was necessary to develop a new algorithm for image reconstruction that would use nonlinear
projections.
Conventional reconstruction using magnetic resonance utilizes linear pro-
jections of the object space. The approach used in the development of this algorithm
using nonlinear projections is similar to the reconstruction of ultrasound images [11]. The
development of this algorithm is contained in this chapter.
3.1
Theoretical Analysis
Table 3.1 contains a description of all the variables used throughout this chapter. For
an arbitrary object space shown in Figure 3-1, the object is represented by a density
distribution or(r, 9) centered around the origin. Transducers are placed on a circle of
radius R. The data a single transducer located at (R, 4) records is the line integral of the
object density around circles centered on (R, 4), or,
pr,(r, 9) =
0
po-(r(p, a;
28
4),
9(p, a; 4))da,
(3.1)
contour of line integral
%for projection PR(pP)
-d epally cy~ 0)
R
r
J/
locus of sensor
locations
Figure 3-1: Schematic of object space and spin density.
where p, the distance from the transducer at (R,#) to a point on the object (r,O), is given
by
p(r, 9; #) = VR2 + r 2 - 2Rr cos(O -
(3.2)
O(r, a; 0) = tan-'Rsn0
i
Rcosq#+ pcosa)
(3.3)
(
and
The complete data set consists of p,,O(r,0) for all q.
Imaging involves reconstructing
o-(r, 0) from the complete set of projection data.
3.1.1
Back Projection
The most elementary method of reconstructing the object is through backprojectionthe reversal of the data acquisition through projection. The back projection algorithm
consists of reconstructing each point (r, 0) in the image space by summing all the rays
that pass through that given point. Given a projection set of N projections, pr,4, the back
projection P(r,0) is equal to
p(r, 9) =
1
N-1
E PR (P (r,9, 0i) , #i),
i=0
29
(3.4)
x, y
Rectangular coordinates of a point in the reconstruction plane
r, 0
Polar coordinates of a point in the reconstruction plane
r
Distance of a ray from the origin
6
Angle between a point on a ray and the x axis
a-
Density function of an object
p(r, 0)
Ray sum or projection of an object
p
Distance from the transducer to the point on the object
q
Angle between the transducer and the x axis
R
Distance of the ray from the origin to the transducer
a
Angle between a line parallel to the x axis at the transducer to the ray p
k
Convolution kernel
P(r,6)
Fourier transform of p(r, 6)
K(jw)
Fourier transform of the convolution kernel
u, v
Fourier transform of x, y variables
p(w)
Hankel transform of p(r, 0)
P
Backprojection of the object
P'
Filtered backprojection of the object
P
Fourier transform of the backprojection
P'
Fourier transform of the filtered backprojection
Table 3.1: Definition of terms used in the theoretical development of the reconstruction
algorithm.
30
(a)
(b)
Figure 3-2: The star artifact (a) is produced by backprojection when a few projections
are used to reconstruct the image; when many projections are used, blurring of the image
results (b). Reproduced from [121.
where N is the number of projections and Oi is the ith projection angle. Since the transducers are spaced around a circle of radius R, the angular distance between transducers
in Equation 3.4 results in
is 21. Substituting A0q =
P(r, 0)
=
1
27r
N-i
: PR
i=O
(p
(r, 0, Oi) , Oi) _A0.
(3.5)
As N -+ oc, the reconstruction summation becomes an integral. Substituting Equation 3.2 into the backprojection algorithm results in
P(r, 0) =
27
0
PR VR2 + r 2
-
2Rr cos(9 - 0), 0) d$.
(3.6)
Figure 3-2 illustrates the results of backprojection at the origin. One of the limitations
of the backprojection algorithm is that when a few projections are backprojected, the
star artifact appears, as is shown in Figure 3-2 (a). On the other hand, when many
projections are used, as in Figure 3-2 (b), blurring becomes an issue. Generally, there is a
significant amount of blurring along edges of the object space as a result of backprojection.
Furthermore, normalization is required to correlate the reconstructed image to the original
object density. Straight backprojection is thus not often used as a reconstruction method.
A number of refinements can be made to improve the quality of the backprojected image.
31
Kernel
PSF
Impulse
Figure 3-3: The convolution of the point spread function from standard back projection
with a kernel results in an improvement in the quality of the reconstructed image by
reducing blurring.
3.1.2
Convolution Back Projection
To improve reconstruction via backprojection, the convolution back projection method is
used. The convolution back projection method, which is used in straight line tomographic
reconstruction, involves convolving each projection p,q5(r, 0) with a kernel k(p) prior to
back projection [3]. Figure 3-3 illustrates the effect of convolving the point spread function with the kernel to improve the backprojection. The kernel filters the data, thereby
reducing the star artifact. For the straight line projection method, the optimal kernel is
derived from the Radon transform. This results in a point spread function of 1.
Since
r
nonlinear projections are being used in this reconstruction method, it is necessary to determine the optimal kernel. To do this, the point spread function must be determined
and its reconstruction over the object space must be examined.
In the nonlinear case, when projections are taken along circular arcs, the convolution
back projection algorithm can be written as
P(r, 9) = -
f
27r a
k(p) 0 PR (p (r, 0, #) , #) d#
(3.7)
where 0 denotes convolution and k(p) is the convolution kernel. For an object located in
32
the neighborhood of the origin, when r << R, the first order approximation of
1
12
IX - 1X 2+
2
8
-/1+X -'
~ 1+
13
IX 3 +.,
48(38
(3.8)
can be used to expand the expression of p such that
R2 + r 2
p(r, 0; 4) =
1r
-
2Rr cos(6 - q)
2
SR + 4
- r cos(6 4R
)-
(3.9)
r2
r cos 2(0 -).
4R
(3.10)
If it is assumed that LR is sufficiently small, then higher order terms are negligible, and
p(r,6; ) ~ R - r cos(6 -#).
(3.11)
The reconstruction, therefore, can be expressed as
P(r, 6) =
If P,,q4(jw,
#)
k(p) 9 pr,O(R - r cos(6 - 0), 0))do.
-
27r
(3.12)
and K(jw) are the Fourier transforms of Pr,, and k(p) respectively, then
k(p) 0 Pr,$ =
27r
-oo
K(jw)Prjo, q)ejwPdw,
K-
(3.13)
and the reconstruction can be written as
(r,6)
3.1.3
-
47r2
0
0
K(j2)P7,,(jr,
#)ej(R-rcos(-0))dwd#.
(3.14)
Optimal Reconstruction Filter
One way in which the reconstruction method can be evaluated is by examining the point
spread function (PSF). The point spread function is the reconstruction of a two dimensional impulse located in the object space. Ideally, the PSF should itself be a two dimensional delta function. To make the PSF a two dimensional delta function, the reconstruction is filtered by a function K(jw). This filtering operation is the equivalent of
convolving the reconstruction with a kernel k(p). It is necessary to determine the optimal
K(jw) that would optimize the shape of the PSF.
33
Ay
S
p
p
Q
Figure 3-4: Geometry for calculating the point spread function.
In the object space shown in Figure 3-4, a point object, 0, is located on the x axis at
a radius r,. The object space is surrounded by a ring of transducers located at a radius
R. The transducers can be used to collect projection data and subsequently reconstruct
an image of the object. In the neighborhood of the object, the PSF can be found by
reconstructing the image of the object at a point,
Q, that
is located at a distance, d, at
an angle, -y, from 0. For a particular transducer, S, located at (R, 4), the distance from
the transducer to the object is po. The distance from the transducer to the reconstruction
point is p, where
p = Vd2 + p2 - 2dpo cos(y -#).
If d
<
(3.15)
R and ro < R, then the first order approximation given in Equation 3.8 can be
used to approximate
-
p ; po - d cos(y
(1-cos (2 (7--))).
4 po
(3.16)
The angle # can therefore be expressed as
R sin
cos
tan
=
(
(3.17)
If ro < R, then 0 ; 0, and if the reconstruction is done in the neighborhood of the object
34
so that d < po, then
p ~ po - d cos(y - #).
(3.18)
The projection data collected by the transducers will be a set of shifted delta functions
such that
6(x - p).
pr (X,
(3.19)
The Fourier transforms of the projections can therefore be expressed as
(3.20)
) = ejWPo.
Pr(jOW,
The filtered backprojection of the PSF at the point Q, thus becomes
P (d, y;ro)=-
1 ~
2
7
k(p) 9 pR (p, .)d
(3.21)
or, using the Fourier transforms K(jw) and PR(jw, 0) and the approximation of p given
in Equation 3.18,
1
(d, -y; ro) =
2
47r2
j
J
J
27
0
S2
=
-
r
J
K(jw)PR(jw, q)eJWPdwdO
-00
K(jw) {
2
eiwdcos(Y)d} dw
ir
K(jw)Jo(wd)dw.
27r- - o
(3.22)
In polar form, the two dimensional delta function can be expressed as
-6(r)
(X, y) =
(r(3.23)
,
7rlrl'
and it can be shown
J/oo
1
0 (rw)wdw = -6(r)
100
[13]. Therefore, if the convolution kernel k(p) has a Fourier transform
K(jw) =
IwI,
(3.24)
(3.25)
the PSF, P(d, -y; ro), will be a delta function centered at d = 0, thus defining the optimal
filter. It should be noted that this optimal filter is the same filter that is used in conventional straight line projection reconstruction [11]; however, where the filter is spatially
invariant for the straight line projections, the filter is only valid in the neighborhood of
the object near the origin using circular projections.
35
AK(W)
-*C
10C
0
Figure 3-5: Optimal filter for image reconstruction.
3.1.4
Filter Truncation Effects
The filter K(jw) = |wJ is acausal and non-realizable. In practice, this filter emphasizes
high frequencies and would amplify noise elements. To improve the frequency response
of the filtered projections, it is necessary to multiply the filter by an apodizing function
to attenuate high frequencies. A cutoff frequency we can be imposed on the filter by
multiplying K(jw) by a rectangular function so
we
-w
for
K(jw) ={w
0
for JwJ > we
as shown in Figure 3-5.
The imposition of limits on the filter also changes the limits of integration on the
reconstruction, which subsequently causes the point spread function to be expressed as
1
P' (r,6) = -
27r
/
Wc
we
J0 (wr) w Idw.
(3.26)
wJ 0 (wr)dw.
(3.27)
Since the Bessel function is an even function,
P'(r,0) = 1
/
7r 0
This expression can be evaluated using the property
fa rJ(wr)dr =
0
36
Ji(aw)
W
(3.28)
025I)
1
015
0 15
0.1
-1
0.05-
-0.05 -20
-15
-5
-10
D
5
10
15
20
Figure 3-6: Cross section of the PSF reconstructed using a truncated filter.
[13]. Thus, the reconstruction becomes
-
(r, Jiwr) = W
Ir
7rr
wer
(3.29)
The result of this analysis indicates that the point spread function for a point object
at the origin is a Bessel function of the first kind. A cross section of the Bessel function
is shown in Figure 3-6. The width of the main lobe of the Bessel function,w, depends on
the cutoff frequency of the filter such that
W = 7.86
wc
(3.30)
The presence of the sidelobes of the Bessel function will produce artifacts in the reconstructed image. Since the magnitude of the sidelobes of the Bessel function is small
compared to the main lobe, the artifact will have a much lower intensity than the main
signal.
3.1.5
Reconstruction of a Point Object at the Origin
A number of assumptions were made while developing the method for image reconstruction
described in section 3.1.2. The most significant of these is that the object is located near
the origin and is imaged at a point in the neighborhood of the origin. These assumptions
37
limit the application of the reconstruction method. An alternative method for analyzing
the convolution backprojection method of reconstruction was examined that made fewer
assumptions and allowed for the reconstruction of an object at the origin anywhere in the
image space.
For the object in the object space shown in Figure 3-1, the projection of the object is
the line integral of the object density given in Equation 3.1. Since the circular projection
p(r) is radially symmetric, it has a Hankel transform and can be expressed as
j
=
(3.31)
rp(r)Jo(wr)dr.
The projections can also be expressed in terms of the two dimensional Fourier transform
p( x2 + y2 ) -==
(3.32)
P(u, v).
Since
P(u, v) = 2 rp(w)1,=r,
(3.33)
it follows that
P(u, v) =
27r
j
rp(r)J(rv/u 2 + v 2 )dr.
(3.34)
If it is assumed that the object is a point object at the origin, then
p(r) = 6(r - R).
(3.35)
The Hankel transform of this is thus
p(w) = RJ(wR),
(3.36)
which subsequently causes the Fourier transform to become
P(u,v) = 27rRJ( /u 2 + v 2 R).
(3.37)
The shifting property of the two dimensional Fourier transform states
f (X - X0, y - Yo) 4==* F(u, v)e(uxa
38
+VyO)
(3.38)
[13]. The Fourier transform of the object at a point (xO, yo) = (R cos q, R sin #) thus
becomes
2
P(u, v)|,, 0 = 27rRJ0 (v'u + v2R)e-j(u cosO+vsin$)
(3.39)
An angle 0 can be defined such that
U = w cos V)
v = w sin V)
=t
W=
-v
- /U2 + V2.
Equation 3.39 therefore becomes
P(u, v)1,,,, = 27rRJ o (Vu 2 + v2R)e-i(wRcOs~cOs +wR sin#sino)
(3.40)
This expression can alternatively be expressed as
a'2 + v2R)e-iw"c4s(V-O).
P(u, v)I..,yo = 27rR JO(
(3.41)
The back projection of the object at the origin can be expressed as
(x, y) = -
2 7r fo
PR,O(x -
Rcosq, y
-
Rsin)d,
(3.42)
or, using Equation 3.41
1
P(U'V) -27r
2r
fo
27rRJ,( /u 2 + v2R)e-wR cos(O-)dO.
(3.43)
This can be simplified to
P(U, v) = RJo(Vu 2 + v 2 R)
jr
e~wRcos(-)dO.
(3.44)
Since the object is circularly symmetric, then the Bessel identity
JO(r) = 1
27r Jo
39
ejrcos(+-q)dO,
(3.45)
where q is an arbitrary constant [13], can be used to further simplify the expression to
PF(u, v) = R JO(Vu 2 + v 2 R)27rJ0 (wR).
(3.46)
Since
w=
+V2
U2
P(w) = 27rR(J(wR))2 .
(3.47)
The inverse Hankel transform can be shown to be
j0(ar)
++
2W
2
w < 2a
0
w > 2a
[13]. The symmetry property, which says
(3.48)
f (r) ++f(j),
allows the inverse Hankel transform to be expressed as
j2r(aw) +-+
22
7r r v
4_2 -
2
(3.49)
It is thus possible to take the inverse Hankel transform of Equation 3.47 to get an expression for the unfiltered back projection of the image:
k(r)
4R
4R2
r4R -r 2
(3.50)
By implementing a filter on the data, it is possible to optimize the shape of the point
spread function to make it similar to an impulse. Incorporating the optimum filter for
the linear backprojection algorithm (see Section 3.1.3), 1w , into the expression for the
backprojection in Fourier space yields
P'(w) = 27r~wIRJ(wR).
(3.51)
It is possible to approximate the Bessel function J0 (x) as
JO(x)=
4) + O(x)
- cos
7rX
40
(3.52)
[13]. Since Equation 3.51 calls for J,2(wR), which is an even function, it is necessary
to modify the approximation in Equation 3.52 such that it is also an even function. An
equivalent even approximation that can be made is
27rIxIJo2(x) = 4 Heaviside(x) sin 2
+ 4 Heaviside(-x) cos 2
X+
X+
.
(3.53)
Using MAPLE, the inverse Fourier transform of the approximation was found to be
'(r)approx =
2(-2 + ir3(r)r2 - 47r6(r)
(r + 2)(r - 2)7r
(354)
This expression can be simplified by separating it into two separate terms. This results
in
2
2
((r + 2)(r - 2)7
+ 6(r)
.
(3.55)
This expression for the filtered backprojection must be slightly modified since Equation 3.51 is in terms of wR and the inverse Fourier transform was taken with respect to
an arbitrary variable x. It can be shown [14] that if
F(x)
(3.56)
-G(r)
then
F(Rx) =>-G -;(3.57)
R
R
thus, the correct expression for the filtered backprojection is
approx
2 ((R + 2 ) -2(
R
2
) 7).
(3.58)
It can clearly be seen from this form of the inverse Fourier transform that the filtered
backprojection consists of a delta function at the origin and delta-like functions located
at a radius of Irl = 2R from the origin.
An effort was made to find a closed form expression for the point spread function of a
point object arbitrarily located in the object space. The details of the development of a
numerical solution are contained in Appendix A.
41
3.2
Numerical Simulations
The reconstruction method described in Section 3.1.2 was evaluated by creating artificial
projection sets and reconstructing images. Two studies were conducted using the method.
