ABSTRACT
In this chapter, we study another type of representation of multidimensional signals or images, that associates with special partitions that reveal the discrete transforms such as the Fourier, Hadamard, and cosine transforms. We are reminded that the tensor representation of images has been associated with irreducible coverings not being partitions of the discrete lattice of the image domain. For these purposes, a concept of paired representation of images with respect to discrete unitary trans forms is introduced, and a class of the transforms that provide the paired repre sentation is considered. This class of transforms is intersected with the class of transforms providing the tensor representation. But unlike the tensor representa tion of images, the paired representation allows us to distribute the information about the image, as well as the spectral information, by disjoint sets of samples. This peculiarity makes the paired representation the most attractive and, as will be shown in following chapters, it makes possible the construction of effective matrix representations and fast algorithms for computing many multidimensional trans forms. As examples, we consider the paired representations of images with respect to the discrete unitary Fourier, Hadamard, Hartley, cosine, and sine transforms with fundamental periods being rectangular lattices. We also describe in detail the paired representation of images defined on the hexagonal lattices. Such represen tation is considered with respect to the 2-D discrete hexagonal Fourier transforms, but it can be used for other transforms, too.
