5R5. Level Set Methods and Dynamic Implicit Surfaces.Applied Mathematical Sciences, Volume 153. - S Osher (Dept of Math, UCLA) and R Fedkiw (Dept of Comput Sci, Stanford). Springer-Verlag, New York. 2003. 273 pp. ISBN 0-387-95482-1. \$79.95.

Reviewed by K Piechor (Inst of Fund Tech Res, Polish Acad of Sci, ul Swietokrzyska 21, Warsaw, 00-049, Poland).

Many problems of applied sciences can be reduced to determination of “moving fronts” described implicitly by an equation of the type $Φt,x=constant,$ where $t$ is the time, and x is the position of the front. The reviewed book is a sort of “extended” introduction to the core or spirit of these methods and, mostly, to numeric techniques related to them.

The contents of the book can be divided into two parts: Part I comprises Chapters I and II entitled Implicit Functions and Level Set Methods, respectively, and Part II comprising the rest of the book where numerous and diverse applications of the methods to image processing and computational physics are given. Chapter I presents a very elementary mathematical background of the theory of implicit functions. In Chapter II the most important mathematical ideas and numerical techniques of their implementation are presented. They include, among others, Hamilton–Jacobi equations and their numerical treatment, motion in the normal direction, construction of the signed distance function etc.

Some applications, with suitable adaptations, of the level set methods begin in Chapter III. Chapter III itself is devoted to image processing and computer vision problems. The contents of this chapter comprise image restoration, active contour methods, and a sort of multidimensional interpolation known as the reconstruction of surfaces from unorganised data points. The authors focused their attention on mathematical modeling of these problems rather than on the numerical procedures. These are merely mentioned, but numerous graphs and pictures illustrating the power and efficiency of the used methods are given (some of them in full color).

Chapter IV, Computational Physics, is the most interesting from the fluid-dynamicist’s point of view. It covers hyperbolic conservation laws and compressible flows, two-phase compressible flow, shock waves and also detonation and deflagration waves, solid-fluid coupling and many other topics. Every section begins with the presentation of the mathematical model, its specific mathematical features, an explanation on how to apply the level set methods to such a problem, and a thorough discussion of used numeric procedures, paying much attention to their advantages and disadvantages. For example, the peculiarity of solid-fluid coupling consists in different ways of mathematical description of the two media: for solids the Lagrangian description is more convenient, whereas for fluids the Eulerian coordinates are used. How to find a common and an efficient numerical procedure for treating the solid-fluid interface? The answer to this and similar questions according to the modern state of art can be found in the reviewed book. Another application of the method discussed in the book, concerns problems, which usually fly away from the eyes of the fluid mechanics researchers. These problems relate to simulation of flows for computer graphics.

In summary, Level Set Methods and Dynamic Implicit Surfaces, is something between a textbook and a book of reference. Even a beginner in numerics can read it, since every chapter and section starts from basic explanations and definitions, followed by a presentation of the numerical procedures, which are accompanied by precious remarks and comments of people with experience. On the other hand, none of the procedures are set with all details and the necessary rigour. The reader is referred to the original papers, so in this sense it can serve as a valuable book of references. In all aspects, this book is worth being in the library of every student or researcher working in applied mathematics, physics or engineering, but it needs a rather good mathematical preparation.