Ortho-Unitary Transforms, Wavelets and Splines

Abstract

Here we present a new theoretical framework for multidimensional image processing using hypercomplex commutative algebras that codes color, multicolor and hypercolor. In this paper a family of discrete color–valued and multicolor–valued 2–D Fourier–like, wavelet–like transforms and splines has been presented (in the context of hypercomplex analysis). These transforms can be used in color, multicolor, and hyperspectral image processing. In our approach, each multichannel pixel is considered not as an K–D vector, but as an K–D hypercomplex number, where K is the number of different optical channels. Orthounitary transforms and splines are specific combination (Centaurus) of orthogonal and unitary transforms. We present several examples of possible Centuaruses (ortho–unitary transforms): Fourier+Walsh, Complex Walsh+Ordinary Walsh and so on. We collect basis functions of these transforms in the form of iconostas. These transforms are applicable to multichannel images with several components and are different from the classical Fourier transform in that they mix the channel components of the image. New multichannel transforms and splines generalize real–valued and complex–valued ones. They can be used for multichannel images compression, interpolation and edge detection from the point of view of hypercomplex commutative algebras. The main goal of the work is to show that hypercomplex algebras can be used to solve problems of multichannel (color, multicolor, and hyperspectral) image processing in a natural and effective manner.