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Introduction Scalable Computation Informative Priors Conclusion
Bayesian Computational Methodsfor Spatial Analysis of Images
Matthew MooresMathematical Sciences School
Science & Engineering Faculty, QUT
PhD final seminarAugust 1, 2014
Introduction Scalable Computation Informative Priors Conclusion
Acknowledgements
Principal supervisor: Kerrie Mengersen
Associate supervisor: Fiona Harden
Members of the Volume Analysis Tool project team at theRadiation Oncology Mater Centre (ROMC), Queensland Health:
Cathy Hargrave
Mike Poulsen
Tim Deegan
QHealth ethics HREC/12/QPAH/475 and QUT ethics 1200000724
Other co-authors:
Chris Drovandi
Clair Alston
Christian Robert
Introduction Scalable Computation Informative Priors Conclusion
Outline
1 IntroductionImage-Guided RadiotherapyCone-Beam Computed TomographyAims & Objectives of the Thesis
2 Scalable ComputationDoubly-Intractable LikelihoodsPre-computation for ABC-SMCR package bayesImageS
3 Informative PriorsInformative Prior for µj and σ2
j
External FieldExperimental Results
4 Conclusion
Introduction Scalable Computation Informative Priors Conclusion
Objectives
The overall objectives of the research are:
to develop a generative model of a digital image thatincorporates prior information,
to produce a computationally efficient implementation of thismodel, and
to apply the model to real world data in image-guidedradiotherapy and satellite remote sensing.
This reflects the parallel perspectives of statistical methods,computational algorithms, and applied bio- and geo-statistics.
Introduction Scalable Computation Informative Priors Conclusion
Image-Guided Radiotherapy
Image courtesy of Varian Medical Systems, Inc. All rights reserved.
Introduction Scalable Computation Informative Priors Conclusion
Radiotherapy Process
Before Treatment
fan-beamCT
MRI
contourstreatmentplan
QA
Daily Fractions (∼8 weeks)
positionpatient
cone-beamCT
deliverdose
off-lineanalysis
Introduction Scalable Computation Informative Priors Conclusion
Radiotherapy Process
Before Treatment
fan-beamCT
MRI
contourstreatmentplan
QA
Daily Fractions (∼8 weeks)
positionpatient
cone-beamCT
deliverdose
off-lineanalysis
Introduction Scalable Computation Informative Priors Conclusion
Segmentation of Anatomical Structures
Radiography courtesy of Cathy Hargrave, Radiation Oncology Mater Centre
Introduction Scalable Computation Informative Priors Conclusion
Physiological Variability
Distribution of observed translations of the organs of interest:
Organ Ant-Post Sup-Inf Left-Right
prostate 0.1± 4.1mm −0.5± 2.9mm 0.2± 0.9mmseminal vesicles 1.2± 7.3mm −0.7± 4.5mm −0.9± 1.9mm
Volume variations in the organs of interest:
Organ Volume Gas
rectum 35− 140cm3 4− 26%bladder 120− 381cm3
Frank, et al. (2008) Quantification of Prostate and Seminal VesicleInterfraction Variation During IMRT. IJROBP 71(3): 813–820.
Introduction Scalable Computation Informative Priors Conclusion
Cone-Beam Computed Tomography
(a) Fan-beam CT (b) Cone-beam CT
Introduction Scalable Computation Informative Priors Conclusion
Distribution of Pixel Intensity
Hounsfield unit
Fre
quency
−1000 −800 −600 −400 −200 0 200
05000
10000
15000
(a) Fan-Beam CT
pixel intensity
Fre
quency
−1000 −800 −600 −400 −200 0 2000
5000
10000
15000
(b) Cone-Beam CT
Introduction Scalable Computation Informative Priors Conclusion
Specific Aims I
The statistical aims of the research are:
M1 derivation and representation of informative priors forthe pixel labels.
M2 derivation of informative priors for additive Gaussiannoise from a previous image of the same subject.
M3 sequential Bayesian updating of this prior informationas more images are acquired.
The computational aims are:
C1 measuring the scalability of existing methods forBayesian inference with intractable likelihoods.
