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Segmentation Foundations

Segmentation Foundations. Easy Segmentation Tissue/Air (except bone in MR) Bone in CT Feasible Segmentation White Matter/Gray Matter: MRI M.S. White Matter Lesions: MRI. Statistical Classification. Probabilistic model of intensity as a function of (tissue) class Intensity data

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Segmentation Foundations

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  1. Segmentation Foundations • Easy Segmentation • Tissue/Air (except bone in MR) • Bone in CT • Feasible Segmentation • White Matter/Gray Matter: MRI • M.S. White Matter Lesions: MRI

  2. Statistical Classification • Probabilistic model of intensity as a function of (tissue) class • Intensity data • Prior model Classification of voxels [Duda, Hart 78][MRI: MikeVannier late 80s]

  3. p(x|tissue class J) intensity probability density mean for tissue J Measurement Model • Characterize sensor Tissue class conditional model of signal intensity

  4. A bit of notation… • Estimate  by finding the one that maximizes the function f

  5. Maximum Likelihood (ML) Estimation • Estimate parameters to maximize probability of observed data conditioned on parameters . • yo : observed data • p(y|) : Measurement Model • : Model Parameters

  6. p(x|gray matter) p(x|white matter) intensity Example

  7. gray matter white matter threshold Example - revisited

  8. Multiple Sclerosis T2w PDw Provided by S Warfield

  9. Dual Echo MRI Feature Space csf T2 Intensity severe lesions air wm gm PD Intensity

  10. severe csf lesions mild healthy gm wm Detail • MS Lesions are “graded phenomenon” in MRI, and can be anywhere on the curve

  11. Multiple Sclerosis T2w Segmentation PDw Provided by S Warfield

  12. Maximum A-Posteriori (MAP) Estimation • Estimate parameters to maximize posterior probability model parameters conditioned on observed data • Use Baye’s rule – ignore denominator • p() : Prior Model

  13. Provided by S Warfield Multiple Sclerosis PDw T2w kNN SVC

  14. Background: Intensity Inhomogeneities in MRI • MRI signal derived from RF signals… • Intra Scan Inhomogeneities • “Shading” … from coil imperfections • interaction with tissue? • Inter Scan Inhomogeneities • Auto Tune • Equipment Upgrades

  15. ML Estimation – with missing data • x : missing data (true labeling) • y0 : observed intensities •  : (parameters of) bias field

  16. ML Estimation – EM Approach • E []: Expected value under p(x|yo, ) • Take expectation of objective function with respect to the missing data, conditioned on everything we know • x : missing data (true labeling) • y0 : observed intensities •  : (parameters of) bias field

  17. EM Algorithm • General exponential family • Iterate to convergence: M step: E step:

  18. EM Algorithm: Example • Measurement Model • Tissue intensity properties with bias correction • Missing Data • Unknown true classification • Prior Models • Tissue Frequencies • Intensity Correction is Low Frequency • ML estimate of bias

  19. EM-Segmentation E-Step Compute tissue posteriors using current intensity correction. Estimate intensity correction using residuals based on current posteriors. M-Step Provided by T Kapur

  20. EM Segmentation… Seg Result w/o EM Seg Result With EM PD, T2 Data

  21. EM Segmentation… External Surface of Brain

  22. EM Segmentation… WM Surface with EM WM Surface w/o EM

  23. EM Segmentation: MS Example PD T2 Data provided by Charles Guttmann

  24. EM Segmentation: MS Example Seg w/o EM Seg with EM

  25. Prior Probability Models • Simple: Frequency of Tissues • More Interesting: • Powerful Mechanism for Incorporating Domain Knowledge into Segmentation • Tissue properties • Relative Location of Structures • Atlases

  26. Prior Model Example: EM-MF Segmentation • Tina Kapur PhD thesis • EM Segmentation, augmented with • Ising prior of tissue homogeneity • Solved with Mean Field Approxomation • Prior on relative position of organs • Spatially Conditioned Models

  27. Prior Models: Ising Model • Ising Model can capture the phenomenon of piecewise-homogeneity. • Initially used in Statistical Physics to model the magnetic domains in Ferromagnetism. • Used in Medical Image Processing to model the piecewise-homogeneity of Tissue.

  28. Prior Models: Ising Model • Ising Model relaxes spatial independence assumption • Voxels depend conditionally on (only) their neighbors • More probable to agree with neighbor

  29. Define the Neighborhood 1st Order Lattice 2nd Order Lattice 6 Neighbors 26 Neighbors Reduce calculation cost => use 1st order Lattice Neighbors = {East, South, West, North, Up, Down} Provided by K Pohl

  30. Potts Model • Potts model generalizes Ising model so that each lattice site takes on several values (more than two). • Frequently used to model tissues (e.g. White Matter, Gray Matter, CSF, Fat, Air, etc.)

  31. Some Results EM EM-MF Provided by T Kapur

  32. More Results Noisy MRI EM Segmentation EM-MF Segmentation Provided by T Kapur

  33. Posterior Probabilities (EM) White matter Gray matter Provided by T Kapur

  34. Posterior Probabilities (EM-MF) White matter Gray matter Provided by T Kapur

  35. Segmentation of 31 Structures Kilian Pohl PhD (defense several weeks ago)

  36. Segmentation of 31 Structures Lower Front Provided by Kilian Pohl

  37. Segmentation of 31 Structures Superior Temporal Gyrus Provided by Kilian Pohl

  38. The End

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