3.2.1
Calculation of the Point Spread Function
The goal of the first study was to determine the severity of any spatial dependence of
the point spread function. This was done by computing the PSF numerically at various
radial distances from the origin. It was assumed that a point object was located in the
object space. The object space was represented by a 64 x 64 grid. It was assumed that a
ring of 200 transducers was located around a circle of radius R = 50 centered around the
object space (see Figure 3-7). One hundred projections were simulated for each of these
200 transducers. Each of the projections was a line integral of the spin density around
a circular arc of radius r. The radius was varied from 0 to 2R in one hundred steps so
projections could be collected across the object space. If the circular arc passed through
the point object, the projection data consisted of an impulse function; if the arc did not
pass through the object, the line integral was equal to zero. Each of these projections
contained data from one hundred points. This data was then backprojected into the image
plane.
Figure 3-8 shows the effect of filtering the reconstruction in the neighborhood of the
object using the optimal filter K(jw) = w| described in Section 3.1.3. The PSF from the
filtered projection is much narrower than the PSF produced by the unfiltered projection.
The broadening of the main peak that is apparent in Figure 3-8 results in blurring of the
reconstructed image.
The projection data for all subsequent simulations was filtered prior to backprojection. Linear interpolation between points was used during the back projection. Filtering
operations were done using the FFT method with the cut-off frequency set to the Nyquist
rate. The MATLAB scripts used for these simulations are contained in Appendix B.
42
R =50
image space
.
64x6
--.
200 projections each with
100 points and linearly
interpolated for back
projection
--. i
Figure 3-7: Geometry used for point spread function simulation.
Figure 3-9 shows the computed point spread functions at r = 0, 16, and 32. It can be
seen from these figures that there is little dependence of the PSF on the distance from
the origin. Thus, assumptions that were made to calculate the closed form of the PSF in
the neighborhood of the origin may have some validity for objects located far from the
origin.
The analysis using the Hankel transform that was derived in Section 3.1.5 indicated
the presence of a ring of impulse functions located outside of the object space at a location
of r = 2R. The numerical simulation of reconstructing a point object in the object space
was repeated to verify the presence of this ring of impulse functions. The object space
remained a 64 x 64 grid; however the image space was expanded so the ring of delta
functions at r = 2R could be seen. One hundred projections with one hundred points
were created from each of the 200 transducers located on a circle of radius R = 50. This
data was filtered and back projected into the enlarged image space. Figure 3-10 shows
the point spread functions at r = 0 and 16. The ring of delta functions at a radius of
r = 2R is absent from each of these plots. This seems to indicate that the ring of delta
43
I I.
0.8-
0.6
0.4,-
80
48
40
802
0
0
0.1
0.6
44
08w
0.0
0.2
0 0
20
(a)
0.1
0.6
044
02,
080
220
(b)
0.2,
2
20
so
0
(c)
Figure 3-9: Point spread functions for a point object located at r=O (a), r=16 (b), and
r=32 (c).
45
-
-~
~
r=0
0.6
0.4
-
0.2
0
250
250
2000
100
1
50
I(a)
50
0
0
(a)
11
0
P
250
(b)
Figure 3-10: Point spread functions for a point object located at r=0 (a) and r=16 (b)
reconstructed in an enlarged image space.
46
functions is an artifact associated with the numerical approximation of the PSF. The
algorithm does, however, correctly place an impulse at the location of the object. This
would suggest that the integrity of the imaging algorithm was not adversely affected by
the approximations.
3.2.2
Reconstruction of Photographic Images
In the second study that was conducted, the reconstruction algorithm was tested using
photographic images. This would verify the reconstruction method using objects that
were not point objects, and it would also allow for the integrity and the resolution of the
method to be qualitatively assessed. Images were scanned into the computer and stored
in a 256 gray level format. Circular projection data was collected from the images and
stored. This data was then passed to the reconstruction algorithm and a new image was
created. The MATLAB scripts for this study are contained in Appendix B.
Two examples of this simulation are shown in Figure 3-11. Each reconstructed image
is created from a set of 360 transducers. Each transducer collected 200 projections with
200 points apiece. The top pair of images in Figure 3-11 is an MRI image of the saggital
section of a human head. The bottom pair of images was created using a photograph
of grains of rice on a gray background. The projections were formed by summing the
pixel values around the circular arc. The agreement between the original image and the
reconstructed image validates the reconstruction method.
47
ta)
(b)
fc)
(d)
Figure 3-11: Original and reconstructed images of the saggital section of the head and
grains of rice.
48
Chapter 4
Experimental Work: RF Gradient
Imaging
4.1
Introduction
RF gradient imaging utilizes the techniques developed in RFI [4] to create images. Whereas
conventional imaging techniques collect linear projection data, this technique encodes the
object along nonlinear gradients. The spin density of the object can be quantified along
the nonlinear gradients by measuring the nutation frequency of the sample. This information can subsequently be used to reconstruct an image of the object. The theoretical
development of the this technique was examined in Chapter 3. It is subsequently necessary to apply the technique to determine the practicality of the method and to examine its
limitations. This required designing an NMR probe with a suitable geometry to facilitate
data collection and designing a series of experiments that could be used to evaluate the
feasibility of this technique.
49
4.2
Coil Design
It was necessary to build an NMR probe with an RF coil that would produce a spatially
varying RF field. According to the Biot-Savart law, the magnetic field for an infinitely
long wire conducting current is
B
(4.1)
27rr'
where p, is a constant, I is the current, and r is the radial distance from the wire. The
magnetic field thus varies as a function of distance from the wire along circular arcs. An
RF pulse that utilizes this type of RF field has previously been used for reducing image
times [9]. Instead of producing a coil that was a single wire, a single-turn rectangular
RF coil was constructed as shown in Figure 4-1. The coil was created out of a tube that
was bent into a rectangle of width, W, and length, 1. The length of wire closest to the
center of the magnet that is aligned with B, would act as an infinite conductor. The
width of the coil was made as large as possible to minimize the effect of the RF field from
the distal side of the coil. The length of the coil was also maximized to approximate the
assumption of an infinitely long wire carrying current. The experiments were conducted
in a 4.7 Tesla magnet with a 12 cm bore at the Francis Bitter Magnet Laboratory. Tuning
and impedance matching capacitors were included in the circuit, and the coil was tuned
to 200.4 MHz using an RF bridge.
The resonant frequency of the coil is
1
W = 27rf =
(4.2)
where L and C respectively are the inductance and the capacitance of the coil, and
f
is
the frequency of the coil in Hertz. In order to select the resonant frequency of the coil, it
was necessary to adjust both the capacitance and the inductance of the coil. For a single
turn of wire as shown in Figure 4-2, the inductance, L, of the wire is
L = -in-7r
d
(4.3)
where d is the diameter of the wire, and p is the magnetic permeability of air. It was
50
K-W-
D
D
#A
coil
B;
lz
L
r
W
sample
coil
r
sample
tuning capacitor
~C1
Cimpedance
2
capacitor
co-ax cable
matching
co-ax cable
Figure 4-1: Geometry of coil.
I
U..-
I
wI
<-
'I,
dj'
Figure 4-2: Inductance of a single turn of wire (Adapted from [15]).
51
i
found that a coil that maximized W and 1 had too high an inductance to be tuned to
200.4 MHz. The dimensions of the coil were reduced to produce a coil that could be tuned
and matched to the appropriate resonant frequency. In its final configuration, the coil
consisted of a tube of diameter 3.6 mm, with a length of 66 mm, and a width of 52 mm.
A rectangular coil of these dimensions does not adhere to the assumption of an infinite
conductor. This may affect the performance of the coil in two respects. At distances close
to the proximal wire, the field produced by the distal wire is very small compared to that
of the proximal wire; however, at distances far away from the proximal and distal wires,
the magnetic field from the distal wire can effectively cancel the magnetic field that is
produced by the proximal wire since the current in the two wires is conducted in opposite
directions. The other potential complication arises from the transverse wire branching
across the proximal and distal wires. The magnetic fields produced by these sections of
wire may contribute to the magnetization, thereby creating gradients in directions which
are assumed to be subjected to homogeneous magnetic fields. Since the workspace around
the coil is small and the sample is small relative to the length of the coil, it was assumed
that these issues would not significantly affect the model.
4.3
Hardware Design
4.3.1
Base Frame
It was necessary to design a fixture to which the coil could be mounted. This fixture
serves three purposes: 1) It electrically isolates the coil from the magnet; 2) It allows
the coil to be free standing; and 3) It accurately positions the coil in the center of the
magnetic field.
A disk was designed to accurately position the coil within the magnet. A drawing
of the disk is shown in Figure 4-3. The disk can fit inside the bore of the magnet but
prevents radial movement within the magnet. Mounting two of these disks to brass rods
created a free standing structure to which the coil could be mounted (see Figure 4-4). This
52
Figure 4-3: Drawing of disk for mounting probe.
free standing structure could be positioned arbitrarily along the length of the magnet. It
was therefore possible to place the coil in the center of the magnet. By fabricating this
base frame from non-conductive materials, the coil would be electrically isolated from
the rest of the magnet. Engineering drawings of the disk in the fixture are contained in
Appendix C. Holes were added to the disks to allow for tuning tools with extensions to
be attached to the tuning and matching capacitors. This completed the construction of
the NMR probe.
4.3.2
Variable Distance Sample Position
Once the construction of the probe was complete, it was necessary to design various
fixtures that could be used to position samples in the probe for the purposes of experimentation. It was intended that the fixtures be modular so the appropriate configuration
of the probe could be attained for each experiment.
The goal of the first experiment was to characterize the behavior of the magnetic field.
It was expected that the magnetic field would vary inversely with r, the distance from the
coil. Since the nutation frequency varies directly with the magnetic field, it was expected
53
Figure 4-4: Picture of the mounting frame for the RF coil.
that the nutation frequency would also vary inversely with r. To verify this behavior,
it was necessary to measure the nutation frequency of a sample as the position of the
sample changed within the magnetic field. It was also important to know the position of
the sample with respect to the coil so the measurement of the nutation frequency could
be verified by comparing it to predicted values. A mechanism was needed that would
accurately position the sample with respect to the coil. It was also necessary that the
distance from the coil to the sample be adjustable in small increments without significantly
disturbing the system. Incremental changes in the radial position of the sample would
allow for the assessment of the sensitivity of the experimental method.
The sample used in initial experiments was a 3 mm OD glass tube containing a column
of water 6 mm in length. A positioning mechanism was designed that would place the
sample next to the proximal coil conductor and could move the sample radially away from
the coil in discrete, measurable increments over a range of approximately 20 mm. This
54
positioning fixture consisted of three components. The first piece is shown in Figure 4-5.
This piece is mounted directly to the rear disk of the probe. The remaining two pieces
are shown in Figure 4-6. These pieces have a groove that is 0.125" wide and 0.06" deep
through their centers. These pieces are sandwiched together to constrain a brass tube
that has an outer diameter of
5"
inside the groove. The inner diameter of the brass
tube is 1". It is therefore possible to insert the NMR tube containing the sample into the
brass tube. The sample, tube, and sandwiched plates are attached to the first piece of
the fixture (that is mounted to the probe) using two M6 screws (see Figure 4-7). As the
screws are turned, the sandwiched pieces move with respect to the fixed piece, moving the
sample away from the coil along the positioning slot in the mounting disks. Since an M6
screw has a pitch of 1, a full turn of the screw moves the sample a radial distance of 1 mm;
thus, the sample can move in measurable, discrete intervals away from the center of the
coil. Engineering drawings of the elements of this positioning mechanism are contained
in Appendix C.
4.3.3
Multiple Sample Holder
A second experiment was designed to determine the resolution of the experimental method.
The resolution of the method would be determined by the ability of the coil to detect multiple samples as the position of the samples varied with respect to the coil and to each
other. It was therefore necessary to design a fixture that could be used to position multiple
samples in a variety of positions. Figure 4-8 shows a drawing of the fixture that was designed to hold up to ten 3 mm NMR tubes simultaneously. The center to center distance
between adjacent samples is 4 mm. It was possible to determine the resolution of the
imaging method by measuring the frequency response when multiple, identical samples
were placed in various locations. Figure 4-9 shows a picture of the probe with multiple
samples in the fixture. Engineering drawings of the fixture are contained in Appendix C.
55
Figure 4-5: Drawing of fixture used to mount the variable positioning mechanism to the
base of the probe.
Figure 4-6: Drawing of sandwich pieces of variable positioning mechanism.
56
Figure 4-7: Picture of assembled positioning mechanism.
Figure 4-8: Drawing of fixture for imaging multiple samples simultaneously.
57
Figure 4-9: Picture of coil with multiple samples.
4.3.4
Rotating Sample
The next goal was to collect a set of projections for a sample to reconstruct an image of
the sample using the algorithm that had been developed in Chapter 3. The algorithms for
image reconstruction described in Chapter 3 require that projection data be collected by
a ring of transducers. Since a standard NMR spectrometer was used for the experimentation, it was not possible to surround the sample with a ring of transducers. The same
effect can be achieved by rotating a sample around an axis parallel to the transducer. As
the sample moves about the central axis of rotation, the distance between the coil and
the sample and the orientation of the object with respect to the transducer varies. This
creates the effect of a ring of transducers surrounding the object. To achieve this effect,
it was necessary to design a fixture that would rotate a sample around an axis that was
a fixed distance from the coil.
This rotary fixture is a small disk with an axial hole and four holes for holding samples
58
-
EmBbERIF"M
-
Figure 4-10: Drawing of disk in rotary fixture.
(see Figure 4-10). Mounting two of these disks to a central shaft creates a rigid fixture in
which samples can be constrained. The 5" diameter brass shaft which passes through a
central axis of the disks also passes through a plastic gear (see Figure 4-11). The plastic
gear is used to quantify the rotation of the sample with respect to the coil. The position
of the sample correlates to the position of the gear since the sample and the gear both
rotate as the central shaft is rotated. There were four possible positions for the sample,
corresponding to the four sample holes located at radial distances of 4.5, 5.5, 6.5, and 7.5
mm from the central axis of the disk. This allowed for multiple samples at a variety of
positions to be imaged simultaneously.
Mounting blocks were attached to the disks that form the body of the probe. These
blocks had a close clearance hole for the central shaft of the rotary fixture. This served
to position the axis of rotation of the rotary fixture with respect to the coil. Engineering
drawings of the rotary disks and the mounting blocks are contained in Appendix C.
59
(a)
Figure 4-11: Picture of fixture for sample rotation.
4.4
Experimental Setup
Each of the fixtures described in Sections 4.3.2, 4.3.3, and 4.3.4 were used during the
experimental phase of the project. The fixtures were each used to locate and move samples
as was appropriate for a particular experiment. Samples consisted of water columns in 3
mm OD glass tubes (Wilmad Glass Part 305-PS-7). Experiments were conducted on
an Oxford 4.7 Tesla magnet with a Bruker AMX2 console that has been modified for a
horizontal imaging system. The operating console is an SGI 02 host computer running
Bruker XWIN-NMR v. 2.6. For each experiment, the probe with the sample was placed
in the bore of the magnet. The probe was positioned such that the sample was located
in the center of the magnet in both the radial and the longitudinal directions. This was
done because the static background field is homogeneous across the center of the magnet.
There were two primary pulse programs that were utilized during the experimentation.
The first pulse program, termed a zg program (see Appendix D) [16], applied a hard pulse
to the sample and measured the response. The resulting free induction decay is shown
in Figure 4-12. This FID can be Fourier transformed to display the frequency spectrum
60
of the sample, as is shown in Figure 4-13. The zg program was primarily used to match
the resonance frequency of the sample with the resonant frequency of the transmitter.
When the sample is on resonance, the FID takes the form of an exponential. The Fourier
transform of a FID produced by a sample that is on resonance is a peak at zero frequency.
This indicates that the sample and the transmitter are well matched.
The zg program was also used to determine the length of the 900 pulse. Since the
duration of the RF pulse determines the magnitude of the transverse magnetization, the
900 pulse, which maximizes the magnetization in the xy plane, produces a very strong
signal. The strength of the signal corresponds to the number of spins that were excited
and can be measured as the magnitude of the peak in frequency space. The 900 pulse
was found by changing the duration of the hard pulse and measuring the intensity of the
frequency response. When the frequency response was maximized, the 900 pulse had been
determined. This value for the 900 pulse was verified by doubling the pulse length and
checking to see that the resulting 1800 pulse minimized the response.
A second pulse program (see Appendix D) was used to measure the nutation frequency of the sample. The nutation sequence consists of a series of RF pulses applied
on-resonance, each followed by single-point acquisitions. The resulting signal is weighted
by the proton density of the sample and the RF field strength. The pulse program administered a series of 10ps pulses (pi) and measured the response of the sample in between
pulses. Each of the 1024 pulses (11) were separated by a 150ps delay (d20) to allow for
the ringing of the probe to settle. A data point was then recorded in the real and the
imaginary spectra respectively. Phase cycling of the pl pulse was used to permit continuous excitement of the nuclei without allowing for the relaxation of the sample. A diagram
depicting the pulse sequence is shown in Figure 4-14.
Each pulse of the nutation sequence tilts the magnetization vector by an angle, a.