C2 development and implementation of improvedalgorithms for fast, approximate inference in imageanalysis.
Introduction Scalable Computation Informative Priors Conclusion
Specific Aims II
The applied aims are:
A1 To classify pixels in cone-beam CT scans ofradiotherapy patients according to tissue type.
A2 To demonstrate the broad applicability of thesemethods by classifying pixels in satellite imageryaccording to land use or abundance of phytoplankton.
Introduction Scalable Computation Informative Priors Conclusion
Research Progress
1 Moores, Hargrave, Harden & Mengersen (2014). Segmentation ofcone-beam CT using a hidden Markov random field with informativepriors. Journal of Physics: Conference Series 489:012076.
2 Moores & Mengersen (2014). Bayesian approaches to spatial inference:modelling and computational challenges and solutions. To appear in AIPConference Proceedings.
3 Moores, Drovandi, Mengersen & Robert. Pre-processing for approximateBayesian computation in image analysis. Statistics & Computing(Submitted: March 2014, Revised: June 2014).
4 Moores, Hargrave, Harden & Mengersen. An external field prior for thehidden Potts model with application to cone-beam computed tomography.Computational Statistics & Data Analysis (currently in revision).
5 Moores, Alston & Mengersen. Scalable Bayesian inference for the inversetemperature of a hidden Potts model. (In Prep).
6 Moores, Hargrave, Deegan, Poulsen, Harden & Mengersen. Multi-objectsegmentation of cone-beam CT using a hidden MRF with external fieldprior. (In Prep).
Introduction Scalable Computation Informative Priors Conclusion
hidden Markov random field
Joint distribution of observed pixel intensities yi ∈ yand latent labels zi ∈ z:
Pr(y, z|µ,σ2, β) ∝ L(y|µ,σ2, z)π(z|β) (1)
Additive Gaussian noise:
yi|zi=jiid∼ N
(µj , σ
2j
)(2)
Potts model:
π(zi|zi∼`, β) =exp {β
∑i∼` δ(zi, z`)}∑k
j=1 exp {β∑
i∼` δ(j, z`)}(3)
Potts (1952) Proceedings of the Cambridge Philosophical Society 48(1)
Introduction Scalable Computation Informative Priors Conclusion
Inverse Temperature
Introduction Scalable Computation Informative Priors Conclusion
Doubly-intractable likelihood
p(β|z) = C(β)−1π(β) exp {β S(z)} (4)
The normalising constant of the Potts model has computationalcomplexity of O(n2kn), since it involves a sum over all possiblecombinations of the labels z ∈ Z:
C(β) =∑z∈Z
exp {β S(z)} (5)
S(z) is the sufficient statistic of the Potts model:
S(z) =∑i∼`∈L
δ(zi, z`) (6)
where L is the set of all unique neighbour pairs.
Introduction Scalable Computation Informative Priors Conclusion
Expectation of S(z)
exact expectation of S(z) for n=12 and k=
β
E(S
(z))
5
10
15
1 2 3 4
2
3
4
(a) n = 12 & k ∈ 2, 3, 4
exact expectation of S(z) for k=3 and n=
β
E(S
(z))
5
10
15
1 2 3 4
4
6
9
12
(b) k = 3 & n ∈ 4, 6, 9, 12
Figure: Distribution of Ez|β [S(z)]
Introduction Scalable Computation Informative Priors Conclusion
Standard deviation of S(z)
exact standard deviation of S(z) for n=12 and k=
β
σ(S
(z))
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1 2 3 4
2
3
4
(a) n = 12 & k ∈ 2, 3, 4
exact standard deviation of S(z) for k=3 and n=
β
σ(S
(z))
0.0
0.5
1.0
1.5
2.0
2.5
1 2 3 4
4
6
9
12
(b) k = 3 & n ∈ 4, 6, 9, 12
Figure: Distribution of σz|β [S(z)]
Introduction Scalable Computation Informative Priors Conclusion
Approximate Bayesian Computation
Algorithm 1 ABC rejection sampler
1: for all iterations t ∈ 1 . . . T do2: Draw independent proposal β′ ∼ π(β)3: Generate w ∼ f(·|β′)4: if |S(w)− S(z)| < ε then5: set βt ← β′
6: else7: set βt ← βt−1
8: end if9: end for
Grelaud, Robert, Marin, Rodolphe & Taly (2009) Bayesian Analysis 4(2)Marin & Robert (2014) Bayesian Essentials with R §8.3
Introduction Scalable Computation Informative Priors Conclusion
Pre-computation Step
The distribution of the summary statistics f(S(w)|β) isindependent of the observed data y
By simulating pseudo-data for values of β, we can create abinding function φ(β) for an auxiliary model fA(S(w)|φ(β))
This binding function can be reused across multiple datasets,amortising its computational cost
By replacing S(w) with approximate values drawn from ourauxiliary model, we avoid the need to simulate pseudo-data duringmodel fitting.
