bayesImageS: Bayesian computation for medical Image Segmentation using a hidden Potts Model

R packages Medical Imaging Statistical Model Bayesian Computation Experimental Results Conclusion
bayesImageS: Bayesian computation for
medical image segmentation
using a hidden Potts model
Matt Moores
MRC Biostatistics Unit Seminar Series
July 25, 2017

Acknowledgements
Queensland University of Technology, Brisbane, Australia:
Prof. Kerrie Mengersen
Dr. Fiona Harden
Members of the Volume Analysis Tool project team at the
Radiation Oncology Mater Centre (ROMC):
Cathy Hargrave
A/Prof Michael Poulsen, MD
Timothy Deegan
Emmanuel Baveas
QHealth ethics HREC/12/QPAH/475 and QUT ethics 1200000724

Outline
1 R packages
bayesImageS
2 Medical Imaging
Image-Guided Radiotherapy
Cone-Beam Computed Tomography
3 Statistical Model
Hidden Potts model
Informative priors
4 Bayesian Computation
Chequerboard Gibbs sampler
Pseudolikelihood
5 Experimental Results
ED phantom experiment
Radiotherapy Patient Data

Why write an R package?
Portability
Test bed for new statistical methods
Build on existing code
Research impact
Kudos
Hadley Wickham (2015) R packages

Why C++?
Most statistical algorithms are iterative
Markov chain Monte Carlo
Scalability for large datasets
Rcpp
OpenMP
Eigen or Armadillo
Dirk Eddelbuettel (2013) Seamless R and C++ integration with Rcpp

Inline
One function at a time:
§
library ( i n l i n e )
sum_logs ← cxxfunction ( signature ( log_vec = "numeric") , plugin = "RcppArmadillo" , body=’
arma::vec log_prob = Rcpp::as<arma::vec>(log_vec);
double suml = 0.0;
double maxl = log_prob.max();
for (unsigned i=0; i < log_prob.n_elem; i++)
{
if (arma::is_finite(log_prob(i)))
suml += exp(log_prob(i) - maxl);
}
return Rcpp::wrap(log(suml) + maxl);
’)

Annotations
Rcpp wrappers generated automatically:
compileAttributes("myRcppPackage")
R package documentation generated automatically:
roxygenize("myRcppPackage")
§
/ / ’ Compute the effective sample size (ESS) of the particles.
/ / ’
/ / ’ The ESS is a ‘‘rule of thumb’’ for assessing the degeneracy of
/ / ’’ the importance distribution:
/ / ’ deqn {ESS = frac { ( sum_ { q=1}^Q w_q ) ^ 2 } { sum_ { q=1}^Q w_q ^2}}
/ / ’
/ / ’’ @param log_weights logarithms of the importance weights of each particle.
/ / ’’ @return the effective sample size, a scalar between 0 and Q
/ / ’’ @references
/ / ’’ Liu, JS (2001) "Monte Carlo Strategies in Scientific Computing." Springer’
/ / [ [ Rcpp : : export ] ]
double effectiveSampleSize ( NumericVector log_weights )
{
double sum_wt = sum_logs ( log_weights ) ;
double sum_sq = sum_logs ( log_weights + log_weights ) ;
double res = exp (sum_wt + sum_wt − sum_sq ) ;
i f ( std : : i s f i n i t e ( res ) ) return res ;
else return 0;
}

Package Skeleton
Create a new R package:
package.skeleton("myPackage", path=".")
Speciﬁc skeletons for each C++ library:
Rcpp.package.skeleton("myRcppPackage")
RcppArmadillo.package.skeleton("MyArmadilloPackage")
RcppEigen.package.skeleton("MyEigenPackage")

Common Problems
Rcpp parameters are passed by reference (not copied):
Can rely on R for garbage collection
Memory allocation is slower
Can crash R (and Rstudio (and your OS))
R is not thread safe
Cannot call any R functions (even indirectly)
within parallel code!
Drew Schmidt (@wrathematics, 2015) Parallelism, R, and OpenMP

bayesImageS
An R package for Bayesian image segmentation
using the hidden Potts model:
RcppArmadillo for fast computation in C++
OpenMP for parallelism
§
library ( bayesImageS )
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 ( algorithm="pseudo" , bandwidth =0.2)
r e s u l t ← mcmcPotts ( y , neigh , block ,NULL,
55000,5000, priors ,mh)
Eddelbuettel & Sanderson (2014) RcppArmadillo: Accelerating R with
high-performance C++ linear algebra. CSDA 71

