BOOKS - Group Filters and Image Processing

Myoung An, Richard Tolimieri.
• 16 chapters, 314 pages with index. View table of contents.
• One edition available from Psypher Press on August 27, '03.
By combining digital signal processing with group harmonic analysis, the authors provide a framework for extending the collection of translations. The specific translations defined in this work through examples and illustrations are sufficiently broad to include many important applications in signal/image processing, biology and physics.
Starting with a complete treatment of abelian group harmonic analysis, this text develops a unifying conceptual basis for both abelian and nonabelian group harmonic analysis. Throughout the text, the common ground in various cases is presented in a common language and format, and emphasizes more clearly and distinctly the differences when they occur. Based on the DSP developed relative to groups, group filters are defined and compared. The main result is that in contrast to abelian group filters, nonabelian group filters localize in the image domain. A detailed account is given to show that expansions over nonabelian group harmonic analysis can be viewed as combined, local image-spectral image domain expansions analogous to time-frequency expansions.
A chapter is devoted to illustrating the use of the localization property for detecting and localizing the position of geometric objects in image data sets by straightforward matched-filtering operation.
The results in this work stand alone in the sense that they are self-contained but it will be an invaluable addition to the larger field of digital signal and image processing.
The translation-invariance of most classical signal processing transforms and filtering operations is largely responsible for their widespread use. Translation-invariance is crucial for interpreting the results of filtering operations and for efficiently implementing these operations. The usual approach in DSP is to define a transform and then show that it is shift or translation invariant. In this work, the translations are viewed as the primary source of classical DSP operations including convolution and Fourier representation. Once this primacy is established, it is reasonable to ask if there are other collections of index set mappings yielding a new DSP and if this new DSP has applications. This text shows that this is the case.