a topological hierarchy for functions on triangulated surfacesgraphics.cs.ucdavis.edu/~hamann... ·...

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A TopologicalHierarchy for FunctionsonTriangulatedSurfaces

Peer-Timo Bremer� ‡, Member, IEEE, HerbertEdelsbrunner†, BerndHamann

�, Member, IEEE, and

Valerio Pascucci‡, Member, IEEE,

Abstract— We combine topological and geometric methods toconstruct a multi-r esolution representation for functions overtwo-dimensional domains. In a preprocessingstage, we createthe Morse-Smale complex of the function and progressivelysimplify its topology by canceling pairs of critical points. Basedon a simple notion of dependency among these cancellationswe construct a hierarchical data structur e supporting traversaland reconstruction operations similarly to traditional geometry-based representations. We use this data structur e to extracttopologically valid approximations that satisfy error boundsprovided at run-time.

Index Terms— Critical point theory, Morse-Smale complex,terrain data, simplification, multi-r esolution data structur e.

I . INTRODUCTION

T HE efficient constructionof topologically and geomet-rically simplified models is a central problem in visu-

alization. This paper describesa hierarchicaldata structurerepresentingthe topologyof a continuousfunction on a trian-gulatedsurface.An exampleof suchdatais thedistribution ofthe electrostaticpotentialon a molecularsurfaceor elevationdata on a sphere(e.g., the Earth). The complete topologyof the function is computedand encodedin a hierarchythatprovidesfastandconsistentaccessto adaptivetopologicalsim-plifications.Additionally, thehierarchyincludesgeometricallyconsistentapproximationsof thefunctioncorrespondingto anytopologicalrefinement.In thespecialcaseof a planardomain,the function can be thoughtof as elevation and the graphofthe function as a surface in three-dimensionalspace.In thiscaseour framework createsa topology-basedhierarchyof thegeometryof this surface.

A. Motivation

Scientific dataoften consistsof measurementsover a geo-metricdomainor space.We canthink of it asadiscretesampleof a continuousfunction over the space.We are interestedinthe casein which the spaceis a triangulatedsurface(with orwithout boundary).

A hierarchical representationis crucial for efficient andpreferably interactive exploration of scientific data.The tra-ditional approachto constructing such a representationisbasedon progressive datasimplificationdrivenby a numericalmeasurementof theerror. Alternatively, we maydrive thesim-plification processwith measurementsof topologicalfeatures.�Centerfor Image Processingand IntegratedComputing,Departmentof

ComputerScience,University of California, Davis, CA 95616-8562†Departmentof ComputerScience,Duke University, Durham,NC 27708

andRaindropGeomagic,ResearchTrianglePark, NC 27709‡Centerfor Applied Scientific Computing,LawrenceLivermoreNational

Laboratory, Livermore,CA 94551

Such an approachis appropriateif topological featuresandtheir spatial relationshipsare more essentialthan geometricerrorboundsto understandthephenomenaunderinvestigation.An exampleis waterflow overa terrain,which is influencedbypossiblysubtleslopes.Small but critical changesin elevationmayresultin catastrophicchangesin waterflow andaccumula-tion. Thus,our approachis distinctly differentfrom onethat ispurelydrivenby numericalapproximationerror. It ensuresthattopologyof a function is preservedaslong aspossibleduringa simplificationprocess,which is not necessarilythecasewithsimplificationmethodsdriven by approximationerror.

Thereareapplicationsbeyondtheanalysisof measureddata.For example,we may artificially createa continuousfunctionover a surfaceandusethat function to guidethesegmentationof the surfaceinto patches.

B. Relatedwork

The topological analysis of scalar valued scientific datahasbeena long standingresearchfocus.Morse-theory-relatedmethodshave alreadybeendevelopedin the 19th century[1],[2], long beforeMorse theory itself was formulated,and hi-erarchicalrepresentationshave beenproposed[3], [4] withoutmaking use of the mathematicalframework developed byMorseandothers[5], [6]. However, mostof this researchwaslost and has beenrediscovered only recently. Most modernresearchin theareaof multi-resolutionstructuresis geometricand many techniqueshave been developed during the lastdecade.The mostsuccessfulalgorithmsdevelopedin that eraarebasedon edgecontractionasthe fundamentalsimplifyingoperation[7], [8] and accumulatedsquaredistancesto planeconstraintsas the error measure[9], [10]. This work focusedon triangulatedsurfacesembeddedin three-dimensionalEu-clideanspace,which we denoteas

� 3. We find a similar focusin the successive attemptsto include the capability to changethe topologicaltype [11], [12].

