...............................Chapter 6, ``Applications,'' is the author's attempt to satisfy his second goal. From the lengthy development of the mathematical foundations and the title of the book, the reader would expect a wide range of specific applications, possibly providing new insights and methodologies. In my opinion, Lenz fails to effectively demonstrate the importance of group theoretical methods in image processing. In fact, little space is devoted to actual image processing applications. Some theoretical discussions give applications to feature extraction, filter design, and regularities in signal space. Other sections in this final chapter deal with the heat equation, eigenvectors of ``intertwining'' operators, ergodic processes, and neural networks. As for image science, three pages are devoted to image restoration and tomography, two pages to the relation between filtered images and the original input images, one page to image coding, and roughly seven pages to motion analysis. No new techniques or novel insights are demonstrated. With all the mathematical machinery developed by the author, one would have hoped for more.
For those familiar with applications of group theoretical methods in computer vision, the title of this book brings to mind Kanatani's excellent work [2]. It is difficult not to compare these two books, as Kanatani succeeds so well where Lenz fails. Each chapter of Kanatani's book deals with a subject in image science, while the fundamentals of group theory, covering roughly the same material as Lenz's first four chapters, are tucked away in the appendix. For readers interested in applications of group theoretical methods in image processing, I would have to recommend Kanatani's work.