Wavelet Representation of Contour Sets Martin Bertram Daniel E. Laney Mark A. Duchaineau Charles D. Hansen Bernd Hamann Kenneth I. Joy Abstract: We present a new wavelet compression and mul- tiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and com- pactly stores this parametrization rather than function values sam- pled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of co- efficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the cor- responding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to vol- umetric domains, time-varying contours, and level sets of vector fields. Keywords: Contours, Geometry Compression, Isosurfaces, Level Sets, Multiresolution Methods, Wavelets 1 Introduction Scientific visualization methods help us to explore and understand the nature of vast amounts of digital data produced by numerical simulations on supercomputers or by imaging technology like com- puter tomography. Visualizing scalar fields via exploration of their isosurface behavior is one of the most powerful ways to gain in- sight into a physical phenomenon. Our approach is driven by the need to explore very large scalar fields interactively by browsing through their continuous space of contours. In the past, multires- olution methods were developed for the modeling, rendering, and exploration of complicated two-manifold data, e.g., large-scale iso- surfaces [1]. In order to explore the entire contour space of a scalar field more powerful methods are required, as entire families of con- tours have to be extracted, represented, and rendered. The approach we are presenting here is driven by such considerations. We intro- duce a new framework for the multiresolution approximation of a multitude of contours defined by a single scalar field. This frame- work promises to have significant impact on state-of-the-art visual- ization and exploration of truly massive, tera-scale scalar field data. Visualization methods often rely on continuous geometric mod- els representing the relevant topological and geometric features of a data set. Multiresolution modeling techniques, like wavelet trans- forms [3, 16], provide efficient progressive access to local geome- try. Wavelet transforms coupled with progressive coders for quan- tized coefficients are among the most efficient schemes for com- pression, error-driven querying, and progressive transmission of 1 University of Utah, SCI Institute, 50 S. Central Campus Drive, Room 3190, Salt Lake City, UT 84112, USA. 2 University of Kaiserslautern, Department of Computer Science, P.O. Box 3049, D-67653 Kaiserslautern, Germany. 3 Center for Applied Scientific Computing (CASC), Lawrence Livermore National Laboratory, P.O. Box 808, L-561, Livermore, CA 94551, USA. 4 Center for Image Processing and Integrated Computing (CIPIC), De- partment of Computer Science, University of California, Davis, CA 95616- 8562, USA. data defined on regularly gridded domains [15, 18]. When visu- alizing derived quantities or features, such as contours, these need to be extracted from a locally reconstructed geometric model. This extraction process can be very expensive, especially in the case of volume data, since an unknown surface topology needs to be recov- ered. Standard wavelet compression algorithms [18] transform a func- tion into wavelet coefficients of expectedly small absolute values. These coefficients are quantized (rounded to integers) or thresh- olded (selected by magnitude of absolute values) and compressed by a progressive coding scheme like zero trees [15]. When extract- ing contours from highly compressed data altered by a quantization error, there exists no guarantee of obtaining topologically correct contours. When reconstructing data from thresholded or quantized wavelet coefficients, for example, the resulting contours may even have additional components enclosing local extrema of the recon- struction error, see Figures 1 and 2. The wavelet approach presented here overcomes this problem by compressing a parametrization of a field function that is induced by its contours, rather than compressing a field function directly. Our approach also simplifies the topology of represented contours. However, this simplification is performed in an initial step of our algorithm, where a finite set of selected contours, called base con- tours, is extracted. All other contours represented by our method have the topology of a corresponding base contour of the closest isovalue. Since the set of base contours can be chosen arbitrarily, our method introduces a predictable topological error reducing the quantity of topological changes that need to be stored. The topol- ogy of contours represented by our method is invariant under the level of detail such that the topological error is not augmented in coarse representations. Starting with the set of base contours, we construct a coarse mesh structure, the base mesh, covering the domain of the underlying field function. This base mesh is recursively subdivided, and its ver- tices are projected onto intermediate contours. The resulting adap- tive mesh structure is equivalent to a subdivision surface/volume with displacement of vertices correcting the geometry at finer levels of detail. Our algorithm represents these displacements compactly in the form of sparse wavelet coefficients. The contours produced by our subdivision process are either linear or cubic polynomials. We represent a scalar field by a continuous parametrization of its domain that is defined by a subdivision surface/volume. This parametrization is a function mapping a manifold into Euclidean space. In the case of planar contours, our manifold domain has one global parameter specifying the isovalue and one local parameter traversing the corresponding contour. (In the case of isosurfaces of trivariate functions, our manifold has one parameter for the isovalue and two local parameters traversing an isosurface.) The coarsest level of resolution is defined by a base mesh providing both, the manifold topology and a coarse parametrization obtained by recur- sive subdivision. During the subdivision process, geometric detail can be expanded from wavelet coefficients resulting in representa- tions at higher level of resolution. Our representation of contour sets is equivalent to a representa- tion of the underlying field function, but it provides rapid access to every contour at multiple levels of resolution. This is a highly desirable property for real-time visualization of contours, allowing
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Wavelet Representation of Contour Sets
Martin Bertram��� �
DanielE. Laney�
Mark A. Duchaineau�
CharlesD. Hansen�
BerndHamann�
KennethI. Joy�
Abstract: We presenta new wavelet compressionand mul-tiresolution modeling approachfor setsof contours(level sets).In contrastto previous wavelet schemes,our algorithm createsaparametrizationof a scalarfield inducedby its contoursandcom-pactlystoresthis parametrizationratherthanfunctionvaluessam-pledon a regulargrid. Our representationis basedon hierarchicalpolygonmesheswith subdivision connectivity whoseverticesaretransformedinto wavelet coefficients. From this sparsesetof co-efficients,every setof contourscanbe efficiently reconstructedatmultiple levels of resolution. When applying lossy compression,introducinghighquantizationerrors,ourmethodpreservescontourtopology, in contrastto compressionmethodsappliedto the cor-respondingfield function. We provide numericalresultsfor scalarfieldsdefinedon planardomains.Our approachgeneralizesto vol-umetric domains,time-varying contours,and level setsof vectorfields.
