BMME 560 & BME 590I
Medical Imaging: X-ray, CT, and
Nuclear Methods
Tomography Part 3
Today
• Tomography
– Filtered backprojection
Tomographic Reconstruction
t
• The problem
y
p(t,q)
t
f(x,y)
s
q
x
Given p(t,q) for 0<q<p
Find f(x,y)
Simple Backprojection
• An LSI system model for projection followed
by simple backprojection:
f ( x, y)
Projection
p(t ,q )
1
f ( x, y)
Fourier
Transform
t
filter
Simple
backprojection
Inverse
Fourier
Transform
fˆsbp ( x, y)
fˆsbp ( x, y)
Simple Backprojection
• Example
True image
Simple backprojection
Filtered Backprojection
• An LSI reconstruction method:
f ( x, y)
Projection
p(t ,q )
1
f ( x, y)
Fourier
Transform
t
filter
Simple
backprojection
t
filter
t
filter
Inverse
Fourier
Transform
fˆfbp ( x, y)
fˆfbp ( x, y)
Filtered Backprojection
• An LSI reconstruction method:
Filtered Backprojection
f ( x, y)
Projection
p(t ,q )
1
f ( x, y)
Fourier
Transform
t
filter
Simple
backprojection
t
filter
t
filter
Inverse
Fourier
Transform
fˆfbp ( x, y)
fˆfbp ( x, y)
Filtered Backprojection
• Example
True image
Simple backprojection
Filtered backprojection
Filtered Backprojection
• Example
True image
Simple backprojection
Filtered backprojection
Filtered backprojection
• In theory, the filtered backprojection estimate
should be equal to the true image.
fˆfbp ( x, y)  f ( x, y)  1  P(t ,q )
• Our discrete implementation introduces some
errors.
• The biggest problem is noise.
Filtered backprojection
Magnitude
• Look at the frequency-domain property of that
ramp filter
There is a mathematical problem here
Frequency
Filtered backprojection
• Consider the practical problem
– The projection of an object is limited in resolution
by physical constraints.
– The noise is manifested in the detector and its
spectrum is not limited.
• What does the ramp filter do to the signal and
to the noise?
Filtered Backprojection
• So, in practice, we have to truncate (roll-off,
apodize) the ramp filter to control noise.
Magnitude
Ideal ramp filter
Practical ramp filter
Frequency
Key Point
Resolution (FWHM)
• There is an essential and inescapable tradeoff
between noise and resolution in every imaging
system.
Noise variance
Filtered Backprojection
• Example
True image
Filtered backprojection
Noise-free
Filtered backprojection
Noisy
Filtered backprojection
• Three low-pass filter cutoffs
.2 cyc/pix
.15 cyc/pix
.1 cyc/pix
Filtered backprojection
• We have a couple of equivalent ways to
implement filtered backprojection
– Backproject, then apply a 2D ramp filter
– Apply 1D ramp filter to each projection, then
backproject
– Convolve each projection with an approximation
of the PSF of the ramp filter, then backproject
(called convolution backprojection).
Filter of the backprojection
Simple
backprojection
2D
Fourier
Transform
Ramp
filter
Inverse
2D Fourier
Transform
Backprojection of filtered projections
p(t ,q )
1D
Fourier
Transform
Ramp
filter
Inverse
1D Fourier
Transform
Simple
backprojection
p(t , q )
At this point, we have created a
set of modified projections.
Convolution Backprojection
Convolution
Of
projections
Simple
backprojection
p(t ,q )  p(t ,q )* c(t )
The convolution kernel
Convolution Backprojection
• The convolution kernel is an approximation of
the inverse Fourier transform of the ramp filter.
c (t )
c (t ) 
 2  (2p t ) 2
  (2p t ) 
2
2
2
t
This was Cormack’s
original approach.
Convolution Backprojection
• In a discrete implementation, we can take the
IFFT of the ramp filter.
Note the similarity in
shape to the
continuous version.
Sampling
• Since the projection space is two-dimensional,
we have to sample in each dimension:
– Spatial sampling (in t)
– Angular sampling (in q)
• What are the effects of sampling in each?
Sampling
• Spatial sampling: The pixel spacing in the
detector
– Finer sampling = finer resolution in the
reconstruction
• Angular sampling: The number of angles over
the desired arc (or the angular increment
between projections)
Spatial Sampling
1x
1.33x
2x
Projection spacing
Insufficient spatial sampling results in?
