statistical methods for tomographic image reconstruction
TRANSCRIPT
![Page 1: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/1.jpg)
Statistical methods for tomographic image reconstruction
Jeffrey A. Fessler
EECS Department, BME Department, andNuclear Medicine Division of Dept. of Internal Medicine
The University of Michigan
GE CRD
Jan 7, 2000
![Page 2: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/2.jpg)
Outline
• Group/Lab• PET Imaging• Statistical image reconstruction
Choices / tradeoffs / considerations:◦ 1. Object parameterization◦ 2. System physical modeling◦ 3. Statistical modeling of measurements◦ 4. Objective functions and regularization◦ 5. Iterative algorithms
Short course lecture notes:http://www.eecs.umich.edu/ ∼fessler/talk
• Ordered-subsets transmission ML algorithm• Incomplete data tomography
1
![Page 3: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/3.jpg)
Students
• El Bakri, Idris Analysis of tomographic imaging• Ferrise, Gianni Signal processing for direct brain interface• Ghanei, Amir Model-based MRI brain segmentation• Kim, Jeongtae Image registration/reconstruction for radiotherapy• Stayman, Web Regularization methods for tomographic reconstruction• Sotthivirat, Saowapak Optical image restoration• Sutton, Brad MRI image reconstruction• Yu, Feng (Dan) Nonlocal regularization for transmission reconstruction
Collaborations with colleagues in Biomedical Engineering, EECS, Nuclear Engineering, Nu-clear Medicine, Radiology, Radiation Oncology, Physical Medicine, Anatomy and Cell Biol-ogy, Biostatistics
2
![Page 4: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/4.jpg)
Research Goals
• Develop methods for making “better” images(modeling of imaging system physics and measurement statistics)
• Faster algorithms for computing/processing images• Analysis of the properties of image formation methods• Design of imaging systems based on performance bounds
Impact
• ASPIRE (A sparse iterative reconstruction environment) software(about 40 registered sites worldwide)
• PWLS reconstruction used routinely for cardiac SPECT at UM,following 1996 ROC study. (> 2000 patients scanned)
• Pittsburgh PET/CT “side information” scans reconstructed using ASPIRE
3
![Page 5: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/5.jpg)
PET Data Collection
iRay
Radial Positions
An
gu
lar
Po
siti
on
s
Sinogrami = 1
i = nd
nd ≈ (ncrystals)2
4
![Page 6: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/6.jpg)
PET Reconstruction Problem - Illustrationλ(~x) {Yi}
x2 θ
x1 rImage Sinogram
5
![Page 7: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/7.jpg)
Reconstruction Methods(Simplified View)
Analytical(FBP)
Iterative(OSEM?)
6
![Page 8: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/8.jpg)
Reconstruction Methods
BPFGridding
...ART
MARTSMART
...
Algebraic
SquaresLeast Poisson
Likelihood
FBPStatistical
...
ISRA...
CGCD
ANALYTICAL ITERATIVE
OSEM
FSCDPSCD
Int. PointCG
(y = Ax)
EM (etc.)
SAGE
GCA
7
![Page 9: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/9.jpg)
Why Statistical Methods?
• Object constraints (e.g. nonnegativity)• Accurate models of physics (reduced artifacts, quantitative accuracy)
(e.g. nonuniform attenuation in SPECT, scatter, beam hardening, ...)• System detector response models (possibly improved spatial resolution)• Appropriate statistical models (reduced image noise or dose)
(FBP treats all rays equally)• Side information (e.g. MRI or CT boundaries)• Nonstandard geometries (“missing” data, e.g. truncation)
Tradeoffs...• Computation time• Model complexity• Software complexity• Less predictable (due to nonlinearities), especially for some methods
e.g. Huesman (1984) FBP ROI variance for kinetic fitting8
![Page 10: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/10.jpg)
Five Categories of Choices
1. Object parameterization: λ(~x) vs λ2. System physical model: si(~x)3. Measurement statistical model Yi ∼ ?4. Objective function: data-fit / regularization5. Algorithm / initializationNo perfect choices - one can critique all approaches!
