notes on differential geometry and lie groups [lecture notes]

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1、NotesonDierentialGeometryandLieGroupsJeanGallierandJocelynQuaintanceDepartmentofComputerandInformationScienceUniversityofPennsylvaniaPhiladelphia,PA19104,USAe-mail:jean@cis.upenn.educJeanGallierPlease,donotreproducewithoutpermissionoftheauthorsJanuary13,20162TomydaughterMia,mywi

2、feAnne,mysonPhilippe,andmydaughterSylvie.PrefaceThemotivationsforwritingthesenotesarosewhileIwascoteachingaseminaronSpecialTopicsinMachinePerceptionwithKostasDaniilidisintheSpringof2004.IntheSpringof2005,IgaveaversionofmycourseAdvancedGeometricMethodsinComputerScience(CIS610),wit

3、hthemaingoalofdiscussingstatisticsondiusiontensorsandshapestatisticsinmedicalimaging.ThisiswhenIrealizedthatitwasnecessarytocoversomematerialonRiemanniangeometrybutIranoutoftimeafterpresentingLiegroupsandnevergotaroundtodoingit!Then,intheFallof2006Iwentonawonderfulandveryproduct

4、ivesabbaticalyearinNicholasAyache'sgroup(ACSEPIOS)atINRIASophiaAntipolis,whereIlearnedaboutthebeautifulandexcitingworkofVincentArsigny,OlivierClatz,HerveDelingette,PierreFillard,GregoireMalandin,XavierPennec,MaximeSermesant,and,ofcourse,NicholasAyache,onstatisticsonmanifoldsand

5、Liegroupsappliedtomedicalimag-ing.Thisinspiredmetowritechaptersondierentialgeometry,andafterafewadditionsmadeduringFall2007andSpring2008,notablyonleft-invariantmetricsonLiegroups,mylittlesetofnotesfrom2004hadgrownintothemanuscriptfoundhere.LetmegobacktotheseminaronSpecialTopicsi

6、nMachinePerceptiongivenin2004.Themainthemeoftheseminarwasgroup-theoreticalmethodsinvisualperception.Inparticular,KostasdecidedtopresentsomeexcitingresultsfromChristopherGeyer'sPh.D.thesis[75]onscenereconstructionusingtwoparaboliccatadioptriccameras(Chapters4and5).Catadioptriccame

7、rasaredeviceswhichusebothmirrors(catioptricelements)andlenses(dioptricelements)toformimages.Catadioptriccamerashavebeenusedincomputervisionandroboticstoobtainawideeldofview,oftengreaterthan180,unobtainablefromperspectivecameras.Applicationsofsuchdevicesincludenavigation,surveil

8、lanceandvizualization,amongothers.Techni