IG — A Disse

Many applications require a computer representation of 2D shape, usually described by a set of points. The challenge of this representation is that, besides capturing the shape characteristics, it must be invariant to relevant transformations. Invariance to geometric transformations was addressed in the past under the assumption that the points are labeled, i.e., that the shape is characterized by an ordered set of landmarks. However, in many practical scenarios the points are obtained from an automatic process, e.g., edge detection, thus without natural ordering. Obviously, the combinatorial problem of computing the correspondences between the points of two shapes becomes a quagmire when the number of points is large. We avoid this problem by representing shapes in a way that is invariant to the permutation of the landmarks. We map a shape to an analytic function on the complex plane, leading to what we call its analytic signature (ANSIG). We show that different shapes lead to different ANSIGs but that shapes that differ by a permutation of the landmarks lead to the same ANSIG. Thus, ANSIG is a maximal invariant with respect to the permutation group. To store an ANSIG, it suffices to sample it along a closed contour. We further show how easy it is to factor out geometric transformations when comparing shapes. We developed a shape-based image classification system, composed by a webcam and a Matlab software package, based on the ANSIG. Our experiments illustrate the performance of this system on automatic trademark retrieval and object recognition.