Keywords: Contours,GeometryCompression,Isosurfaces,LevelSets,MultiresolutionMethods,Wavelets
1 Intr oduction
Scientificvisualizationmethodshelpusto exploreandunderstandthe natureof vastamountsof digital dataproducedby numericalsimulationsonsupercomputersor by imagingtechnologylikecom-putertomography. Visualizingscalarfieldsvia explorationof theirisosurfacebehavior is oneof the mostpowerful ways to gain in-sight into a physicalphenomenon.Our approachis driven by theneedto explore very large scalarfields interactively by browsingthroughtheir continuousspaceof contours. In the past,multires-olution methodsweredevelopedfor the modeling,rendering,andexplorationof complicatedtwo-manifolddata,e.g., large-scaleiso-surfaces[1]. In orderto exploretheentirecontourspaceof ascalarfield morepowerful methodsarerequired,asentirefamiliesof con-tourshaveto beextracted,represented,andrendered.Theapproachwe arepresentinghereis drivenby suchconsiderations.We intro-ducea new framework for the multiresolutionapproximationof amultitudeof contoursdefinedby a singlescalarfield. This frame-work promisesto havesignificantimpactonstate-of-the-artvisual-izationandexplorationof truly massive, tera-scalescalarfield data.
Visualizationmethodsoftenrely on continuousgeometricmod-els representingtherelevant topologicalandgeometricfeaturesofadataset.Multiresolutionmodelingtechniques,likewavelettrans-forms[3, 16], provide efficient progressive accessto local geome-try. Wavelet transformscoupledwith progressive codersfor quan-tized coefficients are amongthe mostefficient schemesfor com-pression,error-driven querying, and progressive transmissionof
1University of Utah,SCI Institute,50 S. CentralCampusDrive, Room3190,SaltLake City, UT 84112,USA.
2University of Kaiserslautern,Departmentof ComputerScience,P.O.Box 3049,D-67653Kaiserslautern,Germany.
3Centerfor AppliedScientificComputing(CASC),LawrenceLivermoreNationalLaboratory, P.O. Box 808,L-561,Livermore,CA 94551,USA.
4Centerfor ImageProcessingandIntegratedComputing(CIPIC), De-partmentof ComputerScience,Universityof California,Davis, CA 95616-8562,USA.
datadefinedon regularly griddeddomains[15, 18]. When visu-alizing derivedquantitiesor features,suchascontours,theseneedto beextractedfrom a locally reconstructedgeometricmodel.Thisextractionprocesscanbevery expensive, especiallyin thecaseofvolumedata,sinceanunknown surfacetopologyneedsto berecov-ered.
Standardwaveletcompressionalgorithms[18] transformafunc-tion into wavelet coefficientsof expectedlysmall absolutevalues.Thesecoefficients are quantized(roundedto integers)or thresh-olded(selectedby magnitudeof absolutevalues)andcompressedby a progressive codingschemelike zero trees[15]. Whenextract-ing contoursfrom highly compresseddataalteredby aquantizationerror, thereexists no guaranteeof obtainingtopologicallycorrectcontours.Whenreconstructingdatafrom thresholdedor quantizedwaveletcoefficients,for example,theresultingcontoursmayevenhave additionalcomponentsenclosinglocal extremaof the recon-structionerror, seeFigures1 and2.
The wavelet approachpresentedhereovercomesthis problemby compressingaparametrizationof afield functionthatis inducedby its contours,ratherthancompressinga field function directly.Our approachalsosimplifiesthetopologyof representedcontours.However, this simplification is performedin an initial stepof ouralgorithm,wherea finite setof selectedcontours,calledbasecon-tours, is extracted. All othercontoursrepresentedby our methodhave the topologyof a correspondingbasecontourof the closestisovalue. Sincethe setof basecontourscanbe chosenarbitrarily,our methodintroducesa predictabletopologicalerrorreducingthequantityof topologicalchangesthatneedto bestored.The topol-ogy of contoursrepresentedby our methodis invariantunderthelevel of detail suchthat the topologicalerror is not augmentedincoarserepresentations.
Startingwith thesetof basecontours,weconstructacoarsemeshstructure,the basemesh, covering the domainof the underlyingfield function.Thisbasemeshis recursively subdivided,andits ver-ticesareprojectedontointermediatecontours.Theresultingadap-tive meshstructureis equivalent to a subdivision surface/volumewith displacementof verticescorrectingthegeometryatfiner levelsof detail. Our algorithmrepresentsthesedisplacementscompactlyin the form of sparsewaveletcoefficients. Thecontoursproducedby oursubdivision processareeitherlinearor cubicpolynomials.
