A blind definition of shape

Lisani, J. L.; Morel, J. M.; Rudin, L.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 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 $ℐ$, 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 $ℐ$. (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.

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