The Cone beam System on
Princess Margaret Hospital,
University health network.
The C-arm Cone beam System on
Princess Margaret Hospital,
University health network
Summarizing the work in
Local Tomograhy
Dr. Shuangren Zhao
Research Associate
Radiation Physics Department
Princess Margaret Hospital
What is local Tomography?
Ordinary tomography is global since
reconstruction at a point x requires
integrals over lines far from the point x.
Local tomography uses only lines close
to the point x
Compare the local and nonLocal Tomography
Ordinary Tomography
Uses the projection
data far away from the
local region
Local Tomography
Only uses the projection
data closing to the local
region
Why use Local
Tomography?
Only small region of the object is interested.
Large dose of radiation is avoided.
Increase the Speed of the reconstruction
algorithm.
Only the discontinuity of the object function is
interested.
If projection data are truncated, there are
truncated artifacts on non-local Tomography.
Simulation with Truncated data using non Local
Tomography
truncated data, truncated object, truncated effect
What difficulties does Local
Tomography have ?
If f(x) is a solution of equation A f = p, some
different functions for example g(x) satisfies
A g= 0, f(x)+ g(x) is still the solution of this
problem. This kind of functions g(x) are called
null functions. If null function is not zero, the
problem is not unique.
Non Local tomography is with uniqueness.
Local Tomogaphy entails the loss of the
uniqueness.
1. Known Theory for Local
Tomography
(λ f) tomography
(λ f+ μ/λ f) tomography
Pseudolocal Tomography
Wavelet-based local Tomography
Extrapolation for the missing data
Definition of operator λ
F is Fourier Transform operator
Important formula about λ
Non local reconstruction
(λ f) tomography
E.I. Vainberg 1981
K.T. Smith 1985
Cupped reconstruction
Pure local reconstruction
(λ f + μ/λ f) tomography
A Faridani (1992)
With Cup correction
Pseudolocal Tomography
A. I. Katsevich 1996
A. G. Ramm 1996
Using Radon reconstruction
Cut the kernel length of Hibert transform short
What errors are introduced from this method?
Wavelet-based local Tomography
D. Walnut (1992)
F. Rashi-Frarokhi (1997)
Filter is implemented with wavelets. Wavelets are
back projected.
The principle of the wavelet Reconstruction:
with different basis functions
For filter backprojection method, we use δ function as basis
For Wavelet reconstruction, they use wavelet basis functions
A) δ function, B) Harr wavelets, C) Daubchies wavelets
At the backprojection process, δ function spread from a point, however
wavelet will be more localized to a point.
The wavelet method will have exactly same results as filter back
projection method if global projections are available, however if
projection is truncated we could not expected wavelets methods have
the same results as filter backprojection methods.
I will make a simulation with one truncated object to check these theory.
Wavelet method could be checked with one cylinder
simulation truncated data, look at the truncated effect
Simple Extrapolation
The projection data are just extrapolated according to
the value in the boundary of truncated projection
data.
λ reconstruction
Comparing local filter to
non-local filter
(λ f + μ/λ f) <=>filter k=k(l, μ)
2 With Simulation Data
•
•
•
•
•
Truncated data
(f )
(λ f)
(λ f + μ/λ f)
l=4
μ=0.01,0.03,0.05,0.2,1.0
Simulation with Truncated data using non Local
Tomography
Simulation with Truncated data using
Local (λ f + μ/λ f) Tomography
μ=0.0
μ=0.01
Simulation with Truncated data using
Local (λ f + μ/λ f) Tomography
μ=0.03
μ=0.05
Simulation with Truncated data using
Local (λ f + μ/λ f) Tomography
μ=0.2
μ=1.0
Profile for μ=0.0, 0.01, 0.03, 0.05, 0.2,
1.0
The Influence of Truncated projection data
Balance the truncated effect
The Influence of Truncated data
Using non local tomography
(The detector size is just half the size of the view of object), for Cylinder_1
with f= 1 and Cylinder_2 with f= - 1..
Important simulation results:
(1) See the above example we could obtain the results
that if the truncated object is outside of ROI with high frequency,
it will has very small influence to the inside of ROI.
(2) If we filter out only the low frequency,
the out side object should has very small influence to the inside.
λ Local tomography utilize this mechanism.
(3) We could use an “anti object” , which has negative value
and looks like the original object to balance the influence of the
original object. The “anti object” does not have to be the same
as the original object. This method will offer a correction of the
truncation effect.
3. With measured data
•
•
•
•
•
•
not truncated data
(f )
(λ f)
(λ f + μ/λ f)
Truncated data
(f )
(λ f)
(λ f + μ/λ f)
Rabbit White, Mode2_Rump
See shell reconstruction
f, λf, λf +0.02 1/λ f
4. The work I have done (Between the theory
and the implementation)
Proved with understanding that (λ f + μ/λ f) = (λ + μ/λ)
f.
The normalization of the kernel of local tomography
filter
Parallel-beam => fan-beam => cone beam
Match the out put of simulation to the input of our
Iview3D software
Prove with understanding that
(λ f + μ/λ f) =? (λ + μ/λ) f.
Using the left side, the back projection process need to be done
two times. The advantage is that μ could be easily adjust.
If the value of μ is known, we could use the right side. In this
case the back projection process need only to be made once.
