From Gestalt theory to image analysis: a probabilistic approach

285 Pages·2008·7.761 MB·English

by  Agnés Desolneux, Lionel Moisan, Jean-Michel Morel (auth.)

Interdisciplinary Applied Mathematics
Volume 34
Editors
S.S. Antman J.E. Marsden
L. Sirovich S. Wiggins
Geophysics and Planetary Sciences
Imaging, Vision, and Graphics
D. Geman
Mathematical Biology
L. Glass, J.D. Murray
Mechanics and Materials
R.V. Kohn
Systems and Control
S.S. Sastry, P.S. Krishnaprasad
Problems in engineering, computational science, and the physical and biological
sciences are using increasingly sophisticated mathematical techniques. Thus, the
bridge between the mathematical sciences and other disciplines is heavily traveled.
The correspondingly increased dialog between the disciplines has led to the establishment
of the series: Interdisciplinary Applied Mathematics.
The purpose of this series is to meet the current and future needs for the interaction
between various science and technology areas on the one hand and mathematics on
the other. This is done, firstly, by encouraging the ways that mathematics may
be applied in traditional areas, as well as point towards new and innovative areas
of applications; and, secondly, by encouraging other scientifi c disciplines to engage in
a dialog with mathematicians outlining their problems to both access new methods
and suggest innovative developments within mathematics itself.
The series will consist of monographs and high-level texts from researchers working
on the interplay between mathematics and other fi elds of science and technology.
Interdisciplinary Applied Mathematics
Volumes published are listed at the end of this book.
Agne`s Desolneux Lionel Moisan
Jean-Michel Morel
From Gestalt Theory
to Image Analysis
A Probabilistic Approach
A.Desolneux L. Moisan
Universite´ Paris Descartes Universite´ Paris Descartes
MAP5 (CNRS UMR 8145) MAP5 (CNRS UMR 8145)
45, rue des Saints-Pe`res 45, rue des Saints-Pe`res
75270 Paris cedex 06, France 75270 Paris cedex 06, France
[email protected] [email protected]
J.-M. Morel
Ecole Normale Supe´rieure de Cachan, CMLA
61, av. du Pre´sident Wilson
94235 Cachan Ce´dex
France
[email protected]
Editors
S.S. Antman J.E. Marsden
Department of Mathematics Control and Dynamical Systems
and Mail Code 107-81
Institute for Physical Science California Institute of Technology
and Technology Pasadena, CA 91125, USA
University of Maryland [email protected]
College Park, MD 20742, USA
[email protected]
L. Sirovich S. Wiggins
Division of Applied Mathematics School of Mathematics
Brown University University of Bristol
Providence, RI 02912, USA Bristol BS8 1TW, UK
[email protected] [email protected]
ISBN:978-0-387-72635-9 e-ISBN:978-0-387-74378-3
DOI:10.1007/978-0-387-74378-3
Libraryof CongressControlNumber:2007939527
MathematicsSubjectClassification(2000):62H35, 68T45, 68U10
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Preface
Thetheoryinthesenoteswastaughtbetween2002and2005atthegraduateschools
of Ecole Normale Supe´rieure de Cachan, Ecole Polytechnique de Palaiseau, Uni-
versitat Pompeu Fabra, Barcelona, Universitat de les Illes of Balears, Palma, and
UniversityofCaliforniaatLosAngeles.ItisalsobeingtaughtbyAndre`sAlmansa
attheFacultaddeIngeneria,Montevideo.
Thistextwillbeofinteresttoseveralkindsofaudience.Ourteachingexperience
provesthatspecialistsinimageanalysisandcomputervisionfindthetexteasyatthe
computervisionsideandaccessibleonthemathematicallevel.Theprerequisitesare
elementary calculus and probability from the first two undergraduate years of any
sciencecourse.Allslightlymoreadvancednotionsinprobability(inequalities,sto-
chasticgeometry,largedeviations,etc.)willbeeitherprovedinthetextordetailed
inseveral exercises attheendofeachchapter. Wehave always askedthestudents
todoallexercisesandtheyusuallysucceedregardlessofwhattheirscienceback-
groundis.Themathematicsstudentsdonotfindthemathematicsdifficultandeasily
learnthroughthetextitselfwhatisneededinvisionpsychologyandthepracticeof
computervision.Thetextaimsatbeingself-containedinallthreeaspects:mathe-
matics,vision,andalgorithms.Wewillinparticularexplainwhatadigitalimageis
andhowtheelementarystructurescanbecomputed.
