Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-04-30T14:16:30.375Z Has data issue: false hasContentIssue false

A blind definition of shape

Published online by Cambridge University Press:  15 August 2002

J. L. Lisani
Affiliation:
Univ. de les Illes Balears, Cra. de Valldemossa, km 7.5, 07071 Palma, Spain; vdmijlr0@clust.uib.es.
J. M. Morel
Affiliation:
CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan, France; morel@cmla.ens-cachan.fr.
L. Rudin
Affiliation:
Cognitech Inc., 225 S. Lake Avenue, CA-91101 Pasadena, U.S.A.; lenny@cognitech.com.
Get access

Abstract

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 $\cal I$, 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 $\cal I$. (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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Alvarez, L., Guichard, F., Lions, P.-L. and Morel, J.M., Axioms and fundamental equations of image processing: Multiscale analysis and P.D.E. Arch. Rational Mech. Anal. 16 (1993) 200-257.
S. Angenent, G. Sapiro and A. Tannenbaum, On the affine heat flow for nonconvex curves. J. Amer. Math. Soc. (1998).
Asada, H. and Brady, M., The curvature primal sketch. PAMI 8 (1986) 2-14. CrossRef
Brown, L.G., A survey of image registration techniques. ACM Comput. Surveys 24 (1992) 325-376. CrossRef
Caselles, V., Coll, B. and Morel, J.M., Topographic maps and local contrast changes in natural images. Int. J. Comput. Vision 33 (1999) 5-27. CrossRef
V. Caselles, B. Coll and J.M. Morel, Geometry and color in natural images. J. Math. Imaging Vision (2002).
T. Cohignac, C. Lopez and J.M. Morel, Integral and local affine invariant parameter and application to shape recognition, in ICPR94 (1994) A164-A168.
A. Desolneux, L. Moisan and J.M. Morel, Edge detection by Helmholtz principle. J. Math. Imaging Vision (to appear).
F. Dibos, From the projective group to the registration group: A new model. Preprint (2000).
R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. Wiley (1973).
Dudek, G. and Tsotsos, J.K., Shape representation and recognition from multiscale curvature. CVIU 2 (1997) 170-189.
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.
F. Guichard and J.M. Morel, Image iterative smoothing and P.D.E.'s (in preparation).
R.K. Hu, Visual pattern recognition by moments invariants. IEEE Trans. Inform. Theor. (1962) 179-187.
G. Kanizsa, Organization in vision: Essays on gestalt perception, in Praeger (1979).
Krzyzak, A., Leung, S.Y. and Suen, C.Y., Reconstruction of two-dimensional patterns from Fourier descriptors. MVA 2 (1989) 123-140.
C.C. Lin and R. Chellappa, Classification of partial 2-d shapes using fourier descriptors, in CVPR86 (1986) 344-350.
J.L. Lisani, Comparaison automatique d'images par leurs formes, Ph.D. Dissertation. Université Paris-Dauphine (2001).
J.L. Lisani, L. Moisan, P. Monasse and J.M. Morel, Planar shapes in digital images. MAMS (submitted).
J.L. Lisani, P. Monasse and L. Rudin, Fast shape extraction and applications. PAMI (submitted).
Marr, D. and Hildreth, E.C., Theory of edge detection. Proc. Roy. Soc. London Ser. A 207 (1980) 187-217. CrossRef
G. Matheron, Random Sets and Integral Geometry. John Wiley, NY (1975).
W. Metzger, Gesetze des Sehens. Waldemar Kramer (1975).
Moisan, L., Affine plane curve evolution: A fully consistent scheme. IEEE Trans. Image Process. 7 (1998) 411-420. CrossRef
Mokhtarian, F. and Mackworth, A.K., A theory of multiscale, curvature-based shape representation for planar curves. PAMI 14 (1992) 789-805. CrossRef
P. Monasse, Contrast invariant image registration, in Proc. of International Conference on Acoustics, Speech and Signal Process., Vol. 6. Phoenix, Arizona (1999) 3221-3224.
Monasse, P. and Guichard, F., Fast computation of a contrast-invariant image representation. IEEE Trans. Image Processing 9 (2000) 860-872. CrossRef
Okutomi, M. and Kanade, T., A locally adaptive window for signal matching. Int. J. Computer Vision 7 (1992) 143-162. CrossRef
Persoon, E. and Shape di, K.S. Fuscrimination using fourier descriptors. SMC 7 (1977) 170-179.
T.H. Reiss, Recognizing Planar Objects Using Invariant Image Features. Springer Verlag, Lecture Notes in Comput. Sci. 676 (1993).
Rucklidge, W.J., Efficiently locating objects using the Hausdorff distance. Int. J. Computer Vision 24 (1997) 251-270. CrossRef
Sapiro, G. and Tannenbaum, A., Affine invariant scale-space. Int. J. Computer Vision 11 (1993) 25-44. CrossRef
J. Serra, Image Analysis and Mathematical Morphology. Academic Press, New York (1982).
C.H. Teh and Chin R, On image analysis by the method of moments. IEEE Trans. Pattern Anal. Machine Intelligence 10 (1998).