SINTEF - Tensor field visualiza0on – My view on the field … · 2016-01-22 · Geilo winter School – Scien0fic Visualiza0on Ingrid Hotz – Linköping ... Ingrid Hotz …

Tensorfieldvisualiza0on–Myviewonthefield

January2016

GeilowinterSchool–Scien0ficVisualiza0on

IngridHotz–LinköpingUniversity

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Tensor field visualization – My view on the field

IngridHotz

Fromthegeneralperspec0ve…

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IngridHotz

…todigginginthesand

Tensorfieldvisualiza0on–Myviewonthefield

January2016

GeilowinterSchool–Scien0ficVisualiza0on

IngridHotz–LinköpingUniversity

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IngridHotz

Tensors–WhyDoWeCare?

§  Tensorsareeverywhere(Simula0ons,physicaltheories)§  Canhardlybeseenanywhere–nobodycares?

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Ifyou’resiWngatacocktailpartywithabunchofengineers,physicists,andmathema0cians,andyouwanttostartaheateddebate,

Justaskoutloud:“Whatisatensor?”

•  Onepersonwillsaythat,forallprac0calpurposes,atensorisjustafancywordforamatrix.

•  Thensomeoneelsewillpipeupindignantlyandinsistthatatensorisalineartransforma;onfromvectorstovectors.

•  Yetanotherpersonwillsaythatatensorisanorderedsetofnumbers

thattransforminapar;cularwayuponachangeofbasis.

•  Otherfolks(likeus)willstartbabblingabout“dyads”and“dyadics.”

[R. Brannon, Functional and Structured Tensor Analysis for Engineers, UNM Book Draft, 2003]

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•  Vectorvisualiza0on•  Intui0veno0onofavector

•  PrevalentApplica0ons

•  Somedrivingques0ons

•  Longvisualiza;onhistory,wellaccepted

•  Tensorvisualiza0on•  Intui0veno0on?

•  Manydifferentapplica0ons

•  Onlyafewsharedques0ons

•  Novisualiza;onhistory,hardlyknown

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Nocommonlanguageacrossapplica0onsButterminologymostlyrelatedtospecifictensorinvariants

àBasisofourvisualiza0onconcept

EigenvaluesEigenvectors

Oeenscien0stshavenogluehowtoexpectfromthetensordata

adaptedfromtalkbyMiriahMeyergivenatDagstuhl2011??

Data??

Head Computer

Fuzzy Crisp

Questions

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Tensorvisualiza0onisfieldthatiss0llinitsinfancy(maybeexceptforDiffusionTensorImaginghoweverthisdoesnotreallyhappeninthevisualiza0oncommunity)

Overview

I.  TensorsAsmathema0calobjectsAsphysicaldescriptors

II.  Somebasicvisualiza0onmethodsIII. TensorInvariantsforfeaturedefini0onIV. Thestoryofacollabora0on

I.  Tensorsasmathema0calobjects

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Second-ordertensorTisabilinearfunc0on

Vn-dimvectorspace

v1,v2 ∈V2

T v1,v2( )∈!

T :V ⊗V → !

T (v,w) = wT ⋅T ⋅v, ∀v,w∈V , T∈!n×n

Second-ordertensorTaslin.operator

T :V →VT (v) = T ⋅v, ∀v∈V

ThisisnotatensorbutamatrixOeenatensorismixedupwithitsrepresenta0on

T =

t11 … t1n

! " !

tn1 # tnn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ITensorsasmathema0calobjects

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Thetensorisuniquelydeterminedbyitsac0ononallunitvectorsRota0on,deforma0on,reflec0on

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Symmetrictensor

Deforma0on,reflec0on–norota0on

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Posi;vedefinite,symmetrictensor

Deforma0on–norota0on,noreflec0on

S

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SymmetrictensorEigenvectorsandeigenvalues

§  Tensorscanbeanalyzedusinganyconvenientreferenceframe

§  Forspecificreferenceframes,thetensorrepresenta0onbecomesespeciallysimple

à Basisgivenbytheeigenvectorsofthematrix

Tei = λiei

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TherearedifferenttensorDecomposi0on,whichareoeenthebasisforvisualiza0onmethods

§  Polardecomposi0on–stretch-rota0on

Concatena0onofmappings

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Sumofmappings

sij =12tij + t ji( ).

