Spherical-Homoscedastic Shapes

Shape analysis requires invariance under translation, scale and rotation. Translation and scale invariance can be realized by normalizing shape vectors with respect to their mean and norm. This maps the shape feature vectors onto the surface of a hypersphere. After normalization, the shape vectors can be made rotational invariant by modelling the resulting data using complex scalar rotation invariant distributions defined on the complex hypersphere, e.g., using the complex Bingham distribution. However, the use of these distributions is hampered by the difficulty in estimating their parameters, which is shown to be very costly or impossible in most cases. The purpose of this paper is twofold. First, we show under which conditions the classification results obtained with complex Binghams are identical to those obtained with the easy-to-estimate complex Normal distribution. Second, we derive a kernel function which (intrinsically) maps the data into a space where the above conditions are satisfied and, hence, where the normal model can be successfully used. This results in a simple, low-cost algorithm for representing and classifying shapes. We demonstrate the use of this technique in several experimental results for object and face recognition. Comparisons to other statistical shape representation/classification approaches demonstrate the superiority of the proposed algorithms in classification accuracy and computational time.