4x
Angular Sampling
• Rule of thumb for the number of projections
needed:
Number of angles over 180 degrees = ____________
Angular Sampling
256 views
128 views
64 views
32 views
96 views
128x128 image (object: 75 pixel diameter), 360-degree arc
Limitations
• To get filtered backprojection, we made a key
assumption:
– The projections are all perfect Radon projections
• They are complete, i.e., the entire object is viewed in
every projection.
• They are perfect line integrals.
• The sampling is sufficient.
• There is no noise.
Limitations
• In the real world, we do not always meet these
assumptions (tradeoffs):
– Completeness: usually not a problem, but some
special geometries suffer truncation
– Perfect line integrals: Beam hardening, scatter,
metals, nonuniformity, body attenuation (ECT),
detector blurring all degrade this.
– Sampling sufficiency: Acquisition time can be
reduced with fewer angles, larger detector pixels
– Noise: Always with us
Limitations
• Beam hardening example
This object experiences
different energy spectra
depending on the direction
of projection.
Limitations
• Deviations from perfect Radon projections
result in inconsistent data
– The mathematics do not care if the data is
inconsistent; it is still possible to perform filtered
backprojection.
– The interpretation of the results may be a problem
if there are inconsistencies.
Limitations
• Example: Detector is too small
Two-sided truncation
One-sided truncation
Limitations
• We have also assumed that we know the
geometry of the system perfectly.
• Calibration is essential in real systems.
– Hospital staff perform frequent quality control
tests and calibrations to ensure system drifts are
accounted for.
Limitations
• Center-of-rotation error
Perfect
0.5 pixel shift
1 pixel shift
Limitations
• Bottom line
– Inconsistencies are created when real projections
deviate from Radon projections.
• Usually due to the physical limitations or
cost/performance tradeoffs of the system
– Images are reconstructed, inconsistencies are
manifested as recognizable image artifacts.
Other Geometries
• We have discussed only parallel (Radon)
projections so far.
• There are other projection geometries that are
more practical in certain situations.
–
–
–
–
Fan-beam
Cone-beam
Pinhole
Ring
Fan-beam Reconstruction
y
y
f(x,y)
f(x,y)
x
t
x
Fan-beam Geometry
• Most commonly used in CT systems
– X-ray production is focused on a point anyway.
• Also found in SPECT
– Provides magnification, at cost of field of view.
Fan-beam Geometry
• What arc do we need? Is 180 enough?
– Consider: Each fan-beam projection line can be
mapped to a projection line in the Radon space.
– Each projection line maps to a (t,q) pair in the
sinogram space.
Fan-beam Geometry
Sinogram mapping of a 180-degree arc with a fan-beam
90
Each “line” is
one
rotational
position.
Sinogram angle, degrees
60
Redundant
angles
30
0
Missing
angles
-30
-60
-90
-3
-2
-1
0
1
Transverse distance, cm
2
3
Fan-beam Geometry
• Rule for the arc required to obtain complete
sinogram data for fan-beam:
Rotational arc = ________________________
• The minimum arc still requires compensation;
it may be easier to do 360-degrees.
90
90
60
60
Sinogram angle, degrees
Sinogram angle, degrees
Fan-beam Geometry
30
0
-30
-60
30
0
-30
-60
-90
-90
-3
-2
-1
0
1
Transverse distance, cm
180 + 53 arc
2
3
-3
-2
-1
0
1
Transverse distance, cm
360 arc
2
3
Fan-beam Geometry
• Options for analytical reconstruction
– Rebinning (resorting): Estimate parallel-beam data
from fan-beam and use filtered backprojection
– Fan-beam filtered backprojection: Modified
version of FBP specifically for fan-beam
Fan-beam Rebinning
Convert fan-beam data to equivalent parallel-beam data
y
f(x,y)
x
Note that each projection
ray in fan-beam is an
element of the parallelbeam projection set at
some (t,q) pair.
Resort projection
rays and interpolate.
Fan-beam Filtered Backprojection
Fan-beam
projection data
Weighting based
on projection
angle
Modified
projection data
Apply
modified ramp
filter
Reconstructed
image
Backproject along
fan, weighted by
distance from
focus
Modified
projection data
* Also the basis for the Feldkamp algorithm