Final PhD Seminar

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


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


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


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.


Image-Guided Radiotherapy

Image courtesy of Varian Medical Systems, Inc. All rights reserved.


Radiotherapy Process

Before Treatment

fan-beamCT

MRI

contourstreatmentplan

QA

Daily Fractions (∼8 weeks)

positionpatient

cone-beamCT

deliverdose

off-lineanalysis


Radiotherapy Process

Before Treatment

fan-beamCT

MRI

contourstreatmentplan

QA

Daily Fractions (∼8 weeks)

positionpatient

cone-beamCT

deliverdose

off-lineanalysis


Segmentation of Anatomical Structures

Radiography courtesy of Cathy Hargrave, Radiation Oncology Mater Centre


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.


Cone-Beam Computed Tomography

(a) Fan-beam CT (b) Cone-beam CT


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


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.


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.


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).


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)


Inverse Temperature


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.


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)]


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)]


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


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)


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


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


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


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)


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


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


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)


Electron Density phantom

(a) CIRS Model 062 ED phantom (b) Helical, fan-beam CT scanner


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


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


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


Treatment Plan

−50 0 50

15

02

00

25

0

right−left (mm)

po

ste

rio

r−a

nte

rio

r (m

m)


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


External Field II

External field prior for the ED phantom (σ∆ = 7.3mm)


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


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


External Field

Organ- and patient-specific external field (slice 49, 16mm Inf)


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


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


Image Segmentation


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.


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)


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


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