Bayesian computational methods
bayesImageS supports methods for updating the latent labels:
Chequerboard Gibbs sampling (Winkler 2003)
Swendsen-Wang (1987)
and also methods for updating the smoothing parameter β:
Pseudolikelihood (Rydén & Titterington 1998)
Thermodynamic integration (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. 2015)

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

Segmentation of Anatomical Structures
Radiography courtesy of Cathy Hargrave, Radiation Oncology Mater Centre,
Queensland Health

Physiological Variability
Organ Ant-Post Sup-Inf Left-Right
prostate 0.1 ± 4.1mm −0.5 ± 2.9mm 0.2 ± 0.9mm
seminal vesicles 1.2 ± 7.3mm −0.7 ± 4.5mm −0.9 ± 1.9mm
Table: Distribution of observed translations of the organs of interest
Organ Volume Gas
rectum 35 − 140cm3 4 − 26%
bladder 120 − 381cm3
Table: Volume variations in the organs of interest
Frank, et al. (2008) Quantiﬁcation of Prostate and Seminal Vesicle
Interfraction Variation During IMRT. IJROBP 71(3): 813–820.

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

Cone-Beam Computed Tomography
(c) Fan-beam CT (d) Cone-beam CT

Distribution of Pixel Intensity
Hounsfield unit
Frequency
−1000 −800 −600 −400 −200 0 200
050001000015000
(a) Fan-Beam CT
pixel intensity
Frequency
−1000 −800 −600 −400 −200 0 200
050001000015000
(b) Cone-Beam CT

Hidden Markov Random Field
Joint distribution of observed pixel intensities y = {yi}n
i=1
and latent labels z = {zi}n
i=1:
p(y, z|µ, σ2
, β) = p(y|µ, σ2
, z)p(z|β) (1)
Additive Gaussian noise:
yi|zi =j
iid
∼ N µj, σ2
j (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)−1
π(β) exp {β S(z)} (4)
The normalising constant has computational complexity O(nkn):
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.

Informative Prior for µj and σ2
j
0 1 2 3 4
−1000−800−600−400−2000200
Electron Density
Hounsfieldunit
(a) Fan-Beam CT
0 1 2 3 4
−1000−800−600−400−2000200
Electron Density
pixelintensity
(b) Cone-Beam CT

External Field
π(zi|αi, β, zi∼ ) ∝ exp αi(zi) + β
i∼
δ(zi, zj) (7)
Isotropic translation:
αi(zi =j) =
1
nj ν∈j
φ ∆(ϑi, ϑν), µ = 1.2, σ2
= 7.32
(8)
where
ν ∈ j are the image voxels ϑν in object j
φ(x, µ, σ2) is the normal density function
∆(u, v) is the Euclidian distance between the coordinates
of voxel u and voxel v

External Field II
(a) Fan-beam CT (b) External Field

Anisotropy
αi(prostate) ∼ MVN




0.1
−0.5
0.2

 ,


4.12 0 0
0 2.92 0
0 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 0
0 0 1.92




(a) Bitmask (b) External Field

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

Chequerboard Gibbs
A 2D or 3D regular lattice with ﬁrst-order neighbourhood ∂i:
◦ ◦ ◦ ◦ ◦
◦ ◦ • ◦ ◦
◦ • × • ◦
◦ ◦ • ◦ ◦
◦ ◦ ◦ ◦ ◦
can be partitioned into 2 blocks:
• ◦ • ◦ • ◦ • ◦ • ◦
◦ • ◦ • ◦ • ◦ • ◦ •
• ◦ • ◦ • ◦ • ◦ • ◦
◦ • ◦ • ◦ • ◦ • ◦ •
• ◦ • ◦ • ◦ • ◦ • ◦
◦ • ◦ • ◦ • ◦ • ◦ •
• ◦ • ◦ • ◦ • ◦ • ◦
◦ • ◦ • ◦ • ◦ • ◦ •
• ◦ • ◦ • ◦ • ◦ • ◦
◦ • ◦ • ◦ • ◦ • ◦ •
so that z◦ are conditionally independent, given z•
Roberts & Sahu (1997) JRSS B 59(2): 291–317
Winkler (2nd
ed., 2003) Image analysis, random ﬁelds and MCMC methods

