Algebra and geometry of multichannel images. Part 2. Ortho-unitary transforms, wavelets and splines

Abstract

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 that 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 Centaurus (specific combination) of orthogonal and unitary transforms. It is known that Centaurus is a combination of half–man and half–horse. By this reason we can called an ortho–unitary (color) transform as a Centaurus of orthogonal and unitary transforms. We present several examples of possible Centuaruses: Fourier+Walsh, Complex Walsh+Ordinary Walsh and so on. We collect basis functions of these transforms in the form of iconostas (in a Russian orthodox church, the ”Iconostas” is literally the ”Stand of Icons” that rise up at the front of the Sanctuary). 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. They can be used for multichannel images compression, interpolation and edge detection from the point of view of hypercomplex commutative algebras.