During the application of the RF pulse, the magnetization precesses around B 1 . In between pulses, precession about B 1 does not occur; thus, when another pulse is applied,
the magnetization is flipped another angle a with respect to its previous location. The
61
25001-
2000
e 1500
1000
500-
0-
0
0.1
0.2
0.3
0.4
0.5
Time(s)
0.6
0.7
0.8
0.9
1
x 10-
(a)
rnn
0
-500
-1000
-1500
Z -2000
-2500
-3000
-3500
-4000
-450nn
0
0.1
0.2
0.3
0.4
0.5
Time(s)
0.6
0.7
0.8
0.9
1
X 10-3
(b)
Figure 4-12: The real (a) and imaginary (b) components of a free induction decay resulting
from the application of a hard pulse during the zg pulse program.
62
x 10_
3.5 -
3-
2.5-
2
E,
1.5-
0.5-
0
-0.5
-2.5
-2
-1.5
-1
-0.5
0.5
0
Frequency (Hz)
1
1.5
2
2.5
x i0
Figure 4-13: Fourier transform of FID data from the zg sequence.
Real Imaginary
Point
Point
P1
d2
11
Figure 4-14: Schematic of nutation pulse program.
63
magnitude of the signal is recorded as the nutation angle varies with each of the 11 applied
pulses. The variation of the magnetization is the nutation frequency of the sample.
For a single sample, the length of the 900 pulse is related to the nutation frequency
by Equations 2.3 and 2.4. Combining these two equations yields
a = wT,
(4.4)
where w is the nutation frequency, T is the duration of the pulse, and a is the tilt angle.
It is therefore possible to confirm the results of the nutation experiments by checking the
length of the pl pulse. Since the length of the pl pulse is empirically determined, it alone
can not be used as an accurate measure of the nutation frequency. It can, however, be
used as an order of magnitude estimate to verify the results of the nutation experiment.
Once the data in the time domain was Fourier transformed, a phase angle was added to
the data to improve the absorption spectrum. This was necessary since the radiofrequency
receivers heterodyne the signal by mixing the signal EMF with the output from a reference
RF oscillator. Two heterodyned references, each 900 out of phase are used to effectively
detect the magnetization in the x and y directions. Since it is difficult to set the receiver
phase precisely, the phase correction is applied to the Fourier transform of the data. The
Fourier transformed data correlates the spin density to a frequency. To reconstruct an
image from this data, it is necessary to correlate the frequency in space to a distance
r in mm that corresponds to a physical distance of the object from the coil. For the
reconstruction process described in the next chapter, the raw "FID" data was used.
Figure 4-15 shows the real (a) and the imaginary (b) components of the "free induction
decay" that is generated by the nutation sequence. A baseline correction was applied to
the data set prior to Fourier transforming the "FID" to smooth the curve, reduce noise,
and eliminate any stray DC components. Some of the noise present in these data sets
could be reduced by increasing the number of scans and thus the number of data sets
that are averaged. The Fourier transform of the data is shown in Figure 4-16. Since the
data points are acquired on resonance, the precession of the magnetization about B, is
not seen. Instead, the precession of the moment about B 1 is visible in the raw "FID"
64
data. It was expected that the imaginary spectrum of the "FID" would be identical to
the real spectrum. It is possible that the exponential trend in the imaginary spectrum is
caused by the spatial extent of the sample, which is large enough that different levels of
excitation are produced across the sample. As these precess about B1 at their respective
frequencies, the signal decays due to molecular interaction and loss of phase coherence.
The different levels of excitation may interact to produce the exponential trend that is
seen in the imaginary spectrum.
The Fourier transform of the data shows the nutation frequency in the positive and
negative spectrum. Phasing can be added to the Fourier transform of the data set to
produce an antisymmetric plot. After the phasing is added, the frequency of the nutation
peak correlates with the distance between the sample and the coil. The peaks are wider
when the sample is close to the coil than when the sample is far from the coil since the
spatial extent of the sample is significant at close distances and becomes compressed as
the sample moves away from the coil (see Figure 4-17).
65
x 10
-E0
-1
-2
-3
-4
0
0.1
0.2
0.3
0.4
0.5
Time(s)
0.6
0.7
0.8
0.9
1
X 10-
(a)
slO
4.5
4
3.5
3
2.5
2
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time(s)
0.6
0.7
0.8
1
0.9
3
x 10-
(b)
Figure 4-15: Real (a) and imaginary (b) spectra of the "FID".
66
x 108
3
2.5
2
.D
1.5
S 1
0.5
0
-0.5
-2.5
-2
-1.5
-1
-0.5
0
0.5
Frequency (Hz)
1
1.5
2
2.5
x 1o
4
Figure 4-16: Fourier transform of "FID" data from the nutation sequence
X 10
0
E-1
L
-2
-3
-2.5
-2
-1.5
-1
-0.5
0.5
0
Frequency (Hz)
1
1.5
2
2.5
x 10
Figure 4-17: Fourier transform of "FID" data from a dual sample. The sample located
close to the coil produces the broad peak centered at approximately 2500 Hz while a
sample located at a further distance from the coil produces the narrow peak centered at
approximately 1200 Hz.
67
Chapter 5
Experimental Results and Analysis
5.1
Calibration of Frequency to Space
The goal of the first set of experiments was to determine the spatial dependence of the
nutation frequency on the distance from the sample to the coil. The Biot-Savart Law
indicates that the magnitude of the magnetic field falls off as 1.
r . Since the nutation
frequency is a function of the magnetic field, it was expected that the nutation frequency
would fall off as a function of
} as well.
Experimentally, the nutation frequency was measured by setting up the probe with
the fixture described in Section 4.3.2. Initially, the sample was positioned immediately
adjacent to the coil, and data was acquired. The screws on the fixture were then turned a
single revolution; this moved the sample radially away from the coil along the positioning
slot in the disk of the probe mount in a discrete step of 1 mm. This procedure was repeated
until the sample was as far away from the coil as the hardware would permit (about 21
mm). The measured nutation frequency was plotted as a function of the distance from
the center of the coil to the center of the sample. Figure 5-1 shows a log-log plot of one
of these experiments. This experiment was repeated a number of times, and each time,
similar data was obtained. The raw data for the experiment is contained in Appendix E.
The trendline used to characterize the data supports the 1 dependence between the
68
Nutation Frequency vs. Position
4Nutation Frequency
3.1
-Linear (Nutation
Frequency)
. 2.5
2
U.
S1.5
z
y= -1.0206x + 4.2254
R2
1
= 0.9477
0.5
0
~' 0
'
0
''
''
'
1
0.5
Log(Position(mm))
1.5
Figure 5-1: Plot of nutation frequency as a function of distance to the coil.
nutation frequency and the distance between the coil and the sample. The trendline
does not agree with some of the data points that were measured when the sample was
relatively close to the coil. The discrepancies between the trendline and the actual data
were attributed to errors made in positioning the sample with respect to the coil. The
experiment is more sensitive to positioning errors when the sample is close to the coil
than when the sample is further away. This is because the spatial encoding causes the
variation of the nutation frequency between adjacent discrete points to be greater when
the sample is close to the coil than when the sample is far from the coil.
The positioning error can be attributed to a number of sources. Since the sample
is contained in a glass tube which is inserted in a brass tube which was clamped at
its base to the positioning mechanism, an error associated with the displacement of a
cantilever beam exists. An additional source of error in positioning the sample is due
to the low stiffness of the structure. The clearance holes through which the M6 screws
69
pass had sufficient freedom to allow for the entire mechanism to pitch slightly. Any slight
angular displacement was thus magnified by the length of the cantilever to produce a
significant positioning error. These errors are evident in the data points close to the coil
when the distance between the coil and the sample is small. For this reason, the data
points as the sample moves away from the coil are more reliable. From the results of
the first experiments, it was apparent that the nutation frequency was a reliable measure
of the position of a sample with respect to the coil. The data would suggest that the
dependence between the nutation frequency and the distance from the coil is a valid
relationship.
5.2
Resolution
It was then necessary to determine the resolution of the experimental method. The second
set of experiments involved placing multiple samples in different positions to determine
the ability of the method to distinguish between different samples. For this experiment,
the sample holder described in Section 4.3.3 was attached to the base frame. Identical
samples of NMR tubes containing 7 mm long columns of water were used. Initially, a
single sample was placed immediately adjacent to the coil, and nutation data was collected.
A second sample was then placed in the hole in the positioning mechanism neighboring
the first sample. The centers of the two tubes were separated by a distance of 4 mm.
Subsequently, a third tube was added to the fixture. Data was collected prior to the
addition of each successive tube. It was found that the ability of the spectrometer to
distinguish between multiple samples varied with the position of the samples with respect
to the coil and to each other. When the samples were close to the coil, the difference in
the RF field strength between two adjacent positions was quite significant. This caused
the formation of two distinct peaks at the respective nutation frequencies of each of the
two neighboring tubes. This can be seen in Figure 5-2. When the samples were far away
from the coil, the relative RF field strengths were approximately the same. Thus, the
70
x 101
3-
2.5-
CM
Ca
1.5-
0.5-
0
-500
-400
-300
-200
-100
0
100
Frequency (Hz)
200
300
400
500
Figure 5-2: Fourier transform of two water samples located close to the coil.
nutation peaks formed at frequencies that were very close to one another. It was not
possible to distinguish between two neighboring samples when the samples were far away
from the coil. This is shown in Figure 5-3
An additional complication arose from the spatial extent of the samples. Since the
samples were 3 mm in diameter, the nutation peaks had a finite width that correlated
to the diameter of the samples. The width of the peaks was often larger than the difference in nutation frequency as a result of introducing an additional sample. Thus, the
peaks produced by adjacent samples would lie on top of one another and could not be
distinguished. When a sample was placed immediately adjacent to the coil, and another
sample was placed 4 mm away from the coil, the spatial resolution was good enough to
observe both tubes simultaneously; however, two samples separated by a distance of 4
mm located at a distance of 12 mm away from the coil could not be distinguished from
one another.
71
~x 10a
3. 3 .
I
I
I
I
I
I
3-
2.5-
2 -
1.5-
0.5-
0
-500
-400
-300
-200
-100
0
100
Frequency (Hz)
200
300
400
500
Figure 5-3: Fourier transform of two water samples located far from the coil.
Since the magnitude of the RF field falls as a function of r, the NMR signal grows
weaker as the sample moves away from the coil. There are a number of ways in which
this complication could be addressed. Increasing the power of the applied RF pulse would
increase the strength of the RF field, and thus cause a greater excitation and a stronger
signal. This would also result in a broader spectrum where the nutation frequencies are
distributed over a greater space. The nutation frequencies of adjacent samples would then
differ by a greater amount and could be distinguished from one another. Another possible
alternative for increasing the strength of the signal as the sample moved away from the
coil is to increase the number of scans that are added together to produce a data set.
The random noise will average to zero while the signal component will add to produce
a stronger signal. There is a practical limit to increasing the number of scans since the
signal to noise ratio varies as a function of the square root of the number of scans, while
the acquisition time varies directly with the number of scans. The number of scans was
72
quadrupled from 16 to 64, but this produced little change in the resolution of the peaks.
An attempt was made to apply high power pulses; however, arcing across the capacitors
prevented this from being a viable solution.
5.3
Rotation
The low resolution of the method when the sample was moved away from the coil affected
the results of the rotation experiment as well. Initially, a 6 mm long column of water
was put into an NMR tube which was inserted into the positioning hole that was located
7.5 mm from the center of rotation. Nutation data was collected for each position of
the sample as the gear was indexed in 45' increments. It was found that the RF field,
which drops off as 1, was significantly weaker when the sample was rotated away from
the coil, thereby making it difficult to distinguish the signal from the noise due to the
low magnitude of the excitation. It was decided to modify the rotary fixture to allow for
a smaller radius of rotation. Also, since the nutation frequency varied significantly over
small distances, it was decided to reduce the diameter of the sample to reduce the width
of the nutation peak. The rotary fixture was modified so a n" diameter brass tube was
inserted into the second hole in the blocks that were used for locating the original rotary
fixture. A 3 mm NMR tube was inserted into the brass tube. A 1.4 mm OD capillary
tube was superglued to the 3 mm glass tube (see Figure 5-4). As the brass tube was
rotated by indexing the gear, the 1.4 mm tube was rotated around the central axis of the
brass tube. The radius of rotation was thus 2.2 mm.
The pulse program was modified to allow for selective excitation of the sample. The
nutation sequence used in the first set of experiments was modified by first employing
the DANTE sequence to select the on-resonance components. The DANTE sequence is
described in Section 5.3.1. The nutation sequence was executed immediately following
the DANTE sequence so only the precession of the on-resonance components about B1
would contribute to the measured nutation signal.
73
U
~
E~
-
Figure 5-4: Photograph of setup of modified rotation fixture.
5.3.1
DANTE Pulse Program
The DANTE sequence (Delays Alternating with Nutation for Tailored Excitation) allows for the selection of the on-resonance signal while dispersing off-resonance signals [2].
Through the use of a multiple pulse train of short, intense pulses (see Figure 5-5) of duration At spaced T seconds apart, the DANTE sequence flips on-resonance spins a total
flip angle of
a = myB 1 At,
(5.1)
where m is the number of pulses contained in the DANTE sequence [2]. The spacing of
the pulses in the DANTE sequence,
T
can be used to exclude off-resonance signals from
the excitation. As a result, the frequency response of the signal consists of a comb of
sidebands spaced by
of width
T- 1
with each spike in the comb having the form of a sinc function
(mT) 1 .
The DANTE sequence can be used in conjunction with other pulse programs to se74
B 1 (t)
0
B1(w)
A Al AA
W
llI
W
HADIHIIIIHNIIIINII
t?7-
Figure 5-5: DANTE sequence in time and space. Reproduced from [2].
lect components of the magnetization which are on resonance. The nutation sequence
described in Section 4.4 was invoked after the DANTE sequence so that only the nutation
frequency of the on-resonance components of the magnetization was measured. This new
pulse program is contained in Appendix D.
5.3.2
Single Sample Rotation Experiment
An experiment was conducted in which a 9 mm long column of water was put into a 1.4
mm OD capillary tube. A zg pulse was executed on the sample to ensure the sample
was on resonance and to determine the value of the 900 pulse. The 900 pulse was used
in the DANTE pulse program to select the on-resonance components of the magnetization. Nutation data was subsequently collected using the modified nutation program that
incorporated the DANTE sequence. Initially, the capillary tube was positioned so the distance to the coil was minimized. The gear on the shaft was advanced in 300 increments.
Nutation data was collected for each position of the gear. The value of the 900 pulse was
adjusted for each position of the sample as well. The 900 pulse was used in the nutation
experiment to apply hard pulses on resonance in between data acquisitions. Since the
appropriate 90' pulse had been determined for each position of the sample, it was neces75
3 mm tube
0
r
coilsample
Figure 5-6: Geometry of rotation experiment.
sary to calibrate the measured nutation frequencies with the 90' pulse to determine the
variation between the nutation frequency and the distance from the sample to the coil.
This was done by dividing the measured nutation frequency by T, the duration T of the
RF pulse to normalize the measurement of frequency.
A diagram depicting the geometry of the experiment is shown in Figure 5-6. It was
assumed that when the sample was closest to the coil, the angle of rotation was equal
to zero. If the distance from the center of the coil to the axis of rotation is R, and the
distance from the axis of rotation to the center of the sample is r, and 0 is the angle of
rotation, then the distance from the center of the coil to the center of the sample, d, is
given by
d = VR2 + r2-2Rrcos 0.
(5.2)
Since 0, r, and R are known, it is possible to calculate the distance between the sample
and the coil. Since the nutation frequency is inversely proportional to the distance between the coil and the sample, it is possible to predict the nutation frequency for a given
separation by estimating, K, the constant of proportionality. The constant of proportionality will vary from experiment to experiment, depending on the spectrometer settings
and parameters. It is therefore necessary to determine K from the experimental data.
One way to do this would be to assume that the highest measured frequency results when
76
Nutation Frequency vs. Rotation Angle
250
+ ExperimentalData
*--- Theoretical Values
200
C
$ 150
0
0
1
3
4
5
2
Rotation Angle (Radians)
6
Figure 5-7: Plot of nutation frequency as the sample is rotated through a circle.
0 = 00 and that the lowest measured nutation frequency results when 6 = 1800. Since
V=.K
(53)
where v is the nutation frequency, then d = R - r when 0 = 00 and d = R + r when
0 = 180'. This yields two values for K, which can be averaged to yield an experimental
K value. Since the sample was stepped through discrete positions based on the rotation
of the gear, it is possible to calculate d for every angle 6 for which the data was collected.
Equation 5.3 can subsequently be used to predict the nutation frequency for each position
of the sample.
A table with the experimental data and the predicted values for the
experiment is contained in Appendix E. Figure 5-7 shows a plot of the theoretical and
experimental values of the nutation frequency as a function of the angle of rotation of the
sample. The agreement between the two curves validates the assumptions of an infinitely
long conducting coil and confirms the assumption that the nutation frequency can be used
to spatially encode the object space. Furthermore, the method was able to distinguish
77
between relatively small changes in distance with good resolution.
