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**Level set methods and dynamic implicit surfaces.**
*(English)*
Zbl 1026.76001

Applied Mathematical Sciences. 153. New York, NY: Springer. xiv, 273 p. (2003).

This book is an introduction to level set methods and dynamic implicit surfaces. These are powerful techniques for analyzing and computing moving fronts in a variety of different settings. Together with many examples, the book also gives a complete description of numerical aspects which will enable users to quickly apply these techniques to real problems.

The book begins with a presentation of implicit surfaces and their basic properties, and then describes the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus. Topics include the level set equation, Hamilton-Jacobi equation, motion of a surface normal, reinitialization to a signed distance function, extrapolation in normal direction, the particle level set method, and the motion of codimension two (and higher) objects. Part III is concerned with image processing and computer visualization. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from non-organized data points.

The final part IV introduces the reader to computational physics. It begins with one-phase compressible fluid dynamics, then presents two-phase compressible flows with possibly different equations of state, discusses detonation and deflagration waves, and fluid-structure interaction. The incompressible fluid dynamics is presented by a computer visualization of smoke; a description of free surface flows includes computer graphics of water waves. Additional related topics include fully two-phase incompressible flows, and incompressible flames with computer applications to the coupling of compressible and incompressible fluids. Finally, heat fluxes and Stefan problem are discussed.

A student or researcher working in mathematics, computer graphics, science or engineering and interested in dynamically moving fronts which might change their topology or develop singularities, will find this book interesting and useful.

The book begins with a presentation of implicit surfaces and their basic properties, and then describes the level set geometry and calculus toolbox, including the construction of signed distance functions. Part II adds dynamics to this static calculus. Topics include the level set equation, Hamilton-Jacobi equation, motion of a surface normal, reinitialization to a signed distance function, extrapolation in normal direction, the particle level set method, and the motion of codimension two (and higher) objects. Part III is concerned with image processing and computer visualization. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from non-organized data points.

The final part IV introduces the reader to computational physics. It begins with one-phase compressible fluid dynamics, then presents two-phase compressible flows with possibly different equations of state, discusses detonation and deflagration waves, and fluid-structure interaction. The incompressible fluid dynamics is presented by a computer visualization of smoke; a description of free surface flows includes computer graphics of water waves. Additional related topics include fully two-phase incompressible flows, and incompressible flames with computer applications to the coupling of compressible and incompressible fluids. Finally, heat fluxes and Stefan problem are discussed.

A student or researcher working in mathematics, computer graphics, science or engineering and interested in dynamically moving fronts which might change their topology or develop singularities, will find this book interesting and useful.

Reviewer: V.M.Soundalgekar (Thane)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |

68U10 | Computing methodologies for image processing |

68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |

76M27 | Visualization algorithms applied to problems in fluid mechanics |