New Developments in the Visualization and Processing of Tensor Fields

388 Pages·2012·12.56 MB·English

by  Hans Knutsson

Mathematics and Visualization
SeriesEditors
GeraldFarin
Hans-ChristianHege
DavidHoffman
ChristopherR.Johnson
KonradPolthier
MartinRumpf
Forfurthervolumes:
http://www.springer.com/series/4562
•
David H. Laidlaw Anna Vilanova
(cid:2)
Editors
New Developments
in the Visualization
and Processing
of Tensor Fields
With133Figures, 129in color
123
Editors
DavidH.Laidlaw AnnaVilanova
DepartmentofComputerScience DepartmentofBiomedicalEngineering
BrownUniversity,Providence EinhovenUniversityofTechnology
RhodeIsland,USA Einhoven,Netherlands
ISBN978-3-642-27342-1 ISBN978-3-642-27343-8(eBook)
DOI10.1007/978-3-642-27343-8
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2012947651
MathematicalSubjectClassification(2010):68-XX,70-XX,15-XX,76-XX,92-XX
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Preface
In this book, we attempt to capture the excitement and inspiration that has been
generated during a series of Dagstuhl Seminars that have looked at visualization
andprocessingoftensorfields.Thisbookincludescontributionsfromattendeesof
thethirdmeetingheldinJuly2009.Asinthetwoearliervolumes,theauthorsreport
onrecentresearchresultsaswellasopiningonfuturedirectionsfortheanalysisand
visualizationoftensorfields.Topicsrangefromapplicationsoftheanalysisoftensor
fieldstopurerresearchintotheirmathematicalandanalyticalproperties.Oneofthe
goalsofthisseminarwastobringtogetherresearchersfromalongtheaxisbetween
pure and applied research, identifying new multidisciplinary research challenges.
Thisbook,wehope,willcontinuetofurtherthatgoalinabroadercontext.
Themanuscriptisorganizedintosevenparts.Theparts,tosomedegree,increase
indimensionalityandinmathematicalsophistication.Onechangeweimplemented
in this third in a series of Dagstuhl seminars is that it no longer restricts the
underlyingdata modelsto second-ordertensor fields. Research on representations
thatgobeyondsecond-ordertensorsareclearlynecessary,asbecamemoreevident
duringtheseminars.
Part I, “Structure-TensorComputation,”focuses primarily on the generationof
the so-called structure tensor to characterize image structure information. This
information is often needed for feature preserving image processing techniques.
Thechapterspresentgeneralizationsofthemethodologyofclassicstructuretensor
estimationanditsapplicationbeyondscalarfields.
PartII,“Tensor-FieldVisualization,”presentsthreeresultsthatextendourability
togenerate,utilize,andvisualizetensorfields.Onelooksatfabriclikevisualizations
on surfaces. The second uses a Lagrangian metaphor to show the structure of
3D tensor fields. The third presents methods for designing tensor fields that have
differentapplicationsinvisualizationsuchasrealizingsampledtexturefields.
Part III, “Applicationsof Tensor-Field Analysis and Visualization,” offers con-
tributions from researchers in disciplines where the application of analysis and
visualization of second-order tensor fields is relevant. These applied domains
include the simulation of combustion and the measurement of material elasticity
usingmagneticresonanceimaging.
vii
viii Preface
PartIV,“DiffusionWeightedMRIVisualization,”breaksoutonescientificarea
where second-ordertensor fields are often applied. The chaptersin this partshow
techniquesforthespecificstudyofdiffusiontensorfieldsproducedfrommagnetic
resonance imaging, typically of the brain. These tensor-valued images capture
propertiesofthebrainwhitematterand,therefore,itsstructuralconnectivity.Brain
connectivity is complex and difficult to understand. These chapters take a step in
makingthatunderstandingeasier.Thelastchapterespeciallyfocusesonuncertainty
visualizationoffeaturesderivedfromthetensorfield,whichisarecentlyemerging
researchtopic.
PartV,“BeyondSecond-OrderDiffusionTensorMRI,”expandsthecomplexity
oftheunderlyingdatamodelusedintheearlierchapters.Despitethechallengesof
working with second-ordertensor valued images, the use of second-ordertensors
haslimitations.Indiffusionweightedimaging,itimpliesamodeloftheunderlying
anatomy and physics that is not sufficiently accurate in some situations. In this
part, models that are more accurate or provide more information are presented.
In diffusion weighted imaging, they also require more complex acquisition and
processing.Asthechaptersinthispartexemplify,someoftheresearchinthisfield
focusesonlimitingtheextracomplexity.
In Part VI, “Tensor Metrics,” some of the challengesof workingwith sampled
tensorfieldsareaddressed.Tensorfieldsareintrinsicallydefinedoveracontinuous
Euclideanspace, and yet they must be representedfor computationin some finite
way. Typically, this is done by sampling them spatially, keeping only one tensor
for each small region of an image. Furthermore, distance measures are needed
forseveralcommonimageprocessingtechniques,e.g.,interpolation,segmentation,
and registration. However, the distance or metric between tensors is not uniquely
defined.Thethreechaptersinthispartdiscusssomeoftheconsequencesofvarious
choicesforthesedistances.
Finally, in Part VII, “Tensor Analysis,” two mathematical areas are presented
that were judged to have relevance to researchers at this seminar, but which
were not widely known to them in advance. These two chapters offer moderately
deep introductions to the areas with the hope of stimulating cross-disciplinary
collaborations.
Wehopethattheseofferingswill,overallorindividually,inspireinreaderssome