A single data set was acquired for each position of the sample. Since the gear was
moved in a complete circle in 30' increments, twelve data sets were acquired. This is
the equivalent of collecting data from twelve transducers equidistantly spaced around the
object at a distance of R. Each of the "free induction decays" encodes a set of projections
taken across the object space. The Fourier transform of the "FID" is analogous to the
projection data that was created in Section 3.2.
To test the reconstruction algorithm, the convolution backprojection method was applied to the projection data. The real and imaginary components of the spectrum were
separated using MATLAB. The script that was used to do this is contained in Appendix B.
The "FID" data was subsequently detrended so exponential trends and DC components
were removed. This improved the frequency spectrum of the data by reducing the width of
the peaks. The "free induction decays" were then Fourier transformed using zero padding
to increase the resolution of the frequency spectrum. The Fourier transformed projection
sets were subsequently filtered using the ramp filter, which was determined to be the optimal filter for the convolution backprojection algorithm in Section 3.1.3. The MATLAB
script that was used to do this is also contained in Appendix B.
Each of the filtered projection sets was incorporated into a projection array of dimensions NxM, where N is the number of sets of projection data and M is the number of
projections per set. The value for M depended on the resolution of the Fourier space as
a result of the zero padding in the Fourier transform. An image of this matrix is shown
in Figure 5-8. The intensity of each pixel corresponds to the amplitude of the nutation
frequency component for each frequency in the projection. Dark pixels correspond to
low or zero amplitudes for a given frequency in the Fourier transform of a "FID," while
bright pixels are associated with high amplitudes of nutation peaks. Since a 900 pulse
was applied during the nutation pulse program, it was expected that there would not be
a great deal of variation in the observed nutation frequency. Because the pulse length
T was being increased as the magnetic field strength B 1 decreased, the resulting nuta-
78
tion frequency would remain relatively constant. This can be seen in the mapping of the
frequency response of the projections used in the experiment.
2
4
C
0
E
CL
8
12
100
200
300
600
500
400
Nutation Frequency (Pixels)
700
800
900
1000
Figure 5-8: Image of intensity of the Fourier spectrum of each of the projections.
In order to reconstruct the image, it was necessary to map the data collected in the
frequency domain into an equivalent mapping in the spatial domain. It was necessary
to determine the correlation between the nutation frequency spectrum and the spatial
distance between the sample and the coil so that each element of the frequency spectrum
would map to an element in the distance array. A unique mapping needed to be created
for every projection set since the spatial encoding was not uniform across the experiment
due to the variable pulse length in the nutation experiments.
It was noticed that the spectrometer data was contained within the first 512 data
points in the frequency domain. The first 200 points contain dark intensity pixels due to
zero values or low amplitudes of the nutation components. The first points in the last
79
few data sets have high intensity pixels at the beginning of the array; however, this can
be attributed to artifacts in the data. The contribution of these high intensity peaks was
therefore neglected. Thus, it was only necessary to map a portion of the elements of the
frequency spectrum into the spatial domain.
The elements with the highest frequency in the spectrum are produced by objects
that are located infinitely far away from the coil. Conversely, the elements with the
lowest frequency in the spectrum are produced by objects that are located near the coil.
Since the wire used in the coil has a finite diameter, and the measured distance from the
sample to the coil was measured from the center of the coil, the first few elements of the
distance array correspond to positions that lie within the wire of the coil. It was thus
decided that the first few elements of the distance array would be equal to zero. The
calibration that was used was
where
f
(5.4)
k = 512a
f
is the index of the frequency array, k is the index of the spatial array, and a is
the length of the applied RF pulse. This mapping function essentially was normalized by
calibrating the nutation frequency using the length of the 900 pulse.
In this mapping, k varied from 1 to 512 to create a high resolution spatial map of
the frequency domain.
Linear interpolation was used to calculate the contribution of
neighboring pixels in the frequency domain to a particular pixel in the spatial domain.
The MATLAB script used to perform this mapping is contained in Appendix B. The
result of this transformation is shown in Figure 5-9. This figure shows a mapping of
the intensity of the nutation peaks as a function of distance. The movement of the high
intensity pixels is a physical mapping of the relative distance between the sample and
the coil. This correlates directly to the projection data that was used for the numerical
simulations described in Section 3.2.
In Section 3.2, the object space was surrounded by a ring of transducers located on a
circle of radius, R. M projections were collected from each of the N transducers by taking
the line integral of the spin densities around arcs of varying radius. In this sinogram, M
80
2
4
E
ts
0
12
50
100
150
Distance (Pixels)
200
250
Figure 5-9: Spatial mapping of the intensity of the nutation frequency for each projection.
projections collected by N transducers are mapped. The projections were obtained by
mapping the magnitude of the nutation frequency as the magnetic field strength varied
along circular arcs. The projection data shown in the sinogram is dimensionless. This
data array could describe the variation of the nutation frequency for a sample that is
arbitrarily large or small. It is therefore necessary to calculate a dimensionless value for
R that would enable the backprojection of the data for the reconstruction of an image.
One method for selecting R was to determine the length of R in pixel units. The
beginning of the distance array corresponds to a zero separation between the coil and the
sample. Thus, it can be assumed that the sample is located at this location. It is then
necessary to determine the location of the transducers with respect to the sample. The
location of the closest high intensity pixel can be attributed to a transducer located very
close to the sample. It can be assumed that the angle of rotation between the sample and
81
the transducer is 0 = 0; thus, d = R - r. The location of the furthest high intensity pixel
can be attributed to a transducer located far away from the sample. If this transducer
is located at a rotation angle of 0 = 1800, then d = R + r. The location of the ring of
transducers can therefore be determined by averaging these two values of d. Once a value
for the radius of the circle upon which the transducers lie had been determined, it was
then necessary to select the size of the object space.
Since the object space must lie completely within the circle of transducers, the maximum dimension for the object space in pixels is v'2R. When the object space is v'2R,
the transducers pass through the corners of the object space. It is therefore suggested
that an object space that is slightly less than v'2R be selected.
Once the dimensions of the object space had been determined, it was possible to reconstruct an image of the object. The algorithm used to reconstruct images using nutation
data was similar to the algorithm used for the numerical simulations in Section 3.2. One
important difference between the two algorithms is that in the simulations, a scaling factor was used to reconstruct a set of projection data onto any sized image space. For the
actual reconstruction, however, the scaling factor was removed since the characteristic dimension of one pixel was being used to calculate distances. This lent greater importance
to the selection of the object space. Without the scaling factor, it would be possible to
reconstruct a partial image due to an inadequate object space.
Figure 5-10 shows the reconstructed image. The MATLAB script used to reconstruct
the image is contained in Appendix B. The image was normalized so that the intensities of
the pixels in the image ranged from zero to one. The reconstruction shown in Figure 5-10
is of very poor quality. There is a region on the right side of the image space which has
a localized concentration of light pixels. This describes the approximate location of the
object with respect to the center of the image space; however, the image is very grainy,
and there is no definitive image of the sample.
The reconstruction algorithm effectively "smears" each of the reconstruction arcs back
across the image space. As each arc is backprojected, its intensity is added to the intensity
82
of the pixels in the image space. Bright regions of high intensity will result where the
object was located. The arcs that were used in the reconstruction process can clearly be
seen across the image space. Since there was only one sample in the object space, all of
the reconstruction arcs should intersect at the location of the object. This should result
in a bright locus on a dark background. The reconstruction arcs shown in Figure 5-10
do not intersect in a single region that defines the location of the object. This lack of
coherence resulted in the poor image quality seen in the figure.
20
40
60
80
100
120
20
40
60
80
100
120
Figure 5-10: Reconstructed image of a column of water.
An error in the mapping of the frequency domain into the spatial domain could result
in the poor image quality. Since each of the projections was formed using a specific RF
pulse length, the mapping reflected the dependence of the nutation frequency on this
value. The value of the 900 pulse, however, was empirically determined. The variability
associated with this measurement is sufficient to produce a skewed mapping. The error
83
in the mapping would contribute a significant error in the reconstruction.
To address this issue, it was necessary to perform an experiment in which a uniform
pulse width was used in the nutation experiment. This would result in a uniform mapping
which could be used to effectively create an image of the object space.
5.3.3
Single Sample Experiment with Uniform Nutation Pulse
Width
A second experiment was performed using a single sample in the 1.4 mm OD capillary
tube. The procedure for the second experiment was nearly identical to the procedure used
in Section 5.3.2; however, the length of the applied RF pulse during the nutation sequence
was set to a constant value of i 0 ps. The 900 pulse length was still used in the DANTE
sequence; however, where the first experiment also recorded nutation data using the 900
pulse length, this experiment used a constant pulse in the nutation sequence regardless
of the position of the sample. Since the nutation angle, a, varies with both the magnetic
field strength and the length of the applied RF pulse, setting T, the length of pl, to a
constant eliminated one of the variables on which the nutation frequency depended. Any
variation in the nutation frequency would therefore be attributed to the magnetic field
strength and thus, the distance from the coil to the sample.
The gear was indexed in 300 increments and nutation data was collected at each
position. It was expected that the nutation frequency would change as the cosine of the
angle of rotation varied. The final value of the nutation frequency would therefore be
on the order of the initial value. After the gear had been rotated through twelve 300
increments, it was noticed that the nutation frequency had not yet returned to its initial
value.
It was theorized that some amount of slipping was occurring as the gear was
being rotated. An additional four data points were collected, at which point the probe
was removed from the magnet. The sample appeared to be in its original position with
respect to the coil. An examination of the sample showed that the sample was no longer
firmly confined within the brass tube. The rotation of the sample with respect to the
84
brass tube could account for the proposed slipping. It was difficult to determine when the
slipping had begun, so it was assumed that slipping had occurred uniformly throughout
the experiment. Using this assumption, the sample had been moved through sixteen steps
of approximately 22.50.
An experimental value of K was found to be 7430 using the method that was developed
in the previous section. This K value was used to predict the nutation frequency of the
sample as the sample was rotated in a circle. Figure 5-11 shows a plot of the predicted
nutation frequencies and the experimentally obtained values as a function of the angle of
rotation of the sample. The raw data for this experiment is contained in Appendix E.
There is reasonable agreement between these two curves. Although the assumption that
the slipping between the brass tube and the sample occurred uniformly throughout the
experiment is not completely accurate, the general shape of the two curves agree. The
sixteen projection sets correspond to the projections that would have been collected by
sixteen transducers equidistantly spaced around the circle of radius R.
The convolution backprojection method was once again applied to the projection data
to reconstruct an image of the object. The real and imaginary components of the spectrum
were separated using MATLAB. Exponential trends and DC components were removed
from this data set to reduce the line width of the spectral peaks. Zero padding of the
Fourier transform of the "free induction decays" was used to increase the resolution of the
frequency spectrum. The Fourier transformed projection sets were subsequently filtered
using the ramp filter.
Each of the filtered projection sets was incorporated into a projection array of dimensions NxM. An image of this matrix is shown in Figure 5-12. The intensity of each pixel
corresponds to the amplitude of the nutation frequency component for each frequency in
the projection. The broadness of the peak results in a smearing of the intensity over a
range of frequency values. This agrees with the expected frequency response. When the
sample was close to the coil, a high nutation frequency resulted; the peak, however, was
quite broad due to the spatial extent of the sample. A wide bright band produced by a
85
Nutation Frequency vs. Rotation Angle
2500
2000
1500
-- Predicted Values
* Experimental Data
1000
~
500
z
''I
0
0
'
1
''
2
'
3
4
I'
5
6
Rotation Angle (Radians)
Figure 5-11: Comparison of predicted and experimental values of the nutation frequency.
broad peak is expected in the frequency spectrum. When the sample was far away from
the coil, lower nutation frequencies resulted; these peaks were narrower due to the compression of the spatial extent of the sample. In the mapping of the array, this corresponds
to the narrow bright bands that occur at low frequencies.
A sinusoidal trend can be seen in the locations of the high intensity pixels. It was
expected that as the sample was rotated in a circle about its axis of rotation, the nutation
frequency would vary inversely with the distance between the coil and the sample. Since
the distance between the coil and the sample varies with the cosine of the angle of rotation,
it was expected that the nutation peak would move through frequency space in a sinusoidal
fashion. The relative locations of the high intensity pixels of each of the projection sets
confirms this expectation.
In order to reconstruct the image, it was necessary to map the measured nutation
86
E
0
LC 10
50
6
Frequency (Pixels)
Figure 5-12: Mapping of the intensity of the nutation frequency in the frequency domain.
frequency data into the spatial domain by correlating the measured nutation frequency
to the distance between the sample and the coil. Since it is known that a reciprocal
relationship exists between the two parameters, it was simply necessary to determine
the constant of proportionality that would accurately characterize the relationship. Each
element of the frequency spectrum is contained in the frequency array and must map to
a corresponding element in the distance array. It was arbitrarily decided that the fifth
element of the frequency array would map to the final element of the distance array.
Since the nutation frequency decreases as the distance between the sample and the coil
increases, the first few elements of the frequency array would correspond to infinitely large
separations between the coil and the sample. In this mapping, the fifth element of the
array corresponds to a physical separation between the coil and the sample that is greater
than the largest distance between the two elements during the experiment. This ensures
87
that all of the frequencies that are of significance are mapped into the spatial domain.
It was also necessary to determine which element of the frequency domain would map
to the first element of the spatial domain. As described in Section 5.3.3, the first few
elements of the distance array correspond to positions that lie within the coil. For this
reason, the first fourteen elements of the distance array were assigned a -value of zero.
Subsequent elements of the distance array corresponded to measured frequencies in the
frequency domain.
The relationship between the frequency domain and the spatial domain was thus
f=
where
f
1280
,
(5.5)
is the index of the frequency array and k is the index of the spatial array. In
this mapping, k varied from 1 to 256, thereby creating a high resolution spatial mapping
of the range of distances over which nutation data was collected. Linear interpolation
was used to calculate the contribution of neighboring pixels in the frequency domain to a
particular pixel in the spatial domain. The MATLAB script used to perform this mapping
is contained in Appendix B. Figure 5-13 shows a spatial map of the frequency spectrum
that was shown in Figure 5-12. The sinogram shown in Figure 5-13 is a mapping of the
amplitude of the nutation frequency component as a function of d, the distance between
the coil and the sample. The movement of the high intensity pixels is a physical mapping
of the relative distance between the sample and the coil.
Once again, it was necessary to calculate a value for R (in pixels) that would enable
the backprojection of the data for reconstruction of an image. This was done in the same
manner as was used in Section 5.3.3. After the value for the radius of the circle upon
which the transducers lie had been determined, it was then necessary to select the size of
the object space. Thereafter, it was possible to reconstruct an image of the object.
Figure 5-14 shows the reconstructed image. The MATLAB script used to reconstruct
the image is contained in Appendix B. The image was normalized so that the intensities of
the pixels in the image ranged from zero to one. The reconstruction of the object shows
an assortment of arcs crossing the image space. There is a gray smear in the general
88
0
12
14
16
50
100
150
Position (Pixels)
200
250
Figure 5-13: Mapping of the intensity of the nutation frequency component in the spatial
domain.
region around the intersection of the arcs in the right half of the image space. There is
a concentrated region of very high intensity pixels in the center of the smeared region.
These white pixels are enveloped in a lighter, grayer region. This is consistent with the
sample that was used for the experiment. The sample used in the experiment was an
NMR tube of water. It would be expected that the reconstruction of the sample would
consist of a very bright circular region representing the water, surrounded by a lighter,
grayer area due to reconstruction artifacts.
The smeared region in Figure 5-14 is not solely due to artifacts as a result of the
reconstruction process. The paths that the reconstruction arcs make can clearly be seen
in the image. The bright center of the smeared region is the locus of the intersection of
the majority of the reconstruction arcs, and the most likely location of the water inside
89
20
40
60
80
100
20
40
60
80
100
120
Figure 5-14: Reconstruction of object in the image space.
the NMR tube. It is apparent, however, that not all of the arcs pass through this locus.
This may be due to an error in the calculation of R, an error in the angle of rotation, or
another experimental error. Since only twelve projections were collected, misalignment of
the projections has a significant impact on the quality of the reconstruction. In addition to
diminishing the intensity of the center of the sample, the stray arcs erroneously contribute
a spin density to a region which otherwise would not register a signal. If a greater number
of projections had been collected, the low level signal produced by the arcs outside of
the smeared region would have been rendered a -darker color after the image had been
normalized.
More projections would probably also reduce the intensity of the broad
smeared region while increasing the intensity of the central locus. Figure 5-15 shows a
three dimensional plot of the reconstruction of the object space. The broad base is much
wider than was expected. Instead of a sharp change in intensity in the neighborhood of
90
1.21
0.4
0.2,-
-
0
-0.21212-
80
..-
60
'.
..
.-
40.--
6
-60
20
20
0
-:100
.80
4
0
Figure 5-15: Three dimensional view of the image reconstruction.
the signal, a gradual increase in intensity is seen. Perhaps more projections would reduce
the relative intensity of the base. It is also likely that an error in the calibration of R
contributed to the gradual increase in signal.
