Level Set Methods and Dynamic Implicit Surfaces

Stanley Osher, Ronald Fedkiw (auth.)9780387227467, 9780387954820, 0387954821

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. While the book gives many examples of the usefulness of the methods for a diverse set of applications, it also gives complete numerical analysis and recipes, which will enable users to quickly apply the techniques to real problems.

The book begins with the description of implicit surfaces and their basic properties, and then devises 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 itself, Hamilton-Jacobi equations, motion of a surface normal to itself, reinitialization to a signed distance function, extrapolation in the normal direction, the particle level set method, and the motion of codimension two (and higher) objects. Part III is concerned with topics taken from the field of image processing and computer vision. These include the restoration of images degraded by noise and blur, image segmentation with active contours (snakes), and reconstruction of surfaces from unorganized data points. Part IV is dedicated to computational physics. It begins with one-phase compressible fluid dynamics, then two-phase compressible flow involving possibly different equations of state, detonation and deflagration waves, and solid/fluid structure interaction. Next it discusses incompressible fluid dynamics, including a computer graphics simulation of smoke; free surface flows, including a computer graphics simulation of water; and fully two-phase incompressible flow. Additional related topics include incompressible flames with applications to computer graphics and coupling a compressible and incompressible fluid. Finally, heat flow and Stefan problems are discussed.

A student or researcher working in mathematics, computer graphics, science, or engineering interested in any dynamic moving front, which might change it’s topology or develop singularities, will find this book interesting and useful.

Table of contents :
Front Matter….Pages i-xiii
Front Matter….Pages 1-1
Implicit Functions….Pages 3-16
Signed Distance Functions….Pages 17-22
Front Matter….Pages 23-23
Motion in an Externally Generated Velocity Field….Pages 25-39
Motion Involving Mean Curvature….Pages 41-46
Hamilton-Jacobi Equations….Pages 47-54
Motion in the Normal Direction….Pages 55-61
Constructing Signed Distance Functions….Pages 63-74
Extrapolation in the Normal Direction….Pages 75-77
Particle Level Set Method….Pages 79-86
Codimension-Two Objects….Pages 87-93
Front Matter….Pages 95-95
Image Restoration….Pages 97-118
Snakes, Active Contours, and Segmentation….Pages 119-138
Reconstruction of Surfaces from Unorganized Data Points….Pages 139-146
Front Matter….Pages 147-148
Hyperbolic Conservation Laws and Compressible Flow….Pages 149-166
Two-Phase Compressible Flow….Pages 167-188
Shocks, Detonations, and Deflagrations….Pages 189-200
Solid-Fluid Coupling….Pages 201-207
Incompressible Flow….Pages 209-215
Free Surfaces….Pages 217-222
Liquid-Gas Interactions….Pages 223-226
Front Matter….Pages 147-148
Two-Phase Incompressible Flow….Pages 227-238
Low-Speed Flames….Pages 239-247
Heat Flow….Pages 249-258
Back Matter….Pages 259-275