Classification of Shapes via Explainable AI

I define the mathematical concept of shape and shape signature in two dimensions, using parametric polar equations. The signature uniquely characterizes the shape, up to a translation or scale factor. In practical applications, the data set consists of points or pixels located on the shape, rather than the curve itself. If these points are not properly sampled - if they are not uniformly distributed on the curve - they need to be re-weighted to compute a meaningful centroid of the shape, and to perform shape comparisons. I discuss the weights, and then introduce metrics to compare shapes (observed as sets of points or pixels in an image). These metrics are related to the Hausdorff distance. I also introduce a correlation distance between two shapes. Equipped with these metrics, one can perform shape recognition or classification using training sets of arbitrary sizes. I use synthetic data in the applications. It allows you to see how the classifier performs, to discriminate between two very similar shapes, or in the presence of noise. Rotation-invariant metrics are also discussed.