Wood (2010) Nature 466Cabras, Castellanos & Ruli (2014) Metron (to appear)
Introduction Scalable Computation Informative Priors Conclusion
Simulation from f(·|β)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10
15
20
25
30
β
E(S
(z))
(a) Ez|β (S(w))
0.0 0.5 1.0 1.5 2.0 2.5 3.00
12
34
β
σ(S
(z))
(b) σz|β (S(w))
Figure: Approximation of S(w)|β using 1000 iterations ofSwendsen-Wang (discarding 500 as burn-in)
Swendsen & Wang (1987) Physical Review Letters 58
Introduction Scalable Computation Informative Priors Conclusion
Piecewise linear model
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10
00
01
50
00
20
00
02
50
00
30
00
0
β
ES
(z)
(a) φµ(β)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
05
01
00
15
02
00
25
03
00
35
0
β
σS
(z)
(b) φσ(β)
Figure: Binding functions for S(w) | β with n = 56, k = 3
Introduction Scalable Computation Informative Priors Conclusion
Scalable ABC-SMC for the hidden Potts model
Algorithm 2 ABC-SMC using precomputed fA(S(w)|φ(β))
1: Draw N particles β′i ∼ π0(β)
2: Draw N ×M statistics S(wi,m) ∼ N(φµ(β′i), φσ(β′i)
2)
3: repeat4: Update S(zt)|y, πt(β)5: Adaptively select ABC tolerance εt6: Update importance weights ωi for each particle7: if effective sample size (ESS) < Nmin then8: Resample particles according to their weights9: end if
10: Update particles using random walk proposal(with adaptive RWMH bandwidth σ2
t )11: until naccept
N < 0.015 or εt < 10−9 or t ≥ 100
Introduction Scalable Computation Informative Priors Conclusion
Accuracy of posterior estimates for β
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
β
po
ste
rio
r d
istr
ibu
tio
n
(a) pseudo-data (M=50)
0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
β
po
ste
rio
r d
istr
ibu
tio
n
(b) pre-computed (M=200)
Introduction Scalable Computation Informative Priors Conclusion
Improvement in runtime
Pseudo−data Pre−computed
0.5
1.0
2.0
5.0
10
.02
0.0
50
.01
00
.0
algorithm
ela
pse
d t
ime
(h
ou
rs)
(a) elapsed (wall clock) time
Pseudo−data Pre−computed
51
02
05
01
00
20
05
00
algorithm
CP
U t
ime
(h
ou
rs)
(b) CPU time
Introduction Scalable Computation Informative Priors Conclusion
bayesImageS
An R package for Bayesian image segmentationusing the hidden Potts model:
RcppArmadillo for fast computation in C++
OpenMP for parallelism�l i b r a r y ( bayes ImageS )p r i o r s ← l i s t ("k"=3,"mu"=rep ( 0 , 3 ) , "mu.sd"=sigma ,
"sigma"=sigma , "sigma.nu"=c ( 1 , 1 , 1 ) , "beta"=c ( 0 , 3 ) )mh ← l i s t ( a l g o r i t h m="pseudo" , bandwidth =0.2)r e s u l t ← mcmcPotts ( y , ne igh , b lock , NULL ,
55000 ,5000 , p r i o r s , mh)
Eddelbuettel & Sanderson (2014) RcppArmadillo: Accelerating R withhigh-performance C++ linear algebra. CSDA 71
Introduction Scalable Computation Informative Priors Conclusion
Bayesian computational methods
bayesImageS supports methods for updating the latent labels z:
Chequerboard updating (Winkler 2003)
Swendsen-Wang (1987)
and also methods for updating the inverse temperature β:
Pseudolikelihood (Ryden & Titterington 1998)
Path Sampling (Gelman & Meng 1998)
