Symmetry and its Applications in Multidimensional Signal Processing

Symmetry is an important aspect of nature. It has been widely studied and used to simplify the analysis and design of physical systems as well as to add beauty and balance to them. This concept has been applied to abstract entities, as well as in the fields of quantum mechanics and crystallography. One- dimensional systems can only have a limited number of symmetries. Two-and higher dimensional systems may exhibit many types of symmetry. These may be used to advantage in reducing the complexity of design and implementation of such systems. In this talk, we will be concerned with various types of symmetries that may be present in two- or three- dimensional filters. Since the frequency and impulse response functions of these filters are related, symmetry in one function induces certain form of symmetry in the other. We will discuss these interdependencies and explore their applications. In particular, we will discuss the design of 2-D FIR and IIR filters by employing various types of symmetries in the frequency response of these filters. The usefulness of symmetry in reducing the complexity in 2-D FFT algorithm will also be considered. In addition, the question of stability of two- and higher dimensional systems, both with and without nonessential singularities of the second kind, will be examined.