Data-Specific Anisotropic Mexican Hat Wavelets for Structure-Preserving Image Processing

Abstract

This paper proposes a novel approach for structure-sensitive image processing based on the rigorous mathematical derivation of data-specific anisotropic Mexican hat wavelets (DAM). Our DAM is derived from the negative first-order derivative of the fundamental solution of heat diffusion equation with respect to time, which not only shares similar properties with Mexican hat wavelet but also intrinsically embeds the image-specific properties. Through the scale-aware DAM transform and its inverse transform, we are capable of conducting structure-sensitive image processing. Our key idea is to represent the images as undirected graphs, whose edge weights are governed by the normalized intensity/color differences within the local neighboring pixel window. Based on the rigorous theory of global graph Laplacian and heat diffusion, our original DAM can also encode the local/global structure of images. We employ the Krylov subspace technique to reduce the computational cost of our DAM transform. Furthermore, aiming at various structure-preserving image processing applications such as filtering, detail enhancement, tone manipulation, and stylization, we conduct comprehensive experiments and make quantitative comparisons with other state-of-the-art methods, which demonstrate the versatility and superiority of our method.