S = 12T + TT( ),

Symmetricpart

aij =12tij − t ji( ),

A =12T − TT( ),

An0-symmetricpart


§  Symmetricandan0-symmetricpart

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§  Manydifferentmeasuresforanisotropyavailable


§  Isotropicscaling,anisotropicdeforma0on(deviator)

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Examples�  Eigenvalues�  Determinant�  Trance�  Allfunc0onsthatonlydependontheeigenvalues

Tensor Invariants

En00esthatdonotdependontherepresenta0onProper0esinherenttothetensor

ITensorasphysicaldescriptors

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Firstorderdescriptorsofthedependenceoftwovectorfields

àfirsttermoftheTaylorexpansion

Theyprovideamoreorlessgoodapproxima0onofthereality

v(w) = T ⋅w + higher order terms

v,w

TensorsareeverywhereinPhysicsandEngineering

§  Notbecausetheworldislinear§  Butbecauselinearissimple

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Gradient tensor

Stress and strain tensor

x

z

y

σy σz

σx σx

τxy

σy σz τyx

Metric tensor

Curvature tensor

Diffusion tensor


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StressTensor

Geo-andMaterialSciences

1.  Solidblock,withtwoappliedforces2.  Implantdesign,stresssimula0oninhumanbone3.  Notchedblockwithexternalforces

Images: Kratz, ZIB


3

2

1

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Metric,Curvature,Stress

GeneralRela0vity•  Simula0onofgravita0onalfieldofa

rota0ngblackhole,respec0velyneutronstar

Images: Benger, Kratz, ZIB

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DiffusionTensor–Medicine

•  Imagingmethod:basedonmagne0cresonancetomography(MRT)measuringthediffusionofwatermoleculesin0ssues

•  Applica0on:Reconstruc0onofneuralfibersinhumanbrain(tractography)

Textured slice 3D Fiber tracking Glyph representations

Images: Kratz, Breßler, Hotz, ZIB

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StructureTensor–ImageAnalysis

Image: Kratz, ZIB

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§  Indefinite§  E.g.stresstensor(forces/

area)§  TraceTàvolumechange

Describesadeforma;on

§  Posi0vedefinite§  E.g.deforma0ontensor§  DeterminantdetTàvolume

change

Describesageneratorofadeforma;on

v ' = T ⋅v v ' = etT ⋅v≈ 1+ tT( ) ⋅v

DifferentCharactersofTensors-Examples


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•  Playafundamentalroleintheunderstandingofapplica0onspecifictensors

•  Everyapplica0onhasitsowninvariantsthatareespeciallyimportant.Thyoeencomewithadomainspecificlanguage

àRelevantinvariantsshouldguidethechoiceofthevisualiza0on

Tensor Invariants

IISomebasicvisualiza0onmethodsWhathasvisualiza0oncurrentlytooffer

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IISomebasicvisualiza0onmethods

Commonprac;ce

•  ColorRepresenta0onofderivedscalars,e.g.trace

•  2Dslicesandsurfaces

Stresssimula0on,construc0onpieceM.Stommel,UniSaarbrücken B.Hirschberger(2004),

UniKaiserslautern

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Thetypicalglyph:Ellipsoid

LocalMethods-Glyphs

•  Geometricobjectsrepresen0ngtensorcharacteris;cs

•  Here:Ellipses•  Mostfrequentlyusedvisualiza0on

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Manydifferentglyphtypesinuse

Whentouse?

•  Evalua0onofdataquality•  Visualiza0onoverview•  Probing

[Schultz2010,Kratz2014]

ExamplefromMaterialSciences

LocalMethods-Glyphs

Glyphplacement

�  Howmanyglyphs?�  Wheretoplacethem?

Glyphdesign

�  Whatshouldberepresented?�  Howshoulditbe

represented?

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Howshoulditberepresented–generaldesignguidelines•  Preserva0onofsymmetry

e.g.eigenvectorshavenoorienta0on,isotropictensorshavenotdis0nguisheddirec0on

•  Con0nuitysimilartensorsshouldhavesimilarglyphrepresenta0ons

•  Disambiguatetensorswithdifferentvaluesshouldbereflectedbydifferentglyphs

•  Useglyphsthathavebeenusedinthecommunitybefor[SchultzandKindlmann2010]

Whatshouldberepresented–applica0onspecificguidelines•  Useapplica0onspecificinvariantsforthedesign

LocalMethods-Glyphs

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[Kindlmann2004]

LocalMethods-Glyphs

ResolvingPercep0onalIssuesSuperquadrics

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[Schulz2010]

LocalMethods-Glyphs

ResolvingPercep0onalIssuesSuperquadrics

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Images:Kratz,BresslerZIB,Amira

LocalMethods-Glyphs

Glyphplacement

�  Howmanyglyphs?�  Wheretoplacethem?