In the field of flow visualizationtopological analysisandtopology basedsimplification are basedon the work by Hel-manandHesselink[13]. They proposeda structuresimilar totheMorsecomplex to analysisvectorfieldsandlatermethodsto simplify this complex have beendeveloped[14], [15], [16].Unfortunately, computingsucha complex relieson numericalintegrationalonginherentlyunstableregionsof thevectorfieldand is therefore limited to relatively small and clean datasets.For the simpler caseof piece-wiselinear scalarvaluedfunctions(whosegradientsdefinea piece-wiseconstantflow-field) we computethe topology in a symbolic mannerwhichis robustevenin degeneratecases.Therefore,we cancomputeMorse complexes for data sets with tens of thousandsof

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critical pointscomparedto hundredsof critical pointsin com-monly usedvector fields [14], [15], [16]. Unlike the methodin [14] we maintaina consistentgeometricapproximationofthetopologyanddo not createhigher-ordercriticalitiesasit isdonein [15]. Additionally, our error boundis directly linkedto theapproximationerror, seeSectionV-A, andwe provide amulti-resolutionhierarchyratherthana simplificationstrategy.

To remove (spurious) topological featuresfrom all levelsets simultaneously, we interpret the critical points of thefunction as the culprits responsiblefor topological featuresthatappearin thelevel sets[17], [18]. While sweepingthroughthe level setswe seethat critical points indeedstart and endsuchfeatures,andwe may usethe lengthof the interval overwhich a featureexists as a measureof its importance.Forthe specialcaseof two-dimensionalheightfields this measurewas first proposedby Horman [19] and later adoptedbyMark [20]. We use the more generalconceptof persistenceintroducedin [21], where the Morse-Smalecomplex of thefunction domainoccupiesa centralposition. Its constructionand simplification is studiedfor 2-manifoldsin [22] and for3-manifoldsin [23].

C. Results

We follow theapproachtakenin [22], with somecrucialdif-ferencesandextensions.Given a piecewise linear continuousfunction over a triangulateddomain,we

1. constructa decompositionof thedomaininto monotonicquadrangularregionsby connectingcritical points withlines of steepestdescent;

2. simplify thedecompositionby performinga sequenceofcancellationsorderedby persistence;and

3. turn the simplificationprocessinto the constructionof ahierarchicalmulti-resolutiondatastructurewhoselevelscorrespondto simplified versionsof the function.

The first two stepsarediscussedin [22], but the third stepisnew. Nevertheless,this papermakes original contributions toall threestepsand in the applicationof the datastructuretoconcretescientificproblems.Thesecontributionsare

(i) a modificationof the algorithm of [22] that constructsthe Morse-Smalecomplex without the use of handleslides;

(ii) the simplification of the complex by simultaneousap-plication of independentcancellations;

(iii) a numerical algorithm to approximatethe simplifiedfunction;

(iv) a shallow multi-resolutiondatastructurecombiningthesimplified versionsof the function into a single hierar-chy;

(v) an algorithm for traversingthe datastructurethat com-binesdifferentlevelsof the hierarchyto constructadap-tive simplifications;and

(vi) the applicationof our methodto variousdatasets.The hallmark of our methodis the fusion of the geometricandtopologicalapproachesto multi-resolutionrepresentations.Theentireprocessis controlledby topologicalconsiderations,and the geometricmethodis usedto realizemonotonicpathsandpatches.The latterplaysa crucialbut sub-ordinaterole inthe overall algorithm.

I I . BACKGROUND

We describean essentiallycombinatorialalgorithm basedon intuitions provided by investigationsof smoothmaps.Inthis section,we describethe necessarybackground,in Morsetheory [6], [24] and in combinatorialtopology [25], [26].

A. Morse functions

Throughoutthis paper, � denotesa compact2-manifoldwithout boundaryand f : ��

denotesa real-valuedsmoothfunction over � . Assuming a local coordinatesystemat apoint a �� , we computetwo partial derivatives and call acritical when both are zero and regular otherwise.Examplesof critical points are maxima( f decreasesin all directions),minima( f increasesin all directions),andsaddles( f switchesbetween decreasingand increasing four times around thepoint).

Using the local coordinatesat a, we computethe Hessianof f , which is the matrix of secondpartial derivatives. Acritical point is non-degenerate when the Hessianis non-singular, which is a property that is independentof thecoordinatesystem. According to the Morse Lemma, it ispossibleto constructa local coordinatesystemsuch that fhasthe form f � x1 x2 �� f � a� x2

1 x22 in a neighborhoodof

a non-degeneratecritical point. The numberof minus signsis the index of a and distinguishesthe different types ofcritical points:minimahave index 0, saddleshave index 1, andmaximahave index 2. Technically, f is a Morsefunctionwhenall its critical points are non-degenerateand have pairwisedifferentfunctionvalues.Mostof thechallengesin ourmethodare rooted in the needto enforcetheseconditionsfor givenfunctionsthat do not satisfy themoriginally.