A contour map of the intensity of the pixels is shown in Figure 5-16. From this map,
the coordinates describing the approximate location of the bright locus can be determined
to be (68, 92). If the center of the object space is located at (60, 60), then the sample was
located at a radial distance corresponding to 33 pixels. Since the value of R is known,
it is possible to calibrate the image space and relate it to the physical dimensions of the
experiment. The image reconstruction was performed using a radius of R = 95. Since
it is known that R = 5.9 mm, it is possible to calibrate approximately 14 pixels to 1
mm. It therefore can be concluded that the sample was located at a radius of 2.05 mm
away from the center of object space. The center of the object space corresponds to the
axis of rotation of the sample. It is therefore expected that the location of the sample
91
120
110
... .....
0
100
..
....
... ....
90
......
......
..
.........
80
70
60
50
40
..
--..
- ..
-.
30
20
0
11~
10
10
20
30
0b
A
40
50
60
70
80
90
100
110
120
Figure 5-16: Contour of the image reconstruction.
would be equal to the value of r for the experiment. The measured value of r for the
experiment was 2.1 mm. The agreement between the experimentally determined value
and the actual value for the location of the sample is surprisingly accurate. This shows
that the reconstruction algorithm is capable of locating the object in the correct place in
the image space; however, the resolution with which the object is imaged is not acceptable.
A simulation of the reconstruction of the sample was performed to asses the quality of
the reconstruction. It was assumed that an object with a diameter of ten pixels was located
at the point (68, 92). Projection data was collected for a set of sixteen transducers. This
data was filtered using the optimum filter and then backprojected over the image space.
Figure 5-17 shows the reconstruction of the object space as a result of the simulation.
It can be seen from this figure that the paths of the arcs are visible in the simulated
reconstruction. The simulated reconstruction does not have the broad region resulting
92
1
20
40
60
80
100
120
Figure 5-17: Simulation of reconstruction of an object using sixteen projections.
from smearing of the data. This may potentially be due to the fact that the value of R
that was used for the reconstruction was precisely equal to the value of R that was used
to collect the projection data. A three dimensional view of the reconstruction, as seen
in Figure 5-18, shows a relatively low level of signal across the image plane with a sharp
increase in intensity in the region of the object. It was noted that the intensity is not
uniform across the object.
Another possible explanation for the poor quality of the reconstructed image is that
the magnetic field produced by the distal wire of the coil is not negligible. It was initially
assumed that the distal wire was sufficiently far away that only the magnetic field from
the proximal wire would contribute to the magnetic field strength seen by the object. The
magnetic field would thus fall off as
as predicted by the Biot-Savart law. If, however,
the effect of the magnetic field produced by the distal wire has a significant effect on
93
If-
0.6
04,
.S
0.2,.-0.1
-0.2 ,.
-0.4: .120100
..
-
..
80- ..
60
.
..--
'
-
2
----
100
80
..-
4
40
20
20
0
0
4
0
Figure 5-18: Three dimensional view of simulated reconstruction.
the magnetic field strength across the object space, then the 1r approximation will no
longer be valid. This would effect the quality of the reconstruction since the projections
would incorrectly be backprojected over the object space. For two wires carrying current
in opposite directions that are separated by a distance W, the magnetic field at a point
located at a radius r from the proximal wire is
B = 1(5.6)
27r
(r
W+ r
As the object moves with respect to the two wires, the contribution of each of the wires
on the magnetic field strength varies. A simulation of the magnetic field produced by two
wires was performed. Figure 5-19 shows the magnetic field lines produced by two wires
separated by a distance, W, the approximate distance between the two wires in the coil.
The diagram in Figure 5-19 is scaled so 1 pixel is approximately 2 mm. From this figure,
it is evident that the effect of th'e distal wire is fairly significant in the region in which
the sample was placed.
94
40-
35 -
30-
20-
15-
10-
5
10
15
20
25
30
35
40
45
Figure 5-19: Simulation of magnetic field produced by two wires.
To account for the effect of the distal wire on the reconstruction algorithm, an alternative spatial encoding scheme was necessary. Instead of having a
dependence and
creating a spatial distribution once the frequency variation was known, the new method
of reconstruction involved mapping each pixel in the image space to find the appropriate
value for the field strength using a simulation of the magnetic field. Once the strength of
the magnetic field was known for a given pixel, the magnitude of the nutation frequency
component for that magnetic field strength was summed to reconstruct an image of the
object space.
The filtered backprojection method for reconstruction provides a high-quality image
efficiently; however, it requires uniformly sampled data in both the linear translational
and angular directions [17]. Since the encoding of the object no longer consists of equally
spaced elements, the filtering operation can not be performed in the frequency domain
95
-I
prior to backprojection.
An alternative imaging method that can be employed is the
backprojection filtering algorithm. The backprojection filtering algorithm assumes that
the backprojected image is subjected to the 1 smearing given by the PSF. This image
can be filtered to reduce smearing; however, there are some inherent practical limitations
associated with this procedure. Truncation of the two dimensional blurred image is required to generate an image that is the same size as the original object. This results in
a loss of information. Also, discontinuity in the slope of the filter function results in a
ringing artifact [17].
20
40
60
80
100
120
Figure 5-20: Reconstruction of sample using the contribution of the proximal and the
distal wires.
Figure 5-20 shows the reconstructed image as a result of the new algorithm. The effect
of filtering the backprojected image can be seen in the high intensity bands that frame
the boundary of the image space. The reconstructed image, however, is of much higher
96
quality than the image shown in Figure 5-14. Whereas many of the reconstruction arcs in
Figure 5-14 did not pass through the object, all of the reconstruction arcs in Figure 5-20
contribute to the object. This indicates that the effect of the distal wire accounted for
the distortion in the field that resulted in the poor quality of the reconstructed image.
One possible way to reduce the effect of the filtering operation is to increase the
size of the reconstructed image. Since the filtering process produces noise at the edge
of the image space, the object space, which is centered in the image space, will not be
significantly affected. Thus, the integrity of the image will not be diminished by the
filtering operation. The projections were backprojected into a reconstruction space of 256
x 256 since it was recommended that the reconstruction space be at least twice the size
of the image space [17]. After filtering the backprojected image, the central 128 x 128
pixels were extracted from the reconstruction to form the image shown in Figure 5-21.
20
40
60
80
100
120
20
40
60
80
100
120
Figure 5-21: Reconstruction of image using an expanded reconstruction space.
97
4
3.5,
3
2.5
-
an ,
15-
0.5
-
0
150
100.
O1'20
100
140
60
50
0
20
0
40
Figure 5-22: Three dimensional view of the filtered backprojection.
The quality of the reconstructed image is much improved. The image of the sample is
much more coherent than the image shown in Figure 5-14. The smeared region is smaller
in area. Also, the central, brighter area of the smear is more pronounced and more
uniform. A closer examination of the high intensity bright region showed a circular region
that was approximately 10 pixels in diameter. This would suggest that the bright region
is approximately 0.62 mm in diameter. This is slightly smaller than the measured value
of the inner diameter of the tube of approximately 0.8 mm. The edges of the sample may
have been blurred by the reconstruction. The gray area surrounding the high intensity is
due to the blurring of the projections. This blurring is an artifact of the backprojection
process. A three dimensional view of the reconstruction, as shown in Figure 5-22, shows
that the gray area has a much lower intensity and a much wider span than the bright
peak of the object. It therefore may be possible to optimize the filter to improve the
reconstruction further.
98
5.3.4
Two Sample Rotation Experiment
A second 1.3 mm capillary tube was added to the fixture at approximately 90' from the
first capillary tube. The second tube was attached to the 3 mm tube with a 1.3 mm
spacer so the radius of rotation between the center of the glass tube to the central axis of
rotation was approximately 3.45 mm. A column of water approximately 9 mm in length
was put into the outer capillary tube. This column of water will be referred to as Sample
2.
The same volume of water was also put into the 1.4 mm capillary tube that had
been used in the previous experiment. This will be referred to as Sample 1. Figure 5-23
shows a schematic of the geometry of this experiment. The central glass tube was initally
positioned such that Sample 2 was immediately adjacent to the coil. The procedure for
this experiment was almost identical to the procedure used in the experiment described in
Section 5.3.3. The zg pulse program was executed to match the resonant frequency of the
transmitter and the sample and to determine the length of the 900 pulse. Nutation data
was collected using the pulse program that utilized the DANTE sequence. The length of
the pulse used during the nutation experiment was a constant of p1
=
10pts.
3 nm tube
\
R
sarnple 2
r2V
dI
s ample I
Figure 5-23: Schematic of geometry of two sample rotary experiment.
99
The gear on the shaft was indexed in 150 increments. The zg pulse program was
executed each time the sample was moved, and nutation data was collected for each
position of the gear. Since there were two water tubes in the sample, it was expected
that two distinct nutation peaks would be apparent in the spectrum. A figure of a typical
frequency spectrum is shown in Figure 5-24. As the sample was rotated, the two peaks
would move to reflect the changing position of their respective water column with respect
to the coil. The measured nutation frequencies were plotted as a function of the angle of
rotation (see Figure 5-25).
X 106
1.8[.
1.61.
1.4
1.2
0)
1
CIS
0.810.6
0.41g
0.2=
0
0.5
1
1.5
Nutation Frequency (Hz)
2
2.5
x 104
Figure 5-24: Frequency spectrum of a typical "FID" of a sample containing two columns
of water.
It is evident in this figure that there are two distinct curves in the nutation data that
was produced by the two samples. One of the curves resembles a cosine curve, while the
other curve is more sinusoidal. The amplitude of the cosine curve is significantly larger
100
than the amplitude of the sine curve. This is consistent with the expected results. Since
the two samples were attached to the central 3 mm glass tube approximately 900 apart,
the resulting nutation frequencies should be about 900 out of phase with one another.
Also, since Sample 2 has a spacer between the central tube and the tube containing the
water sample, it has a larger radius of rotation than Sample 1, which has the water column
attached directly to the central tube. Thus, the range of the variation of the nutation
frequencies were expected to be larger for Sample 2 than for Sample 1 since Sample 2
traverses a greater distance from the coil. Since the initial position of Sample 2 was
immediately next to the coil, it would produce the highest measured nutation frequency
for the experiment, which would subsequently decrease as the sample was rotated away
from the coil. Since Sample 1 initially was positioned approximately 90' away from the
coil, the nutation frequency produced by that sample was expected to increase. It can
therefore be concluded that the cosine curve with the higher amplitude was produced by
Sample 2 while the sinusoidal curve was a result of the rotation of Sample 1.
Once the data produced by the two tubes were separated, it was possible to calculate
a K value based on the data. It was expected that the K value resulting from the
two data sets would be identical since the data was collected in a single experiment. The
experimental values of K for the two data sets were found to be 5600 and 6600 for Samples
1 and 2 respectively. The calculation of K assumes that the highest measured nutation
frequency is produced when the sample has an angle of rotation of 6 = 0' and that the
lowest measured nutation frequency results when the angle of rotation is 0 = 1800. The
initial positions of the two samples, however, is not definitively known. It is possible that
the addition of a phase angle would produce greater agreement between the calculated
values of K for the two data sets. The calculated K values were used to predict the
nutation frequency as a function of the angle of rotation. Figure 5-26 shows a comparison
of the predicted nutation frequencies and the experimental data. The raw data from the
experiment is contained in Appendix E. The agreement between the predicted nutation
frequency and the experimental data is good in the regions where the samples were far
101
4500
4000
3500
-
8 3000
2500
+
a
U2000-,
C 1500 P
.0 1000
z 500
M
"a
0
1
3
2
4
5
6
Rotation Angle (Radians)
Figure 5-25: Predicted and experimental nutation frequencies as a function of rotation
angle.
away from the coil. The variation between the predicted and the experimental data may
be a result of the effect of the magnetic field produced by the distal wire.
A mapping of the magnetic field as a result of current flowing in opposite directions
through two wires separated by a distance W was created. This mapping was identical to
the mapping produced in Section 5.3.3, and was used to create the backprojection of the
image in a similar manner to the method described in the previous section. The filtering
operation was performed after the backprojection of the image had been complete. Once
again, the effects of filtering the backprojected image led to artifacts in the image; this
was partially remedied by enlarging the reconstruction space.
Another alternative is to backproject the object into the enlarged image space and then
102
Experimental Results
4500
4000
3500
3000
a, 2500
2000
1500
0
1000
z 500
0
A
N
Experimental Data
Sample 1
A14.
A6
0
1
2
3
4
5
Angle of Rotation (radians)
6
Predicted Nutation
Frequency Sample
1
SExperimental Data
Sample 2
------- Predicted Nutation
Frequency Sample
2
Figure 5-26: Comparison of predicted nutation frequencies and measured values.
impose a windowing function which would smooth the backprojected image. Filtering of
the smooth backprojected image will not produce the high frequency components along
the boundary of the image space due to sudden, sharp transitions at the boundary. This
approach was used to reconstruct an image of the two samples. It was decided that the
image space would be a 128 x 128 array. The projections were thus backprojected into
a reconstruction space of 256 x 256. A windowing function was subsequently applied to
the enlarged backprojected space. The windowing function smoothed the backprojection
by imposing a cosine curve over the region surrounding the central object space. This
reduced the effect of most of the noise on the filtered reconstructed image. Finally, two
dimensional filtering was performed to extract the image from the reconstruction.
Figure 5-27 shows an image of the reconstructed object. It can be seen that the quality
of the images of the two samples varies. Sample 1, which was closer to the center of the
103
OR Magw"-
Aiw-"'
*
-
20
40
60
80
100
120
20
40
60
80
100
120
Figure 5-27: Reconstruction of two tube sample
image space, is located in the central region of the right half of the reconstruction space.
Sample 2, which had the spacer between the central tube and the water column, is shown
at the bottom of the reconstructed image. There is a significant amount of blurring in
the image. Cloudy regions define the general location of the two samples; however it
is difficult to distinguish the precise location of the water sample inside of the smeared
region. There is a patch of bright pixels located in the center of the image of Sample
1. This patch is approximately 6 pixels in diameter. There is a suggestion of a similar
region in the region that defines Sample 2, but the intensity of that locus is significantly
lower than the intensity of the bright spot in Sample 1. This is most likely due to the
normalization of the image after the reconstruction and filtering had occurred. The bright
regions of 6 pixels correspond to a water sample approximately 0.4 mm in diameter. This
is smaller than the inner diameter of the tube. Some blurring of the edge of the water
104
-
,
sample may have resulted in the disparity between the actual value and the experimental
measure of the size of the water sample.
The approximate coordinates of the bright patches are (103, 70) and (58,117) for
Samples 1 and 2 respectively. Since the center of the image space is' approximately (64, 64),
the radial distance to the two samples is approximately 39 pixels and 53 pixels. Since 95
pixels correspond to approximately 5.9 mm, the two bright patches are at approximately
2.4 mm and 3.3 mm from the center of the image. These distances can be compared to
measured r values of 2.2 mm and 3.4 mm for Samples 1 and 2 respectively. Although it
is difficult to determine the actual location of the sample within the blurred region, the
accuracy of the method is reasonable.
The image shown in Figure 5-27 was subjected to a windowing function prior to filtration. There is not a great difference between an image of this figure and the backprojection
that was created by filtering the image in the expanded image space without imposing
a windowing function on it. This is because the effect of the noise due to blurring is
most severe at the edges of the image space. By enlarging the image space, the object
space, which is in the center of the reconstruction space, is not significantly affected by
the filtering noise.
105
Chapter 6
Discussion
The initial results of the experimental phase of the research are very promising. It was
found that the model used in the theoretical development was not accurate for the coil that
was designed. The effect of the magnetic field produced by the distal wire proved to be a
significant source of error in the initial reconstruction. Accounting for the magnetic field
of the distal wire significantly increased the fidelity of the image; however, blurring in the
neighborhood of the object is still an issue. Furthermore, because the spatial mapping
of the object in frequency space is not equally spaced over the spectrum, convolution
of the kernel prior to backprojection is not possible. The reconstruction, therefore, must
subsequently be performed using the backprojection filtering method instead of the filtered
backprojection method that was developed in Chapter 3. The resolution of images that
are reconstructed using backprojection filtering is lower than the resolution of images
that are reconstructed using filtered backprojection. There are a number of questions
that arose during the experimental process. Further work needs to be done to resolve
many of the issues. A brief discussion of some of the issues follows.
106
6.1
Coil Design
It is possible that by creating a different type of coil, by altering the coil geometry or
by redesigning the coil to be an infinite conductor, the spatial mapping described in
Chapter 3 would be applicable. This would eliminate some of the issues associated with
filtering the backprojected image and may allow for the implementation of alternate image
reconstruction techniques. Also, the coil that was designed for the experiments in this
work was intended to create one and two dimensional images. Further investigation into
spatially encoding the spin densities in three dimensions using coils such as the cone
coil [10] may potentially allow for even more applications of this technology.