We representa scalarfield by a continuousparametrizationofits domainthat is definedby a subdivision surface/volume. Thisparametrizationis a function mappinga manifold into Euclideanspace.In thecaseof planarcontours,ourmanifolddomainhasoneglobal parameterspecifyingthe isovalueandonelocal parametertraversingthecorrespondingcontour. (In thecaseof isosurfacesoftrivariatefunctions,ourmanifoldhasoneparameterfor theisovalueand two local parameterstraversingan isosurface.) The coarsestlevel of resolutionis definedby a basemeshproviding both, themanifoldtopologyanda coarseparametrizationobtainedby recur-sive subdivision. During thesubdivision process,geometricdetailcanbeexpandedfrom waveletcoefficientsresultingin representa-tionsathigherlevel of resolution.
Our representationof contoursetsis equivalentto a representa-tion of the underlyingfield function, but it provides rapid accessto every contourat multiple levels of resolution. This is a highlydesirablepropertyfor real-timevisualizationof contours,allowing
Figure1: Contoursof a ������� piecewisebilinearly interpolatedsliceof a volumedataset,taken from a numericalsimulationof aRayleigh-Taylor instability.
for interactively changingisovaluesandrenderingmultiple trans-parentisosurfacesat once. Our representationprovidesadditionalflexibility for algorithmsprocessingcontourswith thegoal of im-proving the underlyingfield function. For example,constrainedfairing of all contoursof a field function is a non-trivial operationthatbecomesfairly simplewhenusingourapproach.
2 Related Work
Multiresolutioncontouringschemesextract isosurfacesfrom hier-archicalscalarfield representationsproviding multiple levelsof de-tail. Weberet al. [19] presentanefficient constructionmethodforcrack-freeisosurfacesfrom adaptively refinedhexahedraldomains.A similarapproachusingahierarchicaloctreestructurefor interac-tive view-dependentcontouringis presentedby Westermannet al.[20]. A real-timerenderingapproachfor multiple transparentiso-surfacesreconstructedfrom atetrahedralgrid hierarchyisdescribedby Gerstner[4].
Woodetal. [21] useasurfacewavefrontpropagationmethodforconstructinga coarsebasemeshapproximatingan isosurfacewithcorrecttopology. Their approachprovidesa semi-regular triangu-lar subdivision hierarchyof anisosurfacethat is usefulfor waveletcompression.In previous work, we have constructedquadrilateralbasemesheswith subdivision hierarchythatwereusedfor waveletcompressionof isosurfaces[1]. Our wavelet constructionfor sub-division surfaces[2] generalizesto higher dimensions,e.g., vol-umesof manifold topology, like level setsand time-varying sur-faces.Waveletconstructionsfor subdivision surfaceswereinitiallydescribedby Lounsberyetal. [8, 16].
Whenusingwavelet approachesfor geometrycompression[6],it becomesimportantto constructsmoothsurfaceparametrizationsby improving theregularityof controlmeshes.For trianglemeshes,suchregular parametrizationsareconstructedby the MAPSalgo-rithm describedby Lee et al. [7]. Similar algorithmsneedto bedevelopedfor pseudo-regular meshingof three-dimensionallevelsets. A multiresolutionapproachfor matchingcontoursdefinedon differentcutting planesis presentedby Meyers[10]. Efficientmeshingalgorithmsfor level setsaredescribedby Sethian[14].
To our knowledge,previous methodshave not attemptedto re-parametrizesetsof contoursfor the purposeof wavelet compres-sion. Hence,our generalapproachis innovative, combiningindi-
Figure2: Contoursof the scalarfield shown in Figure1, recon-structedform thresholdedwavelet coefficients(12 percent),usinga linear splinewavelet. This compressedrepresentationdoesnotpreserve thetopologyof extractedcontours.
vidual techniquesfrom differentfields,suchascontourextraction,meshgeneration,andsubdivision surfacewavelets.
3 Adaptivel y Representing Contour Sets
This sectiondescribesour novel multiresolutionapproachfor setsof contours.We describeour algorithmin thecontext of bivariatescalarfields andprovide extensionsto volumetricdomains,time-varyingcontours,andlevel sets.
3.1 Overview of the Algorithm
Ouralgorithmfirst constructsacoarsebasemeshinducedby certainbasecontours.Thismeshis thenregularlysubdivided,andthenewverticesareprojectedontointermediatecontours.Finally, we useawavelettransformfor compressionandmultiresolutionmodelingofthis meshstructure,definingsmoothsetsof contoursby recursivesubdivision. Our algorithmconsistsof the following stepsthatareillustratedin Color Plates(a–f):
1. Extractionof a prescribedsetof basecontours, using,for ex-ample,uniformly distributed isolevels. This setof contoursdefinesthe topologyof all intermediatecontoursrepresentedby ourscheme.
2. Samplingbaseverticesdistributeduniformly with respecttoarc lengthsfrom theextractedbasecontours.Theseverticeswill representthecoarsestlevel of detail for our parametriza-tion. Hence,thesetof basecontoursselectedin step1 shouldnotbetoodense.