B. Morse-Smalecomplexes

Assuminga Riemannianmetric and an orthonormallocalcoordinatesystem,the gradient at a point a of the manifoldis the vectorof partial derivatives.The gradientof f forms asmoothvector field on � , with zeroesat the critical points.At any regular point we have a non-zerogradientvector, andwhen we follow that vector we trace out an integral line,which startsat a critical point andendsat a critical point whiletechnicallynot containingeitherof them.Sinceintegral linesascendmonotonically, the two endpointscannotbe the same.Becausef is smooth,two integral lines areeither disjoint orthe same.The setof integral lines coversthe entiremanifold,exceptfor thecritical points.ThedescendingmanifoldD � a ofa critical point a is the setof pointsthat flow toward a. Moreformally, it is the union of a andall integral lines that endata. For example,the descendingmanifold of a maximumis anopendisk, that of a saddleis an openinterval, and that of aminimumis thepoint itself. Thecollectionof stablemanifoldsis a complex, in the sensethat the boundaryof a cell is theunion of lower-dimensionalcells. Symmetrically, we definethe ascendingmanifold A � a of a as the union of a and allintegral lines that startat a.

For the next definition, we need an additional non-degeneracy condition,namelythat ascendingand descending

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manifolds that intersectdo so transversally. For example, ifan ascending1-manifold intersectsa descending1-manifoldthen they cross.Due to the disjointnessof integral lines, thisimplies that the crossingis a single point, namelythe saddlecommonto both. Assumingthat this transversality propertyis satisfied,we overlay the two complexes and obtain whatwe call the Morse-Smalecomplex, or MS complex, of f .Its cells are the connectedcomponentsof the intersectionsbetweenascendinganddescendingmanifolds.Its verticesarethe verticesof the two overlayedcomplexes, which are theminimaandmaximaof f , togetherwith thecrossingpointsofascendinganddescending1-manifolds,which arethe saddlesof f . Each1-manifold is split at its saddle,thus contributingtwo arcsto the MS complex. Eachsaddleis endpointof fourarcs,which alternatelyascendanddescendaroundthe saddle.Finally, eachregion hasfour sides,namelytwo arcsemanatingfrom a minimumandendingat two saddlesandtwo additionalarcscontinuingfrom the saddlesto a commonmaximum.Itis genericallypossiblethat the two saddlesare the same,inwhich casetwo of the four arcsmerge into one. The regionlies on both sides of the merged arc so it makes sensetodouble-countand to maintain that the region has four sides.An exampleis shown in Fig. 1.

minimum

maximum

saddle

Fig. 1. A sampleMS complex.

C. Piecewiselinear functions

Functions occurring in scientific applications are rarelysmooth and mostly known only at a finite set of pointsspreadout over a manifold. It is convenient to assumethatthe function haspairwisedifferentvaluesat thesepoints.Weassumethat the points are the verticesof a triangulationKof � , andwe extend the function valuesby piecewise linearinterpolationappliedto the edgesandtrianglesof K. The starof a vertex u consistsof all simplices (vertices,edgesandtriangles)that containu, and the link consistsof all facesofsimplicesin thestarthataredisjoint from u. Sincethesurfacedefinedby K is a 2-manifold, the link of every vertex is atopological circle. The lower star containsall simplices inthe star for which u is the highestvertex, and the lower linkcontainsall simplicesin the link whoseendpointsare lowerthanu. Notethatthelower link is thesubsetof simplicesin thelink thatarefacesof simplicesin thelower star. Topologically,the lower link is a subsetof a circle. We definewhatwe meanby a critical point of a piecewise linear function basedonthe lower link. As illustrated in Fig. 2, the lower link of amaximumis the entire link and that of a minimumis empty.In all other cases,the lower link of u consistsof k � 1 � 1connectedpieces,eachbeinganarcor possiblya singlevertex.

minimum saddle maximum splitting of 2−fold saddleregular point

Fig. 2. Classificationof a vertex basedon relative heightof verticesin itslink. The lower link is marked black.

The vertex u is regular if k � 0 anda k-fold saddleif k � 1.As illustratedin Fig. 2 for k � 2, a k-fold saddlecanbe splitinto k simpleor 1-fold saddles.

D. Persistence

Werequireanumericalmeasureof theimportanceof criticalpoints that can be usedto drive the simplification of a MScomplex. For this purpose,we pair up critical pointsandusethe absolutedifferencebetweentheir heightsas importancemeasure.To construct the critical point pairs, we imaginesweepingthe 2-manifold � in the direction of increasingheight.This view is equivalentto sortingtheverticesby heightand incrementallyconstructingthe triangulationK of � onelower starat a time. The topologyof the partial triangulationchangeswhenever we add a critical vertex, and it remainsunchangedwheneverwe adda regularvertex. Exceptfor someexceptionalcasesthat have to do with the surfacetype of � ,eachchangeeither createsa componentor an annulusor itdestroys a component(by merging two) or an annulus(byfilling the hole). We pair a vertex v that destroys with thevertex u that createdwhat v destroys. The persistenceof uandof v is thedelaybetweenthe two events:p � f � v�� f � u .An algebraicjustificationof this definitionanda fastalgorithmfor constructingthe pairscanbe found in [21].