6.2
Magnetic Susceptibility
Another aspect of the coil design which could potentially impact the process is the material
of the wire used in the coil. The introduction of the coil into the magnet locally distorts
the magnetic field. These inhomogeneities can be reduced by shimming gradient coils
to homogenize a localized area of the magnetic field. One potential means of reducing
the effect of the coil on the magnetic field is to use "zero susceptibility" wire for the
construction of the coil. Zero susceptibility wire is composed of strands of wire that are
each made of different materials. Since each material has a distinct value describing its
magnetic susceptibility, it is possible to create a composite wire that has zero susceptibility
by summing the susceptibilities of each of the component strands of wire in the bundle.
The composite wire will have almost no effect on the homogeneity of the magnetic field
in the vicinity of the sample.
Eliminating local disturbances due to the introduction of the coil in the magnet may
potentially reduce the need for shimming of the static gradient fields to homogenize the
field in the area around the sample. If shimming of the gradients is not necessary, then
the time required for the set up of the experiment will be significantly reduced. Also, the
variability of the magnetic field during the experiment will be reduced. This will further
107
standardize the data acquisition process.
6.3
Spatial Markers
Refinements to the reconstruction algorithm could potentially improve the quality of the
images produced by this method. The spatial mapping that was performed to convert
the nutation data from the frequency domain to the spatial domain was a critical step
in the reconstruction process. The value of R, the radius of the ring of transducers, was
determined from this mapping. The quality of the image reconstruction was very dependent on the value that was chosen for R. If the R value was too large, the backprojected
arcs would not intersect to form an image. If R was too small, arcs would intersect to
form multiple low-intensity phantom peaks. Either of these situations would result in a
blurred image of diminished integrity.
One possible method of determining R might be to put a small marker at the center of
the axis of revolution. Since the marker is coaxial with the axis of revolution, a constant
signal would result in the frequency domain. This constant signal in the frequency domain
would map to the correct value of R in the spatial domain regardless of the mapping that
was used to convert from frequency to space. Furthermore, this would also provide the
scale for the image. Since it is known that the center of the image is located at a certain
distance from the transducer, an appropriately sized object space may be selected. This
serves to reduce another variable in the process.
One potential complication may arise from the fact that it may not be possible to
place a marker at the center of the axis of rotation. The reconstruction algorithm is most
accurate when the sample is located in the neighborhood of the origin. Thus, it may be
disadvantageous to put a marker at the location that would be reconstructed with the
highest fidelity. Also, it may be physically impossible to put the marker in the center of
the object space due to the nature of the object.
An alternative method of marking the space would be to put markers on the edge of
108
the object space. If multiple markers are placed symmetrically about the axis of rotation,
it will be possible to calculate the value of R. This might be a more viable option for
determining R since there is greater flexibility in the placement of the markers. Also, this
alternative leaves the center of the object space available for the placement of an object.
6.4
Number of Projections
One of the biggest differences between the simulations of the reconstruction algorithm and
the actual implementation was the number of projections that were used to reconstruct
the sample. The high fidelity of the PSF in Chapter 3 can partially be attributed to
the number of projections used in the reconstruction. During the experimentation, the
number of teeth in the gear limited the number of projections that could be collected. As
the complexity of the samples increases, the number of projections required to produce
a quality image will increase as well. To accommodate this need, it will be necessary to
design a device that can be used to rotate the sample in fine, discrete intervals. Such
a device would have to satisfy a number of requirements.
In addition to being non-
ferromagnetic, the magnetic susceptibility of the device will have to be considered. If it
is located at a distance from the sample and the center of the magnet, then the effect
of the device on the homogeneity of the magnetic field may not be critical. If, however,
the device is located in the vicinity of the sample, local distortions in the magnetic field
could affect the spatial encoding of the object. The device would also have to fit inside
the bore of the magnet. It would need to accurately position the sample by maintaining a
fixed angular position. Backlash, slippage, or positioning errors would have to be minimal
to maintain a high quality image. Also, the device would need to be unaffected by high
frequency noise or vibrations.
Automating the motion of the sample in between data acquisitions would minimize
the disturbance of the system. In the process that was used in this work, after the sample
was moved, the tuning and matching of the probe was verified. The gradient coils were
109
subsequently shimmed to homogenize the field. By automating the movement of the
sample, it may be possible to reduce the disturbance of the sample and reduce the time
required for setting up between the acquisition of data for each projection set.
110
Chapter 7
Conclusion
This work has demonstrated that there is potential for imaging by using the radiofrequency gradients. The image reconstruction algorithm shows that it is possible to spatially
encode a two dimensional object space using the nutation frequency. Since the nutation
frequency varies with the magnetic field, a nonlinear mapping of the spatial variation of
the magnetic field can be used to backproject the object. This is true regardless of the
shape of the magnetic field lines. It is, however, necessary to know how the magnetic field
varies in space in order to implement this method. The accuracy with which the field
is mapped determines the accuracy of the image. Using this method no longer makes
it a requirement that the magnetic field be uniform or homogeneous. Instead, the only
requirement is that the magnetic field variation be known.
The implications of the experimental work are clear; this method has the potential of
being quite significant. There are a great number of issues that need to be resolved to
determine the resolution of the method and the ability of the method to produce quality
images. The most significant of these requires building an apparatus that would enable a
greater number of projections to be collected. This would increase the resolution of the
image and allow for a qualitative comparison of RF gradient imaging and conventional
imaging methods. A redesign of the coil to simulate an "infinite" conductor would also
be a worthwhile endeavor. This would allow for the filter kernel to be convolved with
111
the projection data prior to backprojection, thereby improving the quality of the image.
The general case, however, involves the reconstruction of an arbitrary object in a known
magnetic field to be imaged using the backprojection filtering method. Improving the
quality of the image in the general case requires the development of post processing
techniques that can be applied to the backprojected image.
Spatially encoding the object using the nutation frequency of the object has many
advantages.
This method potentially allows for imaging using permanent magnets or
a different geometry to facilitate imaging of objects that might not be imaged using
conventional magnets. Furthermore, if the field can be mapped accurately, there is no
longer a need for static gradient coils. Without rapidly switching gradient coils, the power
source required for the magnet is significantly reduced. Thus, it may be possible to reduce
some of the hardware and the power supplies that are required for conventional systems.
112
Appendix A
Reconstruction Using the Hankel
Transform for an Arbitrary Object
In section 3.1.5, an object located at the origin was reconstructed at any point in the image
space. In this section, an object located at an arbitrary location in the object space is
reconstructed. For an object located at a distance a from the origin (see Figure A-1), the
Hankel transform of a projection can be expressed as
P(w) = j
rp(r)Jo(wr)dr.
(A.1)
The projections can also be expressed in terms of the two dimensional Fourier transform
p( x2 + y 2 ) 4-=> P(U, v).
(A.2)
Since
P(u, v)
=
27rp(w)f_.y2 +v,
(A.3)
it follows that
P(u, v) = 27rj rp(r)J,(r Vu 2 + v 2 )dr.
(A.4)
If it is assumed that the object is a point object located arbitrarily in the object space,
where p is the distance from the transducer to the object, then
p(r) = 6(r - p).
113
(A.5)
(Ros
,R
Si
R
p
r
'I
Figure A-1: Geometry used for reconstructing a point object located off the origin.
The Hankel transform of this is thus
P(w) = pJO(Wp),
(A. 3)
which subsequently causes the Fourier transform to become
P(u,v) = 27rpJ(wp).
(A. 7)
The shifting property of the two dimensional Fourier transform states
f (x - x, y - y,) -<= F(u, v) e(a+YO).
(A. 3)
The Fourier transform of the object at a point (x,, y,) = (p cos#, p sin#) thus becomes
P(u, v)x,,,, = 2,rpJo(wp)e,-(up cos 8+vp sin3 )
An angle 0 can be defined such that
* = w cos
* = w sino
)
V)= tan(U
114
(A. 9)
w = 1/32 +V2.
Equation A.9 thus becomes
P(u, v) I .,y. = 27rpJo(wp)e-j(pw
cos Vcos /+pw sin Osin /)
(A.10)
This expression alternatively be expressed as
(A.11)
P(u,v) Ix.,yo = 27rpJo(wp)e-Pcosf-).
The back projection of the object on the x-axis can be expressed as
y) = I(x, pp,/(X - pcos3, y - psin)d3,
=27
P (,Y
f
(A.12)
or, using expression A.11
P(u, v) =
j
(A.13)
pJO(p)eiwPcs(Vb-O)d.
It can be seen from Figure A-1 that
cos3 =R cos$ - a
p
(A.14)
and
sin 0 = R sin q.
(A.15)
p
Expanding the exponent in equation A.13 and substituting these expressions for cos # and
sin 8 results in
P(U, v) =
ji
pJo(Cp)e-wP(cos P(Rs,-a)+sin
( Rsin
))
(A.16)
0)dO
(A.17)
This can be simplified to
P(U, V)
=
2r pJo(wp)e-jw(R cos V cos *+R sin 0 sin $--a cos
or,
P(u, v) = ejwa cos V
I
27r
pJ(wp)e-jwR cosvP-)dO.
115
(A.18)
U
w
V
Figure A-2: Geometry for Graf's Theorem [181.
It is possible to take the partial derivative of both sides of equation A.14 to determine an
expression for d# in terms of d#. This results in
R 2 - aR cos d
d3=
(A.19)
dq.
2
This can be substituted in Equation A.17 to yield
P (u, v)
-
ejwacos
j
27rj ,p),--jwR
J
p
o
cos(P-O)(R2
p
(
- aR cos
f
-.
do.
Graf's Theorem [18] may be used to obtain an alternate expression for JO(wp).
(A.20)
Graf's
Theorem states that
JV(w)cos(vx) =
J,+k(u)Jk(v)cos(ka),
E
k=-oo
where Figure A-2 shows the relationship between u, v, w, a, and X.
Figures A-1 and A-2 shows that for the image reconstruction,
V =0
w =p
U
=R
v
= a
116
(A.21)
A comparison of
Equation A.21, therefore, can be rewritten as
JO(P) =
(A.22)
,
Jk(R)J(a)cos(k#).
k=-oo
The geometry associated with similar triangles permits
00
JO(Pp)=
5
(A.23)
Jk(wR)Jk(wa) cos(k).
k=-oo
Equation A.23 can alternatively be expressed as
Jo(wp)
=
Jo(wR)Jo(wa) + 2 E Jk(wR)Jk(wa) cos(k).
(A.24)
k=1
Substituting Equation A.24 into Equation A.20 yields
P (u, v)
=
J0 (wR) Jo(wa)e wa cos
ejwa cos
(R 2 - aR cos 0) do
+
p
2
jwR cos(V-O) (R - aR cos 0)
cos(O-0)
2-jwR
b f27
00
2
Jk(wR)Jk(wa) cos(k)
(A.25)
dq.
(k=1
A closed form solution for the Fourier Transform of this expression could not be found. It
was thus not possible to find a general solution for the filtered backprojection of an arbitrary object in the object space at any point within the reconstruction space. Numerical
solutions for the filtered backprojection, however, are possible.
117
Appendix B
MATLAB Scripts
Numerical Simulations
B.1
B.1.1
Creating Circular Projections
%
imp-proj Create a set of impulse projections around circular arcs
%
Useage:
%
where
proj = imp-proj(im-size, Radius, n-proj, npoints,x,y)
im-size - size of image
Radius - radius of the circular sampling path
n-proj - number of projections taken around the circle
n-points - number of points in each projection.
x - the x coordinate of the impulse
y - the y coordinate of the impulse
%
Returns:
proj(n-proj, n-points) - an array of projection data.
%
Notes:
The projections are taken at angular intervals of 2*pi/n-proj,
118
10
around a circle of radius Radius. A spatial distance of
2*Radius/n-points is assumed for sampling along the projection.
%0
Radius should be large enough so that the sampling path
%
encompasses the image,ie Radius > 0.707*im-size
function sum = imp-proj(im-size, Radius, n-proj, n-points,x,y)
20
sum=zeros (n-proj,n-points);
ctheta=cos (linspace(O, 2*pi,n_proj));
stheta=sin(linspace(0,2*pi,n proj));
for T = 1:n-proj
x-org = im-size/2 + Radius*ctheta(T);
y-org = im-size/2 + Radius*stheta(T);
distance = sqrt((x - x-org)^2 + (y - y-org)^2);
k= round (distance*n-points/ (2*Radius));
sum(T,k) =1;
30
end
119
Convolving the Filter Kernel
B.1.2
%
Function: fft-cornv(proj,n-p roj,n-samples,filter,d)
% INPUT ARGS:
filter - a string specifying the filter
classic ramp filter
%
'ram-lak'
%
'shepp-logan' Shepp-Logan filter
'cosine'
Cosine window weighted ramp filter
%
'hamming'
%
'hann'
Hamming window weighted ramp filter
Han window weighted ramp filter
%
len
- the length of the projections
%
d
- the fraction of frequencies below the nyquist
%/
1o
which we want to pass
% OUTPUT ARGS: filt
- the filter to use on the projections
function cproj = fft conv (proj ,nproj ,nsamples,filter,d)
p = proj I;
20
len = size(p,1);
H = designFilter(filter,n-samples, d);
p(length(H),1)=0;
cproj = fft(p);
% Zero pad projections
% p holds fft of projections
for i = 1:size(p,2)
cproj(:,i) = cproj(:,i).*H; %frequency domain filtering
end
120
% p is the filtered projections
cproj = real(ifft(cproj));
cproj(ien+1:end,:) =
cproj = cproj' ;%
Truncate the filtered projections
30
% Sub-Function: designFilter
% Returns the Fourier Transform of the filter which will be
% used to filter the projections
% INPUT ARGS:
filter - a string specifying the filter
classic ramp filter
%'ram-lak'
'shepp-logan' Shepp-Logan filter
/
Cosine window weighted ramp filter
%'cosine'
%
'hamming'
%
'hann'
Hamming window weighted ramp filter
Han window weighted ramp filter
/
len
- the length of the projections
%
d
- the fraction of frequencies below the nyquist
%0
which we want to pass
% OUTPUT ARGS: filt
-
the filter to use on the projections
function filt = designFilter(filter, len, d)
order = max(64,2^nextpow2(2*en));
% First create a ramp filter - go up to the next highest
% power of 2.
filt = 2*(O:(order/2))./order;
121
40
w = 2*pi*(O:size(filt,2)-1)/order;
% frequency axis up to Nyquist
switch filter
case
ram-lak'
% Do nothing
case
shepp-logan
60
% be careful not to divide by 0:
filt(2:end) = filt(2:end)
.*
(sin(w(2:end)/(2*d)) ./(w(2:end)/(2*d)));
.*
cos(w(2:end)/(2*d));
.*
(.54 + .46 * cos(w(2:end)/d));
case 'cosine'
filt(2:end) = filt(2:end)
case 'hamming'
filt(2:end) = filt(2:end)
case 'hann'
filt(2:end) = filt(2:end) .*(1+cos(w(2:end)./d)) / 2;
otherwise
error(' Invalid filter selected. ');0
end
filt(w>pi*d) = 0;
filt = [filt'
; filt(end-1:-1:2)'];
% Crop the frequency response
% Symmetry of the filter
122
B.1.3
Back Projecting an Image
%
back-proj Back project a set of circular arc projections into an image plane.
%
Useage: image = back-proj(proj, im.size, Radius, n-proj, n-points)
%
where proj(n-proj,n-points) - a set of circular arc projections.
%//
im-size
-
size of image to be created
%/
Radius
-
radius of the circular sampling path used to create
projections
%0
%
nproj - number of projections taken around the circle
%0
n-points - number of points in each projection.
%0
10
%
Returns: image(im-size, im-size) - an image created by back projection.
%
Notes:
The projections are assumed to be at angular intervals of
2*pi/n-proj around a circle of radius Radius. A spatial distance of
2* Radius/n-points is assumed for sampling along the projection.
No attempt is made to normalize the image.
function image = back proj(projim -size, Radius,n-proj,n-points)
image = zeros(im-size);
20
mid = (im-size-1)/2;
xcircle = mid + Radius*cos (linspace (0,2*pi* (nproj-1) /nproj ,nproj));
ycircle
=
mid + Radius*sin(linspace(0,2*pi*(n-proj-1)/n-proj,n-proj));
scale = n-points/(2*Radius);
% Iterate over all pixels
for row=l:im-size
y = im-size-row;
123
row
for col = 1:im-size
x =
col-1;
30
for T = 1:n-proj
distance = sqrt((xcircle(T) - x)^2 + (ycircle(T) - y)~2)*scale;
idist = fix(distance);
delta = distance-idist;
if idist < n-points;
image(row,col) = image(row,col) + proj(T,idist) +
delta* (proj (T,idist+1) - proj(T,idist));
end
end
end
40
end
% Normalize the
image
%image = image/max(image(:));
124
B.1.4
Backprojection Over an Enlarged Image Space
% bigback-proj Back project a set of circular arc projections into an enlarged
image plane.
% Useage: recon = bigback-proj(proj, im-size, reconsize, Radius, n-proj, n-points)
%
where proj(n.proj,n-points) - a set of circular arc projections.
im-size - size of image
reconsize - the size of the reconstruction image >4* Radius
Radius
-
nproj
-
radius of the circular sampling path used to create projections
number of projections taken around the circle
n-points - number of points in each projection.
