The Discrete Symmetry Transform in Computer

Symmetry plays a relevant role in perception problems. For example , peaks of brain activity are measured in correspondence with visual patterns showing symmetry. Relevance of symmetry in vision was already noted by KK oler in 1929. Here, properties of symmetry operators, usually applied in computer vision are reviewed. A new algorithm to measure local symmetries is proposed. it is easy to parallelize, because it will be expressed as product of local ltering operators. Its utility to vision problems is tested on four applications: the search of area of interest in active vision; the segmenta-tion of complex visual patterns; the classiication of sparse images; the analysis of movement. 1 Deenition of symmetry An object is said to exhibit symmetry if the application of certain isometries, called symmetry operators, leaves it unchanged while permuting parts. The letter A, for instance, remains unchanged under reeection, the letter Z under half-turn, and the letter H under both reeection and half-turn, the circle has circular symmetry around its centre (see Figure 1a,b,c,d). Moreover, given an object in a 2D space, it exhibits a symmetry respect to an axis x, if x divides the object in two mirror-like components. For example, is a symmetry axis of the letter A (Figure 1a), and are two symmetry axes for the letter H (Figure 1c), the circle has an innnite number of symmetry exes (Figure 1d). The deenition of symmetry can be extended to objects in a 3D space, by including planes and axes of symmetry. It is of great interest to study the relationship between the symmetries of an object in the 3D space and in