1. BWH test pattern. Google “BWH SMPTE Judy” for jpeg copies of BWH & SMPTE test patterns

2. SMPTE test pattern

3. Image Reconstruction from Projections
J. Anthony Parker, MD PhD
Beth Israel Deaconess Medical Center
Boston, Massachusetts
Talk is a bit dry, basic sciency, but Tom keeps asking me to do it.
There are a small number of useful pearls, watch for them.
Caveat Lector

4. Projection Single Slice Axial What’s a projection?
This is what mathematicians say.
Given object and angle, it is integrals through the object.

5. Single Axial Slice: 3600 collimator
Ignoring attenuation, SPECT data are projections
Gamma camera rotates 360o around object.
Data it collects is projections (ignoring attenuation)

6. Attenuation: 180o = 360o x Tc-99m keV 150 100 80 60 50
htl(140 keV) ≈ 4 cm
This is a large patient, 47 cm
140 keV Tc-99m photons are highly attenuated, 0.1%
Cardiac 180o collection is special. In general, 360o

7. Cardiac Perfusion Data Collection Special Case - 180o
Axial
Coronal / Sagittal
Multiple simultaneous axial slices
One horizontal line from each of these images, makes one axial slice in the last slide.
Notice arms are moving; heart is fairly still.

8. Dual-Head General-Purpose Gamma Camera: 900 “Cardiac” Position
2 heads: 900 rotation = 1800 data 1 2
Phillips nee ADAC SPECT camera with attenuation correction
Simultaneous emission and transmission

9. Inconsistent projections “motion corrected”
Tech did motion correction wrong.
Note walls don’t line up, hurricane.
Tall is about how technology can improve imaging, but remember it can introduce new errors.

10. Original data Original data gives a much better reconstruction.
Some people on read bulls eye -> tech essential reads study

11. Single Axial Slice: 3600
One line from each projection image. More usual 360o rotation.
Blob becomes 1-D function.
Marked zero so you can follow what happens with rotation.

12. Sinogram: Projections Single Axial Slice60
projection angle x
Display projections as lines in an image.
First line projection to top.
Blob follows a sine wave. Called sinogram; commonly used.

13. Uniformity & Motion on Sinogram
1 head
24 min
2 heads
12 min
Can use sinogram to detect motion and uniformity.
I don’t think sinogram is particularly insightful.

14. Reconstruction by Backprojection
Backprojection tails
We are going to make backprojection the 1st step in reconstruction.
Projections are 1, 2, and 3.
No information in the projection about where the data came from, so put it back uniformly in all the possible sources.
Artifact from projection lines.

15. Backprojection 2 projections 2 objects
If we have two blobs and only two projections, backprojection gives us 4 blobs.

16. projection tails merge resulting in blurring
More projections give us a better image, but in is a blurred out representation of the original.

17. Projection -> Backprojection of a Point
lines add at
the point
tails spread point out
If we start out with an object which consists of a single point, projection followed by backprojection will make 1/r.
Infinite number of projections.
Each point in the image is replaced with this function. (Convolution; linear shift invariance.)

18. Projection -> Backprojection
more backprojection angles, more like object

19. Projection -> Backprojection Shepp-Logan Phantom
This is a typical brain type of mathematical phantom.
Ignore limited gray scale in back projection.
Again, backprojection makes a blurs the original.

20. Projection->Backprojection Smooths
Smooths or “blurs” the image
(Low pass filter)
((Convolution with 1/r))
Nuclear Medicine physics
Square law detector adds pixels
-> always blurs
Different from MRI (phase)
Key point about nuclear medicine physics is that you always sum the magnitude of pixels.
Detection always smoothes.
Different in MRI: pixels add and subtract. Doesn’t smooth.

21. (Projection-Slice Theorem)
“k-space (k,)”
detail
low
frequency
spatial frequency domain
spatial domain
2D Fourier
transform
Object is defined in spatial domain. Equivalent representation as spatial frequencies.
Polar coordinates in spatial frequency domain often k and theta. So, spatial frequency domain often called k-space, especially in MRI.
Projection in the object domain is mathematically equivalent to a line of data in k-space, often called a slice.
Any engineers? Parenthesis are for the engineers. This is called the projection-slice theorem.

