On similarity

The objective quantification of similarity between two mathematical or physical structures, from scalars to graphs, constitutes a central issue in the physical sciences and technology. In the present work, we develop a principled and systematic approach that adopts the Kronecker delta function of two scalar real values as the prototypical reference for fully strict similarity quantification. We then consider other approaches, namely the cosine similarity, correlation, Sørensen–Dice, and Jaccard indices, and show that they provide successively more strict similarity quantifications. Multiset-based generalizations of these indices to take into account real values are then adopted in order to progressively extend the indices to multisets, vectors, and functions in real spaces. Several important results are reported, including the multiset formulation of similarity indices, as well as the formal derivation of the Jaccard index from the Kronecker delta function. When generalized to real functions, the described similarity indices become respective functionals, which can then be employed to obtain operations analogous to convolution and correlation. Complete application examples involving the recognition of patterns through template matching between two 1D functions as well as the identification of multiples instances of objects in 2D scalar fields (images) in presence of noise are also reported which well-illustrate the potential of the proposed concepts and methods. The characterization of the eigenmodes of successive convolutions are also addressed, with interesting results substantiating the enhanced potential of the coincidence index for accurate and stable similarity quantification.