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Motivation                   Method 1             Method 2              Extensions               Conclusion               ...
Motivation            Method 1        Method 2   Extensions   ConclusionOutline       1     Motivation              Cone-B...
Motivation      Method 1       Method 2        Extensions    ConclusionX-Ray Computed Tomography             (a) Fan-Beam ...
Motivation                                 Method 1                              Method 2                                 ...
Motivation                Method 1          Method 2          Extensions        ConclusionitkBayesianClassifierImageFilter...
Motivation                 Method 1          Method 2           Extensions        ConclusionitkBayesianClassifierImageFilt...
Motivation       Method 1      Method 2        Extensions    ConclusionResult (k-means GMM)             (a) Fan-Beam CT   ...
Motivation                Method 1          Method 2          Extensions       ConclusionPrior             4   matrix pik ...
Motivation    Method 1            Method 2               Extensions   ConclusionResult (with prior)                    (a)...
Motivation   Method 1              Method 2               Extensions   ConclusionResult (with diffusion)                (a)...
Motivation         Method 1              Method 2                Extensions          Conclusionhidden Markov random field  ...
Motivation                                 Method 1                    Method 2                                 Extensions...
Motivation         Method 1          Method 2            Extensions          Conclusionexternal field                      ...
Motivation                Method 1       Method 2          Extensions          Conclusionexternal field II      Prior proba...
Motivation   Method 1              Method 2              Extensions   ConclusionJacobian matrix                           ...
Motivation            Method 1              Method 2             Extensions          Conclusionhybrid model      Chen & Me...
Motivation           Method 1         Method 2          Extensions   ConclusionSummary      Two Bayesian approaches to med...
Motivation            Method 1           Method 2          Extensions         ConclusionReferences I             P. Teo, G...
Motivation            Method 1           Method 2          Extensions         ConclusionReferences II             S.J. Fra...
Motivation           Method 1      Method 2      Extensions   ConclusionAcknowledgements      Bayesian Research & Applicat...
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Informative Priors for Segmentation of Medical Images

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There is an abundance of prior information available for image-guided radiotherapy, making it ideally suited for Bayesian techniques. I will demonstrate some results from applying the method of Teo, Sapiro & Wandell (1997) to cone-beam computed tomography (CT). A previous CT scan of the same object forms the prior expectation. The posterior probabilities of class membership are smoothed by diffusion, before labeling each pixel according to the maximum a posteriori (MAP) estimate. The effect of the prior and of the smoothing is discussed and some potential extensions to this method are proposed.

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Informative Priors for Segmentation of Medical Images

  1. 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. 2. Motivation Method 1 Method 2 Extensions ConclusionOutline 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. 3. Motivation Method 1 Method 2 Extensions ConclusionX-Ray Computed Tomography (a) Fan-Beam CT (b) Cone-Beam CT
  4. 4. Motivation Method 1 Method 2 Extensions ConclusionDistribution 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. 5. Motivation Method 1 Method 2 Extensions ConclusionitkBayesianClassifierImageFilter 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. 6. Motivation Method 1 Method 2 Extensions ConclusionitkBayesianClassifierImageFilter 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. 7. Motivation Method 1 Method 2 Extensions ConclusionResult (k-means GMM) (a) Fan-Beam CT (b) Cone-Beam CT
  8. 8. Motivation Method 1 Method 2 Extensions ConclusionPrior 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. 9. Motivation Method 1 Method 2 Extensions ConclusionResult (with prior) (a) Prior (b) Likelihood (c) Posterior
  10. 10. Motivation Method 1 Method 2 Extensions ConclusionResult (with diffusion) (a) 5 iterations (b) 10 iterations (c) 50 iterations (d) 1000 iterations
  11. 11. Motivation Method 1 Method 2 Extensions Conclusionhidden 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. 12. Motivation Method 1 Method 2 Extensions Conclusioninformative 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. 13. Motivation Method 1 Method 2 Extensions Conclusionexternal 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. 14. Motivation Method 1 Method 2 Extensions Conclusionexternal 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. 15. Motivation Method 1 Method 2 Extensions ConclusionJacobian matrix 2 Figure: discrete Laplacian
  16. 16. Motivation Method 1 Method 2 Extensions Conclusionhybrid 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. 17. Motivation Method 1 Method 2 Extensions ConclusionSummary 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. 18. Motivation Method 1 Method 2 Extensions ConclusionReferences 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. 19. Motivation Method 1 Method 2 Extensions ConclusionReferences 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. 20. Motivation Method 1 Method 2 Extensions ConclusionAcknowledgements Bayesian Research & Applications Group at QUT Radiation Oncology Mater Centre: Emmanuel Baveas Rebecca Owen Timothy Deegan Steven Sylvander John Baines Dr. Michael Poulsen

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