Wewishtoemphasizewhywearepublishingthesenotesinamathematicscol-
lection.Themainquestiontreatedinthiscourseisthevisualperceptionofgeometric
structure.Wehopethisisathemeofinterestforallmathematiciansandallthemore
ifvisualperceptioncanreceive–uptoacertainlimitwecannotyetfix–afullymath-
ematical treatment. In these lectures, we rely on only four formal principles, each
one taken from perception theory, but receiving here a simple mathematical defi-
nition. These mathematically elementary principles are theShannon-Nyquist prin-
ciple, the contrast invariance principle, the isotropy principle and the Helmholtz
principle.Thefirstthreeprinciplesareclassicalandeasilyunderstood.Wewilljust
statethem along withtheir straightforwardconsequences. Thus, thetext ismainly
dedicated to one principle, the Helmholtz principle. Informally, it states that there
is no perception in white noise. A white noise image is an image whose samples
v
vi Preface
areidenticallydistributedindependentrandomvariables.Theviewofawhitesheet
of paper in daylight gives a fair idea of what white noise is. The whole work will
be to draw from this impossibility of seing something on a white sheet a series of
mathematicaltechniquesandalgorithmsanalyzingdigitalimagesand“seeing”the
geometricstructurestheycontain.
Most experiments are performed on digital every-day photographs, as they
presentavarietyofgeometricstructuresthatexceedsbyfaranymathematicalmod-
eling and are therefore apt for checking any generic image analysis algorithm. A
warning to mathematicians: It would be fallacious to deduce from the above lines
thatweareproposingadefinitionofgeometricstructureforallrealfunctions.Such
a definition would include all geometries invented by mathematicians. Now, the
mathematician’srealfunctionsare,fromthephysicalorperceptualviewpoint,im-
possibleobjectswithinfiniteresolutionandthatthereforehaveinfinitedetailsand
structuresonallscales.Digitalsignals,orimages,aresurelyfunctions,butwiththe
essential limitation of having a finite resolution permitting a finite sampling (they
are band-limited, by the Shannon-Nyquist principle). Thus, in order to deal with
digitalimages,amathematicianhastoabandontheinfiniteresolutionparadiseand
stepintoafiniteworldwheregeometricstructuresmustallthesamebefoundand
proven. They can even be found with an almost infinite degree of certainty; how
sureweareofthemispreciselywhatthisbookisabout.
The authors are indebted to their collaborators for their many comments and
corrections, and more particularly to Andre`s Almansa, Je´re´mie Jakubowicz, Gary
Hewer, Carol Hewer, and Nick Chriss. Most of the algorithms used for the exper-
iments are implemented in the public software MegaWave. The research that led
to the development of the present theory was mainly developed at the University
Paris-Dauphine(Ceremade)andattheCentredeMathe´matiquesetLeursApplica-
tions,ENSCachanandCNRS.Itwaspartiallyfinancedduringthepast6yearsby
theCentreNational d’Etudes Spatiales, theOffice ofNaval Research, and NICOP
undergrantN00014-97-1-0839andtheFondationlesTreilles.Wethankverymuch
BernardRouge´,DickLau,WenMasters,RezaMalek-Madani,andJamesGreenberg
fortheirinterestandconstantsupport.TheauthorsaregratefultoJeanBretagnolle,
NicolasVayatis,Fre´de´ricGuichard,IsabelleGaudron-Trouve´,andGuillermoSapiro
forvaluablesuggestionsandcomments.