3. Constructinglinks betweenbaseverticeson adjacentbasecontoursandrelaxingtheselinks by moving the basepointson their correspondingcontours.
4. Filling the spacebetweenadjacentbasecontoursand theirlinks with convex polygonsthathave low numbersof edges.The resulting basemeshserves as coarsestlevel of detail,defining a smoothparametrizationof contourswhen recur-sively subdivided (using,for example,Catmull-Clarksubdi-vision [9]).
Figure3: Dyadicrefinementof aclosedandanopencontourcom-ponent
∆ � ��
�
�
�Figure4: Constructinglinks betweencontours��� and ��� by fol-lowing thegradientstartingat basevertices.Someverticescannotbelinkeddueto field regionsof zerogradient.
5. Regular subdivision of the basemeshby insertingnew ver-ticesat thecentroidof everypolygonandin themiddleof ev-ery edge.Theverticesobtainedby subdivision areconnectedto define a quadrilateralmeshstructurethat is recursivelyrefined. Insteadof applying stationarysubdivision, whichwould resultin incorrectintermediatecontours,thenew ver-ticesareprojectedontocorrespondingcontours.This subdi-visionprocessterminatesataresolutionslightly finer thanthegrid resolutionof theunderlyingfield function.
6. Subdivision-surfacewavelets[1, 2] areusedto generatea hi-erarchyof continuousparametrizations.The differencesbe-tweenindividual levelsof detailarecompactlyrepresentedbywavelet coefficients. Data compressioncan be achieved bythresholdingor by encodingquantizedcoefficients[11, 15].
3.2 Constructing Base Meshes
As a first stepwe extracta finite setof basecontoursusinga stan-dardapproach.Thecorrespondingisovaluescanbeuniformly dis-tributedor they canbemoredenselysampledin certainregionsofinterest.Thenwedefinebaseverticesby re-samplingthebasecon-toursat approximatelyequidistantintervals of arc length � . Thevalueof � dependson thenumber��� of dyadicrefinementlevels,seeFigure3, thatwe will computeandon thefinestsamplingdis-tance � , which shouldbe slightly smallerthanthe edgelengthoftheregulargrid definingthefield function.Hence,we use
����� �"!���# (3.1)
All boundarypointsof contoursneedto bebasevertices,suchthatthe basecontourscan be completelygeneratedby dyadic refine-ment.Additionally, werequireeverycontourcomponentto haveatleastthreebasevertices,to avoid degeneratebasepolygons.
Thenext taskis to fill thespacebetweenevery adjacentpair ofbasecontours,say � � and ��� , with convex polygons. Therefore,it is desiredto connectmatchingpairsof baseverticesfrom both
$
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%
&Figure5: Relaxinga link betweenbaseverticeson contours���and ��� by displacingtheseverticesalongtheir contours.
Figure6: Splittinganon-convex polygon.Left: invalid split; right:correctsplit.
contours,which will improve thefairnessof our final parametriza-tion. We useNewton-iterationto propagatethebaseverticesof ���onto the contour ��� . In eachstepof this iteration,the movementof a vertex is restrictedto a maximaldistanceof � , to avoid diver-gencedueto shallow gradients.Someverticeswill not convergetothecontour ��� , sincethey maygetstuckat local extremaor zero-gradientareas,andthe iterationmust terminateafter a prescribednumberof steps. Thoseverticesthat converge to contour ��� arelinked to the closestbasepoint on ��� andtheir initial positiononcontour � � is restored.To find all possiblelinks, this stepof thealgorithmis repeatedwith theverticesof ��� , iteratingtowardscon-tour ��� , seeFigure4. Baseverticeslocatedon theboundaryof thedatasetaresimply connectedby traversingthis boundary.
The length of the individual links betweenevery pair of baseverticesis minimizedby an iterative procedure,allowing thebaseverticesto move a certaindistancealongtheir correspondingcon-tours, seeFigure5. Here, we restrict the maximal displacementof a vertex to the value ' ( , to avoid coincidenceof adjacentbasevertices.This stepis necessaryto improve smoothnessof thefinalparametrizationandto avoid intersectionsof polygonstripsdefinedby thebaseverticesof every contourcomponent.Isolatedcompo-nents,“islands”,areconnectedby oneadditionallink to theclosestbasevertex on its surroundingcontourcomponent.
Themeshstructureresultingfrom thisprocedurealreadydefinesasetof closedpolygonscoveringthescalarfield domain.However,somepolygonsmay still be very large andnon-convex andneedto be subdivided further. Additionally, we needrepresentthesepolygonsexplicitly. For this purpose,we traverseevery polygonin counter-clockwiseorientationof edgesandrecordtheparticipat-ing basevertices.We useevery basevertex asa startingpoint forconstructinga potentialpolygon. Every edgein themeshhastwoassociatedflagsfor traversalin eachdirection,whicharesetwhenapolygonis constructed.Theseflagsaretestedfor every traversaltoavoid multiple constructionsof thesamepolygon.Theconstructedpolygonsarethenrecursively split until they areconvex andcon-sist of no morethanfive edges.Splitting a polygonis performedby connectinga pair of close,non-adjacentvertices,avoiding self-intersectionsandaugmentationof theenclosedregion in caseof anon-convex polygon,seeFigure6. Theresultingsetof polygonsisaconvex tessellationof thedomain,thebasemesh.