Exchange Algorithm (Murray, Ghahramani & MacKay 2006)
Approximate Bayesian Computation (Grelaud et al. 2009)
Sequential Monte Carlo (ABC-SMC) with pre-computation(Del Moral, Doucet & Jasra 2012; Moores et al. 2014)
Introduction Scalable Computation Informative Priors Conclusion
Electron Density phantom
(a) CIRS Model 062 ED phantom (b) Helical, fan-beam CT scanner
Introduction Scalable Computation Informative Priors Conclusion
Regression Adjustment
0 1 2 3 4
−1
00
0−
80
0−
60
0−
40
0−
20
00
20
0
Electron Density
Ho
un
sfie
ld u
nit
(a) Fan-Beam CT
0 1 2 3 4
−1
00
0−
80
0−
60
0−
40
0−
20
00
20
0
Electron Density
pix
el in
ten
sity
(b) Cone-Beam CT
Introduction Scalable Computation Informative Priors Conclusion
Distribution of Pixel Intensities
Hounsfield units
De
nsity
−1000 −500 0 500 1000
0.0
00
0.0
01
0.0
02
0.0
03
0.0
04
0.0
05
0.0
06
(a) Fan-beam CT
Pixel intensity
De
nsity
−1000 −500 0 500 1000
0.0
00
0.0
01
0.0
02
0.0
03
0.0
04
(b) Cone-beam CT
Introduction Scalable Computation Informative Priors Conclusion
Priors for additive Gaussian noise
Tissue Type Density π(µj)
gas 0.63 -889.74adipose 3.17 -155.03RECT WALL 3.25 29.04BLADDER 3.39 76.75SEM VES 3.40 81.48PROSTATE 3.45 99.25muscle 3.48 110.99spongy bone 3.73 197.75dense bone 4.86 595.37
Introduction Scalable Computation Informative Priors Conclusion
Treatment Plan
−50 0 50
15
02
00
25
0
right−left (mm)
po
ste
rio
r−a
nte
rio
r (m
m)
Introduction Scalable Computation Informative Priors Conclusion
External Field
p(zi|zi∼`, β,µ,σ2, yi) =exp {αi,zi + π(αi,zi)}∑kj=1 exp {αi,j + π(αi,j)}
π(zi|zi∼`, β)
(7)Isotropic translation:
π(αi,j) = log
1
nj
∑h∈j
φ(∆(h, i)|µ∆ = 1.2, σ2
∆ = 7.32) (8)
where
nj is the number of voxels in object j
h ∈ j are the voxels in object j
∆(u, v) is the Euclidean distance between the coordinates ofpixel u and pixel v
µ∆, σ2∆ are parameters that describe the level of spatial
variability of the object j
Introduction Scalable Computation Informative Priors Conclusion
External Field II
External field prior for the ED phantom (σ∆ = 7.3mm)
Introduction Scalable Computation Informative Priors Conclusion
Anisotropy
αi(prostate) ∼ MVN
0.1−0.50.2
,4.12 0 0
0 2.92 00 0 0.92
(a) Bitmask (b) External Field
Introduction Scalable Computation Informative Priors Conclusion
Seminal Vesicles
αi(SV) ∼ MVN
1.2−0.7−0.9
,7.32 0 0
0 4.52 00 0 1.92
(a) Bitmask (b) External Field
Introduction Scalable Computation Informative Priors Conclusion
External Field
Organ- and patient-specific external field (slice 49, 16mm Inf)
Introduction Scalable Computation Informative Priors Conclusion
Preliminary Results
−300 −250 −200 −150
15
02
00
25
03
00
right−left (mm)
po
ste
rio
r−a
nte
rio
r (m
m)
(a) Cone-Beam CT
−300 −250 −200 −150
15
02
00
25
03
00
right−left (mm)
po
ste
rio
r−a
nte
rio
r (m
m)
(b) Segmentation
Introduction Scalable Computation Informative Priors Conclusion
ED phantom experiment
27 cone-beam CT scans of the ED phantom
Cropped to 376× 308 pixels and 23 slices(330× 270× 46 mm)
Inner ring of inserts rotated by between 0◦ and 16◦
2D displacement of between 0mm and 25mmIsotropic external field prior with σ∆ = 7.3mm