Contents
Preface............................................................ v
1 Introduction................................................... 1
1.1 GestaltTheoryandComputerVision .......................... 1
1.2 BasicPrinciplesofComputerVision .......................... 3
2 GestaltTheory................................................. 11
2.1 BeforeGestaltism:Optic-GeometricIllusions................... 11
2.2 GroupingLawsandGestaltPrinciples ......................... 13
2.2.1 GestaltBasicGroupingPrinciples ...................... 13
2.2.2 CollaborationofGroupingLaws ....................... 17
2.2.3 GlobalGestaltPrinciples.............................. 19
2.3 ConflictsofPartialGestaltsandtheMaskingPhenomenon ....... 21
2.3.1 Conflicts ........................................... 21
2.3.2 Masking ........................................... 22
2.4 QuantitativeAspectsofGestaltTheory ........................ 25
2.4.1 QuantitativeAspectsoftheMaskingPhenomenon ........ 25
2.4.2 ShannonTheoryandtheDiscreteNatureofImages ....... 27
2.5 BibliographicNotes ........................................ 29
2.6 Exercise .................................................. 29
2.6.1 GestaltEssay ....................................... 29
3 TheHelmholtzPrinciple ........................................ 31
3.1 IntroducingtheHelmholtzPrinciple:ThreeElementary
Examples ................................................. 31
3.1.1 ABlackSquareonaWhiteBackground ................. 31
3.1.2 BirthdaysinaClassandtheRoleofExpectation.......... 34
3.1.3 VisibleandInvisibleAlignments ....................... 36
3.2 TheHelmholtzPrincipleandε-MeaningfulEvents .............. 37
3.2.1 AFirstIllustration:PlayingRoulettewithDostoievski ..... 39
3.2.2 AFirstApplication:DotAlignments.................... 41
3.2.3 TheNumberofTests ................................. 42
vii
viii Contents
3.3 BibliographicNotes ........................................ 43
3.4 Exercise .................................................. 44
3.4.1 BirthdaysinaClass .................................. 44
4 EstimatingtheBinomialTail .................................... 47
4.1 EstimatesoftheBinomialTail................................ 47
4.1.1 Inequalitiesfor (l,k,p) .............................. 49
B
4.1.2 AsymptoticTheoremsfor (l,k,p)=P[S k]........... 50
l
B ≥
4.1.3 ABriefComparisonofEstimatesfor (l,k,p)............ 50
B
4.2 BibliographicNotes ........................................ 52
4.3 Exercises ................................................. 52
4.3.1 TheBinomialLaw ................................... 52
4.3.2 Hoeffding’sInequalityforaSumofRandomVariables..... 53
4.3.3 ASecondHoeffdingInequality ........................ 55
4.3.4 GeneratingFunction.................................. 56
4.3.5 LargeDeviationsEstimate............................. 57
4.3.6 TheCentralLimitTheorem............................ 60
4.3.7 TheTailoftheGaussianLaw .......................... 63
5 AlignmentsinDigitalImages ................................... 65
5.1 DefinitionofMeaningfulSegments ........................... 65
5.1.1 TheDiscreteNatureofAppliedGeometry ............... 66
5.1.2 TheAContrarioNoiseImage.......................... 67
5.1.3 MeaningfulSegments ................................ 70
5.1.4 DetectabilityWeightsandUnderlyingPrinciples .......... 72
5.2 NumberofFalseAlarms .................................... 74
5.2.1 Definition .......................................... 74
5.2.2 PropertiesoftheNumberofFalseAlarms................ 75
5.3 OrdersofMagnitudesandAsymptoticEstimates ................ 76
5.3.1 SufficientConditionofMeaningfulness.................. 77
5.3.2 AsymptoticsfortheMeaningfulnessThresholdk(l) ....... 78
5.3.3 LowerBoundfortheMeaningfulnessThresholdk(l) ...... 80
5.4 PropertiesofMeaningfulSegments ........................... 81
5.4.1 ContinuousExtensionoftheBinomialTail............... 81
5.4.2 DensityofAlignedPoints ............................ 83
5.5 AboutthePrecision p ....................................... 86
5.6 BibliographicNotes ........................................ 87
5.7 Exercises ................................................. 91
5.7.1 ElementaryPropertiesoftheNumberofFalseAlarms ..... 91
5.7.2 AContinuousExtensionoftheBinomialLaw ............ 91
5.7.3 ANecessaryConditionofMeaningfulness ............... 92
Contents ix
6 MaximalMeaningfulnessandtheExclusionPrinciple ............. 95
6.1 Introduction ............................................... 95
6.2 TheExclusionPrinciple ..................................... 97
6.2.1 Definition .......................................... 97
6.2.2 ApplicationoftheExclusionPrincipletoAlignments...... 98
6.3 MaximalMeaningfulSegments ..............................100
6.3.1 AConjectureAboutMaximality .......................102
6.3.2 ASimplerConjecture ................................103
6.3.3 ProofofConjecture1UnderConjecture2 ...............105