Anisotropicsampling

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Par0clebasedandgeometricmethods onplanes,in3d,onsurfaces

[Feng2008] [Kindlmann2006] [Kratz2013]

LocalMethods-Glyphs

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§  Integrallinessimilartostreamlines

§  Followoneeigenvectorfield

Tensorlines

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Direc0onfieldisnotorientable.Itisnotavectorfield!!

Tensorlines

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Dick2009

Frey2001

Tensorlines

Gravita0onalfieldoftwocollidingneutronstars

Images:Zobel,ZIB

Implantdesign

Forceappliedtosolidblock

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TensorsfromdiffusionMRITrackingofneuralfibers•  Majortensorlinescanbeusedto

approximatefibers•  Linetracingonlyforregionsof

highanisotropy

Tensorlines–Fibertracking

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IISomebasicvisualiza0onmethodsTextures

FabricTexture

Samplingbasedtextures

HyperLIC-2dkernel

SplaWngglyphs[Benger2006]

FabricTexture

Topologybasedtextures

[Pang2003] [Hotz2004] [Eichelbaum2013]

Kratz2011 Auer2012

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Textures–Principaldirec0onsandvalues•  Beambendunderitsownload.•  Simula0onrespec0ngevolvingdamage,30mesteps•  Data:LouiseKellogUniversityofCalifornia.

Data:LouiseKellogg,DepartmentofGeology,UniversityofCalifornia,Images:LouisFengUCD.

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Topologicalstructure§  Segmenta0onofdomaininareasofuniformdirec0onalbehavior§  Similartovectorfieldtopology–butdifferentbasicstructures

Topology

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OnSurfaces[Auer2012]

§  TopologybasedSegmenta0on

§  TopologybasedTextures[Auer2013]

Tensorfielddesign[Zhang2007]

3Dtopology,cri0callines[Zheng2005]

Topology

Computa0on,Simplifica0on,Tracking[Tricoche2001-2003]

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Sometechnicalstuff

•  Glyphsfromallperspec0ves•  Technicalgeneraliza;onofvectorfieldvisualiza;onmethods

•  Textures

Whatiss;llmissing

•  Applica0ons–linkformvisualiza0ontechniquestophysicalinterpreta0ons•  No0onoffeatures•  Ques0ons

Muchworkontensorprocessing

•  Interpola0on/reconstruc0on•  Morphology•  Topology

Whathasbeendonesofar?

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IIITensorInvariantsforfeaturedefini0on

•  Exploara;veframeworksuppor0ngmul0pleapplica0onswithdifferentques0ons

àDomainspecificfeaturespaces

•  Structuringthedata

�  Managecomplexityofthedata

�  Highlighttrends

�  Pointatcri0cal/atypicalbehavior

àDataatlasusingathumbnaillike

representa;on

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•  Explora;veframeworksuppor0ngmul0pleapplica0onswithdifferentques0ons

àDomainspecificfeaturespaces

•  Structuringthedata

�  Managecomplexityofthedata

�  Highlighttrends

�  Pointatcri0cal/atypicalbehavior

àDataatlasusingathumbnaillike

representa;on

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Tensorinvariant

•  Arethelanguageoftheapplica0ons

Tensorisuniquelydefinedby•  3eigenvalues à pointin‘shape-space’•  3eigenvectors à pointin‘direc;on-space’

Useapplica0onspecificinvariantstoparameterizetheshapespace

à  shapedescriptorsIi(λ1,λ2,λ3)

à  basisforthedefini0onsta0s0cviews,glyphdesign,similaritymeasure

Tensorinvariant

•  Arethelanguageoftheapplica0onsTensorinvariant

•  Arethelanguageoftheapplica0ons

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TheStressTensorandFailureModels

ExampleforfailureanalysisinmechanicalengineeringàMohrCircle

τmax

c

τ0

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Example•  Orderedshapespace

λ1 ≥ λ2 ≥ λ3

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Example–Coulombfailure•  Parametrizedby�  Mohrcenter

�  MaxShear

�  Shapefactor

c = λ1 + λ32

τ = λ1 − λ32

R = λ1 − λ2λ1 − λ3

∈[0,1]

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Clusterinfeaturespace Clusterrepresenta0ves•  Reynoldsglyphs

Spa0alview:Topviewofdata


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Clusterinfeaturespace Clusterrepresenta0ves•  Mohrcircles

Spa0alview:Topviewofdata


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Top-viewofthetwo-pointload


Clustering•  Meanshie•  Fullfeaturespace+spa0alcoordinates

Clustersta0s0cs•  Clustersize

•  Direc0onalhistogramλ1

xy

-z

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Two-pointload

Spa;al-view


Clustering• Meanshie•  Featurespace(shear,shape)+spa0alcoordinates

Clustersta;s;cs

•  Clustersize

•  Direc0onalhistogram

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Feature Space Selection §  Change of basis results in another set

of shape descriptors §  Determines

§  Exploration space §  Metric, similarity measure §  Glyph design

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TensorVisualiza0onDrivenMechanicalComponentDesignorThestoryofacollabora0on

