Image Moments and Moment Invariants

Abstract

Moments are widely used to describe mathematical functions due to their numerical simplicity and numerous physical interpretations. The concept has been used for many years in diverse areas such as mechanics and statistics and has been successfully applied to pattern recognition, registration, object matching, data compression, and image understanding. In statistics, moments have been used to describe the shape characteristics of probability density or mass functions. The idea that moments could be used to characterize 2D image functions is attributed to Hu [7], who defined the 2D geometric moment and developed 2D geometric moment invariants for pattern recognition. Since then, many types of moments and their invariants have been proposed. The introduction of orthogonal moments by Teague [15], which was based on the theory of orthogonal polynomials, revolutionized visual pattern recognition and led to the development of efficient algorithms for shape analysis. The success of orthogonal moments, which are obtained from the projection of the 2D image function onto the higher order orthogonal polynomials, is due to their ability to capture higher order nonlinear structures of image functions. Perhaps, the greatest advantage of geometric and orthogonal moments is that they may be used to derive moment invariants. These invariants are computed by subjecting the original moments to certain affine transformations that possess invariance to scaling, shifting, and rotation of a pattern. This robustness to geometric distortions makes moment invariants appealing as features for pattern recognition. Starting from the definition of geometric moments, this chapter proceeds with generalized definition of orthogonal moments, details of some popular orthogonal moments, and formulation of their geometric invariants.