22. Spatial Frequency Basis Functions
f(u,v) ≠ 0, single u,0
f(u,v) ≠ 0, single 0,v
f(u,v) ≠ 0, single u = v
High school: you learned about basis functions for vector spaces
f(u) not zero for some u, f(v) not zero for some v
Sum, f(u,v) not zero for some u = v
MRI samples f(u,v) directly in k-space

23. Projection -> Backprojection: k-space
(Density of
slices is 1/k)
(Fourier Transform of 1/r <-> 1/k)
one projection
multiple projections
In k-space, a projection is equivalent to a line of data.
If we have an infinite number of projections we will have lines which all cross at zero. Over sample low frequencies <- square law detector.
The result in k-space is just like the result in object space. People often skip over the fact that this is a special case.

24. Image Reconstruction: Ramp Filter
Projection -> Backprojection
blurs with 1/r in object space
k-space 1/k ( 1/r<-> 1/k)
Ramp filter
sharpen with k
(windowed at Nyquist frequency) k
Ramp filter increases high frequencies

25. Ramp Filter in the Image Domain
(Fourier transform of
windowed ramp)
Replace each point with this kernel
Negative side lobe
In the image domain each point in the projection-backprojection image is replaced with this kernel.
(Convolve the projection-backprojection image with this function.)
Negative side lobe, typical of sharpening function. Unsharp masking in Photoshop.

26. Negative Side Lobes: Sharpening
Negative side-lobes enhance edges / increase high frequencies

27. Filtered Backprojection: Negative Side Lobes Eat Projection Tails
Replace each point in the projections with the kernel, you get the red functions.
Negative side lobes eat away the trails left by the other projections. Filtered projections all reinforce at the location of the blob.

28. Filtered Backprojection
Here is filtered backprojection.
Final image is equal to the object. No blurry edges. No background.

29. Effect of Limited Projections
This shows the effects of too few projections.
In the object, and especially around the outside you see projections which have not be completely eaten by negative side lobes.

30. Effect of Noise
Reconstruction noise is high spatial frequency, i.e. detail. Looks wormy. Especially see the first image.
As we reduce the noise (more counts) first see large structures, then fine structures.
Remember we over sample low spatial frequencies <- square law detector.

31. Shows the whole process.

32. Mathematically Equivalent Implementations
Backproject filtered projections
“Filtered backprojection”
Filter backprojected projections
“Fourier method”
Some people try to make a distinction between filtered backprojection as on the last slide
and filtering (sharpening) the smooth projection->backprojection image.
Mathematically, these are equivalent.

33. Low Pass Times Ramp Filter
Butterworth
– noise
Ramp –
reconstruct
Red shows a low pass filter, a so called Butterworth filter.
Butterworth filter has two parameters, a cut off defined as the 50% point, and an order which determines the rate of transition.
The ramp (reconstruction) filter modified by the LPF follows the ramp at low frequencies, but smoothly becomes zero at high frequencies.

34. SPECT Filter: Ramp - Reconstruction Low Pass Filter - Noise Reduction
Noise and resolution
Exact reconstruction
SPECT filter (A or B) are the combination of a ramp (reconstruction) filter and a LPF.

35. Reconstruction from Projections
Projection: data collection
Backprojection: 1st step in reconstruction
projection->backprojection:
blurs <-> 1/k
Sharpen: ramp filter multiplies by k
Noise reduction: low pass filter (LPF)
LPF also decreases image detail

36. Reconstruction Kernel with LPF
(Fourier transform
of filter)
LPF makes hot
blobs fatter
hot blobs
eat adjacent
objects
LPF stretches the reconstruction kernel.
The center becomes fatter.
The negative side lobes move out farther. Side effect is that hot blobs eat adjacent objects.

37. Inferior wall defect ?
Is the hot liver eating the inferior wall or is there a real defect?

38. Iterative (OSEM) Reconstruction
Iterative reconstruction doesn’t have as much of a problem with hot objects eating adjacent.
But, still can’t tell what is going on. The hot liver is smeared into the inferior wall.

39. 3 hour delay shows defect Better data always wins
3 hour delay with decreased liver activity shows there is a reversible inferior wall defect.
Pearl: Better data always wins over processing.

