Integrated Active Contours for Texture Segmentation

1. Integrated Active Contours for Texture Segmentation Chen Sagiv1, Nir Sochen1, Yehoshua Y. Zeevi2 Applied Mathematics, University of Tel-Aviv Electrical Engineering, Technion, Haifa MSRI, Women in Math, January 2005

2. Texture Segmentation in general

3. Texture Segmentation in general • Selection of a Texture Representation Space • Extraction of Texture Features • The introduction of a measure on the Texture Features • Defining an Objective Function and solving the optimization problem

4. Outline: • The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

5. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

6. The Beltrami Framework (Sochen, Kimmel and Malladi 96,98,2000) An image = a 2D surface embedded in Rn A parametric description of a surface: S=( X(x,y), Y(x,y), Z(x,y) ) In canonical coordinates: S = ( x, y, I(x,y) )

7. How to measure distances on the image manifold ? The metric of a grayscale image:

8. The Beltrami Framework: Applications • Segmentation • g serves as an edge detector • Non – Linear Diffusion • E(I) = Area( surface ) = • where g = det( metric ) • De-noising process = Area minimization • It = - E

9. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

10. Intensity Based Classical Snakes Kass, Witkin, Terzopoulos 1988 Let C(p) = { x(p),y(p) }, be a parametrzied curve, p  [0,1] C(p) y x Euler-Lagrange Steepest Descent

11. The Theory of Curve Evolution • Euclidean length is given by: • Reduce the length by: Ct=kN • k - Euclidean curvature • N - normal Length =

12. Intensity based Geodesic Snakes ( Caselles, Kimmel & Sapiro 1995) The Gradient Descent equation: b(C) is the stopping term:

13. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

14. Why is the metric useful for edge detection ?

15. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

16. Level-set formulation: Osher-Sethian Taken from: A Fast Introduction to Level Set Methods, web page of J.A. Sethian, Dept. of Mathematics, Univ. of California, Berkeley, California The original front Front lies in x-y plane The level-set function front is intersection of Surface and x-y

17. Level-set formulation Let :R be a level set function which embeds the contour C={ x   | (x) = 0 } Where H( ) denotes the Heaviside function:

18. Level-set formulation (Kimmel) Let :R be a level set function which embeds the contour C={ x   | (x) = 0 } Where H( ) denotes the Heaviside function:

19. Geodesic Snakes Demo Evolution of 

20. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

21. 2D Gabor Wavelets The Gabor feature space: Wmn (x,y) = hmn (x,y) * I(x,y)

22. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

23. Texture based Snakes Sagiv, Sochen & Zeevi We replace b(gradient) by b(texture gradient). We calculate the image responses to the Gabor-Morlet wavelet, . We describe the result as a two dimensional manifold embedded in the spatial feature space: texture gradient = 1/det(Riemannian metric of the surface)

24. Why is the metric useful ?

25. Why is the metric useful ?

26. Level-Set Texture geodesic snakes The generalization to texture is straightforward:

27. Active Contours for Texture Segmentation

28. Active Contours for Texture Segmentation

29. Active Contours for Texture Segmentation

30. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

31. Active Contours without Edges Chan & Vese Let :R be a level set function which embeds the contour C={ x   | (x) = 0 } Minimizing: results in piecewise constant segmentation

32. The benefits of the edge-less active contours model Taken from: Active Contours without Edges for Vector-Valued Images, T. F. Chan, B. Y. Sandberg, and L.A. Vese Journal of Visual Communication and Image Representation 11, 130–141 (2000)

33. Chan-Vese for Texture Sandberg, Chan & Vese Define features Where h is a Gabor filter The segmentation functional is

34. Chan-Vese segmentation (1)

35. Chan-Vese segmentation (2)

36. Active Contours with Edges Active Contours without Edges

37. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

38. Unifying edge and region base techniques We generalize the gray-value formalism of Kimmel (2003). The following functional takes both region and edge information into consideration: Here the two regions are competing while the Length of the interface is weighted by the texture gradient Sagiv, Sochen & Zeevi

39. Active Contours with Edges Active Contours without Edges

40. Active Contours with Edges Active Contours without Edges IAC

41. Active Contours with Edges Active Contours without Edges IAC

43. The Beltrami Framework • Geodesic Snakes • Why is the metric important for edge detection ? • Level Set Formalism • Gabor Feature Space • Active Contours in the Gabor Space • Active Contours without Edges • Integrated Active Contours • Summary

44. The main contributions: • Derivation of an edge indication function in the Gabor feature space of images • Comparison of the edge-based approach and the edge-less based approach • Integration of both approaches in the Gabor Feature Space

45. Thank You

46. Segmentation using the structure tensor (Rousson, Brox, Deriche)

47. Intensity Based Classical Snakes Kass, Witkin, Terzopoulos 1988 Let C(p) = { x(p),y(p) }, be a parametrzied curve, p  [0,1] C(p) y x Euler-Lagrange Steepest Descent