Chequerboard Gibbs II
Algorithm 1 Chequerboard sampling for z
1: for all blocks b do
2: for all pixels i ∈ b do
3: for all labels j ∈ 1 . . . k do
4: Compute λj ← p(yi | zi = j)π(zi = j | zi∼ , β)
5: end for
6: Draw zi ∼ Multinomial(λ1, . . . , λk)
7: end for
8: end for

Gibbs sampler in C++
§
void gibbsLabels ( const arma : : umat & neigh , const std : : vector <arma : : uvec> & blocks ,
arma : : umat & z , arma : : umat & alloc , const double beta ,
const arma : : mat & log_ x f i e l d )
{
const Rcpp : : NumericVector randU = Rcpp : : r u n i f ( neigh . n_rows ) ;
for ( unsigned b=0; b < blocks . size ( ) ; b++)
{
const arma : : uvec block = blocks [ b ] ;
arma : : vec log_prob ( z . n_cols ) ;
#pragma omp p a r a l l e l for private ( log_prob )
for ( unsigned i =0; i < block . size ( ) ; i ++)
{
for ( unsigned j =0; j < z . n_cols ; j ++)
{
unsigned sum_neigh = 0;
for ( unsigned k=0; k < neigh . n_cols ; k++)
{
sum_neigh += z ( neigh ( block [ i ] , k ) , j ) ;
}
log_prob [ j ] = log_ x f i e l d ( block [ i ] , j ) + beta∗sum_neigh ;
}
double t o t a l _ l l i k e = sum_logs ( log_prob ) ;
double cumProb = 0.0;
z . row ( block [ i ] ) . zeros ( ) ;
for ( unsigned j =0; j < log_prob . n_elem ; j ++)
{
cumProb += exp ( log_prob [ j ] − t o t a l _ l l i k e ) ;
i f ( randU [ block [ i ] ] < cumProb )
{
z ( block [ i ] , j ) = 1;
a l l o c ( block [ i ] , j ) += 1;
break ;

Pseudolikelihood (PL)
Algorithm 2 Metropolis-Hastings with PL
1: Draw proposal β ∼ q(β |β◦)
2: Approximate p(β |z) and p(β◦|z) using equation (9):
ˆpPL(β|z) ≈
n
i=1
exp{β i∼ δ(zi, z )}
k
j=1 exp{β i∼ δ(j, z )}
(9)
3: Calculate the M-H ratio ρ = ˆpPL(β |z)π(β )q(β◦|β )
ˆpPL(β◦|z)π(β◦)q(β |β◦)
4: Draw u ∼ Uniform[0, 1]
5: if u < min(1, ρ) then
6: β ← β
7: else
8: β ← β◦
9: end if
Rydén & Titterington (1998) JCGS 7(2): 194–211

Pseudolikelihood in C++
§
double pseudolike ( arma : : mat & ne , arma : : uvec & e ,
double b , unsigned n , unsigned k )
{
double num = 0.0;
double denom = 0.0;
#pragma omp p a r a l l e l for reduction ( + :num, denom)
for ( unsigned i =0; i < n ; i ++)
{
num=num+ne ( e [ i ] , i ) ;
double tdenom =0.0;
for ( unsigned j =0; j < k ; j ++)
{
tdenom=tdenom+exp ( b∗ne ( j , i ) ) ;
}
denom=denom+log ( tdenom ) ;
}
return b∗num−denom ;
}

Approximation Error
PL for n = 12, k = 3 in comparison to the exact likelihood
calculated using a brute force method:
0 1 2 3 4
6810121416
β
µ
exact
pseudolikelihood
(a) Expectation
0 1 2 3 4
0.00.51.01.52.02.5
β
σ
exact
pseudolikelihood
(b) Standard deviation