I I I . MORSE-SMALE COMPLEX

We introducean algorithmfor computingthe MS complexof a function f definedovera triangulationK. In particular, wecomputethe ascendinganddescending1-manifolds(paths)off startingfrom the saddles,and usethemto partition K intoquadrangularregionswhich definethe MS complex.

A. Path construction

Startingfrom eachsaddle,weconstructtwo linesof steepestascentand two lines of steepestdescent.We do not adoptthe original algorithm proposedin [22] and follow actuallines of maximal slope insteadof edgesof K. In particular,we split triangles to create new edgesin the direction ofthe gradient.We modify this basicstrategy to avoid regionswith disconnectedinterior andregionswhoseinterior doesnottouchbothsaddles.Without themodificationsuchregionsmaybe createdbecausef is not smooth and integral lines canmerge. Fig. 3(a) shows one suchcase,wherepathsmerge atjunctionsanddisconnectthe interior of a region into two. Themodificationthat eliminatesthe two undesiredconfigurationsconsistsof disallowing two paths to merge if they are ofdifferent type; seeFig. 3(b). Two pathsare still allowed tomerge if they are both ascendingor both descending.If twopaths are not allowed to merge we split one edge of the

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triangulationandintroducea new samplewith function valuethat preserves the structureof the MS complex but locallyavoids the junction. Fig. 4 shows the repeatedapplicationof this strategy to avoid a junction. Practically, this situationrarely occursand it can be shown that in the worst casethenumberof trianglesintroducedis linearin thesizeof themesh.

maximumminimum

saddle junction

Fig. 3. Portionof the MS complex of a piecewise linear function.Sincethegradientis not continuous,(solid) ascendingand (dotted)descendingpathscanmeetin junctionsandsharesegments.(Left) Complex with no restrictionson sharingsegments.The greenregion touchesonly onesaddle,and the redoneis disconnected.(Right) Only pathof thesametypecanmeet.Theinteriorof eachregion is connectedand touchesboth saddles.

Fig. 4. Triangle split to keeptwo pathsseparated.Solid red lines indicatetwo portionsof pathsalreadycomputed.(Left) The red circle is the currentextremumof a paththat would follow the red dottedline. (Middle) The pathis extendedsplitting a first triangle. (Right) Sincethe two pathswould stillintersect,a secondtriangle is split.

After computingall paths,we partitionK into quadrangularregions forming the cells of the MS complex. Specifically,we grow eachquadranglefrom a triangle incident to a saddlewithout ever crossinga path.

In degenerateareasof � , whereseveral verticesmay havethe samefunction value, the greedychoicesof local steepestascent/descentmay not work consistently. We addressthisproblemusingthe simulationof simplicity, or (SoS)[27]. Weorient eachedgeof K in the direction of ascendingfunctionvalue.Vertex indicesareusedto breakties on flat edgessuchthat the resulting directedgraph has no cycles. Using theseorientations,the searchfor the steepestpath is transformedto a weighted-graphsearchandfunction valuesareonly usedas preferences.Thus,our algorithm is robust even for highlydegeneratedatasetsas the oneshown in Fig. 5.

(a) (b)Fig. 5. MS complex of degeneratedata set. The “volcano” is createdbya sin() function that is flat both inside the “crater” and at the foot of themountain.(a) Originally computedMS complex. A large numberof criticalpointsis createdby eliminatingflat regionsusingsimulationof simplicity. (b)The samecomplex after removal of “topological noise.”

Let T �� F E V � be the triangulationof � ;initialize the MS complex, M � /0;initialize the setsof pathsandcells, P � C � /0;initialize SoSto direct the edgesof T;S � FINDSADDLES � T ;S � SPLITMULTIPLESADDLES� T ;SORTBYHEIGHT � S ;forall s � S in ascendingorderdo

COMPUTEASCENDINGPATH � Pendfor;forall s � S in descendingorderdo

COMPUTEDESCENDINGPATH � Pendfor;while thereexists untouchedf � F do

GROWREGION � f p0 p1 p2 p3 ;CREATEMORSECELL � C p0 p1 p2 p3

endwhile;M � CONNECTMORSECELLS � C .

Fig. 6. Sequenceof high-level operationsusedto createan MS complex.Whenwe grow a cell from a triangle f we encounterthefour boundarypaths.p0 to p3, which are then incorporatedinto a half-edgerepresentationof thecell

B. Diagonalsand diamonds

The central element of our data structure for the MScomplex is the neighborhoodof a simple saddleor, equiva-lently, the halves of the quadranglesthat sharethe saddleasone of their vertices.To be more specific about the halves,recall that in the smooth caseeach quadrangleconsistsofintegral lines that emanatefrom its minimum and end at itsmaximum.Any one of theseintegral lines can be chosenasdiagonal to decomposethequadrangleinto two triangles.Thetrianglessharinga given saddleform the diamondcenteredat the saddle.As illustrated in Fig. 8(a), each diamond isa quadranglewhose verticesalternatebetweenminima andmaximaaroundthe saddlein its center. It is possiblethat twoverticesarethesameandtheboundaryof thediamondis gluedto itself alongtwo consecutive diagonals.