40. What’s Good about FPB
Ramp filter exactly reconstructs projection
Efficient
(Linear shift invariant)
(FFT is order of n log(n)
n = number of pixels)
“Easily” understood

41. New Cardiac Cameras Solid state - CZT: $$$, energy resolution
scatter rejection, dual isotope
Pixelated detector: count rate &
potential high resolution
poorer uniformity
Non-uniform sampling: sensitivity
potential for artifacts
Special purpose design
closer to patient: system resolution
upright: ameliorates diaphragmatic attenuation
SPECT typically not count rate limited; intrinsic resolution typically not major issue in system resolution
Also small foot print may help sitting.
For me, major issues are improved sensitivity and improved system resolution.
$$$ hidden in expensive RFID

42. D-SPECT - initial fast scan to find heart region-of-interest, ROI
Heart ROI sampled for longer time.
Hi (sic) Lo (sic)

43. Collimator Resolution*
Single photon imaging (i.e. not PET)
Collimators: image formation
Sensitivity / resolution trade-off
Resolution recovery hype
“Low resolution, high sensitivity ->
image processing = high resolution”
Reality - ameliorates low resolution
Steve Moore: “Resolution: data = target object”
Can do quick, low resolution image
* not resolution from reduced distance due to design
Single photon imaging requires collimators for image formation -> sensitivity / resolution trade off
Hype: high resolution can be recovered with image processing. Reality: ameliorated
Steve Moore says that collimation should be at resolution of the object being imaged
D-SPECT uses low resolution for quick localization imaging

44. Dual Head: Non-Uniform Sampling

45. Activity Measurement: Attenuation
keV
150
100 80 60 50
htl(140 keV) ≈ 4 cm
CT measures attenuation
SPECT measures activity, attenuation complicates SPECT
Surprising how well we can do without AC

46. Attenuation Correction: Simultaneous Emission (90%) and Transmission (10%)
Gd-153 rods
T1/2 240 d
e.c. 100%
97 keV 29%
103 keV 21%
2 heads: 900 rotation = 1800 data
Phillips nee ADAC SPECT camera with attenuation correction
Dale Bailey spent 6 mo at BIH
Simultaneous emission and transmission

47. Semi-erect: Ameliorates Attenuation

48. Leaning Forward, < 500 Pounds
Digirad shows large patient
Chair takes large patients
Nukes doesn’t like large patients

49. Digirad: Patient Rotates X-ray Attenuation Correction

50. CT: Polychromatic Beam -> Dose
keV
150
100 80 60 50
This is a large patient, 47 cm
Lower energy X-rays have provide very little information
Increase dose without benefit: Ideal mono-energetic X-rays

51. X-ray Tube Spectra X-ray tube: electrons on Tungsten or Molybdenum
bremsstrahlung
characteristic X-rays
e- interaction:
- ionization
- deflection
X-ray tube: electrons on Tungsten or Molybdenum
Characteristic X-rays from ionization from charge particles
Bremsstrahlung from electron slowing in tissue

52. Digirad X-ray Source: X-rays on Lead
74W
82Pb
X-rays interaction
- ionization
- no 10 bremsstrahlung
X-rays not electrons hit lead. No direct direct bremsstrahlung
Lead high Z than tungsten, higher energy characteristic X-rays

53. Digirad X-ray Spectrum

54. Detectors D-SPECT Digirad Crystals CZT CsI(Tl) Photodetector NA PD
Energy resolution 6% 10%
Count rate 1.35 x x 106
Heads 9 3
FOV (cm) 15.7 x x 21.2
Total cm
X-rays (blank) none 20 x 106

55. New Cardiac Cameras D-SPECT CardiArc Digirad GE
Detector CZT* NaI(Tl) CsI(Tl) CZT*
Electronics SS* PMT PD*? SS*
Pixelated Y N Y Y
Collimation holes slits*? holes pinholes
Non-uniform Y* Y* ~N Y*
Limited angle Y Y N ~N
Closer to pt Y Y Y ~N
AC N CT? CT* CT
Position ~semi semi erect supine
D-SPECT BWH, Digirad BIDMC
Advantages: CZT, PD?, slits?, non-uniform sampling, Digirad CT
PD versus APD, Digirad, modestly converging collimators

