Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Iterative Reconstruction
Methods in Computed
Tomography
J. Webster Stayman
Dept. of Biomedical Engineering, Johns Hopkins University
Power of Iterative Reconstruction
FBP reconstruction
450 mAs
360 angles/360°
AAPM 2013 – Indianapolis, IN
Iterative Reconstruction
10 mAs
100 angles/200°
1
Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Learning Objectives
o Fundamentals of iterative methods
• Approach to forming an iterative algorithm
• Identify particular classes of methods
o Advantages of iterative approaches
• Intuition behind what these algorithms do
• Flexibility of these techniques
o Images produced by iterative methods
• Differences from traditional reconstruction
• Image properties associated with iterative
reconstruction
Iterative Reconstruction
o What is iterative reconstruction?
Projection
P
j ti
Data
I
Image
Volume
Make
M
k Image
I
“Better”
o How can we make the image better?
• Get a better match to the data

Requires a data model
• Enforce desirable image properties

Encourage smoothness, edges, etc.
Desirable
Image
Properties
• Need a measure of “better”
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Building an Iterative Technique
o Define the objective
•
Find the volume that best fits the data and desired image
g q
quality
y
volume  arg max  data, model & image_properties 
ˆ  arg max  y, y    & f    
o Devise an algorithm that solves the objective
̂
• Iteratively solve for 
• Decide when to stop iterating (and how to start)
Objective Function
Optimization Algorithm
Reconstruction Choices
Image
Properties
Forward
Model Driven
Regularized
PICCS
ART
ART
Penalized
Likelihood
Maximum
Likelihood
MAP
Image
Restoration
PWLS
Veo
SAFIRE
Image
Denoising
iDose
IRIS
ASIR AIDR
Statistical Models
*Disclaimer: The exact details of commercially available reconstruction methods are not known by the author.
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Model-based Approaches
o Transmission Tomography Forward Model
• Projection Physics, Beer’s Law
o Need a Parameterization of ~
Parameterization of the Object
o Continuous-domain object
o Want finite number of parameters
o Choices of basis functions:
• Point Samples - Bandlimited
• Contours – Piecewise constant
• Blobs, Wavelets, “Natural Pixels,”…
o Voxel Basis
1 2 3 4 5 6 7 8 9 1011
12 


 p
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Projection
o Linear operation
o Discrete-Discrete for
o parameterized
pa a
d problem
p ob
yi
o System matrix
j‐1 j j 1 j
2
Forward Model
o Mean measurements as a function of parameters
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Aside: Backprojection
Projection
j
Backprojection
A Simple Model-Driven Approach
 
o Forward Model y  
o Objective and Estimator
ˆ  arg min y  y   
I 0 exp  A 
 y
  A
 I0 
ˆ  arg min  log 
1
ˆ  arg min A  l   AT A  AT l
2
P j ti B k j ti
Projection-Backprojection
B k j ti
Backprojection
o For a squared-distance metric
• Many possible algorithms
• Can be equivalent to ART, which seeks
AAPM 2013 – Indianapolis, IN
A  l
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Flexibility of Iterative Methods
1
ˆ  arg min A  l   AT A  AT l
2
o For a well-sampled, parallel beam case
1
1
AT Ax  **x   AT A  x  F 1   2 D  **x
r
o For other cases
1
 AT A  AT l
o Iterative methods implicitly handle the geometry
• Find the correct inversion for the specific geometry
• Cannot overcome data nullspaces (needs complete data)
More Complete Forward Models
o More physics
• Polyenergetic beam, energy-dependent
attenuation
• Detector effects, finite size elements, blur
• Source effects, finite size element
• Scattered radiation
o Noise
• Data statistics
• Quantum noise – x-ray photons
• Detector Noise
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
A Simple Estimation Problem
pdfs
p(y1)
3 Random Variables
Different std dev (1,2,3)
p(y2)
p(y3)
Best way to estimate 

realizations
y1
y2
y3
Maximum Likelihood Estimation
o Find the parameter values most likely to be responsible for
the observed measurements.
o Likelihood Function
L  y;    p  y1 , y2 , , y N 1 , 2 , ,  M 
o Maximum Likelihood Objective Function
ˆ  arg max L  y;  
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
ML for the Simple Example
3
Likelihood function: L  y,    p  yi |     p  yi |  
i 1
p(y1)
p(y2)
p(y3)
Maximize over 
p  yi |   
1
2 i2
2
exp   21 2  yi    
i


ˆ  arg max L  y;  