C. Thealgorithm

We computethe descendingpathsstartingfrom the highestsaddleandtheascendingpathsstartingfrom thelowestsaddle.Thus, when two pathsaim for the sameextremum, the onewith higher persistence(importance)is computedfirst. Theboundaryof the dataset is artificially taggedas a path. Thecompletealgorithmis summarizedin Fig. 6.

IV. HIERARCHY

Our main objective is the design of a hierarchical datastructurethat supportsadaptive coarseningand refinementofthedata.In this section,we describesucha datastructureanddiscusshow to useit.

A. Cancellations

We useonly oneatomicoperationto simplify theMS com-plex of a function, namelya cancellationthat eliminatestwocritical points.The inverseoperationthat createstwo critical

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pointsis referredto asananti-cancellation. In orderto canceltwo critical points they mustbe adjacentin the MS complex.Only two possible combinationsarise: a minimum and asaddleor a saddleand a maximum.The two configurationsaresymmetric,andwe can limit the discussionto the secondcase,which is illustratedin Fig. 7.

Fig. 7. Portion of the graph of a function before (left) and after (right)cancellationof a maximum(red) anda saddle(green).

Let u be the saddleand v the maximum of the canceledpair, and let w be the other maximum connectedto u. Werequire w �� v and f � w�� f � v ; otherwise,we prohibit thecancellationof u andv. We view the cancellationasmergingthreecritical points into one,namelyu, v, w into w. All pathsendingat eitheru,v, or w areremovedandwe adaptthe localgeometry to the new topology, as describedin Section V.Subsequently, all pathsthatwereconnectedto eithermaximumare recomputed.In other words,we connectevery saddleontheboundaryof thegeometricallyadaptedregion to theuniquemaximumwithin theregion.To avoid excessivesplittingof thetriangulationwe restrict the re-computedpathsto shareedgesof the triangulation.There are several reasonsfor requiringf � w�� f � v : it implies that all recomputedpaths remainmonotonicandensuresthatwe do not eliminateany level sets,exceptthat theonesbetweenf � u and f � v aresimplified.Wemay think of a cancellationas deleting the two descendingpathsof u andcontractingthe two ascendingpathsof u.

B. Noderemoval

Weconstructthemulti-resolutiondatastructurefrom bottomto top.ThebottomlayerstorestheMS complex of thefunctionf , or, to be moreprecise,the correspondingdecompositionofthe 2-manifold into diamonds.Fig. 8(b) illustratesthis layer

(a) (b) (c)

Fig. 8. (a) The (dotted)portion of an MS complex and the (solid) portionof the correspondingdecompositioninto diamonds.(b) Portion of the datastructure(solid) representingthe pieceof the decompositioninto diamonds(dotted).(c) Cancellationgraph(solid) of the decompositioninto diamonds(dotted).

by showing each diamond as a node with arcs connectingit to neighboringdiamonds.Each node has degree four, butthere can be loops starting and ending at the samenode.Acancellationcorrespondsto removing anodeandre-connecting

its neighbors.When this node is sharedby four differentarcs we can connect the neighborsin two different ways.As illustrated in Fig. 9, this operation correspondsto thetwo different cancellationsmerging the saddlewith the twoadjacentmaximaor the two adjacentminima. There is onlyoneway to removea nodesharedby a loopandtwo otherarcs,namelyto deletethe loop andconnectthe two neighbors.

(A) (B)

(a)

(b) (c)

Fig. 9. The four-sideddiamond(a) canbe zippedup in two ways: from topto bottom (b) or from left to right (c). A folded diamond(A) canbe zippedup in only oneway (B).

To constructthe hierarchy by repeatedcancellations,weuse the algorithm in [21] to match critical points in pairs� s1 v1 � � s2 v2 �� sk vk , with persistenceincreasingfromleft to right. Let Q j betheMS complex obtainedafter thefirstj cancellations,for 0 � j � k. We obtain Q j 1 by modifyingQ j and storing sufficient information so we can recover Q jfrom Q j 1. The hierarchy is completewhen we reach Qk.We call each Q j a layer in the hierarchy and representitby activating its diamondsas well as neighbor and vertexpointers and de-activating all other diamondsand pointers.To ascendin the hierarchy(coarsenthe quadrangulation)wede-activate the diamondof sj 1; to descendin the hierarchy(refinethe quadrangulation)we activate the diamondof sj ! 1.Activating and de-activating a diamondrequiresupdatingofonly a constantnumberof pointers.

