Itgivesmegreatpleasuretoeditthisbook. Thegenesisofthisbookgoes backtotheconferenceheldattheUniversityofBolognainJune1999,on collaborativeworkbetweentheUniversityofCaliforniaatBerkeleyandthe UniversityofBologna. Theoriginalideawastoinvitesomespeakersatthe conferencetosubmitarticlestothebook. Thescopeofthebookwaslater- hancedand,inthepresentform,itisacompilationofsomeoftherecentwork usinggeometricpartialdi?erentialequationsandthelevelsetmethodology inmedicalandbiomedicalimageanalysis. Thesynopsisofthebookisasfollows:Inthe?rstchapter,R. Malladi andJ. A. Sethianpointtotheoriginsoftheuseoflevelsetmethodsand geometricPDEsforsegmentation,andpresentfastmethodsforshapes- mentationinbothmedicalandbiomedicalimageapplications. InChapter 2,C. OrtizdeSolorzano,R. Malladi,andS. J. Lockettdescribeabodyof workthatwasdoneoverthepastcoupleofyearsattheLawrenceBerkeley NationalLaboratoryonapplicationsoflevelsetmethodsinthestudyand understandingofconfocalmicroscopeimagery. TheworkinChapter3byA. Sarti,C. Lamberti,andR. Malladiaddressestheproblemofunderstanding di?culttimevaryingechocardiographicimagery. Thisworkpresentsvarious levelsetmodelsthataredesignedto?tavarietyofimagingsituations,i. e. timevarying2D,3D,andtimevarying3D. InChapter4,L. VeseandT. F. Chanpresentasegmentationmodelwithoutedgesandalsoshowextensions totheMumford-Shahmodel. Thismodelisparticularlypowerfulincertain applicationswhencomparisonsbetweennormalandabnormalsubjectsis- quired. Next,inChapter5,A. EladandR. Kimmelusethefastmarching methodontriangulateddomaintobuildatechniquetounfoldthecortexand mapitontoasphere. Thistechniqueismotivatedinpartbynewadvances infMRIbasedneuroimaging. InChapter6,T. DeschampsandL. D. Cohen presentaminimalpathbasedmethodofgroupingconnectedcomponentsand showcleverapplicationsinvesseldetectionin3Dmedicaldata. Finally,in Chapter7,A. Sarti,K. Mikula,F. Sgallari,andC. Lamberti,describean- linearmodelfor?lteringtimevarying3Dmedicaldataandshowimpressive resultsinbothultrasoundandechoimages. IoweadebtofgratitudetoClaudioLambertiandAlessandroSartifor invitingmetoBologna,andlogisticalsupportfortheconference. Ithank thecontributingauthorsfortheirenthusiasmand?exibility,theSpringer mathematicseditorMartinPetersforhisoptimismandpatience,andJ. A. Sethianforhisunfailingsupport,goodhumor,andguidancethroughthe years. Berkeley,California R. Malladi October,2001 Contents 1 FastMethodsforShapeExtractioninMedicaland BiomedicalImaging...1 R. Malladi,J. A. Sethian 1. 1Introduction...1 1. 2TheFastMarchingMethod...3 1. 3ShapeRecoveryfromMedicalImages...6 1. 4Results...10 References...13 2 AGeometricModelforImageAnalysisinCytology...19 C. OrtizdeSolorzano,R. Malladi,,S. J. Lockett 2. 1Introduction...19 2. 2GeometricModelforImageAnalysis...20 2. 3SegmentationofNuclei...22 2. 4SegmentationofNucleiandCellsUsingMembrane-RelatedProtein Markers...31 2. 5Conclusions...37 References...38 3 LevelSetModelsforAnalysisof2Dand3D EchocardiographicData...43 A. Sarti,C. Lamberti,R. Malladi 3. 1Introduction...43 3. 2TheGeometricEvolutionEquation...45 3. 3TheShock-TypeFiltering...46 3. 4ShapeExtraction...49 3. 52DEchocardiography...52 3. 62D+timeEchocardiography...53 3. 73DEchocardiography...56 3. 83D+timeEchocardiography...58 3. 9Conclusions...59 References...61 4 ActiveContourandSegmentationModelsusing GeometricPDE'sforMedicalImaging...63 T. F. Chan,L. A. Vese 4. 1Introduction...63 4. 2DescriptionoftheModels...64 4. 3ApplicationstoBio-MedicalImages...68 4. 4ConcludingRemarks...68 References...7 0 VIII Contents 5 SphericalFlatteningoftheCortexSurface...77 A. Elad(Elbaz),R. Kimmel 5. 1Introduction...77 5. 2FastMarchingMethodonTriangulatedDomains...80 5. 3Multi-DimensionalScaling...80 5. 4CortexUnfolding...84 5. 5Conclusions...86 References...86 6 GroupingConnectedComponentsusingMinimalPath Techniques...91 T. Deschamps,L. D. Cohen 6. 1Introduction...91 6. 2MinimalPathsin2Dand3D...93 6. 3FindingContoursfromaSetofConnectedComponentsR...96 k 6. 4FindingaSetofPathsina3DImage...102 6. 5Conclusion...103 References...104 7 NonlinearMultiscaleAnalysisModelsforFilteringof 3D+TimeBiomedicalImages...107 A. Sarti,K. Mikula,F. Sgallari,C.