3D CAD-modelVirtual evaluation:

Simulation +Tensor line visualization

Rapid Prototyping and experimental validation:

Test setupBoundary geometry:

Design space definitionPart design: Hand sketch

guided by tensor lines

Kratz,Schoeneich,Zobel,Burgeth,Scheuermann,Hotz,Stommel,PacificVis2014

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IVStoryofaCollabora0on

Star0ngpoint•  Unspecificgoals•  Different

language

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FirstExperiments•  Everybodyshows

whattheyhave

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Visualiza0on–Framework

Images: Kratz, ZIB, Amira

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Engineering–ProductDevelopmentProcess

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TensorsinMaterialSciencesEngineering–ProductDevelopmentProcess

Adds a completely new perspective to the topic

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Engineering–ProductDevelopmentProcess

ConstraintsRequirementSpeci1icationsDesign4rulesExperience

Initial'structure'

Structure4optimization

Introduce4visualization4using4tensor4lines

to4support4this4process

Experimental'validation

CAD7Model

FE7'simulationvalidation

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FirstExperiments•  Everybodyshows

whattheyhave

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!"#$%&'(#%$)*+,(&*-*#%./*0(1(0'%("#$

!"#$#%&'($)*+$*),'

-./,)#0,"$%&'1%&#2%$#3"

4567832,&

9-7'(#0*&%$#3"1%&#2%$#3"

2($,'3(4'%("#5,$(#65%*#$"&53(#*$%"5$,//"&%5%7*

"/%(-(4'%("#5/&"0*$$

.%&,0%,&*5"/%(-(4'%("#8'$*95"#5*:/*&(*#0*5'#959*$(6#5&,3*$


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Hypothesisresul0ngfromfirstexperimentsanddiscussions

•  Thetensorlinesforastresstensorfieldarerelatedtothemajorloadpathsfromtheopera;ngloadstothefixa;on

pointsofatechnicalpart

•  Tensorlinescanbeusedtosupportthedesignofreinforcementstructures

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Design of a break lever •  Buildreinforcementstructureonbasis

oftensorlinevisualiza0on•  Combina;onofanautoma;canalysis

withtheexpert’sknowledge

•  Manageablebutrealis0ccase

Evalua;on

•  Comparisontoreferencestructure–samevolume

•  Numericalvalida0on•  Experimentalvalida0on

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Designoftheribstructure

•  Newgeometriesbasedontensorvisualiza0on

•  Noaddi0onalstepsintheworkflow

Stress Tensor Visualization

CAD - Model Vis supported manual design

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Numericalandexperimentaltests

•  Newdesigns(red,green,yellow)•  Referencegeometry(blue)

àAllnewgeometriesperformedbexerasthereferencegeometry

3D CAD-modelVirtual evaluation:

Simulation +Tensor line visualization

Rapid Prototyping and experimental validation:

Test setupBoundary geometry:

Design space definitionPart design: Hand sketch

guided by tensor lines

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IVStoryofaSuccessfulCollabora0on

AdvancesinbothfieldsCon0nuingStory•  Manyexi0ngques0ons

andideas•  Muchfun•  Proposalsubmixed

Tensors as mathematical objectsproperties and theorems

Tensors as physical descriptors

Tensor invariantsas means for

communication

Visualizationgeneric <-> specific

Relevance and meaning of mathematical properties and theorems is application dependent

•  AnisotropicSamplingofPlanarandTwo-ManifoldDomainsforTextureGeneraAonandGlyphDistribuAon,Kratz,Baum,Hotz,TVCG,2013

•  Three-DimensionalSecond-OrderTensorFields:ExploratoryVisualizaAonandAnisotropicSampling(phdthesis),AndreaKratz,2013

•  VisualizaAonandAnalysisofSecond-OrderTensors:MovingBeyondtheSymmetricPosiAve-DefiniteCase,Kratz,Auer,Stommel,Hotz,ComputerGraphicsForum-StateoftheArtReports,2013

•  TensorInvariantsandGlyphDesign,Kratz,Auer,Hotz,Visualiza0onandProcessingofTensorsandHigherOrderDescriptorsforMul0-ValuedData(Dagstuhl'11),Springer,2014

•  hVp://www.tensorvis.org/

SINTEF - Tensor field visualiza0on – My view on the field … · 2016-01-22 · Geilo winter School – Scien0fic Visualiza0on Ingrid Hotz – Linköping ... Ingrid Hotz …

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