C. Independentcancellations

We generalizethe notion of a layer in the hierarchy topermit view-dependentsimplifications.The key concepthereis the possibility to interchangingtwo cancellations.Themost severe limitation to interchangingcancellationsderivesfrom the assignmentof extremaas verticesof the diamondsand from re-drawing the pathsending at theseextrema. Tounderstandthis limitation, we introducethecancellationgraphwhoseverticesare the minima and maxima.Fig. 8(c) showsanexampleof sucha graph.For eachdiamond,thereexistsanedgeconnectingthe two minimaandanotheredgeconnectingthe two maxima.Thereareno loopsandthereforesometimesonly oneedgeperdiamond.Zippingupadiamondcorrespondsto contractingone of the edgesand deleting the other, if it

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exists. Oneendpointof the edgeremainsasa vertex and theother disappears,implying that the diamondsthat sharethesecondendpointreceive a new vertex. A specialcaseariseswhen a diamondsharesboth endpoints:the connectingedgethat would turn into a loop is deleted.

Two cancellationsin a (possibly simplified) MS complexare interchangeablewhen it is irrelevant in which order thetwo operationsareappliedto the datastructure.For example,the two cancellationszipping up the samediamondare notinterchangeablesinceonepreemptsthe other. In general,twocancellationsare interchangeablewhen their diamondsshareno vertex, a conditionwe refer to asbeing independent. Notethattwo interchangeablecancellationsarenotnecessarilyinde-pendent.Eventhoughindependenceis themorelimiting of thetwo concepts,it offers sufficient flexibility in choosinglayersto support the adaptationof the representationto externalconstraints,suchas the biasedview of the data.

Whenwe canperforma relatively largenumberof indepen-dent cancellationswe have more freedomgeneratinglayersin the multi-resolutiondatastructure.Ideally, we would liketo identify a large independentset and iterate to constructashallow hierarchy. However, in the worst case,every pair ofcancellationsis dependent,which makesthe constructionof ashallow hierarchyimpossible.As illustratedin Fig. 11(a),sucha configurationexistsevenfor thesphereandfor any arbitrarynumber of vertices. Nevertheless,worst-casesituations areunlikely to arise as they require a large number of foldeddiamonds.Specifically, it is possibleto prove that every MScomplex without folded diamondsimplies a linear numberofindependentcancellations.

V. GEOMETRIC APPROXIMATION

After eachcancellation,we createor changethe geometrythat locally definesf . We pursuethreeobjectives:the approx-imation must agreewith the given topology, the error shouldbe small, and the approximationshouldbe smooth.

A. Error bounds

We measurethe error as the differencebetweenfunctionvalues at a point. It is convenient to think of the graphof f as the geometry and this differenceas the (vertical)distancebetweenthe original and the simplified geometryat the location of the point. The persistenceof the criticalpoints involved in a cancellationimplies a lower bound onthe local error. We illustrate this connectionfor the one-dimensionalcasein Fig. 10(a).Recallthat thepersistencep ofthemaximum-minimumpair is thedifferencein their functionvalues.Any monotonicapproximationof the curve betweenthe two critical points has an error of at least p" 2. We canachieveanerrorof p " 2, but only if weacceptaflat segmentforthis portionof thecurve,seetheredcurve in Fig. 10(a).Whenit is allowed to exceedp" 2, smootherapproximationswithoutflat segmentsarepossible,suchasthegreencurve in thesamefigure. Note that the above describesonly the error betweenthe two functionsbeforeand after the one cancellation.Theerror causedby the compositionof two or morecancellationsis more difficult to analyzeand will not be discussedin thispaper.

#$ %&%'()

*&*+,&,&,&,&,&,&,&,&,,&,&,&,&,&,&,&,&,,&,&,&,&,&,&,&,&,,&,&,&,&,&,&,&,&,-&-&-&-&-&-&-&--&-&-&-&-&-&-&--&-&-&-&-&-&-&--&-&-&-&-&-&-&-{p

(a) (b)Fig. 10. Geometryfitting for paths:(a) One-dimensionalcancellationandseveral monotonic approximations.(b) Local averaging used to constructsmoothly varying monotonic approximations.Slopesof neighboringedgesare combinedwith the original slope,and the function valuesare adjustedaccordingly(edgenormalsareshown).

B. Data fitting

......

(a) (b)

Fig. 11. (a)MS complex on thespherewith pairwisedependentcancellations.(b) One-dimensionalsmoothingwith (blue) error constraintsand prescribedendpointderivatives. (Left) Initial configuration;(right) Constructedsolution.

We know that monotonic patchesexist, provided we aretolerant to errors. Our goal is therefore to find monotonicpatchesthat minimize some error measure.A large bodyof literature deals with the more general topic of shape-constrainedapproximation[28], [29]. The generalproblemis to construct the smoothestinterpolant to a set of inputdata while observingsome shapeconstraints(e.g., convex-ity, monotonicity, and boundaryconditions).However, mostpublishedwork usespenalty functions insteadof tight errorbounds.Additionally, the techniquesare typically describedfor tensorproductsetting,andthe definitionsof monotonicityfor the bivariatecasevary anddiffer from the onewe use.

