Table Of ContentLecture Notes in Mathematics 2090
CIME Foundation Subseries
Martin Burger
Andrea C.G. Mennucci
Stanley Osher
Martin Rumpf
Level Set
and PDE Based
Reconstruction
Methods in Imaging
Cetraro, Italy 2008
Editors: Martin Burger, Stanley Osher
Lecture Notes in Mathematics 2090
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,
Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year
foryear,tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
UniversitàdiFirenze UniversitàdiFirenze
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Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’UniversitàedellaRicerca)
Martin Burger Andrea C.G. Mennucci
(cid:2)
Stanley Osher Martin Rumpf
(cid:2)
Level Set and PDE Based
Reconstruction Methods
in Imaging
Cetraro, Italy 2008
Editors: Martin Burger
Stanley Osher
123
MartinBurger AndreaC.G.Mennucci
InstituteforComputational ScuolaNormaleSuperiore
andAppliedMathematics Pisa,Italy
UniversityofMünster
Münster,Germany
StanleyOsher MartinRumpf
DepartmentofMathematics InstituteforNumericalSimulation
UniversityofCalifornia UniversityofBonn
LosAngeles,CA,USA Bonn,Germany
ISBN978-3-319-01711-2 ISBN978-3-319-01712-9(eBook)
DOI10.1007/978-3-319-01712-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013950139
MathematicsSubjectClassification(2010):94A08,35Q94,68U10,65K10,35J20,45Q05,53C44
(cid:2)c SpringerInternationalPublishingSwitzerland2013
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Preface
The present volume collects the lecture notes of courses given at the CIME
SummerSchoolinCetraroinSeptember2008.Theschoolappearedtobeahighly
successful event with around 50 participants from all over the world, enjoying
boththescientificcontentofthecoursesandtheatmosphereandopportunitiesfor
discussionsamongparticipantsandwiththespeakers.
Mathematical imaging and inverse problems are two of the fastest growing
disciplinesinappliedandinterdisciplinarymathematics,withastrongoverlapsince
many image analysis and reconstruction problems are effectively formulated as
inverse problems. A key issue in such problems is to preserve and analyze edges
orshapesintheimage.Thosearetakencarebythemethodscoveredinthesummer
school, which had the particular aim of providing a unified picture to methods
of image reconstruction and analysis, on the one hand, and techniques for shape
reconstruction and analysis, on the other hand. The key steps connecting those
are methods based on level set representations and geometric partial differential
equations, which effectively operate on the level sets (respectively, discontinuity
set)ofanimageaswellasonasingleshape.Thesubjectoflectureswerechosento
covertheseaspectsandprovideanoutlooktootherrelevanttopics:
• Computational methods for nonlinear PDE and L1 techniques: Stanley Osher
(UCLA).
• Totalvariationandrelatedmethods:MartinBurger(Münster).
• Theuseoflevelsetmethodsininverseproblems:OliverDorn(Manchester).
• Variational methods in image matching and motion extraction: Medical and
biologicalapplications:MartinRumpf(Bonn).
• Metrics of curves in shape optimization and analysis: Andrea C.G. Mennucci
(SNSPisa).
In addition, few young participants already involved in research related to the
coursetopicsweregivenanopportunitytopresenttheirworkinshortertalks.
ThelectureseriesbyStanleyOshergaveageneraloverviewtomoderncomputa-
tionalmethodsforimagereconstructionandanalysis,includingBregmaniterations,
v
vi Preface
sparsity-based methods with `1-type functionals, and local and nonlocal total
variationmethodsforimagedenoising.
MartinBurgerfurtherexpandedontotalvariationtechniquesanddiscussedthe
analysisoftotalvariationmethods,furtherdetailsonBregmaniterationsandinverse
scalespacemethods,andalsoapplicationsinbiomedicalimaging.
The lectures by Oliver Dorn gave an overview of level set methods for the
reconstruction of shapes in inverse problems. He discussed basics about level set
representation of shapes and the construction of computational methods to solve
appropriate variational formulations, including the necessary shape calculus. A
major part of the lecture series was devoted to several applications modeled by
inverseproblemsforpartialdifferentialequations.
MartinRumpfmovedfromsingleimagestopairsofimages.Themajortasksthen
consistoffindingdeformationsoftheimagedomainsthatmaketheimageslookas
similar as possible. The correspondingobjectivesare either matchingif interested
inresultingimages,andmotionestimationifinterestedinthedeformation.Exten-
sions of the framework to shapes and their matching respectively averaging were
presentedaswell.
Andrea C.G. Mennucci provided an overview of metrics on curves, linking
this topic originating in differential geometry to very applied problems in image
processing and shape optimization. He showed that by choosing differentmetrics
methodsfortaskssuchasobjectsegmentationinimagescanbeimproved.
Inthelecturenotesmosttopicsofthefirsttwolectureswereunifiedinachapter
providing a comprehensive introduction to the zoo of total variation and related
methods.Morespecializedtopicsrelatedtophotoncountdatafrequentlyarisingin
imagingwereunifiedwiththecontentsofparticipanttalksbyChristophBruneand
AlexSawatzkyinthesecondchapter.ThethirdchaptercoversthecontentsofMartin
Rumpf’slectureseries,thefourthchaptertheonesofAndreaC.G.Mennucci’s.We
hopethatreadersfindthematerialasinterestingaswedoandenjoythebreadthof
coveredtopicsaswellasthedepthofdetailinthesinglechapters.