9 component Potts model
8 different tissue types, plus water-equivalent backgroundPriors for noise parameters estimated from 28 fan-beam CTand 26 cone-beam CT scans
Introduction Scalable Computation Informative Priors Conclusion
Image Segmentation
Introduction Scalable Computation Informative Priors Conclusion
Quantification of Segmentation Accuracy
Dice similarity coefficient:
DSCg =2× |g ∩ g||g|+ |g|
(9)
where
DSCg is the Dice similarity coefficient for label g
|g| is the count of pixels that were classified with thelabel g
|g| is the number of pixels that are known to trulybelong to component g
|g ∩ g| is the count of pixels in g that were labeled correctly
Dice (1945) Measures of the amount of ecologic association between species.Ecology 26(3): 297–302.
Introduction Scalable Computation Informative Priors Conclusion
Results
Tissue Type Simple Potts External Field
Lung (inhale) 0.507± 0.053 0.868± 0.011Lung (exhale) 0.169± 0.006 0.839± 0.008Adipose 0.048± 0.006 0.713± 0.041Breast 0.057± 0.017 0.748± 0.007Water 0.123± 0.134 0.954± 0.004Muscle 0.071± 0.004 0.758± 0.016Liver 0.075± 0.011 0.662± 0.033Spongy Bone 0.094± 0.020 0.402± 0.175Dense Bone 0.013± 0.001 0.297± 0.201
Table: Segmentation Accuracy (Dice Similarity Coefficient ±σ)
Introduction Scalable Computation Informative Priors Conclusion
Discussion
Contributions of this thesis:
M1 External field prior for representing spatialinformation in the hidden Potts model
M2 Regression model for adjusting priors for the noiseparameters µj and σ2
j
C2 Pre-computation for ABC-SMC leads to two ordersof magnitude faster computation
A1 Application to cone-beam CT scans of the EDphantom and radiotherapy patient data from theRadiation Oncology Mater Centre
Not discussed in this talk:
M3 Sequential Bayesian updating of the external fieldprior
C1 Scalability experiments with other algorithms fordoubly-intractable likelihoods
A2 Application to satellite remote sensing
Introduction Scalable Computation Informative Priors Conclusion
Ongoing & Future Work
Complete the analysis of the patient data and submit journalarticle to ANZ J. Stat.
Model object boundaries (eg. for bony anatomy) and spatialcorrelation between objects
Model spatially-correlated noise and artefacts in cone-beamCT scans
Collaboration with Antonietta Mira & Alberto Caimo (USI,Switzerland) on pre-computation for ERGM
ED phantom inserts
Tissue Type Electron Density Diameter(×1023/cc) (cm)
Lung (inhale) 0.634 3.05Lung (exhale) 1.632 3.05Adipose 3.170 3.05Breast 3.261 3.05Water 3.340 *Muscle 3.483 3.05Liver 3.516 3.05Spongy Bone 3.730 3.05Dense Bone 4.862 1.00
Table: Properties of the CIRS Model 062 ED phantom
* overall dimensions are 33cm× 27cm× 5cm
Cone-beam CT reconstructed images
Half-fan acquisition mode: FOV 450mm × 450mm × 137mm(Kan, Leung, Wong & Lam 2008)
reconstructed from 650-700 projections (Varian .HND files)
512 × 512 pixels with 2mm slice width (70-80 slices)
∼ 20 million voxels
70-80MB DICOM image stack
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