Geometric Partial Differential Equations and Image Analysis
Guillermo SapiroCambridge University Press2001xxv +385 pp.ISBN 0-521-79075-1hardback, £40.00
As its title suggests, this introduces a rather formidable body of mathematical theory having valuable applications in image processing and computer vision. The first chapter of 70 pages is devoted to providing basic mathematical background, and it begins by recommending nine books that should be on the shelves of anyone involved in this topic. Everything is presented in a lucid style, but it is clear that the topic has numerous ramifications,indicated by the many references to the bibliography (of 430 items) for elaboration of particular aspects.
Despite this, in the first part of the chapter the treatment is self-sufficient and accessible without enormous demands on the reader's mathematical background, but when it comes to a section on "Differential Invariants and Lie Group Theory" the author acknowledges that familiarity with concepts of "manifold" and "smooth function" is necessary, and that in this section results are stated without proofs (but with references to where proofs can be found). The chapter includes an overview of numerical computational techniques, and the orientation is firmly towards producing practical algorithms.
The methods described depend on the "evolution" of geometric curves(in two dimensions) and surfaces (in three dimensions). The curves or surfaces may be "level sets" of higher-dimensional surfaces, and results obtained by J.A. Sethian are included. (Sethian's book, Level Set Methods and Fast Marching Methods, was reviewed in Kybernetes, volume 29, no. 2, pp. 242-244. )An impressive example of the use of such a method is illustrated on page 93,where the starting data is the 3-D coordinates of unconnected points on the surfaces of two interlinked tori, and the method allows a surface to collapse from being an enveloping sphere to one con. forming to the surfaces of the tori. The shapes of 3-D objects may be smoothed according to either volume-preserving or surface- preserving flows.
Methods are described for segmentation of images. An algorithm is described that will maintain a segmentation, originally introduced manually, while the segregated parts move and change shape. An example is the following of two hands in motion, but a perhaps more serious application is to the tracking of structures through successive microscope images of slices of biological tissue.
There are many references to biomedical image processing, and one application is mentioned where a particular object (the image of a neuron) is separated out from a complicated messy image. Other methods are known for this, but the one described allows the user to guide the process by indicating some points on the outline of the required object. This is a useful feature that can obviate the reporting of irrelevant components of the image.
Various ways of treating photographic images are described, some of them describable as "denoising". A way of converting coloured pictures to monochrome versions is discussed, with setting of brightness levels to indicate the implicit 3-D structure much better than could be done by straight colour conversion. In the final chapter on "Additional Theories and Applications",methods are described for removing superimposed text, or unwanted objects, from photographs, with the image completed as though the text or object had never been there. An example is shown of someone diving from a high building, where the original picture shows that his legs are securely bound and fastened to a bungee, but in the "doctored" version no trace of the binding or bungee can be seen. The derivation of shape from shading is also treated.
All of these intriguing and valuable possibilities, and more, stem from the mathematical theory mentioned earlier and this book will certainly be accepted as a standard text. It is written with course teaching in mind, and there are exercises for the student following each chapter. A student who had worked through everything here would be very well equipped to develop powerful image processing software.
Alex M. Andrew