L  y,    0


ˆ 

3
yi
i 1  i2
3
1
i 1  i2
ML for Tomography
o
o
o
o
Need a noise model
Depends on the statistics of the measurements
Depends on the detection process
Common choices
• Poisson – x-ray photon statistics
• Poisson-Gaussian mixtures – photons + readout noise
• Gaussian (nonuniform variances) – approx. many effects
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Gaussian Likelihood Function
o Marginal Likelihoods
y 
li   log  i  p  li |   
 I0 
o Likelihood
N

2 i2
N
L  l |    p  l |     p  li |    
i 1
i 1

2
exp   21 2 li   A i 


i
1
1
2 i2


2
exp   21 2 li   A i 


i
o Log-Likelihood
N

l L  l |      21 2 li   A i
log
i 1
i

2
o Objective Function and Estimator
1
ˆ  arg max log L  l |     AT D  1  A  AT D  1  l
  
 

2
i
2
i
Iterative Algorithms
o Plethora of iterative approaches
•
•
•
•
Expectation Maximization – General Purpose Methodology
Expectation-Maximization
Gradient-based Methods – Coordinate Ascent/Descent
Optimization Transfer – Paraboloidal Surrogates
Ordered-Subsets methods
o Properties of iterative algorithms
•
•
•
•
Monotonicity
True convergence
g
Speed
Complexity
AAPM 2013 – Indianapolis, IN
10
Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Statistical Reconstruction Example
o Test Case
•
•
•
•
•
Single slice x-ray transmission problem
512 x 512 0.3 mm volume
400 detector bins over 180 angles (360 degrees)
Poisson noise: 1e5 counts per 0.5 mm detector element
SO: 380 mm, DO: 220 mm
o Reconstruction
•
•
•
•
•
Voxel basis: 512 x 512 0.3 mm voxels
Maximum-likelihood objective
EM-type algorithm
Initial image – constant value
Lots of iterations
ML-EM Iterations
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
FBP vs ML-EM Comparison
Enforcing Desirable Properties
o FBP
• Filter designs – cutoff frequencies
o Iterative methods
•
•
•
•
Modify the objective to penalize “bad” images
Discourage noise
Preserve desirable image features
Other prior knowledge
o Image-domain Denoising
ˆ  arg max F      R   
o Model-based Reconstruction
ˆ  arg max F  y,     R   
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Local Control of Image Properties
o Pairwise Penalty
• Penalize the difference between neighboring voxels
R       w jk   j   k 
j k N
o Quadratic Penalty
R       w jk   j  k 
2
j k N
Penalized-Likelihood Example
o Forward Model
•
•
•
•
•
Fan-beam Transmission Tomography
512 x 512 0.3 mm volume
400 detector bins over 180 angles (360 degrees)
Poisson: {1e4,1e3} counts per 0.5 mm detector element
SO: 380 mm, DO: 220 mm
o Objective Function
• Poisson Likelihood
• Quadratic, first-order Penalty
o Algorithm
• Separable Paraboloidal Surrogates
• 400 iterations – well-converged
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
FBP vs PL– 1e4 Counts
FBP vs PL– 1e4 Counts
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
FBP vs PL – 1e3 Counts
FBP vs PL – 1e3 Counts
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Other Penalties/Energy Functions
o Quadratic penalty
• Tends to enforce smoothness throughout image
• Increasingly penalizes larger pixel differences
o Non-quadratic penalties
• Attempt to preserve edges in the image
• Once pixel differences become large allow for decreased
penalty (perhaps a relative decrease)
o Flexibility: Even more penalties
• Wavelets and other bases, non-local means, etc.
Nonquadratic Penalties
o Choices
• Truncated Quadratic
• Lange Penalty
• P-Norm
Truncated Quadratric
AAPM 2013 – Indianapolis, IN
Lange
P-Norm
16
8/5/2013
Lan
nge 1e3 Counts
Lan
nge 1e3 Counts
Iterative Reconstruction Methods in CT
J. Webster Stayman
AAPM 2013 – Indianapolis, IN
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Closer Look at Image Properties
o Test Case
•
•
•
•
•
Single slice x-ray transmission problem
480 x 480 0.8 mm volume
1000 detector bins over 360 angles (360 degrees)
Poisson noise: 1e5 counts / 0.76 mm detector element
SO: 600 mm, DO: 600 mm
o Reconstruction
•
•
•
•
•
Penalized-likelihood objective
Shift-Invariant Quadratic Penalty
Separable paraboloidal surrogates
200 iterations
Voxel basis: 480 x 480 0.8 mm voxels
FBP vs PL, Noise Properties
x 10
5
FBP
-3
ˆ  yˆ  yˆ y 
Penalized-Likelihood
-4
x 10
8
0.03
6
4
50
50
3
100
100
2
150
150
1
200
200
0
250
250
300
300
-2
350
350
0.01
-4
400
400
450
450
-1
-2
-3
-4
-5
100
200
300
400
o Noise in FBP
• Shift-variant variance
• Shift-variant covariance
AAPM 2013 – Indianapolis, IN
0.025
4
2
0.02
0
0.015
0.005
0
-6005
100
200
300
400
-8
o Noise in Quadratic PL
• Relatively shift-invariant
variance (in object)
• Shift-variant covariance
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
FBP vs PL, Resolution Properties