Hence the speed of the calculation is increased in practical use.
In general these two sides are not equal. However if the
operators λ and 1/λ are linear they will be equal. The problem is
whether or not they are linear. I found that the two operator will
be linear if and only if the interpolation of the back projection is
linear. As we know, the interpolation of our software Iview3D is
linear. So that for our λ tomography reconstruction the right side
of the above formula could be used.
The normalization of the kernel of local
tomography filter
λ f, 1/λ f and f have different units. It is not required that the
reconstruction of those 3 functions have similar values. However
our software Iview3D could only show the reconstruction value
between 0 to 1. This suggest that any reconstruction values
should be at this range.
A Faridani use μ=47 to balance two operator λ and 1/λ. It is clear
he did not normalize the three functions suitable. If λ and 1/λ are
normalized in the same order then μ should be in the range
0 ~ 1. I made this normalization, which is shown in the
following.
Nomalization
for μ=0.03, the reconstructed value is
close to the value of phantom
Parallel-beam=> fan-beam=>cone beam
The theory of λ local tomography is written for parallel beam case.
Parallel beam and cone beam cases still have some differences.
Implementation for his theory to the cone beam case is required to find
out whether or not some correction necessary.
After checking the theory, I find that the reconstruction formula for the
cone beam case is just the same as the parallel beam case. We are
very lucky.
However these does not mean cone beam λ tomography are easy. The
λ tomography began in 1981. The first cone-beam
implementation was reported at 2000[3]
Match the output of the simulation to
the input of our Iview3D
The simulation program we have is not designed for our
IView3D software. Hence the output has a different format. We
need to match the simulation output results to our IView3D
software and produce correct header files. The format
transformation is done with Matlab. Now it could be used but still
it still has problem.
The size of variables U and V of simulated program could not be
greater than 512 (the program will crash). The size of variable V
and U could not take different number than 128, otherwise the
object leave the the center of the view.
5. The work I am doing now
Speeding up the Local Tomography
Optimization of the kernel of the local filter
Iterated extrapolation to the projection data
Correction of the influence of the truncation effect
Reconstruction with general Hankel Transform
Optimization of the kernel of local filter
Optimize the kernel in Fourier frequency domain or in
wavelet frequency domain. For a given length of the
kernel we could modify the kernel so that it is the
closest kernel in frequency domains to the Ram-Lak
filter.
Optimize the kernel with the goal function in the
spatial domain. For a given length of the kernel we
could modify the kernel so that the reconstructed
object is as close to the phantom as possible.
Optimize the kernel so it is not sensitive to noises.
Iterated extroplation
The truncated objects are obtained by first
reconstruction
The extrapolation data are calculated from the
truncated objects.
The second reconstruction is done through the
truncated data together with the extrapolated
projection data.
Iterated correction
Given a negative value to the truncated object that is calculated
from the first reconstruction.
The truncated projection is calculated from the negative object.
Correction to the reconstruction is calculated from the above
calculated projections by some reconstruction algorithm.
The final reconstruction is obtained by adding up the first
reconstruction and the correction.
Since the corrections are only with low frequency, low
frequency reconstruction algorithm could be used.
Balance the truncated effect
The Influence of Truncated data
Using non local tomography
(The detector size is just half the size of the view of object), for Cylinder_1
with f= 1 and Cylinder_2 with f= - 1..
General Hankel Transform
Hankel transform could be used for parallel beam
reconstruction. General Hankel transform is
developed for fan beam reconstruction, (which is one
of my work in Julich research center of Germany).
General Hankel transform could be utilized If the
projection and back projection process are in a
iterated way. Using General Hankel transform, the
calculations of the algorithms could be faster,
especially if the object has only low angle-frequency.
Conclusions
Different technologies could be utilize to
implement the local tomograpy
(1) Filter out low frequency: λ tomography
(2) Choose different basis function: wavelet
methods
(3) simple extrapolation, iterated extrapolation
(4) iterated correction.
Some of the above technology could be
implemented together.
Acknowledgments
Douglas J. Moseley for offering the input projection data
simulation program and Matlab program to see the results of
IView3D
Steve M. Ansell for developing the software interface
Sami Siddique for the data acquisition
Graham A. Wilson for the working environment
Jeffrey H. Siewerdsen for introducing me the basic knowledge xray equipment
David A. Jaffray for introducing the technology--reconstruction
with large size. Because of this technology, I could see
immediately the influence of the truncation, the shadow. The
shadow is produced by the truncated object outside VOI.
References
[1]A Faridani, “Local Tomography”, SIAM J. APPL.
MATH Vol 32, No 2, pp459-484, April 1992
[2]F. Rashid-Farrokhi, “Reconstruction
singularities……”, Math. And Comut. Modelling,
18(1993), pp. 109-138
[3] P. Huabsomboon, 3D Filtered Backprojection
Algorithm for local Tomography, M.S. Paper, Dept. of
Mathematics, Oregon State University, Corvallis, Or
97331, U.S.A., (2000).
[4] S.R. Zhao, H. Halling: “Image Reconstruction for
fan beam tomography using a new interal trasform
pair”. International Symposium on Computerized
Tomography in Novosibirsk, Russia August 10-14,
1993. Abstracts ed. M.M. Lavrentev, p125.