Generalized correlation for shape alignment

Ordered set-valued functions are introduced in this study to represent arbitrary shapes in a simple manner. The arithmetic of ordered set-valued functions is meaningless; hence, a novel concept, namely, generalized correlation, is proposed to handle shape alignment problems. Generalized correlation is an extension of correlation from single-valued functions to ordered set-valued functions. It is especially designed for shape alignment but not for a general mathematical definition. Moreover, efficient algorithms are proposed to compute generalized correlation in 2D and 3D cases. Experiments indicate that the proposed alignment methods converge rapidly and globally. Furthermore, generalized correlation is used handle registration problems to demonstrate its potential in computer vision tasks.