Using Procrustes distance and shape space for automatic target recognition

Equating objects based on shape similarity (for example scaled Euclidean transformations) is often desirable to solve the Automatic Target Recognition (ATR) problem. The Procrustes distance is a metric that captures the shape of an object independent of the following transformations: translation, rotation, and scale. The Procrustes metric assumes that all objects can be represented by a set of landmarks (i.e. points), that they have the same number of points, and that the points are ordered (i.e., the exact correspondence between the points is known from one object to the next). Although this correspondence is not known for many ATR problems, computationally feasible methods for examining all possible combinations are being explored. Additionally, most objects can be mapped to a shape space where translation, rotation, and scaling are removed, and distances between object points in this space can then form another useful metric. To establish a decision boundary in any classification problem, it is essential to know the a prior probabilities in the appropriate space. This paper analyzes basic objects (triangles) in two-dimensional space to assess how a known distribution in Euclidean space maps to the shape space. Any triangles whose three coordinate points are uniformly distributed within a two-dimensional box transforms to a bivariate independent normal distribution with mean (0,0) and standard deviations of 2 in Kendall shape space (two points of the triangle are mapped to {-1/2,0} and {1/2,0}). The Central Limit Theorem proves that the limit of sums of finite variance distributions approaches the normal distribution. This is a reasonable model of the relationship between the three Euclidean coordinates relative to the single Kendall shape space coordinate. This paper establishes the relationship between different objects in the shape space and the Procrustes distance, which is an established shape metric, between these objects. Ignoring reflections (because it is a special case), the Procrustes distance is isometric to the shape space coordinates. This result demonstrates that both Kendall coordinates and Procrustes distance are useful features for ATR.