This document summarizes two Bayesian approaches to medical image segmentation - k-means with posterior diffusion and hidden Markov random field models. It discusses potential extensions, such as using an external field to incorporate organ size and position variability, or a hybrid level set model. The conclusions discuss references for further information on these methods.

1. Motivation Method 1 Method 2 Extensions Conclusion
Informative Priors for Segmentation
of Medical Images
Matt Moores1,2 , Cathy Hargrave3 , Fiona Harden2
& Kerrie Mengersen1
1 Discipline of Mathematical Sciences, Queensland University of Technology
2 Discipline of Medical Radiation Sciences, Queensland University of Technology
3 Radiation Oncology Mater Centre, Queensland Health
Bayes on the Beach, 2011

2. Motivation Method 1 Method 2 Extensions Conclusion
Outline
1 Motivation
Cone-Beam Computed Tomography
2 Method 1
k-means with posterior diffusion
3 Method 2
hidden Markov random field
4 Extensions
5 Conclusion

3. Motivation Method 1 Method 2 Extensions Conclusion
X-Ray Computed Tomography
(a) Fan-Beam CT (b) Cone-Beam CT

4. Motivation Method 1 Method 2 Extensions Conclusion
Distribution of Pixel Intensity
15000
15000
10000
10000
Frequency
Frequency
5000
5000
0
−1000 −800 −600 −400 −200 0 200 0 −1000 −800 −600 −400 −200 0 200
Hounsfield unit pixel intensity
(a) Fan-Beam CT (b) Cone-Beam CT

5. Motivation Method 1 Method 2 Extensions Conclusion
itkBayesianClassifierImageFilter
1 estimate µ using k-means
2 estimate σ 2 for each cluster
(mixing proportions are assumed equal)
3 create a matrix y∗ :
for each pixel yi and each cluster Ck ∼ N(µk , σk ),
yik = p(yi |µk , σk )
6
5
4 classify each pixel yi according to the largest value of yik

6. Motivation Method 1 Method 2 Extensions Conclusion
itkBayesianClassifierImageFilter
1 estimate µ using k-means
1 select initial values for µ
2 assign each pixel y to the nearest µk
3 recalculate each µk by averaging over the members of k
4 repeat steps 2 & 3 until none of the pixel assignments change
2 estimate σ 2 for each cluster
(mixing proportions are assumed equal)
3 create a matrix y∗ :
for each pixel yi and each cluster Ck ∼ N(µk , σk ),
yik = p(yi |µk , σk )
6
5
4 classify each pixel yi according to the largest value of yik

7. Motivation Method 1 Method 2 Extensions Conclusion
Result (k-means GMM)
(a) Fan-Beam CT (b) Cone-Beam CT

8. Motivation Method 1 Method 2 Extensions Conclusion
Prior
4 matrix pik representing the prior probability of pixel i
belonging to cluster k
then pixel classification is based on the posterior pik × yik
but:
this has no effect on the number of clusters, nor on their
parameters µk and σk
can’t use the posterior from one classification as the prior for
another, unless the clusters are the same

9. Motivation Method 1 Method 2 Extensions Conclusion
Result (with prior)
(a) Prior (b) Likelihood
(c) Posterior

10. Motivation Method 1 Method 2 Extensions Conclusion
Result (with diffusion)
(a) 5 iterations (b) 10 iterations
(c) 50 iterations (d) 1000 iterations

11. Motivation Method 1 Method 2 Extensions Conclusion
hidden Markov random field
Joint distribution of observed intensities y and unobserved labels z:
p(y, z|µ, τ ) ∝ p(y|µ, τ , z)p(z) (1)
1
yi |µj , τj , zi = j ∼ N µj , (2)
τj
 
 N 
p(z) = C(β)−1 exp αi (zi ) + β wij f (zi , zj ) (3)
 
i=1 i∼j
simple Potts model (without external field):
 
 
p(z) = C(β)−1 exp β I(zi = zj ) (4)
 
i∼j

12. Motivation Method 1 Method 2 Extensions Conclusion
informative prior for µ and τ
200
200
0
0
−200
−200
Hounsfield unit
pixel intensity
−400
−400
−600
−600
−800
−800
−1000
−1000
0 1 2 3 4 0 1 2 3 4
Electron Density Electron Density
(a) Fan-Beam CT (b) Cone-Beam CT

13. Motivation Method 1 Method 2 Extensions Conclusion
external field
N
In equation (3) earlier, the term exp i=1 αi (zi ) defines an
external field.
Figure: manual contours of the organs of interest.