We did not adapt standardtechniquesfor our purposes.Instead,we decidedto usea multi-stageiterative approachtoconstructthegeometrythatspecifiesthesimplifiedrepresenta-tion of f . It providesa smoothC1-continuousapproximationwithin a specifiederror bound along the boundariesof thequadrangularpatchesanda similar approximationbut withoutobservingan error bound in the interior of the patches.Thepathsare constructediteratively by smoothingthe gradientsalong the edgesand post-fitting the function values,as illus-trated in Fig. 10(b). During eachiteration, we first computethe new gradientof an edgeas a convex combinationof itsgradient and the gradientsof the adjacentedges.We thenadjust the function valuesat the verticesto realize the newgradients.During an iteration, we maintain the error boundat the vertices and make sure that the completedpath ismonotonic.In addition, the gradientat the critical points isset to zero.

The techniqueperformswell in practicealthoughit con-verges slowly. Sampleresultsare shown in Fig. 11(b). The

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interior of the quadrangularpatchesaremodifiedby applyingstandardLaplacian smoothing to the function values [30].During eachiteration, the value at a vertex is averagedwiththoseof its neighbors.Since the boundariesare monotonic,this procedureconverges to a monotonic solution for thepatchinterior. We summarizethe stepsof the geometryfittingprocess:

1. Find all pathsaffectedby a cancellation;2. usethe gradientsmoothingto geometricallyremove the

canceledcritical points;3. smooththe old regionsuntil they aremonotonic;4. erasethepathsandre-computenew pathsusingthenew

geometry;5. use one-dimensionalgradient smoothing to force the

new pathsto comply with the constraints;and6. smooththe new regionsuntil all pointsareregular.

The reasonfor repeatinggradientsmoothingin Step5 is thisone: The pathsconstructedin Step 4 are not guaranteedtosatisfytherequirederrorbounds.In practice,wedonotsmoothuntil the geometryconverges, insteadperform a predefinednumberof stepsor smoothuntil all constrains(monotonicityanderror bounds)aremet (whichever takeslonger).Basedonour experiments,theconstrainsaretypically easilysatisfiedinwhich casea constantnumberof smoothingstepsis performedafter eachcancellation.Currently, the geometricfitting is thecomputationalbottleneckof the algorithm. However, we arenot aware of faster methodsto createtopologically correctapproximations.

VI . REMESHING

While traversing the hierarchy we want to interactivelydisplaygeometrythat agreeswith the currenttopologyof thegraphof f . Thus,we mustdeterminea triangularmeshwithineachquadrangularregion. The triangulationfor eachregionshouldbe independentof all other regions to allow us to usetopology other than that encounteredduring the creationofthe hierarchy.

A. Path smoothing

We first examine the geometricdatawe want to approxi-mate.Without modificationsthe algorithmsusedto computepaths tend to create “non-smooth” paths, see Fig. 12(a).Theseare visually not pleasingand difficult to approximate.Therefore,we modify the data slightly in order to createsmootherpaths.We smoothpath vertices,except the criticalpoints,in � (in thecaseof heightfields thexy-plane)withoutflipping triangles,while preservingtheir functionvalue.Again,we use Laplacian smoothing that is modified at junctions,seeFig. 12(b). Here, we first averageall incoming vertices(basedon the directionof the paths)andall outgoingverticesseparately. The averagesareusedto updatethe junction.Thisstrategy reducesthechangein directionbetweenthe incomingand outgoing edgesrather than minimizing the changeofdirections betweenall edges.The result is a more “flow-like” structure,as shown in Fig. 12(c). No vertex can leaveits original triangle strip, and, assuminga sufficiently densebasemesh,theoverall changein positionis minor andcritical

pointsarenever moved. In practice,oneor two iterationsaresufficient to significantly improve pathshape.

./ 011/2

1/41/3

1/3

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23original position

incoming/outgoing positionsintermediate averages of all new position

(a) (b) (c)

Fig. 12. Path smoothing:(a) A typical path structurewithout smoothing.(b) Smoothingappliedat junctions.All incoming and outgoingverticesareaveragedseparately. The averagesare used to update the position of thejunction. (c) The pathstructureof (a) after two smoothingsteps.

B. Parametrization

To enablefastandversatilerenderingof thedataandreducememory requirementswe remesheach quadrangularregionusinga regularstructure.For this purpose,we usemean-valuecoordinatesasproposedin [31]. We mapthe boundaryof theregion to the boundaryof oneor multiple unit squares.In theinterior of the region (which, at this point, is representedasaportionof thetriangulationK), we usethefactthateachvertexcan be expressedas a convex combinationof its neighbors.The coefficients in this combination can be computedbysolving a sparselinear system.Given the parametrizationontheboundary, we usethesecoefficientsto mapinterior verticesto the parameterspace,thuscompletingthe parametrization.

Next, we sampletheparameterspaceon a uniform grid anduseits preimageon � asa new meshfor the region. Thegridresolutionis chosenbasedon a given error boundevaluatedalong boundary paths, which, by construction, follow thedirectionof maximumchangein function value.Specifically,werefineeachpathuntil it satisfiestheerrorboundandchoosethegrid resolutionto matchthemaximumresolutionalongthefour boundarypaths.

