Affine invariant detection: Edge maps, anisotropic diffusion, and active contours

Acta Applicandae Mathematicae

Volumn 59, Issue 1, 1999, Pages 45-77

  Olver, Peter J ^a   Sapiro, Guillermo ^b   Tannenbaum, Allen ^b  

^a University of Minnesota   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Active contours; Affine gradient; Affine invariant detection; Affine scale space; Denoising; Geodesics; Gradient flows; Invariant metrics; Segmentation

Indexed keywords

ALGORITHMS; IMAGE ENHANCEMENT; IMAGE SEGMENTATION; OBJECT RECOGNITION; OPTIMIZATION; TOPOLOGY;

ACTIVE CONTOURS; AFFINE GRADIENT; AFFINE INVARIANT DETECTION; GRADIENT FLOWS; IMAGE DENOISING;

EDGE DETECTION;

EID: 0033203926     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1006295328209     Document Type: Article

References (86)

1. Alvarez, L., Guichard, F., Lions, P. L. and Morel, J. M.: Axioms and fundamental equations of image processing, Arch. Rational Mech. 123 (1993), 199-257.
2. Alvarez, L., Lions, P. L. and Morel, J. M.: Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal. 29 (1992), 845-866.
3. Angenent, S.: Parabolic equations for curves on surfaces, Part II. Intersections, blow-up, and generalized solutions, Ann. of Math. 133 (1991), 171-215.
4. Angenent, S., Sapiro, G. and Tannenbaum, A.: On the affine heat flow for non-convex curves, J. Amer. Math. Soc. 11 (1998), 601-634.
5. Ballester, C., Caselles, V. and Gonzalez, M.: Affine invariant segmentation by variational method, SIAM J. Appl. Math. 56 (1996), 294-325.
6. Blake, A. and Yuille, A.: Active Vision, MIT Press, Cambridge, 1992.
7. Blaschke, W.: Vorlesungen über Differentialgeometrie II, Springer, Berlin, 1923.
8. Bookstein, F. L.: Fitting conic sections to scattered data, Comput. Graph. Image Process. 9 (1979), 56-71.
9. Calabi, E., Olver, P. J. and Tannenbaum, A.: Affine geometry, curve flows, and invariant numerical approximations, Adv. Math. 124 (1996), 154-196.
10. Calabi, E., Olver, P. J., Shakiban, C., Tannenbaum, A. and Haker, S.: Differential and numerical invariant signature curves applied to object recognition, Internat. J. Computer Vision 26 (1998), 107-135.
11. Caselles, V., Catte, F., Coll, T. and Dibos, F.: A geometric model for active contours, Numer. Math. 66 (1993), 1-31.
12. Caselles, V., Kimmel, R. and Sapiro, G.: Geodesic active contours, Internat. J. Computer Vision 22(1) (1997), 61-79. Also in Proc. ICCV, Cambridge, MA, June 1995.
13. Caselles, V., Kimmel, R. and Sapiro, G.: Geodesic active contours, Internat. J. Computer Vision 22(1) (1997), 61-79. Also in Proc. ICCV, Cambridge, MA, June 1995.
14. Caselles, V., Kimmel, R., Sapiro, G. and Sbert, C.: Minimal surfaces: A three dimensional segmentation approach, IEEE-PAMI 19(4) (1997), 394-398.
15. Caselles, V., Kimmel, R., Sapiro, G. and Sbert, C.: Three dimensional object modeling via minimal surfaces, in: Proc. ECCV, Cambridge, UK, April 1996.
16. Chen, Y. G., Giga, Y. and Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749-786.
17. Chopp, D.: Computing minimal surfaces via level set curvature flows, J. Comput. Physics 106 (1993), 77-91.
18. Cohen, L. D.: On active contour models and balloons, CVGIP: Image Understanding 53 (1991), 211-218.
19. Cohen, I., Cohen, L. D. and Ayache, N.: Using deformable surfaces to segment 3D images and infer differential structure, CVGIP: Image Understanding 56 (1992), 242-263.
20. Cohignac, T., Lopez, C. and Morel, J. M.: Integral and local affine invariant parametrizations and applications to shape recognition, in: Proc. 12th IEEE Int. Conf. Pattern Recognition, Jerusalem, 1994.
21. Crandall, M. G., Ishii, H. and Lions, P. L.: User's guide to viscosity solutions of second order partial linear differential equations, Bull. Amer. Math. Soc. 27 (1992), 1-67.
22. Dubrovin, B. A., Fomenko, A. T. and Novikov, S. P.: Modern Geometry - Methods and Applications I, Springer-Verlag, New York, 1984.
23. Epstein, C. L. and Gage, M.: The curve shortening flow, in: A. Chorin and A. Majda (eds), Wave Motion: Theory, Modeling, and Computation, Springer-Verlag, New York, 1987.
24. Evans, L. C. and Spruck, J.: Motion of level sets by mean curvature, I, J. Differential Geom. 33 (1991), 635-681.
25. Faugeras, O.: On the evolution of simple curves of the real projective plane, C.R. Acad. Sci. Paris 317 (1993), 565-570.