Choices impact:• Image spatial resolution• Image noise• Quantitative accuracy• Computation time• Memory• Algorithm complexity
9
![Page 11: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/11.jpg)
Choice 1. Object Parameterization
Radioisotopespatial distribution→ λ(~x)≈ λ(~x) =
np
∑j=1
λ j bj(~x) ←Series expansion“basis functions”
02
46
8
0
2
4
6
8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
x2
µ 0(x,y)
02
46
8
0
2
4
6
8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Object λ(~x) Pixelized approximation λ(~x)10
![Page 12: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/12.jpg)
Basis FunctionsChoices• Fourier series• Circular harmonics• Wavelets• Kaiser-Bessel windows• Overlapping disks• B-splines (pyramids)
• Polar grids• Logarithmic polar grids• “Natural pixels”• Point masses• pixels / voxels• ...
Considerations• Represent object λ(~x) “well” with moderate np
• system matrix elements {ai j} “easy” to compute• The nd×np system matrix: A= {ai j}, should be sparse (mostly zeros).• Easy to represent nonnegative functions
e.g., if λ j ≥ 0, then λ(~x)≥ 0, i.e. bj(~x)≥ 0.
11
![Page 13: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/13.jpg)
Point-Lattice Projector/Backprojector
λ1 λ2
ith ray
ai j ’s determined by linear interpolation
12
![Page 14: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/14.jpg)
Point-Lattice Artifacts
Projections (sinograms) of uniform disk object:
θ
0◦
45◦
135◦
180◦
r r
Point Lattice Strip Area
13
![Page 15: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/15.jpg)
Choice 2. System ModelSystem matrix A= {ai j} elements:
ai j = P[decay in the jth pixel is recorded by the ith detector unit]
Physical effects• scanner geometry• solid angles• detector efficiency• attenuation• scatter• collimation
• detector response• dwell time at each angle• dead-time losses• positron range• noncolinearity• ...
Considerations• Accuracy vs computation and storage vs compute-on-fly• Model uncertainties
(e.g. calculated scatter probabilities based on noisy attenuation map)• Artifacts due to over-simplifications
14
![Page 16: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/16.jpg)
“Line Length”System Model
“Strip Area”System Model
λ1 λ2
ai j4= length of intersection
ith ray
λ1
λ j−1
ai j4= area
ith ray
15
![Page 17: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/17.jpg)
Sensitivity Patterns
nd
∑i=1
ai j ≈ s(xj) =nd
∑i=1
si(xj)
Line Length Strip Area
16
![Page 18: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/18.jpg)
Forward- / Back-projector “Pairs”Forward projection (image domain to projection domain):
E[Yi] =
∫si(~x)λ(~x)d~x=
np
∑j=1
ai j λ j = [Aλ]i, or E[Y] = Aλ
Backprojection (projection domain to image domain):
A′y=
{nd
∑i=1
ai jyi
}np
j=1
Often A′ is implemented as By for some “backprojector” B 6= A′
Least-squares solutions (for example):
λ= [A′A]−1A′y 6= [BA]−1By
17
![Page 19: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/19.jpg)
Mismatched Backprojector B 6= A′ (3D PET)λ λ (PWLS-CG) λ (PWLS-CG)
(64×64×4) Matched Mismatched18
![Page 20: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/20.jpg)
Horizontal Profiles
0 10 20 30 40 50 60 70−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
MatchedMismatchedλ(
~x)
19
![Page 21: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/21.jpg)
Choice 3. Statistical ModelsAfter modeling the system physics, we have a deterministic “model:”
Y ≈ E[Y] = Aλ+ r.
Statistical modeling is concerned with the “ ≈ ” aspect.
Random Phenomena• Number of tracer atoms injected N• Spatial locations of tracer atoms {~Xk}N
k=1• Time of decay of tracer atoms {Tk}N
k=1• Positron range• Emission angle• Photon absorption
• Compton scatter• Detection Sk 6= 0• Detector unit {Sk}
ndi=1
• Random coincidences• Deadtime losses• ...