Münster,Germany MartinBurger
LosAngeles,CA StanleyOsher
Contents
AGuidetotheTVZoo .......................................................... 1
MartinBurgerandStanleyOsher
EM-TVMethodsforInverseProblemswithPoissonNoise ................. 71
Alex Sawatzky, Christoph Brune, Thomas Kösters,
FrankWübbeling,andMartinBurger
VariationalMethodsinImageMatchingandMotionExtraction .......... 143
MartinRumpf
MetricsofCurvesinShapeOptimizationandAnalysis..................... 205
AndreaC.G.Mennucci
vii
A Guide to the TV Zoo
MartinBurgerandStanleyOsher
Abstract Totalvariationmethodsandsimilarapproachesbasedonregularizations
with `1-type norms (and seminorms) have become a very popular tool in image
processing and inverse problems due to peculiar features that cannot be realized
with smooth regularizations.In particulartotal variation techniqueshad particular
successduetotheirabilitytorealizecartoon-typereconstructionswithsharpedges.
Duetoanexplosionofnewdevelopmentsinthisfieldwithinthelastdecadeitisa
difficulttasktokeepanoverviewofthemajorresultsinanalysis,thecomputational
schemes,andtheapplicationfields.Withtheselecturesweattempttoprovidesuch
anoverview,ofcoursebiasedbyourmajorlinesofresearch.
Wearefocusingonthebasicanalysisoftotalvariationmethodsandtheextension
of the original ROF-denoising model due various application fields. Furthermore
weprovideabriefdiscussionofstate-of-theartcomputationalmethodsandgivean
outlooktoapplicationsindifferentdisciplines.
1 Introduction
Reconstructingand processingimageswith appropriateedgesis of centralimpor-
tance in modern imaging. The development of mathematical techniques that
preserve or even favour sharp edges has become a necessity and created various
interesting approaches. The two most succesful frameworks are two variational
approaches:totalvariationmodelsontheonehandandmodelswithexplicitedges
M.Burger((cid:2))
InstituteforComputationalandAppliedMathematics,UniversityofMünster,
Münster,Germany
e-mail:[email protected]
S.Osher
DepartmentofMathematics,UCLA,LosAngeles,USA
M.Burgeretal.(eds.),LevelSetandPDEBasedReconstructionMethodsinImaging, 1
LectureNotesinMathematics2090,DOI10.1007/978-3-319-01712-9__1,
©SpringerInternationalPublishingSwitzerland2013
2 M.BurgerandS.Osher
intheframeworkofMumfordandShah[140]ontheother.Thelatterusuallylead
to various difficulties in the analysis (cf. [138]) and numerical realization due to
the explicit treatment of edges and arising nonconvexity, consequently they have
foundlimitedimpactinpracticalapplicationsbeyondimagesegmentation.Inthese
lectures notes we will discuss various developments for total variation methods,
whichcanbeformulatedasconvexvariationalmethods.Awholezooofapproaches
tothemodelling,analysis,numericalsolution,andapplicationshasdevelopedinthe
lasttwodecades,throughwhichweshalltrytodevelopaguide.
The starting pointof total variation(TV) methodshas been the introductionof
avariationaldenoisingmodelbyRudinetal.[163],consistinginminimizingtotal
variationamongallfunctionswithinavariancebound
Z
TV.u/!min subjectto .u(cid:3)f/2dx (cid:4)(cid:3)2: (1)
u (cid:2)
IntroducingaLagrangemultiplier(cid:4)itcanbeshownthatthisapproachisequivalent
totheunconstrainedproblemofminimizing
Z
(cid:4)
E .u/WD .u(cid:3)f/2dxCTV.u/; (2)
ROF
2
(cid:2)
in the following often referred to as the ROF model. In subsequent years this
model was generalized for many imaging tasks and inverse problems and found
applicationsindifferentareas.
We will providea more detailed motivationfortotal variationregularizationin
a rather general setup in Sect.2. In Sects.3–5 we provide an overview of various
aspects in the analysis of total variation regularization. In Sect.6 we discuss the
concepts of Bregman iterations and inverse scale space methods, which allow to
compensatesystematic errorsof variationalmethods,e.g. contrastloss in the case
ofTVregularization,andgaveanotherboosttoresearchinthisareainrecentyears.
InSects.7and8wediscusssomevariantsofthemodels,withchangesconcerning
thefidelityterminSect.7andtheregularizationterminSect.8.InSect.9wediscuss
someapproachesforthenumericalsolutionofthevariationalproblems.Section10
is devoted to geometric aspects of total variation minimization and the relaxation
of segmentationproblemsinto convexmodelsfor functionsof boundedvariation.
We then proceed to applications, which we mainly incorporate as further links to
literatureinSect.11.Finallywepresentsomeopenquestionsinthemodellingand
analysisofTVmethodsinSect.12.
2 TheMotivation for TVandRelated Methods
InthefollowingweprovidebasicmotivationsforthegeneralsetupofTVmethods
withrespecttoforwardoperators,datafidelityterms,andregularization.