26. Faugeras, O.: Cartan's moving frame method and its application on the geometry and evolution of curves in the Euclidean, affine, and projective planes, in: J. L. Mundy, A. Zisserman and D. Forsyth (eds), Applications of Invariance in Computer Vision, Springer-Verlag, New York, 1994, pp. 11-46.
27. Faugeras, O. and Keriven, R.: Scale-spaces and affine curvature, in: R. Mohr and C. Wu (eds), Proc. Europe-China Workshop on Geometrical Modeling and Invariants for Computer Vision, 1995, pp. 17-24.
28. Forsyth, D., Mundy, J. L., Zisserman, A. and Brown C. M.: Projectively invariant representations using implicit curves, Image Vision Comput. 8 (1990), 130-136.
29. Forsyth, D., Mundy, J. L., Zisserman, A., Coelho, C., Heller, A. and Rothwell, C.: Invariant description of object representation and pose, IEEE PAMI 13 (1991), 971-991.
30. Fua, P. and Leclerc, Y. G.: Model driven edge detection, Machine Vision Appl. 3 (1990), 45-56.
31. Gage, M. and Hamilton, R. S.: The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), 69-96.
32. Grayson, M.: The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285-314.
33. Grayson, M.: Shortening embedded curves, Ann. of Math. 129 (1989), 285-314.
34. Guggenheimer, H. W.: Differential Geometry, McGraw-Hill, New York, 1963.
35. Gurtin, M. E.: Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Univ. Press, New York, 1993.
36. Kass, M., Witkin, A. and Terzopoulos, D.: Snakes: Active contour models, Internat. J. Comput. Vision 1 (1988), 321-331.
37. Kichenassamy, S., Kumar, A., Olver, P. J., Tannenbaum, A. and Yezzi, A.: Gradient flows and geometric active contour models, in: Proc. ICCV, Cambridge, MA, June 1995, pp. 810-815.
38. Kichenassamy, S., Kumar, A., Olver, P. J., Tannenbaum, A. and Yezzi, A.: Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996), 275-301.
39. Kimia, B. B., Tannenbaum, A. and Zucker, S. W.: Toward a computational theory of shape: An overview, Lecture Notes in Comput. Sci. 427 (1990), 402-407.
40. Kimia, B. B., Tannenbaum, A. and Zucker, S. W.: Shapes, shocks, and deformations, I, Internat. J. Comput. Vision 15 (1995), 189-224.
41. Kimmel, R.: Invariant framework for differential affine signatures, in: Proc. ICPR '96.
42. Kimmel, R., Amir, A. and Bruckstein, A. M.: Finding shortest paths on surfaces using level sets propagation, IEEE-PAMI 17(1) (1995), 635-640.
43. Kimmel, R., Kiryati, N. and Bruckstein, A. M.: Sub-pixel distance maps and weighted distance transforms, J. Math. Imaging and Vision, Special Issue on Topology and Geometry in Computer Vision, 6 (1996), 223-233.
44. Lindeberg, T.: Scale-Space Theory in Computer Vision, Kluwer Acad. Publ., Dordrecht, 1994.
45. Lindeberg, T. and Garding, J.: Shape-adapted smoothing in estimation of 3D depth cues from affine distortions of local 2D structures, in: Proc. ECCV, Stockholm, Sweden, May 1994.
46. Malladi, R. and Sethian, J. A.: A unified approach to noise removal, image enhancement, and shape recovery, IEEE Trans. Image Processing 5 (1996), 1554-1568.
47. Malladi, R., Sethian, J. A. and Vemuri, B. C.: Evolutionary fronts for topology independent shape modeling and recovery, in: Proc. of the 3rd ECCV, Stockholm, Sweden, 1994, pp. 3-13.
48. Malladi, R., Sethian, J. A. and Vemuri, B. C.: Shape modeling with front propagation: A level set approach, IEEE Trans, on PAMI 17 (1995), 158-175.
49. Malladi, R., Sethian, J. A. and Vemuri, B. C.: A fast level set based algorithm for topology independent shape modeling, J. Math. Imaging and Vision 6 (1996), 269-289.
50. McInerney, T. and Terzopoulos, D.: Topologically adaptable snakes, in: Proc. ICCV, Cambridge, MA, June 1995.
51. Mumford, D. and Shah, J.: Optimal approximations by piecewise smooth functions and variational problems, Comm. Pure Appl. Math. 42 (1989), 577-685.
52. Mundy, J. L. and Zisserman, A. (eds): Geometric Invariance in Computer Vision, MIT Press, 1992.
53. Niessen, W. J., ter Haar Romeny, B. M., Florack, L. M. J. and Salden, A. H.: Nonlinear diffusion of scalar images using well-posed differential operators, in: Proceedings CVPR, IEEE Press 1994, pp. 92-97.
54. Olver, P. J.: Applications of Lie Groups to Differential Equations, 2nd edn, Springer-Verlag, New York, 1993.
55. Olver, P. J.: Equivalence, Invariants, and Symmetry, Cambridge Univ. Press, Cambridge, UK, 1995.
56. Olver, P. J., Sapiro, G. and Tannenbaum, A.: Differential invariant signatures and flows in computer vision: A symmetry group approach, in [62].
