1. Statistical Methods for Image Reconstruction
MIC 2007 Short Course
October 30, 2007 Honolulu

2. Topics Lecture I: Warming Up (Larry Zeng)
Lecture II: ML and MAP Reconstruction (Johan Nuyts)
Lecture III: X-Ray CT Iterative Reconstruction (Bruno De Man)
Round-Table Discussion of Your Interested Topics

3. Larry Zeng, Ph.D. University of Utah
Warming Up
Larry Zeng, Ph.D.
University of Utah

4. Problem to solve: Computed Tomography with discrete measurements
Continuous
Unknown
Image

5. Analytic Algorithms For example, FBP (Filtered Backprojection)
Treat the unknown image as continuous
Point-by-point reconstruction (arbitrary points)
Regular grid points are commonly chosen for display
Continuous
Unknown
Image

6. Iterative Algorithms For example, ML-EM, OS-EM, ART, GC, …
Discretize the image into pixels
Solve imaging equations AX=P
X = unknowns (pixel values)
P = projection data
A = imaging system matrix P
aij X

7. Example x4 x3 x2 x1 p4 p3 p2 p1
AX=P

8. Example 5 4 x1 x2 3 AX=P Rank(A) = 3 < 4 System not consistent
No solution x3 x4 2

9. Solving AX=P
In practice, A is not invertible (not square, or not full rank)
A generalized inverse is used X = A†P
Moore-Penrose inverse A†=(ATA)-1AT
AA†A=A
A†AA†=A†
Least-squares solution (Gaussian noise model)

10. Least-squares solution4 3 5 4
2.25
1.75 3 4 2 3
1.75
1.25

11. The most significant motivation of using an iterative algorithm: Usually, iterative algorithms give better reconstructions than analytical algorithms.

12. Computer Simulations Examples
(Fessler)
FBP = Filtered Backprojection (Analytical)
PWLS = Penalized data-weighted least squares (Iterative)
PL (or MAP) = Penalized likelihood (Iterative)

13. Why are iterative algorithms better than analytical algorithms?
The system matrix A is more flexible than the assumptions in an analytical algorithm
It is a lot harder for an analytical algorithm to handle some realistic situations

14. Attenuation in SPECT g

15. Attenuation in SPECT Photon attenuation is spatially variant
Non-uniform atten. modeled in an iterative algorithm system matrix A (1970s)
Uniform atten. in analytic algorithm (1980s)
Non-uniform atten. in analytic algorithm (2000s)

16. Distance-Dependent Blurring
Close
Far

17. Distance-Dependent Blurring
System matrix A models the blurring P X

18. Distance-Dependent Blurring
Analytic treatment uses the Frequency-Distance Principle (approximation)
Near
Far
Near
Far
Slope ~ Distance

19. Truncation (Analytic) Old FBP algorithm does not allow truncation
(Iterative) System matrix only models measured data, ignores unmeasured data
(Analytic) New DBP (Derivative Backprojection) algorithm can handle some truncation situations

20. Scatter
Scattered
Primary

21. Scatter
(Iterative) System matrix A can model scatter using “effective scattering source” or Monte Carlo
(Analytic) Still don’t know how, other than preprocessing data using multiple energy windows

22. In physics and imaging geometry modeling, analytic methods are behind the iterative ones.

23. Analytic algorithm — “Open-loop system”
Data
Filtering
Backprojection
Image

24. Iterative algorithm — “Closed-loop system”
Compare
Image
Projection
Data A AT
Update
Backprojection

25. Main Difference
Analytic algorithms do not have a projector A, but have a set of assumptions.
Iterative algorithms have a projector A.

26. Under what conditions will the analytic algorithm outperform the iterative algorithm?
The analytical assumptions (e.g., sampling, geometry, physics)
System matrix A is always an approximation (e.g., the pixel model assumes a uniform activity within the pixel)

27. Noise

28. Noise Consideration Iterative: Objective function (e.g., Likelihood)
Iterative: Regularization (very effective!)
Analytic: Filtering (assumes spatially invariant, not as good)

29. Modeling noise in an iterative algorithm (thus Statistical Algorithm)
Example: A one-pixel image (i.e., total count is the only concern)
3 measurements:
1100 counts for 10 sec. (110/s)
100 counts for 1 sec. (100/s)
15000 counts for 100 sec. (150/s)
Counts per second?

30. m1 = 1100, s2(m1)=1100,
x1=1100/10=110,
s2(x1)= s2(m1)/(10)2 = 1100/100=11
m2 = 100, s2(m2)=100,
x2=100/1=10,
s2(x2)= s2(m2)= 100
m3 = 15000, s2(m3)=15000,
x3=15000/100=150,
s2(x3)= s2(m3)/(100)2 = 15000/10000=1.5

31. Objective Function

32. Generalization
If you have N unknowns, you need to make a lot more than N measurements.
The measurements are redundant.
Put more weights on the measurements that you trust more.

33. Geometric Illustration
Two unknowns x1, x2
Each measurement = a linear eqn = 1 line
We make 3 measurements; we have 3 lines

34. Noise Free (consistent)x2 x1

35. Noisy (inconsistent) x2 e2 e3 e1 x1

36. If you don’t have redundant measurements, the noise model (e. g
If you don’t have redundant measurements, the noise model (e.g., s2) does not make any differences x2 x1

37. In practice, # of unknowns ≈ # of measurements
However, iterative methods still outperform analytic methods
Why?

38. Answer: Regularization & Constraints
Helpful constraints:
Non-negativity (xi ≥ 0)
Image support (xk = 0, x in K)
Bounds of values (e.g. transmission map)
Prior

39. Some common regularization methods (1) — Stop early
Why stop? Doesn’t the algorithm converge?
Yes, in many cases (e.g., ML-EM) it does.
When an ML-EM reconstruction becomes noisy, its likelihood function is still increasing.
The ultimate solution (e.g., ML-EM) solution is too noisy; we don’t like it.