oftheenthusiasmandenergythattheseminarbroughttoitsparticipants.
Providence,RI,USA DavidH.Laidlaw
Eindhoven,TheNetherlands AnnaVilanova
Contents
PartI Structure-TensorComputation
Structure Tensor Estimation: Introducing Monomial
QuadratureFilterSets........................................................... 3
HansKnutsson,Carl-FredrikWestin,andMatsAndersson
1 Introduction.................................................................... 4
1.1 ThisChapterPresents................................................... 4
2 MonomialFilters.............................................................. 5
2.1 RadialPart............................................................... 5
2.2 DirectionalMatrix ...................................................... 6
2.3 MonomialFilterMatrices............................................... 7
3 MonomialsLinkOrder,ScaleandGradients ................................ 9
3.1 TheGradientOperatorIncreasesOrderandShiftsScale ............. 11
4 MonomialFilterResponseMatrices ......................................... 12
4.1 SignalClasses........................................................... 13
5 MonomialStructureTensors.................................................. 15
5.1 TwoSimpleExamples.................................................. 15
5.2 GeneralStructureTensorConstruction ................................ 16
5.3 MonomialQuadrature .................................................. 17
5.4 PhaseInvariance ........................................................ 17
5.5 TensorPositivity ........................................................ 17
6 StructureTensorVariations................................................... 18
6.1 TheStructureTensor,T ................................................ 18
q
6.2 TheGradientTensor,T ................................................ 19
G
6.3 TheBoundaryTensor,T ............................................... 19
B
6.4 TheEnergyTensor,T .................................................. 20
E
6.5 GradientEnergyTensor,T ........................................... 20
GE
6.6 Spatial2ndOrderPolynomialTensor.................................. 20
6.7 SphericalHarmonics.................................................... 21
6.8 SumofMonomialTensors.............................................. 21
7 TableofStructureTensorRelatedAlgorithms............................... 21
ix
x Contents
8 HigherOrderStructureTensors .............................................. 22
8.1 Non-simpleSignals..................................................... 23
9 Conclusion..................................................................... 24
References......................................................................... 24
AdaptationofTensorVotingtoImageStructureEstimation ............... 29
Rodrigo Moreno,Luis Pizarro, Bernhard Burgeth,Joachim
Weickert,MiguelAngelGarcia,andDomenecPuig
1 Introduction.................................................................... 30
2 TensorVoting.................................................................. 32
2.1 StickTensorVoting...................................................... 33
2.2 ThePlateTensorVoting................................................ 34
2.3 TheBallTensorVoting ................................................. 35
3 RelationshipsBetweentheStructureTensorandTensorVoting............ 35
3.1 Similarities .............................................................. 35
3.2 Differences .............................................................. 36
4 TensorVotingforStructureEstimation ...................................... 37
4.1 Gray-ScaleImages...................................................... 37
4.2 ColorandVector-ValuedImages....................................... 38
4.3 Tensor-ValuedImages .................................................. 39
5 ExperimentalResults.......................................................... 42
6 ConcludingRemarks.......................................................... 47
References......................................................................... 48
Edge-EnhancingDiffusionFilteringforMatrixFields ...................... 51
BernhardBurgeth,LuisPizarro,andStephanDidas
1 Introduction.................................................................... 52
2 Edge-EnhancingDiffusion.................................................... 54
3 BasicDifferentialCalculusforMatrixFields................................ 55
4 TheGeneralisedStructureTensorS forMatrixFields.................... 57
G
4.1 ADiffusionTensorDforMatrixFields............................... 58
5 Edge-EnhancingDiffusionFilteringforMatrixFields...................... 58
6 NumericalIssues.............................................................. 59
7 Experiments ................................................................... 60
8 ConcludingRemarks.......................................................... 66
References......................................................................... 66
PartII Tensor-FieldVisualization
Fabric-LikeVisualizationofTensorFieldDataonArbitrary
SurfacesinImageSpace......................................................... 71
Sebastian Eichelbaum,Mario Hlawitschka, Bernd Hamann,
andGerikScheuermann
1 MotivationandRelatedWork................................................. 72
2 Method......................................................................... 73
2.1 InitialNoiseTextureGeneration ....................................... 74