10
% Returns: recon(reconsize, reconsize) - an image created by back projection.
% Notes:
The projections are assumed to be at angular intervals of
2*pi/n-proj around a circle of radius Radius. A spatial distance of
2*Radius/n-points is assumed for sampling along the projection.
No attempt is made to normalize the image.
function recon = bigback-proj (proj,im-size,reconsize, Radius,n proj ,n.points)
recon = zeros(reconsize,reconsize);
mid = (reconsize-1)/2;
xcircle = mid + Radius*cos (linspace (0,2*pi* (n-proj -1) /nproj,nproj));
ycircle = mid + Radius*sin(linspace (0,2*pi* (n-proj -1) /nproj,nproj));
scale = n-points/(2*Radius);
%Iterate over all pixels
for row=1 :reconsize
125
20
y = reconsize-row;
row
for col = 1:reconsize
30
x = col-1;
for T = 1:n-proj
distance = sqrt((xcircle(T) - x)^2 + (ycircle(T) - y)^2)*scale;
idist = fix(distance);
delta = distance-idist;
if idist>O & idist < n.points
recon(row,col) = recon(row,col) + proj(T,idist); %+
delta*(proj(Tidist+1) - proj(Tidist));
end
end
40
end
end
% Normalize the image
recon
=
recon/max(recon(:));
126
B.1.5
Collecting Projections from an Image
% photoproj Create a set of projections around circular arcs from an image
%
Useage:
%
where
proj = photo-proj(image, im-size, Radius, n-proj, n-points)
image - a square image of size im-size x im-size
%1/
im-size
-
size of image
%/
Radius
-
radius of the circular sampling path
%O
nproj - number of projections taken around the circle
%0
n-points - number of points in each projection.
10
%0
%
Returns:
proj(n-proj, n-points) - an array of projection data.
%
Notes:
The projections are taken at angular intervals of 2*pi/n-proj,
%0
around a circle of radius Radius. A spatial distance of
%0
2*Radius/ n-points is assumed for sampling along the projection.
%0
Radius should be large enough so that the sampling path
%0
encompasses the image,ie Radius > 0.707*im-size
function sum = photo proj(image, im-size, Radius, n-proj, n-points)
ds = 1;
% Set the interval to one pixel
r = linspace(2*Radius/n-points,
2*Radius, n-points);
angle = linspace(O, 2*pi*(nproj-1)/nproj, n-proj);
x-org = (im-size-1)/2 + Radius*cos(angle);
y-org = (im-size-1)/2 + Radius*sin(angle);
sum = zeros(n-proj);
127
20
for T = 1:n-proj
T
30
for rad = 1:n-points
n-steps = pi*r(rad)/ds;
comega = cos(linspace(angle(T) + pi/2, angle(T) +
1.5*pi*(n-steps-1)/n-steps, n-steps));
somega = sin(linspace(angle(T) + pi/2, angle(T) +
1.5*pi*(n-steps-1)/n-steps, n-steps));
s= 0;
for i = 1:n-steps
ix = round(r(rad)*comega(i) + x-org(T));
iy = round(r(rad)*somega(i) + y-org(T));
40
row = im-size - iy;
col = ix+1;
if (row>O) & (row<=im-size) & (col>O) & (col<=im-size)
s = s + image(row,col);
end
end
sum(T,rad) = s;
end
end
50
128
B.2
Image Reconstruction
Data Extraction
B.2.1
% fid-read Reads the FID data and separates the real and the imaginary data
% Useage: function [Re, Im]=fidread(file, td)
% where
file - string name of the file that is being opened by fidread
%
td- time domain of the acquisition *MUST BE A POWER OF TWO*
% Returns: Re
Im
-
vector containing real data
-
vector containing imaginary data
function [Re, Im]==fidread(file, td)
fl=fopen(file,' r ,' b');
10
fid=fread(fl, ' int32');
fclose(fl);
Re=zeros(td,1);
Im=Re;
for j=1:td,
Re(j)=fid(2*j-1);
Im(j)=fid(2*j);
end
129
B.2.2
Data Processing
function out = fidprocess(Re,Im, top)
ri = detrend(Re);
il = detrend(Im);
f = abs(fft(rl + i*il));
for j= 2:top
out(j) = f(j) + f(514-j);
end;
%out=f(1:top);
out=out/max(out(5:top));
130
B.2.3
Single Sample Rotation Data
% Script to extract FID data for Dec 20, 2000
function [data,proj] = fidgetnewl;
top = 100;
[Re02,Im02]=fidread( f id2l',512);
p02=fidprocess(ReO2, Im02, top);
[Re04,Im04]=fidread( f id23',512);
p04=fidprocess(ReO4, Im04, top);
10
[Re06,Im06]=fidread( 'fid25',512);
p06=fidprocess(ReO6, Im06, top);
[Re08,Im08]=fidread('fid271,512);
p08=fidprocess(ReO8, Im08, top);
[RelO,ImlO]=fidread(' f id29',512);
plO=fidprocess(ReO, ImlO, top);
[Re12,Im12]=fidread(' f id3l',512);
20
p12=fidprocess(Re12, Im12, top);
[Re14,Im14]=fidread(' fid33 ,512);
p14=fidprocess(Re14, Im14, top);
[Re16,Im16]=fidread(' f id35 ',512);
p16=fidprocess(Rel6, Im16, top);
131
[Rel8,Iml8]=fidread('fid37' ,512);
p18==fidprocess(Rel8, Imi8, top);
30
[Re20,Im20]=fidread(' f id39' ,512);
p20=fidprocess(Re2O, Im20, top);
[Re22,Im22]=fidread(' f id4l',512);
p22=fidprocess(Re22, Im22, top);
[Re24,Im24]=fidread(' f id431,512);
p24=fidprocess(Re24, Im24, top);
40
proj=[p02; p04; p06; p08; plO; p12; p14; p16; p18; p20; p22; p24];
j
for
= 1:12
proj(j,2)=O;
proj (j,3)=0;
end;
% Create a sinogram in the frequency domain:
for
j
= 1:256
50
for k = 1:3
map(j,k) = (j-1)/255;
end;
end;
figure;
132
colormap(map);
image(255*proj);
% Create a sinogram in the spatial domain
a = [33; 35; 37; 50; 65; 75; 70; 75; 65; 55; 50; 40];
60
data = zero's(12,256);
for
j
=
1:12
for k = 10: 256
dist = 10*k/(a(j)*1280*4);
freq = 1/dist;
intf = fix(freq);
if intf < 1024
intf, j,k
fracf = freq - intf;
data(j,k) = proj(j,intf + 1) + fracf*(proj(j,intf + 2)-proj(j,intf + 1));
end;
end;
end;
figure;
colormap(map);
image(255*data);
133
70
B.2.4
Single Sample with Uniform Pulse Data
% Script to extract FID data for Dec 21, 2000
function data = fidgetnewl;
top = 100;
[Re02,Im02]=fidread(' f id02' ,512);
p02=fidprocess(ReO2, Im02, top);
[Re04,Im04]=fidread(' f id05' ,512);
p04=fidprocess(ReO4, Im04, top);
10
[Re08,Im08]=fidread(' f id08' ,512);
p08=fidprocess(ReO8, Im08, top);
[Re10,Im10]=fidread(' fidO 1,512);
p1O=fidprocess(Re1O, ImlO, top);
[Re12,Im12]=fidread(' f id12',512);
p12=fidprocess(Re12, Im12, top);
[Re14,Im14]=fidread(' f id14',512);
20
p14=fidprocess(Rel4, Im14, top);
[Re16,Im16]=fidread('f id16',512);
p16=fidprocess(Rel6, Im16, top);
[Re18,Im18]=fidread(' f id18' ,512);
p18=fidprocess(Re18, Im18, top);
134
[Re20,Im2O]=fidread( 'fid201,512);
p20=fidprocess(Re20, Im20, top);
30
[Re22,Im22]=fidread(' f id22 1,512);
p22=fidprocess(Re22, Im22, top);
[Re24,Im24]=fidread(' f id24' ,512);
p24=fidprocess(Re24, Im24, top);
[Re26,Im26]=fidread( 'fid261,512);
p26=fidprocess(Re26, Im26, top);
40
[Re28,Im28]=fidread(' f id28',512);
p28=fidprocess(Re28, Im28, top);
[Re30,Im30]=fidread(' fid30',512);
p30=fidprocess(Re3O, Im30, top);
[Re32,Im32]=fidread(' f id321,512);
p32=fidprocess(Re32, Im32, top);
[Re34,Im34] =fidread(' f id34' ,512);
50
p34=fidprocess(Re34, Im34, top);
proj=[p02; p04; p08; plo; p12; p14; p16; p18; p20; p22; p24; p26; p2 8 ; p30; p3 2 ; p3 4 ];
135
N=16;
j
for
=
1:N
proj (j,2)=0;
proj (j,3)=0;
end;
60
% Create a sinogram in
for j = 1:256
the frequency domain
for k = 1:3
map(j,k) = (j-1)/255;
end;
end;
figure;
colormap (map);
image(255*proj);
% Create a sinogram in the spatial domain
70
data = zeros(N,256);
for
j
=
1:N
for k = 14: 256
dist = k/1280;
freq = 1/dist;
intf = fix(freq);
fracf = freq - intf;
data(j,k) = proj(j,intf + 1) + fracf*(proj(j,intf + 2)-proj(j,intf + 1));
end;
end;
80
figure;
colormap(map);
image (255 *data);
136
B.2.5
Mapping the Magnetic Field
function
B = BiotSavart(imsize, R,alpha, D)
x
=
R*cos(alpha);
yO
=
R*sin(alpha);
x1 = (R+D)*cos(alpha);
yl = (R+D)*sin(alpha);
mid = (imsize+1)/2;
for xa = 1: imsize
for ya = 1: imsize
dxO = (xa-mid) - xO;
dyG = (ya-mid) - yO;
10
dxl = (xa-mid) - x1;
dyl = (ya-mid) - yl;
RO = sqrt(dxO^2 + dyO^2);
RI = sqrt(dx1^2 + dy1^2);
cosO = dxO/RO;
sinO = dyO/RO;
cosi = dxl/R1;
sinI = dyl/Ri;
MagBO = 1/RO;
MagBi = 1/Ri;
Bx = -MagBO*sinO
By =
20
+ MagBl*sinl;
MagBO*cosO - MagBl*cosl;
B(xa,ya) = sqrt(Bx^2 +-By^2);
end;
end;
137
B.2.6
Backprojection
% BS-back-proj Back project a set of nonlinear projections into an image plane.
% Useage: image = BS-back-proj(proj, imsize, Radius, D, K, n-proj, n-points)
% where
proj(n-proj,n-points) - a set of circular arc projections.
%0
im-size
-
size of image to be created
Radius
-
radius of the circular sampling path used to create projections
D - separation between the two conductors
%0
K - constant of proportionality from Biot-Savart law
%O
n-proj - number of projections taken around the circle
%
n-points - number of points in each projection.
% Returns:
10
image(im-size, im-size) - an image created by back projection.
function image = bs back-proj (proj,im-size,Radius, D K ,nproj ,npoints)
image = zeros(im-size);
mid = (im-size+1)/2;
for T = 1:n-proj
alpha = 2*pi *(T-1)/n-proj;
b = BiotSavart(im-size, Radius, alpha, D);
b = K*b;
% Iterate over all pixels
for row=1:im-size
%
row
for col = 1:im-size
f = b(row,col);
fl = fix(f);
138
20
delta = f -fl;
if fl < n-points;
image(row,col)
=
image(row,col) + proj(T,fl) +
delta*(proj(T,fl+l) - proj(T,fl));
end
end
end
% newim(image);
end
% Normalize the image
image = image/max(image(:));
139
30
B.2.7
Filtering of the Backprojection
function image = filtim(in,imsize, rmax);
z= fftshift(fft2(in));
im = zeros(imsize,imsize);
mid = imsize/2 + 1;
for row= 1:imsize
for col = 1:imsize
r = sqrt((row-mid)^2 + (col-mid)^2);
if r < rmax
im(row,col) = r* z(row, col);
else
10
im(row,col) = 0;
end
end
end
image = abs(ifft2(ifftshift(im)));
140
B.2.8
Two Sample Data
% Script to extract FID data for Dec 22, 2000
function data = fidgetnewl;
top = 255;
[Re02,Im02]=fidread(' f id021,512);
p02=fidprocess(ReO2, Im02, top);
[Re04,Im04]=fidread(' f id04',512);
p04=fidprocess(ReO4, Im04, top);
10
[Re06,Im06]=fidread(' f id06',512);
p06=fidprocess(Re6, Im06, top);
[Re08,Im08]=fidread(' f id08 ',512);
p08=fidprocess(ReO8, Im08, top);
[RelO,ImlO]=fidread(' f id1 ',512);
p1O=fidprocess(ReO, ImlO, top);
[Re12,Im12]=fidread('f id12' ,512);
20
p12=fidprocess(Rel2, Im12, top);
[Re14,Im14]=fidread(' f id14 ',512);
p14=fidprocess(Rel4, Im14, top);
[Re16,Im16]=fidread(' fid16',512);
p16=fidprocess(Re16, Im16, top);
141
[Re18,Im18]=fidread(' f id181,512);
p18=fidprocess(Rel8, Im18, top);
30
[Re20,Im20]=fidread( 'fid20',512);
p20=fidprocess(Re2O, Im20, top);
[Re22,Im22]=fidread(' f id22',512);
p22=fidprocess(Re22, Im22, top);
[Re24,Im24]==fidread('f id241,512);
p24=fidprocess(Re24, Im24, top);
40
[Re26,Im26]=fidread(' f id26 1,512);
p26=fidprocess(Re26, Im26, top);
[Re28,Im28]=fidread(' f id28 1,512);
p28=fidprocess(Re28, Im28, top);
[Re30,Im30]=fidread(' f id30 1,512);
p30=fidprocess(Re3O, Im30, top);
[Re32,Im32]=fidread(' f id321,512);
50
p32=fidprocess(Re32, Im32, top);
[Re34,Im34]=fidread(' f id34 ',512);
p34=fidprocess(Re34, Im34, top);
142
[Re36,Im36]=fidread('f id36' ,512);
p36=fidprocess(Re36, Im36, top);
[Re38,Im38]=fidread(' f id38',512);
p38=fidprocess(Re38, Im38, top);
60
[Re40,Im4]=fidread(' fid40',512);
p40=fidprocess(Re4O, Im40, top);
[Re42,Im42]=fidread(' f id42',512);
p42=fidprocess(Re42, Im42, top);
[Re44,Im44]=fidread(' f id44',512);
p44=fidprocess(Re44, Im44, top);
70
[Re46,Im46]=fidread( 'fid46' ,512);
p46=fidprocess(Re46, Im46, top);
[Re48,Im48]=fidread(' f id48',512);
p48=fidprocess(Re48, Im48, top);
[Re50,Im50]=fidread(' f id50 1,512);
p50=fidprocess(Re5O, Im50, top);
data=[p02; p04; p06; p08; plO; p12; p14; p16; p18; p20; p22; p24; p26; p2 8 ;
p30; p32; p34; p36; p38; p40; p42; p44; p4 6 ; p48];
for
j
= 1:24
data(j,2)=0;
143
80
data(j,3)=O;
end;
% Create a sinogram in the frequency domain:
for
j
= 1:256
for k = 1:3
90
map(j,k) = (j-1)/255;
end;
end;
figure;
colormap (map);
image(255*data);
144
B.2.9
Windowing the Enlarged Image Space
function im = window-im(input,imsize, rl, r2)
im = zeros(imsize);
mid = (imsize+1)/2;
deIr = r2-rl;
for row= 1:imsize
for col = 1:imsize
r = sqrt((row-mid)^2 + (col-mid)^2);
if r < ri
im(row,col) = input(row,col);
elseif (r > ri) & (r <= r2)
10
im(row,col) = input(row,col)* 0.5*(cos(3.14159*(r-rl)/delr) + 1);
end
end
end
145
Appendix C
Engineering Drawings
146
2
For Educati
al
Use Only
I .970
I.969
II
I Ii
01
01
%
B
%
B
I1110
I
If
I
1% 1
C
C
0.25
0
R.09375
0
I 575
0
1 969
0
0
0.25
-o
L&J
L.
0..25
NAJ
-o
I .969
L.
0
0
LL.
A
0.25
04.74
[e-.50
DESIGNER
IS-Jan-RI
Rainuka Gpt
Gupta
TLRAiN
TOLERANCES
TIT
'/- 0.005
angles +/0.5
MATERIAL
Departmenf of Mechanical Engineering
71 Mas. Ave. Cambridge, MA 02139
TITLE
deg
[
Accal
INTERPRET
PER ANSI
ALL DIMENSIONS IN INCHES
UNLESS SPECIFIED OTHERWISE
2
MassochusetIs Inslifule of Technology
Iy
On
0n I y
U 5e
For Educati 1al Use
Pro/E Drawing
ro
Di5k
File
PROBED I SK
SIZE A
ISCALE
0.500
1 SHEET I OF I
2
For Educati
al
Use Only
-
3.60
3.35
. 75
.60
LLi-
B
C
0
B
\
I
25
0-U
0
I,
V
.25
o
.25
0.20
-I-
/-74
I I
00
0
LAJ.