40. Iterative Reconstruction
An Example (ML-EM)

41. Iterative Reconstruction
An Example (ML-EM)

42. Iterative Reconstruction
An Example (ML-EM)

43. Iterative Reconstruction
An Example (ML-EM)

44. Iterative Reconstruction
An Example (ML-EM)

45. Iterative Reconstruction
An Example (ML-EM)

46. Iterative Reconstruction
An Example (ML-EM)

47. Iterative Reconstruction
An Example (ML-EM)

48. EARLY Stopping early is like lowpass filtering.
Low freq. components converge faster than high freq. components.

49. OR: Over-iterate, then filter

50. Regularization using Prior
What is a prior?
Example:
You want to estimate tomorrow’s temperature.
You know today’s temperature.
You assume that tomorrow’s temperature is pretty close to today’s

51. Regularization using Prior
MAP algorithm = Maximum A Posteriori algorithm = Bayesian

52. How to select a prior? V If V(x) = x2, it enforces smoothness.
Not easy. It is an art.
Example: V
Prior
encouragement
Data matching
Noise model
If V(x) = x2, it enforces smoothness.

53. Edge Preserving Prior V(x)
If V(x) = |x|, it preserves edges and reduces noise. x

54. How does V know where is the edge?
V(x) = |x| increases a lot slower than V(x) = x2.
V(x) = |x| treats small jumps as noise to suppress and large jumps as edges not to suppress as much.
The shape of V(x) is important.
V(x) x

55. What is an ML algorithm?
An algorithm that maximizes the likelihood function.

56. Example: Gaussian Noise
A likelihood function is a conditional probability
Take log
Thus maximize the likelihood function L(X) is equivalent to Least-Squares Problem

57. Least-Squares = ML for Gaussian
An algorithm that minimizes
is an ML algorithm for independent Gaussian noise
One can set up a Poisson likelihood function and find an ML algorithm for that. The well-known ML-EM algorithm is one of such algorithms.

58. What is a MAP algorithm? MAP = maximum a posteriori
An algorithm that maximizes the likelihood function with a prior
For example, an algorithm that minimizes the following objective function is a MAP algorithm

59. What is ill-condition?
Small data noise causes large reconstruction noise
SVD

60. Regularization changes an ill-conditioned problem into a different well-conditioned problem.

61. Example
Big error

62. Assume a prior:x1 and x2 are close
Objective Function:
Note: b is NOT a Lagrange multiplier

63. To minimize
We set
We obtain a different linear problem

64. MAP Solutions Noiseless (d1 = d2= 0)b
0.01
0.1 1 10
100
Cond. #
50000
200 22
5.7 x1
0.978
1.094 x2
22.222
1.178
1.103
1.095

65. MAP Solutions Noisy (d1 = -d2= 0.2)b
0.01
0.1 1 10
100
Cond. #
50000
200 22
5.7 x1
0.978
0.733
1.094
1.095 x2
22.222
66.667
1.178
1.357
1.103
1.122
1.098
1.096

66. Observations Using a prior changes the original problem
For noiseless data, the solutions are different
More stable when there is noise
It is an art

67. Observations
In fact, a closed-form solution exist for this PWLS (penalized weighted least squares) problem:

68. Can analytic algorithm do these?
Not yet.
They don’t even know how to correctly enforce the non-negativity, the support etc.

69. On Condition Numbers AX=P
Error propagation:
k = Conditin #
Trade-off: Introducing b reduces k, but increases ||DA||.

70. More on Condition Numbers AX=P
Error propagation:
k = Conditin #
Trade-offs:
More accurate (i.e., ||DA|| significantly smaller ) modeling makes A less sparse
A may have a larger k
The overall error of X may be reduced (better resolution and lower noise)

71. Ray-Driven & Pixel-Driven Projector(A)/Backprojector(AT)
Ray-driven aij:
Along a projection ray, calculate how much pixel i contributes to detector bin j.
Usually, line-length and trip-area weighting

72. Line-Length Weightingpj
aij
Projection (A): X
Backprojection (AT):

73. Area Weighting pj
aij
Projection (A): X
Backprojection (AT):

74. Rotation-Based Projector/Backprojector
Very easy to implement distance dependent blurring
Fast
Warping is required for convergent beams

75. Pixel-Driven Backprojector
Widely used in analytic algorithms
aij = 1 P
Interpolation is required on the detector X

76. If you don’t know what you are doing, do not use an “iterative algorithm backprojector” (AT), e.g., line-length weighted backprojector, in an analytic algorithm
An “analytic algorithm backprojector” backprojector is not AT. You may not want to use it in an iterative algorithm if you don’t know what you are doing.

77. Some Common Terms Used in Iterative Algorithms
Pixels (Voxels)
Easy to use
Unrealistic high-frequency edges

78. Blobs (no edge, overlap)x y z
Voxel x y z
Blob

79. Natural Pixels Projection path Depends on the collimator geometry
A Natural Pixel
f = Σ wi xi

80. Iterative Methods Summary
Can incorporate
Imaging physics
Irregular imaging geometry
Prior information
But
It is an art to set things up
Complex
Long computation time