  ˆj  y   true 
ˆ y  ˆtruey  true
j 
FBP
Penalized-Likelihood
-3
x 10
2.5
0.023
50
50
0.022
100
100
100
0.021
150
150
150
0.02
200
200
200
0.019
250
250
250
1
0.018
300
300
300
350
350
350
400
400
400
450
450
450
2
1.5
0.017
0.5
50
100
100
150
200
250
300
350
400
400
450
0.016
0.015
0
0.014
50
100
100
150
200
200
250
300
300
350
400
400
450
-0.5
0.013
FBP vs PL, Resolution Properties




  ˆj  y   true 
ˆ y  ˆtruey  true
j 
FBP
Penalized-Likelihood
-3
x 10
2.5
0.023
50
50
0.022
100
100
100
0.021
150
150
150
0.02
200
200
200
0.019
250
250
250
1
0.018
300
300
300
350
350
350
400
400
400
2
1.5
0.017
0.5
450
0.016
0.015
0
0.014
450
450
50
AAPM 2013 – Indianapolis, IN
100
100
150
200
250
300
350
400
400
450
50
100
100
150
200
200
250
300
300
350
400
400
450
-0.5
0.013
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Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
PL Image Properties – Thorax
MTF
NPS
NPS
MTF
1
4
3
2
NPS
NPS
MTF
MTF
Image Properties
o Filtered-Backprojection
• Largely shift-invariant spatial resolution
• Shift-variant, object-dependent noise
o Uniform Quadratic Penalized Likelihood
• Shift-variant, object-dependent spatial resolution
• Shift-variant, object-dependent noise
o Edge-preserving
Edge preserving Penalty Methods
• Shift-variant, object-dependent spatial resolution
• Shift-variant, object-dependent noise
• Noise-resolution properties may not even be locally
smooth
AAPM 2013 – Indianapolis, IN
20
Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Learning Objectives I
o Fundamentals of iterative methods
• Approach to forming an iterative algorithm
 Forward Model
 Objective Function
 Optimization Algorithm
• Identify particular classes of methods
 Model-based vs. Image Denoising approaches
 Statistical vs. Nonstatistical approaches
 Kinds of regularization
Learning Objectives II
o Advantages of iterative approaches
• Intuition behind what these algorithms do
 Fitting reconstruction to observations
 Data weighting by information content
 Importance of regularization
• Flexibility of these techniques
 Arbitrary geometries
 Sophisticated modeling of physics
 General incorporation of desired image properties
through regularization
AAPM 2013 – Indianapolis, IN
21
Iterative Reconstruction Methods in CT
J. Webster Stayman
8/5/2013
Learning Objectives III
o Images produced by iterative methods
• Differences from traditional reconstruction
 Regularization is key
 Image properties are tied to statistical weighting
 Can depend on algorithm when iterative solution
has not yet converged
• Image properties
i associated
i
d with
i h iiterative
i
reconstruction
 Highly dependent on regularization
 Data- and object dependent
 Shift-variant; different noise, texture than FBP
Acknowledgements
I‐STAR Laboratory
Imaging for Surgery, Therapy, and Radiology
www.jhu.edu/istar
Faculty and Scientists
Jeff Siewerdsen
Yoshito Otake
Sebastian Schafer
Adam Wang
Wojtek Zbijewski
Hopkins Collaborators
School of Medicine
John A Carrino (Radiology)
Gary Gallia (Neurosurgery)
A Jay Khanna (Orthopaedic Surgery)
Martin Radvany (Interventional
Neuroradiology)
Doug Reh (Oto H&N Surgery)
Mahadevappa Mahesh (Radiology)
Marc Sussman (Thoracic Surgery)
BME Students
Hao Dang
Grace Gang
Sajendra Nithiananthan
Steven Tilly II
Jennifer
f Xu
School of Engineering
Greg Hager (CS)
Junghoon Lee (CS)
Jerry Prince (ECE)
y
(CS)
( )
Russell Taylor
CS Students
Sureerat Reaungamornrat
Ali Uneri
Support
AAPM 2013 – Indianapolis, IN
National Institutes of Health
Carestream Health
Varian Medical Systems
22