) *
) *) +
) +
Figure7: Regularmeshrefinementneara saddlepoint locatedinthecenterof thefive-sidedpatch.
Figure 8: Mapping intermediateisovalues to verticesdefinedby subdivision. The first refinementstepgeneratesquadrilateralpatchesthathaveeitherone,two, or threeverticeslocatedonabasecontour.
3.3 Regular Mesh Refinement
Oncewe have generatedour basemesh,we apply recursive sub-division usingtherefinementconnectivity of Catmull-Clarksubdi-vision insertingverticesat the centroidsof polygonsandon theiredges.The first subdivision stepgeneratesquadrilateralsthat areregularly refinedin thesubsequentsteps,asillustratedin Figure7.Insteadof applyingstationarysubdivision rulesto computetheco-ordinatesfor verticeson finer levels, we placethem on interme-diatecontours.Thesubdivision processterminatesat a resolutionslightly finer thantheresolutionof theinitial rectilineargrid defin-ing thefield function. This meshhierarchyis thencompressedus-ing wavelets.
Beforewe canproject the new verticesonto intermediatecon-tours,we have to defineanisovaluefor every vertex. After thefirstsubdivision step,theresultingverticesareeitherlocatedon a basecontouror placedin the spacebetweentwo basecontours. In thelattercase,theseverticeswill beassociatedwith theaverageof bothcorrespondingisovalues.For thesubsequentlevelsof regular, recti-linearrefinement,we usethetemplatesillustratedin Figure8: ver-ticeslocatedonedgesareassignedtheaverageisovalueof bothin-cidentvertices.Verticeslocatedinsidea quadrilateralareassignedtheaverageof theminimalandthemaximalisovaluesof thequadri-lateral’s four cornervertices.
Every vertex is projectedonto a contourwith the correct iso-value.For thispurpose,weuseaconstrainedNewton iterationcou-pled with Laplaciansmoothingof the mesh(moving every vertexto the centroidof its neighbors). In every stepof the Newton it-eration,a vertex is propagatedalongthe gradientof the field andsubsequentlyrelaxed orthogonalto the gradientby projectingtheLaplaciandisplacementonto a vector/planeorthogonalto thegra-dient. Again, themaximaldisplacementis limited by thedistance� . Due to the topology simplification imposedby the choiceofbasecontours,someverticescannotbe projectedonto the correctcontour, sincea nearbycomponentof this contourdoesnot exist.In this case,therelaxationpreventsthemeshfrom entangling.Theiterationprocessmustterminateaftera finite numberof steps.
, -
, .
, .
, -
, .
, .Figure9: Topologyof contour � � changesinto topologyof ��� atanintermediatecontour. Thetopologyof this intermediatecontour(left) cannotberepresentedcorrectly(right).
Figure10: Algorithm appliedto a surfacewith two critical points(a saddlepoint anda local minimum). Small regionsof the meshcollapsenearthesepoints,dueto topologicalerror.
Besidescritical points(pointswherecontourtopologychanges,e.g., saddlepointsandlocal extrema),the worst-casescenarioarelong “headlands”in the scalarfield, wherethe meshis eithercol-lapsedor stretchedalonga ridge, seeFigure9. However, the ge-ometricerror of every meshvertex is boundedby onehalf of thesamplingdistancefor baseisovalues.Thebehavior of our meshingalgorithmis shown in Figure10andColor Plates(a–f).
Fromthemeshstructurewehaveconstructed,everycontourcanbe derived immediatelyby linear interpolationof its closestcon-toursthatareexplicitly representedin themesh.Alternatively, wecan use a subdivision scheme,like Catmull-Clark, to refine themeshsmoothly. We do not needto storethe isovaluesassociatedwith everyvertex, sincethesecanberecoveredfrom thebasemesh.
3.4 Subdivision-surface Wavelets
Startingwith ourregularlyrefinedmeshhierarchycomposedof ver-ticeslocatedoncertaincontours,wecanefficiently deriveany setofcontoursusingsubdivision andlinear interpolation. For compres-
/ 0 1 0 2
3
2 23 3
/ 4 1 0 5 6
7 5 8 4 9 4 : ; 6 < 1 5 ; 8 = ; 0 9 : 4> > >
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Figure11: Modelingparadigmof a wavelettransform.
@ AA BB C D B
@ E BF F F F F FG GH
Figure12: Vertex manipulationdefinedby ans-lift operation.
sion purposesand level-of-detail rendering,we needa multireso-lution representationof this meshstructureproviding theseopera-tions:I
Subdivision. This operationdefinesstationarysubdivisionrulesproviding a continuouslimit surfacewhenappliedre-cursively. Themeshverticescorrespondto controlpointsofsmoothbasisfunctions.JExpandingdetail.At every level of refinement,geometricde-tail canbeaddedto a subdivision surface.This detail is com-pactly storedin the form of wavelet coefficients andcanbeexpandedfrom these.KFitting. This operationreversesa subdivision step.Basedonall verticesonafinelevel, theverticesonthenext coarserlevelarepredictedsuchthatthey provide a goodapproximationtothefine level whenapplyingsubdivision,again.LCompactingdetail. Thedifferencebetweentwo levelsof res-olution, i.e., thedisplacementof meshverticeswhenapplyingK
followedby
I, is compactlystoredin form of waveletco-
efficientsthatreplacetheverticesremovedbyK
.