CWi
0
. 07
.240
La.
""
La
A
75
.24
0 20
L5 6
.19
.25
DESIGNER
19-Jan-01
Rainuka Gupfa
MATERIAL
5
I
ifUle of Iechnology
Infteorecoog
.ee*-0.005TTL
B as5e Moun
Aleslof - 0.5 deg
INTERPRET
E
E
T
P
PER ANSI
,T
-Y14.5
ALL DIMENSIONS IN INCHES
UNLESS SPECIFIED OTHERWISE
2
uacuef
Deparlment of Mechanical Engineering
77 MasA. Ave. Cambrige MA 02139
TOLERANCES
ATL
M550CAuse
For Educ at iaIl Use Only
Pro/C Drawing File
BASEFIXTURE
SIZE A
f SCALE 1.000
1 SHEET I OF
I
2
For Educati
al Use Only
3.50
3.1875
2.375
-iEa
1
125 -
B
.44
'>----------
r----------
B
----
a-..---.---------- -0
C
0
3125
-125
0
0.20
.25
56
0
-0C
125
-0
U
7-
LdJ
0
--
A-0
v
75
*0
06
125
20
-0
-J
DESIGNERDEINR 19jon-01
RANCkES
TL
TEACERainuka
TOLERANCES
nov
MATERIAL
+/-
A
For Educat i
al
Use Only
Deportment
17 Mann.
upt
Gupta
0.005
0.deg
INTERPRET
PER ANSI
14.5
ALL DIMENSIONS IN INCHES
"urcc eore rirri iiTUCOWit
2
Massachusetts lnslilute of Technology
ri
of Mechanical Engineering
r
r39
MAA 02139
Ave. Cmbridge,
a
TITLE
ar able Pos I on Mec han i
Pro/E
Drawing File
VAKRPOS I
SIZE A
ISCALE
1.000
SHEET I OF I
2
al
For Educo ti
Use Only
2.50
K
.87
B
5
---------- ------ - r--5
625
a- -
-
-
-
-
-
-
B
~ 60~
-i
0
0
125
. 25
.3125
0
-0-
*
o
.75
0
LO. 12
C
125-
L
006
.125
A
DESIGNER
19-Jan-01
Rainuka GupM
s Inslilue of Technology
hu
Deparment of Mechanical Engineering
.LRACE +110 IT-E7 Ma33. Ave. Cambridge, MA 02139
TITLE
nI-RD
0 5
.0
0
# / E A
A
angles
+/- 0.5 deg
In
MATERIAL
Acelnl
RUIr
O I
ProIE Drawing File
INTERPRET
PER ANSI
Y14.5
TOLERANCES
T
l
Ia
ALL DIMENSIONS IN INCHES
UNLES S~ PECIFIED ATUrDWIC17
2
For Educotij
I Use Only
o~iflon SandwuiIchi
VA R _POS SAND
SUlE
A
SIZEA
00
CAL 11000
IISCALE
ISHEEFT
O I.
1 HEE II01I
2
For Educoti
.98
al
Use Oniy -
.1968
B
B
0
N
N -
-
4 N
-
0
13--E
7 7 9
0
2755
v,
I18 1 1
79.2
0
0
v
-
0
--- -- - -
0
0
1 65
o
AJ
.16
1.65
0
0
0
0
-
*0
LL.
T
.----------
DESIGNER
.98425
8
D
19-Jan-RI
TLRANCES
TOL.ERANCES
X
n5
i ote of
Massa(hu5ll
upt
TITLE
k/G0p01
Depot lin
D77
Mass.
.All +/- 0.005
deg
angles #/-
0.5
MATERIAL
u f
Acelol
s14.5
N
O
2
p
r e H
le
Use Only
MULTHOLDER
E
ALL DIMENSIONS IN INCHES
UNLESS SPECIFIED OTHERWISE
Educat i al
-a
Pro/E Drawing file
INTERPRET
PER ANSI
For
Iednology
of Mechanical Engineering
Ave. Cambridge, MA 01i39
SIZE A
SCALE 1.000
SHEET I OF I
2
For Educati
ol Use Only
0%
0',
B
a
a
I
I
a
a
I
I
I
I
a
a
a
a
a
a
a
a
a
a
~a1
g
1a
*,
~a
1a
1a
aa
aa
Ig I
Ii I
Ii I
I
a~ a
a
Ii
I
I
a* a
a1 a
a* a
aa a
aa a
Ig I
A
%
a
a
a
a
a
a
0
%
%
%
B
%
%0
%
%0
4.50
0
6.50
C
03.00
v,
S0
1.
525
509
020.00
0
----------
03.00
A-
----------
.-
0
La
a.
03.00
A
0 3.00
7 . 50
----19-Jan-01
DESIGNER
03.90
~
Ruinuka Gupta
n/-0.0
f/- 0.005
Angles +/- 0.5 deg
:1
MATERIAL
Dimensions
in mill imeters
2
Musa~husetts Insfilufe of Technology
TOLERANCES
AceIoI
INTERPRET
PER ANSI
Y 14. 5
ALL DIMENSIONS IN INCHES
UNLESS SPECIFIED OTHERWISE
For Educoati AlI Use Only
TIL
Deparlient of Mechanical Engineering
as, Ave. Cambridge, MA CZ139
TITLE
Roain Sape Hle
Pro/E Drawing File
ROTATION
SIZE A
ISCALE 0.091
1 SHEET I OF I
2
Educ at i
For
10
al
Use Only
100
2. 500
I
I
I
I
I
I
I
I
I
B
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
%
%
%
C
%
% %
%%
B
%
%
%
0
0
%
of)
32. 00
C.A3
2.700
0
0
C
-0
2.00
--
-
0
24. 00
LL.
0
L.
A
L
4. 7 50
19-Jon-01
DESIGNER
4.000
1TIL
TOLERANCES
MATERIAL
in mi IImeters
f/Acetal
0.5
Engineering
qe,. MA 0213 9
B
deg
;f
INTERPRET
PER ANSI
Y14. 5
ALLr DINSIONSIN NCHES
,_____
A
r -**'- OTHERWIS
2
Deportnt of Mechanical
1- o~a
mn.f AveM eCambri Ed
- 0.005TITLE
ongles
Dimensions
N05sochuseff5 lostiule of Technology
Rainuka Gupta
For Educati 1al Use Only
MOUNTINGBLOCK
SIZE A
ISCALE 0.059
|
SHEET I OF I
Appendix D
Pulse Programs
D.1
zg Pulse Program
zg
amx-version
1D sequence with high power hard pulse
1 ze
2 dl thi
p1 phl
go=2 ph31
wr #0
exit
phi
10
=
0 2 2 0 1 3 3 1
ph31 = 0 2 2 0 1 3 3 1
;hll/xl: transmitter high power level
;pl
90 degree transmitter high power pulse
;dl
relaxation delay; 1-5*T1
154
Nutation Pulse Program
D.2
nutation
amx version
lo power nutation
1
ze
2
dI tlo
10Ou tlO
3u adc
3
pl:e phi
d20 phO:r
10
3u :x
3u :x
lo to 3 times 11
rcyc=2 ph31
wr #0
exit
ph0=0
phl=0 1 2 3
ph3l=O 1 2 3
155
D.3
DANTE/Nutation Pulse Program
dantnut
amx version
nutation pulse program with selective excitation using the DANTE sequence
1 ze
2 dl tlO
10Ou tlo
3 pl:e phi
d20
pl ph2
10
d20
lo to 3 times 11
d22
adc
3u :x
3u :x
4 p2:e ph3
10Ou phO:r
3u :x
3u :x
20
lo to 4 times td
aq
rcyc=2 ph31
wr #0
exit
156
phO=O
phl=0
ph2=2
30
ph3=0 1 2 3
ph31=0 1 2 3
157
Appendix E
Experimental Data
158
Experiment
Distance (mm)
log(Distance)
Nutation
log(Frequency)
Frequency (Hz)
Number
2
4
0.60
4174.81
3.62
4
5
0.70
2539.06
3.40
6
6
0.78
2514.64
3.40
8
7
0.85
2636.71
3.42
10
8
0.90
2612.30
3.42
12
9
0.95
1806.64
3.26
14
10
1.00
1611.32
3.21
16
11
1.04
1440.44
3.16
18
12
1.08
1318.36
3.12
20
13
1.11
1196.30
3.08
22
14
1.15
1123.06
3.05
24
15
1.18
1049.80
3.02
26
17
1.23
1074.22
3.03
28
19
1.28
781.24
2.89
30
21
1.32
659.17
2.82
Table E.1: Raw data for calibration of measured nutation frequency to spatial distance
between the coil and the sample.
159
Exp.
Gear
Angle
Cos
Calculated
Calculated
Measured
Length of
No.
Tooth
(radians)
(Angle)
Distance
Nutation
Nutation
p1 Pulse
(mm)
Freq (Hz)
Freq (Hz)
(ps)
No.
21
1
0
1
3.423
211.000
210.818
33
23
3
0.524
0.866
3.824
188.873
194.600
35
25
4
0.785
0.707
4.251
169.905
179.459
37
27
6
1.309
0.259
5.272
136.997
135.740
50
29
8
1.833
-0.259
6.247
115.619
111.538
65
31
10
2.356
-0.707
6.982
103.447
94.387
75
33
12
2.880
-0.966
7.373
97.959
97.300
70
35
14
3.403
-0.966
7.373
97.959
101.880
75
37
16
3.927
-0.707
6.982
103.447
107.415
65
39
18
4.451
-0.259
6.247
115.619
120.727
55
41
20
4.974
0.259
5.272
136.997
138.180
50
43
22
5.498
0.707
4.251
169.905
165.475
40
Table E.2: Raw data for single sample rotation experiment with varying applied RF pulse.
160
Experiment
Gear
Angle
Cos of
Calculated
Calculated
Measured
Number
Tooth No.
(radians)
Angle
Distance
Nutation
Nutation
(mm)
Freq (Hz)
Freq (Hz)
2
1
0.000
1.000
3.552
2075.000
2038
5
2
0.393
0.924
3.783
1948.575
2001
8
3
0.785
0.707
4.373
1685.550
1806
10
4
1.178
0.383
5.131
1436.521
1562
12
5
1.571
0.000
5.902
1249.005
1305.5
14
6
1.963
-0.383
6.582
1119.822
1086
16
7
2.356
-0.707
7.109
1036.918
1025
18
8
2.749
-0.924
7.440
990.794
976
20
9
3.142
-1.000
7.552
976.000
1000
22
10
3.534
-0.924
7.440
990.794
988
24
11
3.927
-0.707
7.109
1036.918
1098
26
12
4.320
-0.383
6.582
1119.822
1232.5
28
13
4.712
0.000
5.902
1249.005
1379
30
14
5.105
0.383
5.131
1436.521
1649
32
15
5.498
0.707
4.373
1685.550
1855
34
16
5.890
0.924
3.783
1948.575
1928.5
Table E.3: Raw data for single sample rotation experiment with uniform applied RF
pulse.
161
Experiment
Gear
Angle
Cos
Calculated
Calculated
Measured
Number
Tooth
(radians)
(Angle)
Distance
Nutation
Nutation
(mm)
Freq (Hz)
Freq (Hz)
No.
4.712
0.000
4.744
1184.941
1513
4.974
0.259
4.209
1335.437
1879
5.236
0.500
3.641
1543.816
5.498
0.707
3.070
1830.747
2465
5.760
0.866
2.547
2206.736
2490
6.021
0.966
2.154
2609.235
2661
0.000
1.000
2.003
2806.717
2807
0.262
0.966
2.154
2609.235
2709
0.524
0.866
2.547
2206.736
1904
0.785
0.707
3.070
1830.747
1831
1.047
0.500
3.641
1543.816
1635
1.309
0.259
4.209
1335.437
1318
1.571
0.000
4.744
1184.941
1171
1.833
-0.259
5.224
1076.025
1123
2.094
-0.500
5.635
997.588
976
2.356
-0.707
5.965
942.368
878
2.618
-0.866
6.206
905.705
976
2.880
-0.966
6.353
884.737
854
3.142
-1.000
6.403
877.911
878
3.403
-0.966
6.353
884.737
878
3.665
-0.866
6.206
905.705
927
3.927
-0.707
5.965
942.368
976
4.189
-0.500
5.635
997.588
1098
4.451
-0.259
5.224
1076.025
1171
Table E.4: Raw data for sample 1 of the dual sample rotation experiment.
162
Experiment
Gear
Angle
Cos
Calculated
Calculated
Measured
Number
Tooth
(radians)
(Angle)
Distance
Nutation
Nutation
(mm)
Freq (Hz)
Freq (Hz)
No.
2
1
0
1
1.624
4051.487
4052
4
2
0.262
0.966
1.949
3374.411
3539
6
3
0.524
0.866
2.686
2449.313
2465
8
4
0.785
0.707
3.555
1850.100
1977
10
5
1.047
0.500
4.440
1481.429
1464
12
6
1.309
0.259
5.287
1244.121
1220
14
7
1.571
0.000
6.066
1084.402
1098
16
8
1.833
-0.259
6.756
973.713
976
18
9
2.094
-0.500
7.340
896.175
903
20
10
2.356
-0.707
7.807
842.560
878
22
11
2.618
-0.866
8.147
807.372
805
24
12
2.880
-0.966
8.354
787.386
781
26
13
3.142
-1.000
8.424
780.901
781
28
14
3.403
-0.966
8.354
787.386
781
30
15
3.665
-0.866
8.147
807.372
32
16
3.927
-0.707
7.807
842.560
976
34
17
4.189
-0.500
7.340
896.175
1074
36
18
4.451
-0.259
6.756
973.713
1171
38
19
4.712
0.000
6.066
1084.402
1342
40
20
4.974
0.259
5.287
1244.121
1586
42
21
5.236
0.500
4.440
1481.429
2001
44
22
5.498
0.707
3.555
1850.100
2587
46
23
5.760
0.866
2.686
2449.313
3613
48
24
6.021
0.966
1.949
3374.411
3979
Table E.5: Raw data for sample 2 of the dual sample rotation experiment.
163
Bibliography
[1] Steven E. Harms and David M. Kramer. Fundamentals of magnetic resonance imaging. CRC Critical Review in Diagnostic Imaging, 1985.
[2] Paul T. Callaghan. Principles of Nuclear Magnetic Resonance Microscopy. Oxford
University Press Inc., 1991.
[3] Steve Webb. The Physics of Medical Imaging. IOP Publishing Ltd., 1988.
[4] D.I. Hoult. Rotating frame zeugmatography. Journal of Magnetic Resonance, 1979.
[5] John P. Boehmer, Kenneth R. Metz, and Richard W. Briggs. One-dimensional spatial
localization of spin-lattice relaxation times using rotating-frame imaging. Journal of
Magnetic Resonance, 1985.
[6] F. De Luca, C. Nuccetelli, C. De Simone, and B. Maraviglia. NMR imaging of a solid
by the magic-angle rotating-frame method. Journal of Magnetic Resonance, 1986.
[7] David G. Cory, F.H. Laukien, and W.E. Maas. NMR spectroscopy with radial pulses.
p- and n-type selection COSY. Chemical Physics Letters, 1993.
[8] Wurong Zhang and David G. Cory. Measurement of flow velocities by NMR using a
spatially modulated RF field. Measurement Science and Technology, 1998.
[9] G.A. Barrall, Y.K. Lee, and G.C. Chingas.
Radial RF imaging using a coaxial
resonator. Journal of Magnetic Resonance, 1994.
164
[10] John P. Boehmer, Robert I. Prince, and Richard W. Briggs. The cone coil, and
RF gradient coil for spatial encoding along the B, axis in rotating-frame imaging
experiments. Journal of Magnetic Resonance, 1989.
[11] Avinash Kak and Malcome Slaney. Principlesof Computerized Tomographic Imaging.
IEEE Press, 1988.
[12] Charles E. Swenberg and James J. Conklin.
Imaging Techniques in Biology and
Medicine. Academic Press, Inc., 1988.
[13] Athanasios Papoulis. Systems and Transforms with Applications in Optics. McGraw
Hill, 1968.
[14] George A. Campbell and Ronald M. Foster. Fourier Integrals for PracticalApplications. D. Van Nostrand Co., 1948.
[15] Derek Rowell and David N. Wormley. System Dynamics: An Introduction. Prentice
Hall Inc, 1997.
[16] Bruker Analytik GmbH. XwinNMR Processing Software Manual Part I: General
Features and Data Processing. Bruker, 1997.
[17] Zang-hee Cho, Joie P. Jones, and Manbir Singh. Foundations of Medical Imaging.
John Wiley and Sons, Inc, 1993.
[18] Milton Abramowitz and Irene A. Stegun. Handbook of MathematicalFunctions with
Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., 1965.
165