6.3.4 PartialResultsAboutConjecture2......................106
6.4 ExperimentalResults .......................................109
6.5 BibliographicalNotes.......................................112
6.6 Exercise ..................................................113
6.6.1 StraightContourCompletion ..........................113
7 ModesofaHistogram ..........................................115
7.1 Introduction ...............................................115
7.2 MeaningfulIntervals........................................115
7.3 MaximalMeaningfulIntervals................................119
7.4 MeaningfulGapsandModes.................................122
7.5 StructurePropertiesofMeaningfulIntervals ....................123
7.5.1 MeanValueofanInterval .............................123
7.5.2 StructureofMaximalMeaningfulIntervals...............124
7.5.3 TheReferenceInterval................................126
7.6 ApplicationsandExperimentalResults ........................127
7.7 BibliographicNotes ........................................129
7.8 Exercises .................................................129
7.8.1 Kullback-LeiblerDistance.............................129
7.8.2 AQualitativeaContrarioHypothesis ...................130
8 VanishingPoints ...............................................133
8.1 Introduction ...............................................133
8.2 DetectionofVanishingPoints ................................133
8.2.1 MeaningfulVanishingRegions.........................134
8.2.2 ProbabilityofaLineMeetingaVanishingRegion.........135
8.2.3 PartitionoftheImagePlaneintoVanishingRegions .......137
8.2.4 FinalRemarks.......................................141
8.3 ExperimentalResults .......................................144
8.4 BibliographicNotes ........................................145
8.5 Exercises .................................................150
8.5.1 Poincare´-InvariantMeasureontheSetofLines ...........150
8.5.2 PerimeterofaConvexSet ............................150
8.5.3 Crofton’sFormula ...................................150
x Contents
9 ContrastedBoundaries .........................................153
9.1 Introduction ...............................................153
9.2 LevelLinesandtheColorConstancyPrinciple..................153
9.3 AContrarioDefinitionofContrastedBoundaries ................159
9.3.1 MeaningfulBoundariesandEdges......................159
9.3.2 Thresholds..........................................162
9.3.3 Maximality .........................................163
9.4 Experiments...............................................164
9.5 TwelveObjectionsandQuestions .............................168
9.6 BibliographicNotes ........................................174
9.7 Exercise ..................................................175
9.7.1 TheBilinearInterpolationofanImage ..................175
10 VariationalorMeaningfulBoundaries?...........................177
10.1 Introduction ...............................................177
10.2 The“Snakes”Models .......................................177
10.3 ChoiceoftheContrastFunctiong.............................180
10.4 SnakesVersusMeaningfulBoundaries.........................185
10.5 BibliographicNotes ........................................188
10.6 Exercise ..................................................188
10.6.1 NumericalScheme...................................188
11 Clusters ......................................................191
11.1 Model ....................................................191
11.1.1 Low-ResolutionCurves...............................191
11.1.2 MeaningfulClusters..................................193
11.1.3 MeaningfulIsolatedClusters ..........................193
11.2 FindingtheClusters ........................................194
11.2.1 SpanningTree.......................................194
11.2.2 ConstructionofaCurveEnclosingaGivenCluster ........194
11.2.3 MaximalClusters ....................................196
11.3 Algorithm.................................................196
11.3.1 ComputationoftheMinimalSpanningTree..............196
11.3.2 DetectionofMeaningfulIsolatedClusters ...............197
11.4 Experiments...............................................198
11.4.1 Hand-MadeExamples ................................198
11.4.2 ExperimentonaRealImage...........................198
11.5 BibliographicNotes ........................................198
11.6 Exercise ..................................................201
11.6.1 PoissonPointProcess ................................201
12 BinocularGrouping ...........................................203
12.1 Introduction ...............................................203
12.2 EpipolarGeometry .........................................204
12.2.1 TheEpipolarConstraint...............................204
12.2.2 TheSeven-PointAlgorithm............................204