Themodelingparadigmof suchamultiresolutionrepresentationis illustratedin Figure11. Thesefour operationsdefinea wavelettransformfor subdivision surfaces.We have constructedwaveletsfor bilinearandbicubicsubdivisiongeneralizedto arbitrarymesheswith regularrefinement[2] andusedthebicubicwavelet transformfor multiresolutionmodeling of large-scaleisosurfaces[1]. Wesummarizethe detailsnecessaryto implementthesetransformsintheremainderof this section.
Our wavelet transformsare computedby a few local vertexmanipulations,called lifting operations[17], since they can beusedto manipulatethe shapeof basisfunctions. Consideringapolygon strip composedof vertices MON and its dyadic refinementwith verticesPQN locatedon theedgesMRNSMONUTWV , we definetwo liftingoperations:
s-lift XSY[Z]\_^ : M Na` YbP NScOVad \eM Nfd YbP N # (3.2)
w-lift XSY[Z]\_^ : PQN ` YbMRN d \gPQN d YbMRNhTWVi# (3.3)
Figure13: LinearandcubicB-splinewavelets
jA@
Figure14: Regularly refinedmesh,composedof verticesof typesM (coarseresolutionmesh),P , and k (waveletcoefficients).
An s-lift operationis illustratedin Figure12. This operationma-nipulatescoefficientsassociatedwith scalingfunctionsrepresentingthe individual levels of resolution,anda w-lift operationmanipu-latescoefficients associatedwith waveletsrepresentinggeometricdetail,i.e., displacementsbetweentwo levels.
The operations
Iand
Jdefinethe reconstructionor synthesis,
which is one stepof an inversewavelet transform. The verticesM N representinitially a coarselevel of resolutionand the verticesPQN contain wavelet coefficients. After a reconstructionstep, allverticesrepresentthe next finer level of resolution. An inversewavelet transformis computedby repeatedreconstructionstartingwith a coarsebasepolygon. The reconstructionproceduresforwavelet transformsbasedon dyadicrefinementof linearandcubicB-splinesaredefinedasfollows:
Linear B-spline wavelet reconstruction:
lnmporqtshu X m Vv Ziw ^yxz mporqtshu X V{ Z�w ^y#Cubic B-spline wavelet reconstruction:
lnmporqtshu X m (| Ziw ^yxz mporqtshu X V{ Z�w ^yxlnmporqtshu X Vv Z V{ ^yZThe basisfunctionsof the transformcorrespondingto a waveletcoefficient aredepictedin Figure13.
.The operations
Kand
Lrepresentwavelet decompositionor
analysis, which is theinverseof a reconstructionstep.Thedecom-position formulaefor our one-dimensionalwavelet constructionsare defined by the inverse of every individual lifting operationappliedin reverseorder. Decompositionis definedasfollows:
Linear B-spline wavelet decomposition:
z mporqtshu X m V{ Ziw ^yxlnmporqtshu X Vv Z�w ^y#
}
} }
}~~~
~��
}
} }
}~
~~ ~
Figure15: A two-dimensionals-lift is computedby applyingitsone-dimensionalequivalentto therows andcolumnsof a regularlyrefinedgrid.
}
} }
}~~~
~� �}
} }
}~
~~ ~
Figure16: Two-dimensionals-lift operationperformedin differentorderof vertex updates.
Cubic B-spline wavelet decomposition:l�mporqrsUu X m V{ Zg�"^yxz mporqrsUu X m V{ Z�w ^yxl�mporqrsUu X (| Ziw ^y#A wavelettransformis computedby repeateddecomposition,start-ing with a fine resolutionandterminatingat the coarseresolutionof a basepolygon.
We now describethe generalizationof theselifting operationsto polygonmesheswith regularsubdivision hierarchy. Therefine-mentof a regular, rectilinearmeshis illustratedin Figure14. Theverticescorrespondingto waveletcoefficientsarelocatedon edgesandpolygons(faces)of thecoarsemeshandaredenotedby P , andk , respectively. On a completelyregular mesh,a lifting operationis performedby applyingthecorrespondingone-dimensionaloper-ation to the rows andcolumns,seeFigure15. Insteadof updatingthe vertices M twice in an s-lift operation,we canchangethe or-derof computationsuchthatevery vertex is modifiedonly once,asillustratedin Figure16.
The correspondingtwo-dimensionallifting operationscan bedefinedin anotationwithout indices,where��� denotestheaverageof theverticesof type � thatareadjacentto vertex � (or thatbelongto thecloseststencilaround� ). For example,MR� is themidpointofan edgeand MO� is the centroidof a polygon. Using this notation,thetwo-dimensionallifting operationsaredefinedlike this:
2D s-lift XSY[Z]\_^ :P ` \eP d � Y k � xM ` \ { M m ��Y { k � d ��Y P���# (3.4)
2D w-lift XSY[Z]\_^ :P ` \gP d �_Y M � xk ` \ { k m ��Y { M � d ��Y P � # (3.5)
Theadvantageof theseindex-freedefinitionsis thatthey canbeusedfor irregularmesheswith regularrefinement.notonly applica-ble to regularmeshes.Theselifting operationsarewell definedforextraordinaryvertices(verticesthatdonothavefour incidentedges)andfor arbitrarypolygonsin a basemesh. For a correcttransfor-mationof meshboundaries,we apply the one-dimensionallifting
Figure17: Generalizedbicubicscalingfunctionandwavelet.
operationsto all verticeslocatedon theseboundaries.The over-all wavelet transformis analogousto the one-dimensionaltrans-form, except that for all inner meshverticesequations(3.2) and(3.3) aresubstitutedby equations(3.4) and(3.5), respectively. Atwo-dimensionalscaling function and a wavelet are depictedinFigure17.
