informative priors for segmentation of medical images

Motivation Method 1 Method 2 Extensions Conclusion

Informative Priors for Segmentationof Medical Images

Matt Moores1,2, Cathy Hargrave3, Fiona Harden2

& Kerrie Mengersen1

1 Discipline of Mathematical Sciences, Queensland University of Technology2 Discipline of Medical Radiation Sciences, Queensland University of Technology

3 Radiation Oncology Mater Centre, Queensland Health

Bayes on the Beach, 2011


Outline

1 MotivationCone-Beam Computed Tomography

2 Method 1k-means with posterior diffusion

3 Method 2hidden Markov random field

4 Extensions

5 Conclusion


X-Ray Computed Tomography

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


Distribution of Pixel Intensity

Hounsfield unit

Fre

quen

cy

−1000 −800 −600 −400 −200 0 200

050

0010

000

1500

0

(a) Fan-Beam CT

pixel intensity

Fre

quen

cy

−1000 −800 −600 −400 −200 0 2000

5000

1000

015

000

(b) Cone-Beam CT


itkBayesianClassifierImageFilter

1 estimate µ using k-means

1 select initial values for µ2 assign each pixel y to the nearest µk

3 recalculate each µk by averaging over the members of k4 repeat steps 2 & 3 until none of the pixel assignments change

2 estimate σ2 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)

456 classify each pixel yi according to the largest value of yik


itkBayesianClassifierImageFilter

1 estimate µ using k-means1 select initial values for µ2 assign each pixel y to the nearest µk

3 recalculate each µk by averaging over the members of k4 repeat steps 2 & 3 until none of the pixel assignments change

2 estimate σ2 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)

456 classify each pixel yi according to the largest value of yik


Result (k-means GMM)

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


Prior

4 matrix pik representing the prior probability of pixel ibelonging 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 theirparameters µk and σk

can’t use the posterior from one classification as the prior foranother, unless the clusters are the same


Result (with prior)

(a) Prior (b) Likelihood

(c) Posterior


Result (with diffusion)

(a) 5 iterations (b) 10 iterations

(c) 50 iterations (d) 1000 iterations


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)

yi|µj , τj , zi=j ∼ N

(µj ,

1

τj

)(2)

p(z) = C(β)−1exp

N∑i=1

αi(zi) + β∑i∼j

wijf(zi, zj)

(3)

simple Potts model (without external field):

p(z) = C(β)−1exp

β∑i∼j

I(zi = zj)

(4)


informative prior for µ and τ

0 1 2 3 4

−10

00−

800

−60

0−

400

−20

00

200

Electron Density

Hou

nsfie

ld u

nit

(a) Fan-Beam CT

0 1 2 3 4

−10

00−

800

−60

0−

400

−20

00

200

Electron Density

pixe

l int

ensi

ty

(b) Cone-Beam CT


external field

In equation (3) earlier, the term exp{∑N

i=1 αi(zi)}

defines an

external field.

Figure: manual contours of the organs of interest.


external field II

Prior probabilities αi(zi) for each pixel can be generated bysimulation, 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 inposition, along three axes: anteroposterior (Ant-Post); superoinferior(Sup-Inf); & lateral (Left-Right) relative to the patient.


Jacobian matrix

Figure: discrete Laplacian 52


hybrid model

Chen & Metaxas [6, 7] define the object boundary implicitly as thezero level set of a cost function:

∂φi∂t

=

[λ1Mi +5λ2Pi ·

(5φi‖ 5 φi‖

)− (λ2Pi + λ3)5 ·

(5φi‖ 5 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

5 ·(5φi‖5φi‖

)is the local curvature

(surface smoothness constraint)


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


References I

P. Teo, G. Sapiro and B. Wandell (1997) Creating connectedrepresentations of cortical gray matter for functional MRIvisualization. IEEE Trans. Med. Imag. 16: 852-863.

J. Melonakos, K. Krishnan and A. Tannenbaum (2006)An ITK Filter for Bayesian Segmentation:itkBayesianClassifierImageFilter The Insight Journalhttp://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 usingMarkov 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 longitudinalsetting for analysing sheep CAT scan images. ComputationalStatistics & Data Analysis 51(9): 4282-4296.


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 InterfractionVariation During IMRT. International Journal of RadiationOncology*Biology*Physics 71(3): 813-820.

T. Chen and D. Metaxas (2005) A hybrid framework for 3D medicalimage 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 inImage Guided Radiotherapy. In Proceedings of MICCAI 43-50.

P. Thevenaz, T. Blu & M. Unser (2000) Interpolation Revisited.IEEE Trans. Medical Imaging 19(7): 739–758.


Acknowledgements

Bayesian Research & Applications Group at QUT

Radiation Oncology Mater Centre:

Emmanuel Baveas

Rebecca Owen

Timothy Deegan

Steven Sylvander

John Baines

Dr. Michael Poulsen

informative priors for segmentation of medical images

Health & Medicine