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Level Set Methods and Dynamic Implicit Surfaces Stanley Osher Ronald Fedkiw SPRINGER Applied Mathematical Sciences Volume 153 Editors S.S.Antman J.E.Marsden L.Sirovich Advisors J.K.Hale P.Holmes J.Keener J.Keller B.J.Matkowksy A.Mielke C.S.Peskin K.R.Sreenivasan Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Stanley Osher Ronald Fedkiw Level Set Methods and Dynamic Implicit Surfaces With 99 Figures, Including 24 in Full Color Stanley Osher Ronald Fedkiw Department of Mathematics Department of Computer Science University of California Stanford University at Los Angeles Stanford, CA 94305-9020 Los Angeles, CA 90095-1555 USA USA [email protected] [email protected] Editors: S.S.Antman J.E.Marsden L.Sirovich DepartmentofMathematics ControlandDynamical DivisionofApplied and Systems,107-81 Mathematics InstituteforPhysicalScience CaliforniaInstituteof BrownUniversity andTechnology Technology Providence,RI02912 UniversityofMaryland Pasadena,CA91125 USA CollegePark,MD20742-4015 USA [email protected] USA [email protected] [email protected] Cover photos: Top left and right, hand and rat brain — Duc Nguyen and Hong-Kai Zhao. Center campfire — Duc Nguyen and Nick Rasmussen and Industrial Light and Magic.Lowerleftandcenter,waterglasses—SteveMarschnerandDougEnright. MathematicsSubjectClassification(2000):65Mxx,65C20,65D17,65-02,65V10,73V LibraryofCongressCataloging-in-PublicationData Osher,Stanley. Levelsetmethodsanddynamicimplicitsurfaces/StanleyOsher,RonaldFedkiw p.cm.–(Appliedmathematicalsciences;153) Includesbibliographicalreferencesandindex. ISBN0-387-95482-1(alk.paper) 1.Levelsetmethods. 2.Implicitfunctions. I.Fedkiw,RonaldP.,1968– II.Title. III.Appliedmathematicalsciences(Springer-VerlagNewYorkInc.);v.153 QA1.A647vol.153 [QC173.4] 510s–dc21 [515’.8] 2002020939 ISBN0-387-95482-1 Printedonacid-freepaper. (cid:1)c 2003Springer-VerlagNewYork,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout thewrittenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAv- enue,NewYork,NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms, eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionas towhetherornottheyaresubjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 SPIN10874508 www.springer-ny.com Springer-Verlag NewYork Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Dedicated with love to Katy, Brittany, and Bobbie This page intentionally left blank 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.Wehaveusedthesechapterstoteachintroductorycoursesonthe materialtostudentswithlittlemorethanafundamentalmathbackground. 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 occasionallyomitted.Thesechaptersdoincludethoroughexplanationsand 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 theutilityofthemethodtoadiverse(butbynomeanscomplete)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- lievethatthisbookwillenableuserstoapplythetechniquespresentedhere to real problems. Thelevelsetmethodhasbeenusedinarapidlygrowingnumberofareas, far too many to be represented here. These include epitaxial growth, opti- maldesign,CAD,MEMS,optimalcontrol,andotherswherethesimulation viii Preface ofmovinginterfacesplaysakeyroleintheproblemtobesolved.Asearchof “levelsetmethods”ontheGooglewebsite(whichgaveover2,700responses as of May 2002) will give an interested reader some idea of the scope and utilityofthemethod.Inaddition,someexcitingadvancesinthetechnology havebeenmadesincewebeganwritingthisbook.Wehopetocovermanyof thesetopicsinafutureedition.Inthemeantimeyoucanfindsomeexciting animationsandmovingimagesaswellaslinkstomorerelevantresearchpa- persviaourpersonalwebsites: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 colleagues 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 chapter 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- laborationsanddiscussionswithfartoomanypeopletomention.Wehope 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 Contents Preface vii Color Insert (facing page 146) I Implicit Surfaces 1 1 Implicit Functions 3 1.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Geometry Toolbox . . . . . . . . . . . . . . . . . . . . . 8 1.5 Calculus Toolbox . . . . . . . . . . . . . . . . . . . . . . 13 2 Signed Distance Functions 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Distance Functions . . . . . . . . . . . . . . . . . . . . . 17 2.3 Signed Distance Functions . . . . . . . . . . . . . . . . . 18 2.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Geometry and Calculus Toolboxes . . . . . . . . . . . . . 21 II Level Set Methods 23 3 Motion in an Externally Generated Velocity Field 25 3.1 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . 25