Startingwith the finest-level meshstructureconstructedin theprevious section,we computeour wavelet decompositionrepeat-edlyuntil wereachthebasemesh.Thebaseverticesthenrepresenta coarseapproximationthat is obtainedby subdivision without ex-pansionof detail. All verticesthat are not baseverticescontainwaveletcoefficientsthatcanbeusedto reconstructthesubdivisionlevel wheretheseverticeswereintroduced.For compressionpur-poses,we canquantizethewaveletcoefficientsandcompressthemusing,for example,arithmeticcoding.We notethatall coefficientshave two coordinates,sincethey representpointsandvectorsin theplane.
4 Extensions of our Algorithm
We outline somemodificationsto our algorithm that are neces-sary to representtime-varying contoursand to representthree-dimensionalmanifolds.
4.1 Time-v arying Contour s and Level Sets
Time-varying contoursand level sets,i.e., surfacesevolving overtime like shockwavesandmaterialinterfacesin fluid simulations,can be representedand compressedin the sameway as contoursets.A majordifferenceof time-varyingcurves/surfacesis thattheycanbecomeself-intersectingovertime,whereascontourspropagatelocally in only one direction when their isovaluesare monotoni-cally changed. Our algorithm for constructingbasemeshescan-not be usedfor curves/surfacesof this type, sinceit assumesthatthesetof basepolygonsprovidesa planartessellationwithoutself-intersections.
In general,it ispossibletoconstructmesheswith manifoldtopol-ogyapproximatingtime-varyingobjects.For thispurpose,weneedto constructamappingbetweenobjectsfrom consecutivebasetimesteps. For the caseof one-dimensionalcontours,a multiresolu-tion tiling algorithmis presentedby Meyers [10]. This algorithmconstructspolygonsconnectingcontourson differentplanescorre-spondingto different times steps. We could usethis methodforgeneratingbasemeshesof manifold topology that could then besubdividedrecursively anditeratively displacedontocontoursat in-termediatetimesteps.Level setandefficientmarchingmethodsformeshingtime-dependentsurfacesaredescribedby Sethian[14].
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Figure18: Regularsubdivision of polyhedra.Subdividing a pyra-mid resultsin for hexahedraandonetype-4cell.
4.2 Wavelet Representation of Three-manif olds
In the caseof time-varying surfacesor setsof static isosurfaces,latticescomposedof polyhedralcellsneedto beconstructed,con-nectingtwo surfacecomponentsof consecutive basetime stepsorfilling thespacebetweentwo adjacentbaseisosurfaces.Theselat-ticesarerecursively refinedbyplacingnew verticesinsideeachcell,on every face,andon every edge,seeFigure18. A generalizationof Catmull-Clarksurfacesto this typeof volumetricsubdivision isprovidedby MacCrackenandJoy [9].
Many typesof polyhedra,like prismsand tetrahedra,producehexahedraafter the first subdivision step,allowing for regular re-finement. Unfortunately, somepolyhedra,like pyramids,produceso-calledtype-� cells composedof � � d � verticesand � � faces.Thesereproducetwo type-� cells whensubdivided. To keepthemeshstructuresimple,it is desiredto avoid type-� cells,exceptforthecase����� (hexahedra).
Our wavelet transform generalizesnicely to volumetric (andhigher-dimensional)subdivision, since the individual lifting op-erationscan be computedby a sequenceof vertex-manipulationsfor every typeof vertex, analogouslyto the two-dimensionalcase.Whenappliedto a regularly griddeddomain,theselifting opera-tionsdefinetensor-productbasisfunctions.
5 Results
Wehave implementedandtestedouralgorithmfor scalarfieldsde-fined on planardomains. As an exampledataset, we have useda slice of rich geometricdetail taken from a three-dimensionalnumericalsimulationof a Rayleigh-Taylor instability, courtesyofLawrenceLivermoreNationalLaboratory. The initial slice is de-finedby � ��f� ����������� bytesamplesgivenonaregularlygriddeddomain. We extractedninebasecontoursat uniformly distributedisovalues,re-sampledat a resolution� of half the lengthof a gridedge.Weused� � = 3 levelsof subdivision for meshgeneration.
