1. 02.12.2005 MUSCLE-WP5&7 meeting Priors, Syntax and Semantics in Variational Level-Set Approachs TAU-VISUAL: Nir Sochen & Nahum Kiryati Based on works with Tammy Riklin-Raviv

2. 02.12.2005 MUSCLE-WP5&7 meeting Segmentation as an inference problem Given: a visual data – an image Infer: (projection of) the scene structure

3. 02.12.2005 MUSCLE-WP5&7 meeting Maximum Likelihood ws= World’s State Then: find the ws that best explains the data Ws=argmax P(data|ws) Problem: UFO

4. 02.12.2005 MUSCLE-WP5&7 meeting Bayesian formalism ws= World’s State Then: Find a ws that explains the data and has high probability ws=argmax P(ws|data)= argmax P(data|ws)P(ws)

5. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: General Task: Partition the image to its significant domains Or Find the relevant shapes

6. 02.12.2005 MUSCLE-WP5&7 meeting Segmentation Functional is the input imagewhere is the segmenting boundary is the average color fidelity Compatibility with the prior Smoothness and length

7. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: Assumptions Assumptions: There are objects and background. Object pixels: Normal probability dist. Background pixels: Another normal dist. Boundary: Close smooth curve

8. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: Modelling1 : Foreground : Background : Separating curve

9. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: Modelling2

10. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: steps Step 0: Initialize Step 1: Updating the constants u_in, u_out as the mean Values of the respective regions Step 2: Maximization of the posterior w.r.t. C. Denote E = -log P then argmax P(C, u_in, u_out) = argmin E(C, u_in, u_out) Step 3: while not converge go to step 1

11. 02.12.2005 MUSCLE-WP5&7 meeting Image Segmentation: math (no shape prior yet)

12. 02.12.2005 MUSCLE-WP5&7 meeting LS Framework for MS Let be a level set function which embeds the contour: A piecewise constant segmentation of an input image f can be obtained by minimizing the functional: (3) Where denotes the Heaviside function: (4) Chan & Vese Trans. IP 01

13. 02.12.2005 MUSCLE-WP5&7 meeting LS Framework for MS (Cont.) Chan & Vese 01 Let be a smooth approximation of the Heaviside function. It’s derivative takes the form: (7)

14. 02.12.2005 MUSCLE-WP5&7 meeting LS Framework for MS (Cont) The Euler-Lagrange equation for this functional can be implemented by the following gradient descent: (5) where the scalars are updated in alternation with the level set evolution: are the mean gray value in the input image inside and outside C respectively. (6)

15. 02.12.2005 MUSCLE-WP5&7 meeting Shape prior + transformation 

16. 02.12.2005 MUSCLE-WP5&7 meeting Segmentation with Prior

17. 02.12.2005 MUSCLE-WP5&7 meeting Summary : There is much to gain from exploiting The relation statistics variational approaches F I N