56. Soft Tissue Attenuation: Supine
breast
lung

57. Soft Tissue Attenuation: Prone
breast

58. Soft Tissue Attenuation: Digirad Erect
breast
post

59. Sequential Tidal-Breathing Emission and Average-Transmission Alignment
Sequential emission and transmission images suffer from respiratory or gross patient motion
Tidal breathing average CT better than instantaneous transmission

60. Sensitivity / Resolution Trade-Off
Non-uniform sampling -> sensitivity
Special purpose design -> resolution
Advantages
Throughput at same noise
Patient motion - Hx: 1 head -> 2 head
Cost
Non-uniform sampling -> artifacts
History: 7-pinhole - failed
180o sampling - success
Sequential emission transmission
For me, the key advantage of new cameras is sensitivity from non-uniform sampling and resolution from design.
Can design collimator to trade-off sensitivity / resolution

61. What’s Wrong with Ramp-Filtered Backprojection for SPECT
Noise
Attenuation
Scatter
Depth dependant resolution
New imaging geometries

62. (Noise is White (Poisson) in Projection Space not in Object / Image Space)f n p + ^ H k
f = object
A = imaging system
n = noise
p = projections
H = image reconstruction, FBP
f = estimate of object ^

63. Spatial Frequency of Signal and Noise in Reconstructed Image
Signal energy equal at all spatial frequencies k
Noise energy greater at high spatial frequency

64. Decreasing Signal-to-Noisek
No problem when there is a lot more signal than noise. With increased amounts of noise, the noise swamps the signal.
First at very high frequencies, then at low frequencies.
Noise affects high frequencies first,
then lower frequencies

65. What’s Wrong with Filtered Backprojection, FBP, for SPECT
Can’t model:
Attenuation
Scatter
Depth dependant resolution
New imaging geometries
(Linear shift invariant model)

66. Attenuation and Scatter Correction
Increased density (sub-diaphragmatic)
-> increase in both
attenuation
scatter
Opposite effects on count rate
Need to correct for both simultaneously

67. Solution
Iterative reconstruction
Uses:
Simultaneous linear equations
Matrix algebra
Can model image physics
(Linear model)

68. Projections as Simultaneous Equations (Linear Model)
But, exact solution for a large
number of equations isn’t practical
Projection data can be seen to be equivalent to a set of simultaneous linear equations.
Solution of equations is equivalent to reconstruction.

69. ((Simultaneous Equations))
Under-determined -> extra samples
Ill conditioned -> a priori constraints
Over-determined -> inconsistent
Noise -> more inconsistent
Normal Equations of Linear Algebra

70. Large Set of Simultaneous Equations
Image modeled as a vector, not a matrix
128 by 128 image -> 16,384 pixels
Sinogram: about the same
16,384 by 16,384 matrix is impossible
Solution: iterative reconstruction
(Sparse matrix)
Matrix algebra becomes difficult for small sizes, 6x6. Double precision gets to maybe 10x10.
Solution

71. + Image Acquisition A f n p f = object A = imaging system n = noise
p = projections (sinogram)
f is the object, the patients heart.
A is data collection.
p is the projections, the sinogram.

72. f p A is projection f is the object
A is the the projection operation, the red arrows
p is the projection operation p

73. ^ + Image Reconstruction A f n p H ^
H = image reconstruction, e.g. BP, FBP etc.
f = image (estimate) of object, f ^
H is the backprojection operation.
f hat is the estimate of the object, f.

74. ^ f H is backprojection H is the backprojection operation.
f hat is the estimate of the object.
Since this is backprojection our estimate isn’t very good.

75. How Good is the Projection-> Backprojection Estimate?
Can’t compare f to f (f is unknown)
Project f to make p, estimate of projections
Need model of system, A ^
Can’t compare f hat to f because f is unknown.
Solution: project f hat to and make a sinogram p hat.
But, we need to model data collection, A hat.

76. Produce Estimated Projectionsf n p ^ + H
A hat is just physics which is pretty well known.

77. f ^ A ^
Projection backprojection is not exact, so the image (estimate) of the object is not exact.
Thus, the estimate of the projections will not be equal to the projection data. p ^

78. Compare Projections to Estimatef n p ^ e + - H
If the reconstruction (backprojection) is somewhat accurate, then the errors should be small compared to the original data.
|e| ^
< |p|
e is the error in estimate, f

79. Corrections: Backproject Errorsf n p ^ e + - H
corrections
What should we do with the errors?
We can backproject (reconstruct) the errors to give us correction factors.

