- The Problem:

Representing shapes is a significant problem for vision systems that must recognize or classify objects.

Methods to compare two shape contours based on evaluating global deformations tend to be sensitive to occlusion and fail to account for local deformations (such as articulations) since these deformations may change the global appearance of objects considerably while the entire deformation is concentrated in specific points. A class of methods compares objects by deforming one object into another and evaluating the amount of deformation applied in this process. Guaranteed methods, typically, use dynamic programming (time-warping) to register two contours. These are all string (contour) matching algorithms. The main drawback of these approaches is that they do not account for region information and for symmetries. For example, in the following figures, we have a shape contour

**A**followed by two shapes**B**and**C**obtained by different deformations (stretching) into the first one. Note that the amount of stretching on each one is exactly the same, namely doubling the size of two pieces of straight lines. However, because they are applied in different segments, it causes the third shape to look very different from the first one. All local string deformation methods will fail to distinguish the dissimilarity between them and the first one.

- Our Approach:

We derive a representation for a given shape by investigating its self-similarities, and constructing its shape axis (SA) and shape axis tree (SA-tree). We start with a shape, its boundary contour, and two different parameterizations for the contour. To measure its self-similarity we consider matching pairs of points (and their tangents) along the boundary contour, i.e., matching the two parameterizations. The matching, or self-similarity criteria may vary, e.g., co-circularity, parallelism, distance, region homogeneity. The loci of middle points of the pairing contour points are the shape axis and they can be grouped into a unique tree graph, the SA-tree. The shape axis for the co-circularity criteria is compared to the symmetry axis. An interpretation in terms of object parts is also presented. For details, please see iccv98.ps.gz.

- Examples: (The Java demos will be available soon)