A blind definition of shape
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 863-872

In this note, we propose a general definition of shape which is both compatible with the one proposed in phenomenology (gestaltism) and with a computer vision implementation. We reverse the usual order in Computer Vision. We do not define “shape recognition” as a task which requires a “model” pattern which is searched in all images of a certain kind. We give instead a “blind” definition of shapes relying only on invariance and repetition arguments. Given a set of images , we call shape of this set any spatial pattern which can be found at several locations of some image, or in several different images of . (This means that the shapes of a set of images are defined without any a priori assumption or knowledge.) The definition is powerful when it is invariant and we prove that the following invariance requirements can be matched in theory and in practice: local contrast invariance, robustness to blur, noise and sampling, affine deformations. We display experiments with single images and image pairs. In each case, we display the detected shapes. Surprisingly enough, but in accordance with Gestalt theory, the repetition of shapes is so frequent in human environment, that many shapes can even be learned from single images.

DOI : https://doi.org/10.1051/cocv:2002022
Classification:  62H35
Keywords: image analysis, basic shape elements, contrast invariance, level lines, scale space
@article{COCV_2002__8__863_0,
     author = {Lisani, J. L. and Morel, J. M. and Rudin, L.},
     title = {A blind definition of shape},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {863-872},
     doi = {10.1051/cocv:2002022},
     zbl = {1092.68679},
     mrnumber = {1932976},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__863_0}
}
Lisani, J. L.; Morel, J. M.; Rudin, L. A blind definition of shape. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 863-872. doi : 10.1051/cocv:2002022. http://www.numdam.org/item/COCV_2002__8__863_0/

[1] S. Abbasi and F. Mokhtarian, Retrieval of similar shapes under affine transformation, in Proc. International Conference on Visual Information Systems. Amsterdam, The Netherlands (1999) 566-574.

[2] L. Alvarez, F. Guichard, P.-L. Lions and J.M. Morel, Axioms and fundamental equations of image processing: Multiscale analysis and P.D.E. Arch. Rational Mech. Anal. 16 (1993) 200-257. | MR 1225209 | Zbl 0788.68153

[3] S. Angenent, G. Sapiro and A. Tannenbaum, On the affine heat flow for nonconvex curves. J. Amer. Math. Soc. (1998). | MR 1491538 | Zbl 0902.35048

[4] H. Asada and M. Brady, The curvature primal sketch. PAMI 8 (1986) 2-14.

[5] L.G. Brown, A survey of image registration techniques. ACM Comput. Surveys 24 (1992) 325-376.

[6] V. Caselles, B. Coll and J.M. Morel, Topographic maps and local contrast changes in natural images. Int. J. Comput. Vision 33 (1999) 5-27.

[7] V. Caselles, B. Coll and J.M. Morel, Geometry and color in natural images. J. Math. Imaging Vision (2002). | MR 1892677 | Zbl 0994.68167

[8] T. Cohignac, C. Lopez and J.M. Morel, Integral and local affine invariant parameter and application to shape recognition, in ICPR94 (1994) A164-A168.

[9] A. Desolneux, L. Moisan and J.M. Morel, Edge detection by Helmholtz principle. J. Math. Imaging Vision (to appear). | Zbl 0988.68819

[10] F. Dibos, From the projective group to the registration group: A new model. Preprint (2000). | MR 1836089 | Zbl 1008.68150

[11] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. Wiley (1973). | Zbl 0277.68056

[12] G. Dudek and J.K. Tsotsos, Shape representation and recognition from multiscale curvature. CVIU 2 (1997) 170-189.

[13] O. Faugeras and R. Keriven, Some recent results on the projective evolution of 2d curves, in Proc. IEEE International Conference on Image Processing. Washington DC (1995) 13-16.

[14] F. Guichard and J.M. Morel, Image iterative smoothing and P.D.E.'s (in preparation).

[15] R.K. Hu, Visual pattern recognition by moments invariants. IEEE Trans. Inform. Theor. (1962) 179-187. | Zbl 0102.13304

[16] G. Kanizsa, Organization in vision: Essays on gestalt perception, in Praeger (1979).

[17] A. Krzyzak, S.Y. Leung and C.Y. Suen, Reconstruction of two-dimensional patterns from Fourier descriptors. MVA 2 (1989) 123-140.

[18] C.C. Lin and R. Chellappa, Classification of partial 2-d shapes using fourier descriptors, in CVPR86 (1986) 344-350.

[19] J.L. Lisani, Comparaison automatique d'images par leurs formes, Ph.D. Dissertation. Université Paris-Dauphine (2001).

[20] J.L. Lisani, L. Moisan, P. Monasse and J.M. Morel, Planar shapes in digital images. MAMS (submitted).

[21] J.L. Lisani, P. Monasse and L. Rudin, Fast shape extraction and applications. PAMI (submitted).

[22] D. Marr and E.C. Hildreth, Theory of edge detection. Proc. Roy. Soc. London Ser. A 207 (1980) 187-217.

[23] G. Matheron, Random Sets and Integral Geometry. John Wiley, NY (1975). | MR 385969 | Zbl 0321.60009

[24] W. Metzger, Gesetze des Sehens. Waldemar Kramer (1975).

[25] L. Moisan, Affine plane curve evolution: A fully consistent scheme. IEEE Trans. Image Process. 7 (1998) 411-420. | MR 1669520 | Zbl 0973.94002

[26] F. Mokhtarian and A.K. Mackworth, A theory of multiscale, curvature-based shape representation for planar curves. PAMI 14 (1992) 789-805.

[27] P. Monasse, Contrast invariant image registration, in Proc. of International Conference on Acoustics, Speech and Signal Process., Vol. 6. Phoenix, Arizona (1999) 3221-3224.

[28] P. Monasse and F. Guichard, Fast computation of a contrast-invariant image representation. IEEE Trans. Image Processing 9 (2000) 860-872.

[29] M. Okutomi and T. Kanade, A locally adaptive window for signal matching. Int. J. Computer Vision 7 (1992) 143-162.

[30] E. Persoon and K.S. Fu, Shape discrimination using fourier descriptors. SMC 7 (1977) 170-179. | MR 451923

[31] T.H. Reiss, Recognizing Planar Objects Using Invariant Image Features. Springer Verlag, Lecture Notes in Comput. Sci. 676 (1993). | MR 1239444 | Zbl 0789.68120

[32] W.J. Rucklidge, Efficiently locating objects using the Hausdorff distance. Int. J. Computer Vision 24 (1997) 251-270.

[33] G. Sapiro and A. Tannenbaum, Affine invariant scale-space. Int. J. Computer Vision 11 (1993) 25-44.

[34] J. Serra, Image Analysis and Mathematical Morphology. Academic Press, New York (1982). | MR 753649 | Zbl 0565.92001

[35] C.H. Teh and Chin R, On image analysis by the method of moments. IEEE Trans. Pattern Anal. Machine Intelligence 10 (1998). | Zbl 0709.94543