Table Of Content
Advances in Pattern Recognition
Forfuthervolumes:
http://www.springer.com/series/4205
· ·
Santiago Aja-Ferna´ndez Rodrigo de Luis Garc´ıa
·
Dacheng Tao Xuelong Li
Editors
Tensors in Image Processing
and Computer Vision
123
Editors
Asst.Prof.SantiagoAja-Ferna´ndez Asst.Prof.RodrigodeLuisGarc´ıa
UniversidaddeValladolid UniversidaddeValladolid
EscuelaTe´cnicaSuperiorde EscuelaTe´cnicaSuperiorde
Ingenierosde Ingenierosde
Telecomunicacio´n Telecomunicacio´n
CampusMiguelDelibes CampusMiguelDelibes
47011Valladolid 47011Valladolid
Spain Spain
[email protected] [email protected]
Dr.DachengTao Dr.XuelongLi
BlkN4,02b-67 UniversityofLondon
SchoolofComputerEngineering BirkbeckCollege
NanyangTechnologicalUniversity SchoolofComputerScience&
Singapore639798 InformationSystems
[email protected] MaletStreet
LondonWC1E7HX
Serieseditor
UnitedKingdom
ProfessorSameerSingh,PhD
xuelong [email protected]
ResearchSchoolofInformatics
LoughboroughUniversity
Loughborough,UK
ISSN1617-7916
ISBN978-1-84882-298-6 e-ISBN978-1-84882-299-3
DOI10.1007/978-1-84882-299-3
SpringerDordrechtHeidelbergLondonNewYork
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Preface
Overthepastfewyears,tensorprocessingtoolshavebecomemoreandmorepopu-
larinthefieldsofcomputervisionandimageprocessing,andtensor-valuedimage
modalitieshavealsobeenmorecommonlyemployed,withtheremarkableexample
ofDiffusionTensorMagneticResonanceImaging.
However, tensor applications and tensor processing tools arise from very dif-
ferent areas. This can prevent important advances from rapidly spreading over the
scientificcommunity.Eventhoughnoveldiscoveriescangreatlybenefitmanyhet-
erogeneous fields such as medical image processing or multilinear analysis, these
advances are too often kept within the areas of knowledge where they were first
employed.
Given this fact, a compilation of some of the most recent advances in tensor
processing can be a valuable tool for the scientific community, thus providing a
usefulreferencestateoftheartfortensorprocessingapplicationsaswellasageneral
insightontheareaswheretensorprocessingisbeingsuccessfullyapplied.
The idea behind this book started after the Tensor Workshop held at the IEEE
ComputerSocietyConferenceonComputerVisionandPatternRecognitionheldin
Anchorage, Alaska, in June 2008. This workshop gathered experts from different
fields working on tensor processing, and some of the ideas presented there were
furtherdevelopedtoghetherwiththeresultsoftheTensorWorkshopheldearlierin
Gran Canaria (Spain) in November 2006, which was sponsored by the SIMILAR
NetworkofExcellenceoftheEuropean6thFrameworkProgramme.
Asthisbookcomprisestheoreticaladvancesandapplicationsregardingveryhet-
erogeneous areas of image processing and computer vision, it has been organized
into five parts, which are nevertheless preceeded by an introductory chapter about
thedifferentapplicationsoftheuseoftensorsinsignalprocessing.PartIisdevoted
to the use of tensors and tensor field processing in general. The processing tools
describedinthesechapterscanbeappliedinanumberofdifferentapplications.In
PartII,twotensortechniquesforimageprocessingarepresented.Later,PartIIIfo-
cusesontheuseoftensorsincomputervisionapplications,suchascameramodels
or multilinear applications. As medical imaging is one of the areas that has taken
more advantage of the advances of tensor processing, Part IV is dedicated to this
v
vi Preface
issue,collectingapplicationsfromDiffusionTensorMagneticResonanceImaging
tostraintensorestimationincardiacanalysisorelastographyimaging.Finally,Part
Visdevotedtostorage,visualizationandinterfaceswithtensors,anissueofconsid-
erableimportancesincethisnewdatamodalitypresentsparticularitiesthatrequire
newapproachestotheseotherwisetraditionalproblems.
The preparation of this book has been an arduous and difficult task. We would
like to thank all the authors for their great effort and dedication in preparing their
contributions. Also, all the reviewers, and specially the members of the editorial
boardsoftheTensorWorkshopsatCVPR’08andinGranCanariain2006,deserve
ourutmostgratitude.Thanksalottoallofthem.
