Scale-Space Image Analysis Based on Hermite Polynomials Theory

Abstract

The Hermite transform allows to locally approximate an image by a linear combination of polynomials. For a given scale σ and position ξ, the polynomial coefficients are closely related to the differential jet (set of partial derivatives of the blurred image) for the same scale and position. By making use of a classical formula due to Mehler (late 19^th century), we establish a linear relationship linking the differential jets at two different scales σ and positions ξ involving Hermite polynomials. Pattern registration and matching applications are suggested.

We introduce a Gaussian windowed correlation function K(v) for locally matching two images. When taking the mutual translation parameter v as independent variable, we express the Hermite coefficients of K(v) in terms of the Hermite coefficients of the two images being matched. This new result bears similarity with the Wiener-Khinchin theorem which links the Fourier transform of the conventional (flat-windowed) correlation function with the Fourier spectra of the images being correlated. Compared to the conventional correlation function, ours is more suited for matching localized image features.

The mathematical tools we propose are shown to have attractive computational features. Numerical simulations using synthetic 1D and 2D test patterns demonstrate the advantages of our proposals for signal and image matching in terms of accuracy and low algorithm complexity.