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 25mm
Isotropic external ﬁeld prior with σ∆ = 7.3mm
9 component Potts model
8 different tissue types, plus water-equivalent background
Priors for noise parameters estimated from 28 fan-beam CT
and 26 cone-beam CT scans

Image Segmentation

Quantification of Segmentation Accuracy
Dice similarity coefficient:
DSCg =
2 × |ˆg ∩ g|
|ˆg| + |g|
(10)
where
DSCg is the Dice similarity coefficient for label g
|ˆg| is the count of pixels that were classified with the
label g
|g| is the number of pixels that are known to truly
belong to component g
|ˆg ∩ 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) 0.540 ± 0.037 0.902 ± 0.009
Lung (exhale) 0.172 ± 0.008 0.814 ± 0.022
Adipose 0.059 ± 0.008 0.704 ± 0.062
Breast 0.077 ± 0.011 0.720 ± 0.048
Water 0.174 ± 0.003 0.964 ± 0.003
Muscle 0.035 ± 0.004 0.697 ± 0.076
Liver 0.020 ± 0.007 0.654 ± 0.033
Spongy Bone 0.094 ± 0.014 0.758 ± 0.018
Dense Bone 0.014 ± 0.001 0.616 ± 0.151
Table: Segmentation Accuracy (Dice Similarity Coefﬁcient ±σ)
Moores, et al. (2014) In Proc. XVII Intl Conf. ICCR; J. Phys: Conf. Ser. 489

Preliminary Results
−300 −250 −200 −150
150200250300
right−left (mm)
posterior−anterior(mm)
(a) Cone-Beam CT
−300 −250 −200 −150
150200250300
right−left (mm)
posterior−anterior(mm)
(b) Segmentation
Moores, et al. (2015) CSDA 86: 27–41.

Summary
Informative priors can dramatically improve segmentation
accuracy for noisy data
inverse regression for µ & σ2
external ﬁeld prior for z
It is feasible to use MCMC for image analysis of realistic
datasets
but auxiliary variable methods don’t scale well
requires parallelized implementation in C++ or Fortran
RcppArmadillo & OpenMP are a good combination
faster algorithms are available, such as VB or ICM

Appendix
For Further Reading I
Moores, Hargrave, Deegan, Poulsen, Harden & Mengersen
An external ﬁeld prior for the hidden Potts model with application to
cone-beam computed tomography.
CSDA 86: 27–41, 2015.
Moores & Mengersen
bayesImageS: Bayesian methods for image segmentation using a
hidden Potts model.
R package version 0.3-3
https://CRAN.R-project.org/package=bayesImageS
Moores, Drovandi, Mengersen & Robert
Pre-processing for approximate Bayesian computation in image
analysis.
Statistics & Computing 25(1): 23–33, 2015.
Moores, Pettitt & Mengersen
Scalable Bayesian inference for the inverse temperature of a hidden
Potts model.
arXiv:1503.08066 [stat.CO], 2015.

Appendix
For Further Reading II
Eddelbuettel & Sanderson
RcppArmadillo: Accelerating R with high-performance C++ linear
algebra.
Comput. Stat. Data Anal. 71: 1054–63, 2014.
Bates & Eddelbuettel
Fast and elegant numerical linear algebra using the RcppEigen
package.
J. Stat. Soft. 52(5): 1–24, 2013.
Eddelbuettel
Seamless R and C++ integration with Rcpp
Springer-Verlag, 2013.
Wickham
R packages
O’Reilly, 2015.

Appendix
For Further Reading III
Winkler
Image analysis, random ﬁelds and Markov chain Monte Carlo methods
2nd
ed., Springer-Verlag, 2003.
Marin & Robert
Bayesian Essentials with R
Springer-Verlag, 2014.
Roberts & Sahu
Updating Schemes, Correlation Structure, Blocking and
Parameterization for the Gibbs Sampler
J. R. Stat. Soc. Ser. B 59(2): 291–317, 1997.
Rydén & Titterington
Computational Bayesian Analysis of Hidden Markov Models
J. Comput. Graph. Stat. 7(2): 194–211, 1998.

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