viii Preface
of moving interfaces plays a key role in the problem to be solved. A search of
“level set methods” on the Google website (which gave over 2,700 responses
as of May 2002) will give an interested reader some idea of the scope and
utility of the method. In addition, some exciting advances in the technology
have been made since we began writing this book. We hope to cover many of
these topics in a future edition. In the meantime you can find some exciting
animations and moving images as well as links to more relevant research pa-
pers via our personal web sites: http://graphics.stanford.edu/~fedkiw
and http://www.math.ucla.edu/~sjo/.
Acknowledgments
Many people have helped us in this effort. We thank the following col-
leagues in particular: Steve Marschner, Paul Romburg, Gary Hewer, and
Steve Ruuth for proofreading parts of the manuscript, Peter Smereka and
Li-Tien Cheng for providing figures for the chapter on Codimension-Two
Objects, Myungjoo Kang for providing figures for the chapter on Motion
Involving Mean Curvature and Motion in the Normal Direction, Antonio
Marquina and Frederic Gibou for help with the chapter on Image Restora-
tion, Hong-Kai Zhao for help with chapter 13, Reconstruction of Surfaces
from Unorganized Data Points, and Luminita Vese for help with the chap-
ter on Snakes, Active Contours, and Segmentation. We particularly thank
Barry Merriman for his extremely valuable collaboration on much of the
research described here. Of course we have benefitted immensely from col-
laborations and discussions with far too many people to mention. We hope
these colleagues and friends forgive us for omitting their names.
We would like to thank the following agencies for their support during
this period: ONR, AFOSR, NSF, ARO, and DARPA. We are particularly
grateful to Dr. Wen Masters of ONR for suggesting and believing in this
project and for all of her encouragement during some of the more difficult
times.
Finally, we thank our families and friends for putting up with us during
this exciting, but stressful period.
Los Angeles, California Stanley Osher
Stanford, California Ronald Fedkiw
Preface
Scope, Aims, and Audiences
This book, Level Set Methods and Dynamic Implicit Surfaces is designed
to serve two purposes:
Parts I and II introduce the reader to implicit surfaces and level set
methods. We have used these chapters to teach introductory courses on the
material to students with little more than a fundamental math background.
No prior knowledge of partial differential equations or numerical analysis
is required. These first eight chapters include enough detailed information
to allow students to create working level set codes from scratch.
Parts III and IV of this book are based on a series of papers published
by us and our colleagues. For the sake of brevity, a few details have been
occasionally omitted. These chapters do include thorough explanations and
enough of the significant details along with the appropriate references to
allow the reader to get a firm grasp on the material.
This book is an introduction to the subject. We have given examples of
the utility of the method to a diverse (but by no means complete) collection
of application areas. We have also tried to give complete numerical recipes
and a self-contained course in the appropriate numerical analysis. We be-
lieve that this book will enable users to apply the techniques presented here
to real problems.
The level set method has been used in a rapidly growing number of areas,
far too many to be represented here. These include epitaxial growth, opti-
mal design, CAD, MEMS, optimal control, and others where the simulation
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