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Geometric Edge Detection Techniques for Image Segmentation

Explore classic and new-wave geometric methods for edge detection and integration in image processing, focusing on active contours and segmentation using variational models. Learn about geodesic active contours, snakes, Laplacian active contours, and more. Discover the latest advances in variational measures for classical operators.

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Geometric Edge Detection Techniques for Image Segmentation

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  1. Computer Science Department • Technion-Israel Institute of Technology Geometric Active Contours Ron Kimmel www.cs.technion.ac.il/~ron • Geometric Image Processing Lab

  2. Edge Detection • Edge Detection: • The process of labeling the locations in the image where the gray level’s “rate of change” is high. • OUTPUT:“edgels” locations, direction, strength • Edge Integration: • The process of combining “local” and perhaps sparse and non-contiguous “edgel”-data into meaningful, long edge curves (or closed contours) for segmentation • OUTPUT:edges/curves consistent with the local data

  3. The Classics • Edge detection: • Sobel, Prewitt, Other gradient estimators • Marr Hildreth zero crossings of • Haralick/Canny/Deriche et al. “optimal” directional local max of derivative • Edge Integration: • tensor voting (Rom, Medioni, Williams, …) • dynamic programming (Shashua & Ullman) • generalized “grouping” processes (Lindenbaum et al.)

  4. “nice” curves that optimize a functional of g( ), i.e. nice: “regularized”, smooth, fit some prior information Image Edge Curves Edge Indicator Function The “New-Wave” • Snakes • Geodesic Active Contours • Model Driven Edge Detection

  5. Geodesic Active Contours • Snakes Terzopoulos-Witkin-Kass 88 • Linear functional efficient implementation • non-geometric depends on parameterization • Open geometric scaling invariant, Fua-Leclerc90 • Non-variational geometric flow Caselles et al. 93, Malladi et al. 93 • Geometric, yet does not minimize any functional • Geodesic active contours Caselles-Kimmel-Sapiro 95 • derived from geometric functional • non-linear inefficient implementations: • Explicit Euler schemes limit numerical step for stability • Level set method Ohta-Jansow-Karasaki82,Osher-Sethian 88 • automatically handles contour topology • Fast geodesic active contours Goldenberg-Kimmel-Rivlin-Rudzsky 99 • no limitation on the time step • efficient computations in a narrow band

  6. Laplacian Active Contours • Closed contours on vector fields • Non-variational models Xu-Prince98,Paragios et al.01 • A variational model Vasilevskiy-Siddiqi01 • Laplacian active contours open/closed/robust Kimmel-Bruckstein 01 Most recent: variational measures for good old operators Kimmel-Bruckstein 03

  7. Segmentation

  8. Segmentation • Ultrasound images Caselles,Kimmel, Sapiro ICCV’95

  9. Segmentation Pintos

  10. Woodland Encounter Bev Doolittle 1985 • With a good prior who needs the data…

  11. Segmentation Caselles,Kimmel, Sapiro ICCV’95

  12. Prior knowledge…

  13. Prior knowledge…

  14. Segmentation

  15. Segmentation

  16. Segmentation Caselles,Kimmel, Sapiro ICCV’95

  17. Segmentation • With a good prior who needs the data…

  18. Wrong Prior???

  19. Wrong Prior???

  20. Wrong Prior???

  21. C =tangent p Curves in the Plane • C(p)={x(p),y(p)}, p [0,1] C(0.1) C(0.2) C(0.7) C(0) C(0.4) C(0.8) C(0.95) y C(0.9) x

  22. Arc-length and Curvature s(p)= | |dp C

  23. Calculus of Variations Find C for which is an extremum Euler-Lagrange:

  24. Calculus of Variations Important Example • Euler-Lagrange: , setting • Curvature flow

  25. Potential Functions (g) I(x,y) I(x) Image x x g(x,y) g(x) Edges x x

  26. Snakes & Geodesic Active Contours • Snake model Terzopoulos-Witkin-Kass 88 • Euler Lagrange as a gradient descent • Geodesic active contour model Caselles-Kimmel-Sapiro 95 • Euler Lagrange gradient descent

  27. Maupertuis Principle of Least Action p Snake = Geodesic active contour up to some, i.e • Snakes depend on parameterization. • Different initial parameterizations yield solutions for different geometric functionals 1 y 0 x Caselles Kimmel Sapiro, IJCV 97

  28. Geodesic Active Contours in 1D I(x) Geodesic active contours are reparameterization invariant x g(x) x

  29. Geodesic Active Contours in 2D G *I s g(x)=

  30. Controlling -max Smoothness g I Cohen Kimmel, IJCV 97

  31. Fermat’s Principle In an isotropic medium, the paths taken by light rays are extremal geodesics w.r.t. i.e., Cohen Kimmel, IJCV 97

  32. Experiments - Color Segmentation Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

  33. Tumor in 3D MRI Caselles,Kimmel, Sapiro, Sbert, IEEE T-PAMI 97

  34. Segmentation in 4D Malladi, Kimmel, Adalsteinsson,  Caselles, Sapiro, Sethian SIAM Biomedical workshop 96

  35. Tracking in Color Movies Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

  36. Tracking in Color Movies Goldenberg, Kimmel, Rivlin, Rudzsky, IEEE T-IP 2001

  37. Edge Gradient Estimators Xu-Prince98,Paragios et al.01, Vasilevskiy-Siddiqi01, Kimmel-Bruckstein 01

  38. Edge Gradient Estimators • We want a curve with large points and small ‘s so: • Consider the functional • Where is a scalar function, e.g. .

  39. The Classic Connection Supposeand we consider a closed contour for C(s). We have and by Green’s Theorem we have

  40. The Classic Connection • Therefore: • Hence curves that maximize are curves that enclose all regions where is positive! • We have that the optimal curves in this case are The Zero Crossings of the Laplacian isn’t this familiar?

  41. The Classic Connection • It is pedagogically nice, but the MARR-HILDRETH edge detector is a bit too sensitive. • So we do not propose a grand return to MH but a rethinking of the functionals used in active contours in view of this. • INDEED, why should we ignore the gradient directions (estimates) and have every edge integrator controlled by the local gradient intensity alone?

  42. Our Proposal • Consider functional of the form • These functionals yield “regularized” curves that combine the good properties of LZC’s where precise border following is needed, with the good properties of the GAC over noisy regions!

  43. Implementation Details • We implement curve evolution that do gradient descent w.r.t. the functional Here the Euler Lagrange Equations provide the explicit formulae. • For closed contours we compute the evolved curve via the Osher-Sethian “miracle” numeric level set formulation.

  44. Closed contours EL eq. GAC LZC LZC GAC Kimmel-Bruckstein IVCNZ01

  45. Closed contours EL eq. GAC LZC LZC+eGAC Kimmel-Bruckstein IVCNZ01

  46. Open contours Along the curve b.c. at C(0) and C(L) Kimmel-Bruckstein IVCNZ01

  47. Open contours Kimmel-Bruckstein IVCNZ01

  48. Geometric Measures Weighted arc-length Weighted area Alignment Robust-alignment e.g. Variational meaning for Marr-Hildreth edge detectorKimmel-Bruckstein IVCNZ01

  49. Geometric Measures Minimal variance Chan-Vese, Mumford-Shah, Max-Lloyd, Threshold,…

  50. Geometric Measures Robust minimal deviation

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