20
![Page 22: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/22.jpg)
Statistical Model Considerations
• More accurate models:◦ can lead to lower variance images,◦ can reduce bias◦ may incur additional computation,◦ may involve additional algorithm complexity
(e.g. proper transmission Poisson model has nonconcave log-likelihood)• Statistical model errors (e.g. deadtime)• Incorrect models (e.g. log-processed transmission data)
21
![Page 23: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/23.jpg)
Statistical Model Choices
• “None.” Assume Y− r = Aλ. “Solve algebraically” to find λ.• White Gaussian noise. Ordinary least squares: minimize ‖Y−Aλ‖2
• Non-White Gaussian noise. Weighted least squares: minimize
‖Y−Aλ‖2W =
nd
∑i=1
wi (yi− [Aλ]i)2, where [Aλ]i4=
np
∑j=1
ai j λ j
• Ordinary Poisson model (ignoring or precorrecting for background)
Yi ∼ Poisson{[Aλ]i}
• Poisson modelYi ∼ Poisson{[Aλ]i+ ri}
• Shifted Poisson model (for randoms precorrected PET)
Yi =Yprompti −Ydelay
i ∼ Poisson{[Aλ]i+2ri}−2ri
22
![Page 24: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/24.jpg)
Transmission Phantom
FBP 7hour FBP 12min
Thorax PhantomECAT EXACT
23
![Page 25: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/25.jpg)
Effect of statistical model
OSEM
OSTR
Iteration: 1 3 5 7
24
![Page 26: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/26.jpg)
Choice 4. Objective FunctionsComponents:• Data-fit term• Regularization term (and regularization parameter β)• Constraints (e.g. nonnegativity)
Φ(λ) = DataFit(Y,Aλ+ r)−β ·Roughness(λ)
λ 4= argmaxλ≥0
Φ(λ)
“Find the image that ‘best fits’ the sinogram data”
Actually three choices to make for Choice 4 ...
Distinguishes “statistical methods” from “algebraic methods” for “Y = Aλ.”
25
![Page 27: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/27.jpg)
Why Objective Functions?(vs “procedure” e.g. adaptive neural net with wavelet denoising)
Theoretical reasonsML is based on maximizing an objective function: the log-likelihood• ML is asymptotically consistent• ML is asymptotically unbiased• ML is asymptotically efficient (under true statistical model...)• Penalized-likelihood achieves uniform CR bound asymptotically
Practical reasons• Stability of estimates (if Φ and algorithm chosen properly)• Predictability of properties (despite nonlinearities)• Empirical evidence (?)
26
![Page 28: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/28.jpg)
Choice 4.1: Data-Fit Term
• Least squares, weighted least squares (quadratic data-fit terms)• Reweighted least-squares• Model-weighted least-squares• Norms robust to outliers• Log-likelihood of statistical model. Poisson case:
L(λ;Y) = logP[Y = y;λ] =nd
∑i=1
yi log([Aλ]i+ ri)−([Aλ]i+ ri)− logyi!
Poisson probability mass function (PMF):
P[Y = y;λ] =∏ndi=1e−yi yyi
i /yi! where y4= Aλ+ r
Considerations• Faithfulness to statistical model vs computation• Effect of statistical modeling errors
27
![Page 29: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/29.jpg)
Choice 4.2: RegularizationForcing too much “data fit” gives noisy imagesIll-conditioned problems: small data noise causes large image noise
Solutions:• Noise-reduction methods◦ Modify the data (prefilter or extrapolate sinogram data)◦ Modify an algorithm derived for an ill-conditioned problem
(stop before converging, post-filter)• True regularization methods
Redefine the problem to eliminate ill-conditioning◦ Use bigger pixels (fewer basis functions)◦ Method of sieves (constrain image roughness)◦ Change objective function by adding a roughness penalty / prior
R(λ) =np
∑j=1
∑k∈N j
ψ(λ j−λk)
28
![Page 30: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/30.jpg)
Noise-Reduction vs True RegularizationAdvantages of “noise-reduction” methods• Simplicity (?)• Familiarity• Appear less subjective than using penalty functions or priors• Only fiddle factors are # of iterations, amount of smoothing• Resolution/noise tradeoff usually varies with iteration
(stop when image looks good - in principle)
Advantages of true regularization methods• Stability• Predictability• Resolution can be made object independent• Controlled resolution (e.g. spatially uniform, edge preserving)• Start with (e.g.) FBP image⇒ reach solution faster.
29
![Page 31: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/31.jpg)
Unregularized vs Regularized Reconstruction
ML (unregularized)
Penalized likelihood
Iteration:
(OSTR)
1 3 5 7
30
![Page 32: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/32.jpg)
Roughness Penalty Function Considerations
R(λ) =np
∑j=1
∑k∈N j
ψ(λ j−λk)
• Computation• Algorithm complexity• Uniqueness of maximum of Φ• Resolution properties (edge preserving?)• # of adjustable parameters• Predictability of properties (resolution and noise)
Choices• separable vs nonseparable• quadratic vs nonquadratic• convex vs nonconvex
This topic is actively debated!31
![Page 33: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/33.jpg)
Nonseparable Penalty Function Example
x1 x2 x3
x4 x5
Example
R(x) = (x2−x1)2+(x3−x2)
2+(x5−x4)2
+(x4−x1)2+(x5−x2)
2
2 2 2
2 1
3 3 1
2 2
1 3 1
2 2R(x) = 1 R(x) = 6 R(x) = 10
Rougher images⇒ greater R(x)
32
![Page 34: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/34.jpg)
Penalty Functions: Quadratic vs Nonquadratic
Phantom Quadratic Penalty Huber Penalty
33
![Page 35: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/35.jpg)
Summary of Modeling Choices
1. Object parameterization: λ(x) vs λ2. System physical model: si(x)3. Measurement statistical model Yi ∼ ?4. Objective function: data-fit / regularization / constraints
