American Institute of Physics

# Book Review

## Numerical Methods for Image Registration

Jan Modersitzki
Oxford University Press, New York, 2004 199 pp., \$95.00 hb ISBN 0-19-852841-8

Reviewed by Gurusham Sudhir

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The domain of computer vision and image processing is quite an interesting one—while it is relatively easy for evolved species, such as humans, to perform tasks like object recognition, object matching, or object registration, it can be extremely difficult to do the same using the most modern computers. The primary reason for this difficulty is the complexity faced in defining “the problem” sufficiently to look for a solution. Mathematically, one of the main reasons is the fact that many of the problems in this domain can be classified as illposed. An ill-posed problem is one that does not meet the following three conditions: (1) there exists a solution, (2) the solution is unique, and (3) the solution depends continuously on the input data. Solving such ill-posed problems invariably requires a procedure called “regularization,” where the mathematical formulation of the problem contains an extra term—a regularizer—to make the problem a wellposed one.

In Numerical Methods for Image Registration, author Jan Modersitzki presents a well-focused text on the solutions to a particular problem in the field of computer vision and image processing—image registration. This specific problem deals with techniques for computation of the mapping (transformation) of an object as observed in a template image to the same object as observed in a reference image. Note that even though both the template and reference images contain observations of the same object, the object may have undergone some physical displacement/motion/deformation between the observations for many reasons, depending on the context of the application in which the two observations are made. Modersitzki addresses this problem systematically from a mathematical point of view, classifying the solution approaches into two categories: parametric and nonparametric. The first category addresses solution techniques that express the desired transformation (also called registration) in terms of a finite number of basis functions (so that the problem is typically well-posed). The author presents a general theory covering three state-of-the-art techniques for parametric registration (landmark-based registration, principal axes-based registration, and optimal linear registration, using different intensity- based distance measures) and discusses the relative advantages and disadvantages of these techniques.

In the second (and more challenging) category, Modersitzki presents a unified approach, based on his own prior research, to illustrate how nonparametric registration techniques can be systematically designed using mathematical formulations based on variational techniques. Central to this unified approach is the characterization of the desired transformation (solution) to be the minimizer of a certain functional that embodies a similarity measure (for registration) and a smoothness measure (for regularization). Based on this approach, he illustrates four different techniques—elastic registration, fluid registration, diffusion registration, and curvature registration—and provides a generic numerical treatment for the computation of the solutions in a fast, efficient, and stable manner.

Particularly, Modersitzki shows that the diffusion registration formulation yields an O(N) implementation, whereas all other nonparametric techniques yield an O(N log N) implementation. For all the techniques and throughout the book, Modersitzki illustrates the ideas clearly and presents the results of the experiments using a standard set of two different images of human hands. He conveys essential information in a succinct yet sufficient manner. He provides proofs for most of the pertinent mathematical results and gives appropriate references for the rest. Even though Modersitzki addresses image registration as the central problem in the book, many of the ideas and techniques are relevant to other areas in the same field. I certainly recommend the book for researchers as well as mathematically inclined practitioners in the field of image processing and computer vision.

Gurusham Sudhir is a senior vision engineer at Adept Technology in City of Industry, California. He works in the field of machine vision and image processing.