Our algorithmgenerateda basemeshcomposedof 656verticesand 661 polygons,resulting in 40947vertices(correspondingtowavelet coefficients) after threelevels of subdivision, which cor-respondsto anover-samplingfactorof aboutten. For our waveletrepresentation,weneedto storethecoordinatesof the656basever-tices(their isovaluescanberecordedby groupingverticesof samecontourstogetherinto alist), theconnectivity of thebasemesh,andthe wavelet coefficients, which can be quantizedand encodedathighcompressionrates.Hence,ourover-sampledrepresentationofcontoursmayuselessstoragespacethantheoriginal dataset.Thisbecomescrucialwhenconvertinglarge-scaledatasetsinto our rep-resentation.The computationallyexpensive part of our algorithmis theprojectionof verticesontocontours,whichrequiredlessthanten secondson an SGI � {
workstationusing a 180 MHz R5000processor.
We used our generalizedbilinear and bicubic wavelet trans-formsto computedifferentlevelsof resolution,obtainedby remov-ing waveletcoefficientson thehighest-resolutionlevelsandrecon-
Transform Level No. of coeff. � { -error � V -errornone 3 40947 0.37 0.06bicubic 2 10331 0.65 0.29bicubic 1 2631 1.18 0.72bicubic 0 656 2.61 1.89bilinear 2 10331 0.63 0.29bilinear 1 2631 1.00 0.61bilinear 0 656 1.82 1.29
Table1: Geometricerrorof representedcontoursrelative to incre-mentof isovalueat finestsubdivision level (index three).
structingthe meshat the finest level of refinement,obtainedafterthreesubdivisions.Thereconstructedmeshesaredepictedin ColorPlates(g–l). The finest-resolutionmeshis shown in Color Plate(f). We renderedthesemeshesby assigningthe samecolor to allquadrilateralslocatedbetweeneachpairof adjacentcontoursat thefinestlevel.
The geometricerrorsof all contoursthat are explicitly repre-sentedin the meshat finestlevel areshown in Table1. Theseer-rors representthe differencebetweenthe isovalueassociatedwitha meshvertex andthe real functionvalueof theunderlyingscalarfield at thevertex location. All errorsarerelative to thedifferenceof two adjacentcontoursrepresentedin the finestmesh. An errorlarger thanonemeansthat adjacentcontoursmay be intersectingandthemeshno longerdefinesa uniqueparametrizationof thedo-main. In the caseof lossy compression,this can be avoided byappropriatelychoosinga thresholdfor quantizationof wavelet co-efficients. We notethat this problemdoesnot occurin thecaseoftime-varyingcontours,whereself-intersectionsover time arenatu-ral.
The geometricerror at the finest-resolutionmeshis causedbyregionsof incorrecttopologywhereverticescouldnotbeprojectedonto their correspondingcontour. For themajority of vertices,thegeometricerror is zero,which explainswhy the � V -error (the av-erageof individual errors)is muchsmallerthanthe � { -error (thesquare-rootof theaveragedsquarederrors).
6 Conc lusions
Our approachsupportsthe exploration of scalarfields via theircontours. A key issueof our approachis the constructionof abasemeshof manifold topology that is inducedby a setof orig-inally extractedcontours. This basemeshdefinesa subdivisionsurface/volume from which all intermediatecontourscan be re-constructedefficiently. During this subdivision process,geometricdetail is expandedfrom waveletcoefficientsincreasingthelevel ofdetail.For efficiently representingverylargedatasetsit will becru-cial to selectandconstructalocallyoptimalsetof basecontoursandto blendtheresultinglocal basemeshesto a globalrepresentation.A solutionto this challengingproblemmight be theconsiderationof topologicalcharacteristicsof a field function,like critical pointsandseparatrices,whichcanbeconstructedexplicitly for scalarandvectorfields[12, 13].
Ackno wledg ements
This work wasperformedunderthe auspicesof the U.S. Depart-mentof Energy by University of California LawrenceLivermoreNationalLaboratoryundercontractNo. W-7405-Eng-48.It wasalsosupportedby theNationalScienceFoundationundercontractACI 9624034(CAREERAward), throughtheLargeScientificandSoftware Data Set Visualization(LSSDSV) programundercon-tract ACI 9982251,andthroughthe NationalPartnershipfor Ad-
vancedComputationalInfrastructure(NPACI); theOffice of NavalResearch� undercontractN00014-97-1-0222;the Army ResearchOffice undercontractARO 36598-MA-RIP;theNASA AmesRe-searchCenterthroughanNRA awardundercontractNAG2-1216;the LawrenceLivermoreNationalLaboratoryunderASCI ASAPLevel-2 MemorandumAgreementB347878andunderMemoran-dum AgreementB503159;andthe North Atlantic TreatyOrgani-zation (NATO) undercontractCRG.971628awardedto the Uni-versity of California, Davis. We alsoacknowledgethe supportofALSTOM SchillingRoboticsandSGI.
Wethankthemembersof theScientificComputingandImaging(SCI) Instituteat theUniversityof Utah,thescientistsat theCenterfor Applied ScientificComputing(CASC)at LawrenceLivermoreNationalLaboratory, andthemembersof theCenterfor ImagePro-cessingandIntegratedComputing(CIPIC)at theUniversityof Cal-ifornia, Davis.
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� ¥ � � ¦ � � § �Color Plate.(a)Sliceof trivariatescalarfield; (b) basecontours;(c) linking basevertices;(d) basepolygons;(e)patchesdefinedby fittedmesh;(f) meshwith coloredcontours;(g–i) bicubicwaveletreconstructionsusing10331,2631,and656
coefficients,respectively; (j–l) correspondingbilinearwaveletreconstructions.
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