Boston,MA RodrigodeLuisGarc´ıa
Fall2008 SantiagoAjaFerna´ndez
Contents
Preface ........................................................ v
AReviewofTensorsandTensorSignalProcessing.................... 1
L.Cammoun,C.A.Castan˜o-Moraga,E.Mun˜oz-Moreno,D.Sosa-Cabrera,
B.Acar,M.A.Rodriguez-Florido,A.Brun,H.KnutssonandJ.P.Thiran
PartI TensorsandTensorFieldProcessing
SegmentationofTensorFields:RecentAdvancesandPerspectives ...... 35
RodrigodeLuis-Garc´ıa,CarlosAlberola-Lo´pezandCarl-FredrikWestin
AVariationalApproachtotheRegistrationofTensor-ValuedImages .... 59
SebastianoBarbieri,MartinWelkandJoachimWeickert
QualityAssessmentofTensorImages............................... 79
Emma Mun˜oz-Moreno, Santiago Aja-Ferna´ndez and Marcos Martin-
Fernandez
AlgorithmsforNonnegativeTensorFactorization .................... 105
StefanosZafeiriou
PDE-basedMorphologyforMatrixFields:NumericalSolutionSchemes . 125
BernhardBurgeth,MichaelBreuß,StephanDidas,andJoachimWeickert
PartII TensorsinImageProcessing
SphericalTensorCalculusforLocalAdaptiveFiltering................ 153
MarcoReisertandHansBurkhardt
OnGeometricTransformationsofLocalStructureTensors ............ 179
Bjo¨rnSvensson,AndersBrun,MatsAndersson,andHansKnutsson
vii
viii Contents
PartIII TensorsinComputerVision
Multi-ViewMatchingTensorsfromLinesforGeneralCameraModels... 197
SimoneGaspariniandPeterSturm
BinocularFull-BodyPoseRecognitionandOrientationInferenceUsing
MultilinearAnalysis ............................................. 215
PengandQian
ApplicationsofMultiviewTensorsinHigherDimensions .............. 237
MarinaBertolini,GianMarioBesana,andCristinaTurrini
ConstraintsfortheTrifocalTensor................................. 261
AlbertoAlzatiandAlfonsoTortora
PartIV DiffusionTensorImagingandMedicalApplications
ReviewofTechniquesforRegistrationofDiffusionTensorImaging ..... 273
Emma Mun˜oz-Moreno, Rube´n Ca´rdenes-Almeida and Marcos
Martin-Fernandez
PracticalandIntuitiveBasisforTensorFieldProcessingwithInvariant
GradientsandRotationTangents .................................. 299
GordonL.KindlmannandCarl-FredrikWestin
FromSecondtoHigherOrderTensorsinDiffusion-MRI .............. 315
AurobrataGhoshandRachidDeriche
DT-MRIConnectivityand/orTractography?:TwoNewAlgorithms ..... 335
BurakAcarandErdemYo¨ru¨k
StrainRateTensorEstimationinCineCardiacMRIBasedonElastic
ImageRegistration .............................................. 355
GonzaloVegasSa´nchez-Ferrero,AntonioTrista´nVega,LucilioCordero
Grande,PabloCasasecadelaHiguera,SantiagoAjaFerna´ndez,Marcos
Mart´ınFerna´ndezandCarlosAlberolaLo´pez
StrainTensorElastography:2Dand3DVisualizations ................ 381
Dar´ıo Sosa-Cabrera, Karl Krissian, Javier Gonza´lez-Ferna´ndez, Luis
Go´mez-De´niz, Eduardo Rovaris, Carlos Castan˜o-Moraga and Juan
Ruiz-Alzola
PartV Storage,VisualizationandInterfaces
SimilarTensorArrays–AFrameworkforStorageofTensorArrayData 407
Anders Brun, Marcos Martin-Fernandez, Burak Acar, Emma Munoz-
Moreno, Leila Cammoun, Andreas Sigfridsson, Dario Sosa-Cabrera,
Bjo¨rnSvensson,MagnusHerberthsonandHansKnutsson
Contents ix
UserInterfacestoInteractwithTensorFields........................ 429
SusanaMerino-CaviedesandMarcosMart´ın-Ferna´ndez
T-flash:TensorVisualizationinMedicalStudio ...................... 455
J.Wiklund,V.Nicolas,P.Rondao,M.AnderssonandH.Knutsson
Index .............................................................467
A Review of Tensors and Tensor Signal
Processing
L.Cammoun1,C.A.Castan˜o-Moraga2,E.Mun˜oz-Moreno4,D.Sosa-Cabrera2,
B.Acar5,M.A.Rodriguez-Florido2,3,A.Brun6,H.Knutsson6,J.P.Thiran1
Abstract Tensorshavebeenbroadlyusedinmathematicsandphysics,sincethey
areageneralizationofscalarsorvectorsandallowtorepresentmorecomplexprop-
erties.Inthischapterwepresentanoverviewofsometensorapplications,especially
thosefocusedontheimageprocessingfield.Fromamathematicalpointofview,a
lotofworkhasbeendevelopedabouttensorcalculus,whichobviouslyismorecom-
plexthanscalarorvectorialcalculus.Moreover,tensorscanrepresentthemetricof
avectorspace,whichisveryusefulinthefieldofdifferentialgeometry.Inphysics,
tensors have been used to describe several magnitudes, such as the strain or stress
ofmaterials.Insolidmechanics,tensorsareusedtodefinethegeneralizedHooke’s
law, where a fourth order tensor relates the strain and stress tensors. In fluid dy-
namics, the velocity gradient tensor provides information about the vorticity and
the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies
the notation of the Maxwell equations. But tensors are not constrained to physics
andmathematics.Theyhavebeenused,forinstance,inmedicalimaging,wherewe
can highlight two applications: the diffusion tensor image, which represents how
moleculesdiffuseinsidethetissuesandisbroadlyusedforbrainimaging;andthe
tensorial elastography, which computes the strain and vorticity tensor to analyze
the tissues properties. Tensors have also been used in computer vision to provide
informationaboutthelocalstructureortodefineanisotropicimagefilters.
1SignalProcessingInstituteEcolePolytechniqueFe´de´raledeLausanneSwitzerland
Email:{leila.cammoun,JP.Thiran}@epfl.ch
2CenterforTechnologyinMedicine.DepartmentofSignalsandCommunciations
UniversityofLasPalmasdeGranCanaria.
3CanaryIslandsInstituteofTechnology
Email:{ccasmor,dario,marf}@ctm.ulpgc.es
4LaboratoryofImageProcessing.Univ.deValladolid(Spain)
Email:[email protected]
5Electrical-ElectronicsEng.Dept.BogaziciUniversityIstanbulTurkey
Email:[email protected]
6BiomedicalEngineeringDept.CMIVLinko¨pingsUniversitetSweden
Email:{andbr,knutte}@imt.liu.se
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