14. Motivation Method 1 Method 2 Extensions Conclusion
external field II
Prior probabilities αi (zi ) for each pixel can be generated by
simulation, based on:
geometry of each organ, from the treatment plan
variability in size and position, from published studies
Axis prostate seminal vesicles
Ant-Post x = 0.1, sd = 4.1 mm x = 1.2, sd = 7.3 mm
Sup-Inf x = −0.5, sd = 2.9 mm x = −0.7, sd = 4.5 mm
Left-Right x = 0.2, sd = 0.9 mm x = −0.9, sd = 1.9 mm
Table: Mean x and standard deviation sd of observed [5] variability in
position, along three axes: anteroposterior (Ant-Post); superoinferior
(Sup-Inf); & lateral (Left-Right) relative to the patient.

15. Motivation Method 1 Method 2 Extensions Conclusion
Jacobian matrix
2
Figure: discrete Laplacian

16. Motivation Method 1 Method 2 Extensions Conclusion
hybrid model
Chen & Metaxas [6, 7] define the object boundary implicitly as the
zero level set of a cost function:
∂φi φi φi
= λ1 M i + λ 2 Pi · − (λ2 Pi + λ3 ) ·
∂t φi φi
(5)
where:
Mi is the inflation force (total gradient magnitude)
Pi is the local image force at each pixel
(probability of pixel j belonging to object i)
non-overlapping constraint
φi
· φi is the local curvature
(surface smoothness constraint)

17. Motivation Method 1 Method 2 Extensions Conclusion
Summary
Two Bayesian approaches to medical image segmentation:
k-means with posterior diffusion
(itkBayesianClassifierImageFilter)
hidden Markov random field
(PyMCMC)
Potential extensions to Potts MRF:
external field defined by size and position of objects
hybrid Level Set model

18. Motivation Method 1 Method 2 Extensions Conclusion
References I
P. Teo, G. Sapiro and B. Wandell (1997) Creating connected
representations of cortical gray matter for functional MRI
visualization. IEEE Trans. Med. Imag. 16: 852-863.
J. Melonakos, K. Krishnan and A. Tannenbaum (2006)
An ITK Filter for Bayesian Segmentation:
itkBayesianClassifierImageFilter The Insight Journal
http://hdl.handle.net/1926/160
Strickland, C. M., Denham, R. J., Alston, C. L., & Mengersen, K. L.
(2011) PyMCMC : a Python package for Bayesian Estimation using
Markov chain Monte Carlo. Journal of Statistical Software (In Press)
C. Alston, K. Mengersen, C. Robert, J. Thompson, P. Littlefield, D.
Perry and A. Ball (2007) Bayesian mixture models in a longitudinal
setting for analysing sheep CAT scan images. Computational
Statistics & Data Analysis 51(9): 4282-4296.

19. Motivation Method 1 Method 2 Extensions Conclusion
References II
S.J. Frank, L. Dong, R. J. Kudchadker, R. De Crevoisier, A. K. Lee,
R. Cheung, S. Choi, J. O’Daniel, S. L. Tucker, H. Wang, et al.
(2008) Quantification of Prostate and Seminal Vesicle Interfraction
Variation During IMRT. International Journal of Radiation
Oncology*Biology*Physics 71(3): 813-820.
T. Chen and D. Metaxas (2005) A hybrid framework for 3D medical
image segmentation. Medical Image Analysis 9(6): 547-565.
T. Chen, S. Kim, J. Zhou, D. Metaxas, G. Rajagopal & N. Yue
(2009) 3D Meshless Prostate Segmentation and Registration in
Image Guided Radiotherapy. In Proceedings of MICCAI 43-50.
P. Th´venaz, T. Blu & M. Unser (2000) Interpolation Revisited.
e
IEEE Trans. Medical Imaging 19(7): 739–758.

20. Motivation Method 1 Method 2 Extensions Conclusion
Acknowledgements
Bayesian Research & Applications Group at QUT
Radiation Oncology Mater Centre:
Emmanuel Baveas
Rebecca Owen
Timothy Deegan
Steven Sylvander
John Baines
Dr. Michael Poulsen