Reconstruction Method = Objective Function + Algorithm
5. Iterative algorithmML-EM, MAP-OSL, PL-SAGE, PWLS+SOR, PWLS-CG, . . .
34
![Page 36: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/36.jpg)
Choice 5. Algorithms
Measurements Attenuation ...
Parameters
ModelSystem
IterationΦ
x(n) x(n+1)
Deterministic iterative mapping: x(n+1) =M (x(n))All algorithms are imperfect. No single best solution.
35
![Page 37: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/37.jpg)
Ideal Algorithm
x?4= argmax
x≥0Φ(x) (global maximum)
stable and convergent {x(n)} converges to x? if run indefinitelyconverges quickly {x(n)} gets “close” to x? in just a few iterationsglobally convergent limnx(n) independent of starting imagefast requires minimal computation per iterationrobust insensitive to finite numerical precisionuser friendly nothing to adjust (e.g. acceleration factors)monotonic Φ(x(n)) increases every iterationparallelizable (when necessary)simple easy to program and debugflexible accommodates any type of system model
(matrix stored by row or column or projector/backprojector)Choices: forgo one or more of the above
36
![Page 38: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/38.jpg)
Optimization Transfer Illustrated
0
0.2
0.4
0.6
0.8
1 Φ
φ
Objective ΦSurrogate φ
x(n)x(n+1)
Φ(x)
and
φ(x;
x(n))
37
![Page 39: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/39.jpg)
Convergence Rate: Fast
Fast Convergence
Old
Large StepsLow Curvature
xNew
φ
Φ
38
![Page 40: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/40.jpg)
Slow Convergence of EM
−1 −0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2L: Log−LikelihoodQ: EM Surrogate
l
h i(l)
and
Q(l
;ln )
39
![Page 41: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/41.jpg)
Paraboloidal Surrogates
• Not separable (unlike EM)• Not self-similar (unlike EM)• Poisson log-likelihood replaced by a series of least squares problems.• Maximize each quadratic problem easily using coordinate ascent.
Advantages• Fast converging• Instrinsically monotone global convergence• Fairly simple to derive / implement• Nonnegativity easy (with coordinate ascent)
Disadvantages• Coordinate ascent ... column-stored system matrix
40
![Page 42: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/42.jpg)
Convergence rate: PSCA vs EM
0 2 4 6 8 10400
450
500
550
600
650
700
750
800
Iteration
Obj
ectiv
e F
unct
ion
Φ(x
n)
PSCAOSDPEMDP
41
![Page 43: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/43.jpg)
Ordered Subsets Algorithms
• The backprojection operation appears in every algorithm.• Intuition: with half the angular sampling, the backprojection would look
fairly similar.• To “OS-ize” an algorithm, replace all backprojections with partial sums.
Problems with OS-EM• Non-monotone• Does not converge (may cycle)• Byrne’s RBBI approach only converges for consistent (noiseless) data• ... unpredictable• What resolution after n iterations?• Object-dependent, spatially nonuniform• What variance after n iterations?• ROI variance? (e.g. for Huesman’s WLS kinetics)
42
![Page 44: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/44.jpg)
OSEM vs Penalized Likelihood
• 64×62 image• 66×60 sinogram• 106 counts• 15% randoms/scatter• uniform attenuation• contrast in cold region• within-region σ opposite side
43
![Page 45: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/45.jpg)
Contrast-Noise Results
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Contrast
Noi
se
Uniform image
(64 angles)
OSEM 1 subsetOSEM 4 subsetOSEM 16 subsetPL−PSCA
↙
44
![Page 46: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/46.jpg)
0 10 20 30 40 50 60 700
0.5
1
1.5
x1
Rel
ativ
e A
ctiv
ity
Horizontal Profile
OSEM 4 subsets, 5 iterationsPL−PSCA 10 iterations
45
![Page 47: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/47.jpg)
Noise Properties
Cov{x} ≈[∇20Φ
]−1[∇11Φ]
Cov{Y}[∇11Φ
]T [∇20Φ]−1
• Enables prediction of noise properties• Useful for computing ROI variance for kinetic fitting
IEEE Tr. Image Processing, 5(3):493 1996
46
![Page 48: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/48.jpg)
Summary
• General principles of statistical image reconstruction• Optimization transfer• Principles apply to transmission reconstruction• Predictability of resolution / noise and controlling spatial resolution
argues for regularized objective-function• Still work to be done...