80. corrections e ^

81. Iterative Backprojection Reconstructionf n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
object
data
projection
backprojection
estimate
model
error
estimated
estimate +
backprojected
Instead of exact solution, use approximate iterative solution
Then add the errors to the image (estimate) of the object to obtain a refined estimate.
Subscripts show the whole iterative process.

82. Iterative Reconstructionf n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
Then add the errors to the image (estimate) of the object to obtain a refined estimate.
Subscripts show the whole iterative process.

83. Iterative Reconstructionf n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
relaxation factor 0-1

84. Reconstruction, H, can be Approximatef n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
Our reconstruction operation, backprojection, is not very accurate. It makes a blurry image of the object.
In general, the reconstruction does not have to be very good.
As long as the image is somewhat like the object, that is good enough.

85. Accuracy of Model, A, is Key^
Accuracy of Model, A, is Key A f n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
What is important is that the model is an accurate representation of the data collection process.
Then we get an accurate comparison between the data and the estimated data.

86. Model, A, is Well-known Physics Problem: Model of the Body^
Model, A, is Well-known Physics Problem: Model of the Body
The model of the imaging system is well known physics.
You can accurately model attenuation and scatter if you know the shape of body.
It’s doable, but it’s not easy. My perception is that the various vendors do this with more or less accuracy.
Tc-99m half-tissue layer: 4 cm

87. Attenuation Map Gd-153 Transmission
Map adds noise to reconstruction
and can introduce artifacts
As part of the data collection process we obtain transmission images at the same time as the perfusion images.
Left: woman, prone - heart anterior, considerable chest wall soft tissue.
Right: man, arms down, supine - heart posterior,down in diaphragm.

88. (Ill Conditioned) A f n p
fn-1 ^
pn-1
en-1 fn + - x f0 r H
Estimating

89. Ill Conditioned: Need Stopping Heuristic
Iterative reconstruction is ill conditioned.
Reconstruction is fairly good after about 10 iterations, but then it gets worse; it goods to the ill conditioned portion of the solution.
EM Iterations

90. Iterative Reconstruction is Ill Conditioned
Stop after N iterations
((The ill conditioned portion of the
solution will be similar to
starting conditions
Starting condition often 0))
A priori constraints
Often smoothness

91. Noise: FBP vs EM SD/mean
Note that iterative reconstruction is much less noisy than FBP.
Number in the corner is standard deviation over the mean.
SD/mean

92. Filtered Back Projection (FBP) vs Ordered Subset Estimation Maximization (OSEM)
FDG PET
Iterative reconstruction can be used for PET scanners.

93. Ordered Subset Estimation Maximization (OSEM)
Estimation Maximization (EM) is an iterative
reconstruction method
Ordered subset EM is a variation of EM
(Update estimate for each subset)
Advantage of OSEM: faster
(OSEM has about 1/nsubsets of EM iterations)
Currently it is the principle iterative method for
both SPECT and PET

94. Iterative Reconstruction Noise is “Blobby”
Notice that the noise character of iterative reconstruction is different.
Noise makes blobs. Recall that FBP noise was “wormy”.
Blobs are a problem in oncology where what you are looking for is blobs.

95. What’s Good About Iterative Reconstruction
Able to model:
Data collection, including new geometries
Attenuation
Scatter
Depth dependant resolution
Fairly efficient given current computers
(Iterative solution, e.g. EM, reasonable)
(OSEM is even better)
((OSEM has about 1/nsubsets of EM iterations))

96. What’s Wrong with Iterative Reconstruction
(Complicated by ill conditioned model)
((Estimating projections not object))
Noise character bad for oncology
To model attenuation & scatter
- need to measure attenuation
- adds noise

97. Conclusions Filtered backprojection, FBP Efficient (Models noise)
“Easy” to understand
Iterative reconstruction, OSEM
Moderately efficient
Models noise, attenuation, scatter,
depth dependant resolution,
and new cameras

98. Applause

99. Projection
Projection: 1, 2, 3, and 4 are projection data.

100. Backprojection
Put the projection data back

101. A ^
Since the system is just projection, we will use projection as our estimate of the system.
In this case the model is exactly equal to the system.