57. Olver, P. J., Sapiro, G. and Tannenbaum, A.: Affine invariant edge maps and active contours, Geometry Center Technical Report 90, University of Minnesota, October 1995.
58. Olver, P. J., Sapiro, G. and Tannenbaum, A.: Affine invariant detection: Edges, active contours, and segments, in: Proc. Computer Vision Pattern Recognition, San Francisco, June 1996, pp. 520-525.
59. Olver, P. J., Sapiro, G. and Tannenbaum, A.: Invariant geometric evolutions of surfaces and volumetric smoothing, SIAM J. Appl. Math. 57 (1997), 176-194.
60. Osher, S. J. and Sethian, J. A.: Fronts propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988), 12-49.
61. Pauwels, E. J., Fiddelaers, P. and Van Gool, L. J.: Shape-extraction for curves using geometry-driven diffusion and functional optimization, in: Proc. ICCV, Cambridge, MA, June 1995.
62. Perona, P. and Malik, J.: Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intell. 12 (1990), 629-639.
63. Romeny, B. (ed.): Geometry Driven Diffusion in Computer Vision, Kluwer Acad. Publ., Dordrecht, 1994.
64. Rudin, L. I., Osher, S. and Fatemi, E.: Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), 259-268.
65. Sapiro, G., Kimmel, R., Shaked, D., Kimia, B. B. and Bruckstein, A. M.: Implementing continuous-scale morphology via curve evolution, Pattern Recog. 26(9) (1993), 1363-1372.
66. Sapiro, G. and Tannenbaum, A.: On affine plane curve evolution, J. Funct. Anal. 119(1) (1994), 79-120.
67. Sapiro, G. and Tannenbaum, A.: Affine invariant scale-space, Internat. J. Comput. Vision 11(1) (1993), 25-44.
68. Sapiro, G. and Tannenbaum, A.: Image smoothing based on an affine invariant flow, in: Proceedings of Conference on Information Sciences and Systems, Johns Hopkins University, March 1993.
69. Sapiro, G. and Tannenbaum, A.: On invariant curve evolution and image analysis, Indiana Univ. Math. J. 42 (1993), 95-1009.
70. Sapiro, G. and Tannenbaum, A.: Area and length preserving geometric invariant scale-spaces, IEEE Trans. PAMI 17(1) (1995), 67-72.
71. Sapiro, G., Tannenbaum, A., You, Y. L. and Kaveh, M.: Experiments on geometric image enhancement, in: First IEEE-International Conference on Image Processing, Austin-Texas, November 1994.
72. Shah, J.: Recovery of shapes by evolution of zero-crossings, Technical Report, Math. Dept. Northeastern Univ., Boston, MA, 1995.
73. Soner, H. M.: Motion of a set by the curvature of its boundary, J. Differential Equations 101 (1993), 313-372.
74. Szeliski, R., Tonnesen, D. and Terzopoulos, D.: Modeling surfaces of arbitrary topology with dynamic particles, in: Proc. CVPR, 1993, pp. 82-87.
75. Sethian, J. A.: Curvature and the evolution of fronts, Comm. Math. Phys. 101 (1985), 487-499.
76. Sethian, J. A.: A review of recent numerical algorithms for hypersurfaces moving with curvature dependent flows, J. Differential Geom. 31 (1989), 131-161.
77. Tek, H. and Kimia, B. B.: Image segmentation by reaction-diffusion bubbles, in: Proc. ICCV, Cambridge, MA, June 1995.
78. Terzopoulos, D., Witkin, A. and Kass, M.: Constraints on deformable models: Recovering 3D shape and nonrigid motions, Artif. Intell. 36 (1988), 91-123.
79. Torre, V. and Poggio, T.: On edge detection, IEEE Trans. PAMI 8 (1986), 147-163.
80. Van Gool, L., Moons, T. and Ungureanu, D.: Affine/photometric invariants for planar intensity patterns, in: Proc. ECCV, Cambridge, UK, April 1996, pp. 642-651.
81. Weiss, I.: Geometric invariants and object recognition, Internat. J. Comput. Vision (1993), 207-231.
82. Whitaker, R. T.: Volumetric deformable models: Active blobs, ECRC TR 94-25, 1994.
83. Whitaker, R. T.: Algorithms for implicit deformable models, in: Proc. ICCV'95, Cambridge, MA, June 1995.
84. Yezzi, A., Kichenassamy, S., Kumar, A., Olver, P. and Tannenbaum, A.: Geometric snakes for edge detection and segmentation of medical imagery, IEEE Trans. Medical Imaging 16 (1997), 199-210.
85. Yezzi, A., Kichenassamy, S., Olver, P. and Tannenbaum, A.: A gradient surface approach to 3D segmentation, in: Proceedings of 49th IS&T, Society for Imaging Science and Technology, Springfield, VA, 1996, pp. 305-307.
86. Zhu, S. C., Lee, T. S. and Yuille, A. L.: Region competition: Unifying snakes, region growing, energy/Bayes/MDL for multi-band image segmentation, in: Proc. ICCV, Cambridge, MA, June 1995.