An Open ProblemStill no algorithm with all of the following properties:• Nonnegativity easy• Fast converging• Intrinsically monotone global convergence• Accepts any type of system matrix• Parallelizable
47
![Page 49: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/49.jpg)
Fast Maximum Likelihood Transmission Reconstructionusing Ordered Subsets
Jeffrey A. Fessler, Hakan Erdogan
EECS Department, BME Department, andNuclear Medicine Division of Dept. of Internal Medicine
The University of Michigan
![Page 50: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/50.jpg)
Transmission Scans
Ph
oto
n S
ou
rce
Det
ecto
r B
ins
Each measurement Yi is related to a single “line integral” through the object.
Yi ∼ Poisson
{bi exp
(−
p
∑j=1
ai jµj
)+ ri
}
48
![Page 51: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/51.jpg)
Transmission Scan Statistical Model
Yi ∼ Poisson
{bi exp
(−
p
∑j=1
ai jµj
)+ ri
}, i = 1, . . . ,N
• N number of detector elements• Yi recorded counts by ith detector element• bi blank scan value for ith detector element• ai j length of intersection of ith ray with jth pixel• µj linear attenuation coefficient of jth pixel• ri contribution of room background, scatter, and emission crosstalk
(Monoenergetic case, can be generalized for dual-energy CT)(Can be generalized for additive Gaussian detector noise)
49
![Page 52: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/52.jpg)
Maximum-Likelihood Reconstruction
µ= argmaxµ≥0
L(µ) (Log-likelihood)
L(µ) =N
∑i=1
Yi log
[bi exp
(−
p
∑j=1
ai jµj
)+ ri
]−
[bi exp
(−
p
∑j=1
ai jµj
)+ ri
]
Transmission ML Reconstruction Algorithms• Conjugate gradient
Mumcuoglu et al., T-MI, Dec. 1994
• Paraboloidal surrogates coordinate ascent (PSCA)Erdogan and Fessler, T-MI, 1999
• Ordered subsets separable paraboloidal surrogatesErdogan et al., PMB, Nov. 1999
• Transmission expectation maximization (EM) algorithmLange and Carson, JCAT, Apr. 1984
50
![Page 53: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/53.jpg)
Optimization Transfer Illustrated
0
0.2
0.4
0.6
0.8
1 Φ
φ
Objective ΦSurrogate φ
µ(n)µ(n+1)
Φ(µ)
and
φ(µ;
µ(n))
51
![Page 54: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/54.jpg)
Parabola Surrogate Function
• h(l) = ylog(be−l+ r)− (be−l+ r) has a parabola surrogate: q(n)im• Optimum curvature of parabola derived by Erdogan (T-MI, 1999)• Replace likelihood with paraboloidal surrogate
L(µ(n)) =N
∑i=1
hi
(p
∑j=1
ai jµj
)≥Q1(µ;µ(n)) =
N
∑i=1
q(n)im
(p
∑j=1
ai jµj
)
• q(n)im is a simple quadratic function• Iterative algorithm:
µ(n+1) = argmaxµ≥0
Q1(µ;µ(n))
• Maximizing Q1(µ;µ(n)) over µ is equivalent to (reweighted) least-squares.• Natural algorithms◦ Conjugate gradient◦ Coordinate ascent
52
![Page 55: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/55.jpg)
Separable Paraboloid Surrogate Function
• Parabolas are convex functions• Apply De Pierro’s “additive” convexity trick (T-MI, Mar. 1995)
p
∑j=1
ai jµj =p
∑j=1
ai j
ai
[ai(µj−µ(n)j )
]+[Aµ(n)
]i
where ai4=
p
∑j=1
ai j
• Move summation over pixels outside quadratic
Q1(µ;µ(n)) =N
∑i=1
q(n)im
(p
∑j=1
ai jµj
)
≥ Q2(µ;µ(n)) =N
∑i=1
p
∑j=1
ai j
aiq(n)im
(ai(µj−µ(n)j )+
[Aµ(n)
]i
)
=p
∑j=1
Q(n)2 j (µj), where Q(n)2 j (x)4=
N
∑i=1
ai j
aiq(n)im
(ai(x−µ(n)j )+
[Aµ(n)
]i
)• Separable paraboloidal surrogate function⇒ trivial to maximize (cf EM)
53
![Page 56: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/56.jpg)
Iterative algorithm:
µ(n+1)j = argmax
µj≥0Q(n)2 j (µj) =
µ(n)j +
∂∂µj
Q(n)2 j (µ(n))
− ∂2
∂µ2jQ(n)2 j (µ
(n))
+
=
µ(n)j +
1
− ∂2
∂µ2jQ(n)2 j (µ
(n))
∂∂µj
L(µ(n))
+
=
[µ(n)j +
∑Ni=1(yi/y
(n)i −1)bi exp
(−[Aµ(n)
]i
)∑N
i=1a2i jaic
(n)i
]+
, j = 1, . . . , p
• c(n)i ’s related to parabola curvatures• Parallelizable (ideal for multiprocessor workstations)• Monotonically increases the likelihood each iteration• Intrinsically enforces the nonnegativity constraint• Guaranteed to converge if unique maximizer• Natural starting point for forming ordered-subsets variation
54
![Page 57: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/57.jpg)
Ordered Subsets Algorithm
• Each ∑Ni=1 is a backprojection
• Replace “full” backprojections with partial backprojections• Partial backprojection based on angular subsampling• Cycle through subsets of projection angles
Pros• Accelerates “convergence”• Very simple to implement• Reasonable images in just 1 or 2 iterations• Regularization easily incorporated
Cons:• Does not converge to true maximizer• Makes analysis of properties difficult
55
![Page 58: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/58.jpg)
Phantom Study
• 12-minute PET transmission scan• Anthropomorphic thorax phantom (Data Spectrum, Chapel Hill, NC)• Sinogram: 160 3.375mm bins by 192 angles over 180◦
• Image: 128 by 128 4.2mm pixels• Ground truth determined from 15-hour scan, FBP reconstruction / seg-
mentation
56
![Page 59: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/59.jpg)
Algorithm Convergence
0 5 10 15 20 25 301200
1250
1300
1350
1400
1450
1500
1550
1600
Iteration
Obj
ectiv
e D
ecre
ase
Transmission Algorithms
Initialized with FBP Image
PL−OSTR−1PL−OSTR−4PL−OSTR−16PL−PSCD
57
![Page 60: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/60.jpg)
Reconstructed Images
FBP ML−OSEM−8
2 iterations
ML−OSTR−8
3 iterations
58
![Page 61: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/61.jpg)
Reconstructed Images
FBP PL−OSTR−16
4 iterations
PL−PSCD
10 iterations
59
![Page 62: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/62.jpg)
Segmented Images
FBP ML−OSEM−8
2 iterations
ML−OSTR−8
3 iterations
60
![Page 63: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/63.jpg)
Segmented Images
FBP PL−OSTR−16
4 iterations
PL−PSCD
10 iterations
61
![Page 64: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/64.jpg)
0 2 4 6 8 10 12 140
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
iterations
norm
aliz
ed m
ean
squa
red
erro
r
NMSE performance
ML−OSTR−8ML−OSTR−16ML−OSEM−8PL−OSTR−16PL−PSCDFBP
62
![Page 65: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/65.jpg)
0 2 4 6 8 10 12 140
1
2
3
4
5
6
7
8
iterations
perc
enta
ge o
f seg
men
tatio
n er
rors
Segmentation performance
ML−OSTR−8ML−OSTR−16ML−OSEM−8PL−OSTR−16PL−PSCDFBP
63
![Page 66: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/66.jpg)
Quantitative Results
NMSE
FBP
ML−OSEM
ML−OSTR
PL−OSTR
PL−PSCD
Segmentation Errors
FBP
ML−OSEM
ML−OSTR
PL−OSTR
PL−PSCD
0% 6.5% 0% 5.5%
64
![Page 67: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/67.jpg)
FDG PET Patient Data, PL-OSTR vs FBP
(15-minute transmission scan | 2-minute transmission scan)65
![Page 68: Statistical methods for tomographic image reconstruction](https://reader030.vdocument.in/reader030/viewer/2022012020/616889c4d394e9041f705d54/html5/thumbnails/68.jpg)
Truncated Fan-Beam SPECT Transmission
Truncated Truncated UntruncatedFBP PWLS FBP
66