Definition of a discrete color monogenic wavelet transform

Abstract

In this chapter, we propose to review different approaches for the introduction of a color monogenic wavelet transform. Monogenic wavelets offer a geometric representation of grayscale images through an AM/FM model allowing invariance of coefficients to translations and rotations. The underlying concept of a local phase includes a fine contour analysis into a coherent unified framework. Wavelet based color image processing schemes have mostly been made by using a grayscale tool separately on color channels. In this chapter, we propose to discuss definitions that consider a color (vector) image right at the beginning of the mathematical definition. After a general description of the background of monogenic concept, we review a first approach built from the grayscale monogenic wavelets together with a color extension of the monogenic signal based on geometric algebra. Then, starting from a link with structure tensors, we discuss an alternative nontrivial extension of the monogenic framework to vector-valued signals. The crucial point is that our color monogenic wavelet transform is non-marginal and it inherits the coherent geometric analysis from the monogenic framework. Finally, we address the numerical aspect by introducing an innovative scheme that uses a discrete Radon transform based on discrete geometry.