C. Boundaryparametrization

The parametervaluesof the interior verticesare uniquelydefined by the parametervalues assignedto the boundaryvertices.Therefore,the overall quality of the parametrizationrelieson a favorableboundaryparametrization.The boundaryof a region consistsof critical points, junctions,andstandardpath vertices. Independentlyof the current approximation,the triangulationof a region must always contain its criticalpointsandjunctions.Thecritical pointsrepresenttheextremalfunction values of a region and therefore carry maximalinformation. Junctionsare createdwhen two pathsthat flowtoward the sameextremum merge. Therefore,eachjunctionreplacesa critical point for theregionsharingboththesepaths.To avoid cracksin the meshall adjacentregionsmustcontainthe junction aswell.

To remeshthe path segmentsbetweentheseextraordinarypoints (minima maxima, saddles,and junctions) we applymidpoint subdivision basedon arclength.To further avoidresolutiondependenciesbetweenmeshesof different regions

8

we permit T-junctions (hangingnodes)along boundaries.Inother words,our representationis not a globally conformingtriangulation of � but rather a collection of patches.Eachpatchis triangulatedwith a regular, conformingmesh.We callthecollectioncrack-freewhenthemeshesagreegeometricallyalongboundaries.Nevertheless,pixel-widecracksmayappearduring renderingaspolygonsarerasterizedat fixed precision.A possiblesolution is to “fill-in” the cracksduring renderingas describedby Balazset al. [32]. Unfortunately, we cannotusecontinuoussurfacerepresentationsthat allow T-junctionssuchas the methodin [33], sincethey cannotbe guaranteedto preserve the topology.

Fig. 13. Creatinga parametrizationfor the boundary. (Top-left) Originalregion and local coordinatesystemdefinedby PCA of all boundaryvertices.(Top-right) The verticesare transformedinto the PCA coordinatesystem.Inthis example,we usea single unit squareasbaseshapein parameterspace.The cornersaredefinedby the maximalprojectedlengthonto the lines y 4 xandy 4�5 x. (Bottom-left)The regular meshafter thefirst level of recursivelyfitting the junctions.(Bottom-right)Final remesh.

What is left to defineare the parametervaluesfor specialpoints and the baseshapein parameterspace.An exampleis shown in Fig. 13. We perform a principal componentanalysis(PCA) step in xy-spaceusing all boundaryvertices.The distribution ratio betweenthe two principal directionsdefines the number of consecutive unit squareswe use asparameterdomain.For a rationof 1 : 2 we usetwo unit square,for 1 : 3 threeetc. Next, we transformthe coordinatesof allboundaryverticesinto the PCA coordinateframe using theircentroidasorigin andthe two (normalized)PCA eigenvectorsasbasisvectors.In thecaseof asingleunit squareasparameterspace,we computefor all extraordinaryverticesvi � � xi yi (in thePCA coordinateframe)thescalarproductpi � � xi yi 76� � 1 �� 1 . (For two unit squaresasbaseshapewe would use� � 2 �� 1 etc..) We assignthe lower-left corner in parameterspace(parametervalueP � 0 0 ) to thevertex vLL with maximalprojectedlength pi : LL � maxi � pi , P � vLL 8� � 0 0 . Similarly,we assignthe other three cornersin parameterspaceusingprojections onto � 1 1 , � 1 �� 1 , and � � 1 1 . However, weguaranteethat all cornersare mappedto different vertices.The remaining special vertices are recursively fitted usingarc-lengthparametrization.Oncethe parametervaluesfor allspecialverticesare known the parametervaluesof the pathverticesareassignedusing their arc-lengthvalues.

D. Data layout and rendering

Ratherthan storing a meshfor eachquadrangleexplicitly,we useregulargrids.This approachallows us to usemethodslike the one describedin [34] for rendering purposes.Bystoring each grid in what Lindstrom and Pascucci calledinterleaved embeddedquadtreeswe avoid having to storeconnectivity information, while maintaining high flexibilityduring rendering.This framework can be extendedeasily toadaptive, view-dependentrendering,as well as efficient viewfrustum culling and geomorphing.One disadvantageof thisdata layout is a 33% memory overhead.Another importantaspectis the definition of local error coefficients.As we areworking with many smallergrids, rather than a single high-resolutionone,we must ensurea consistentrenderingacrossboundaries.Sincewe enforcethat sampleson grid boundariesaresharedtheir respective error termsagree.Independentlyofthe error term, a region must always renderall its junctionswhich we guaranteeby settingtheir errorsto infinity.

VI I . RESULTS

We have appliedour algorithmto severaldatasetsincludingterrain data converted from digital elevation models1, two-dimensionalsimulation data, and isosurfaces from variousscientificdatasets.ThePugetSounddatasetwith a resolutionof 1025-by-1025is representedby two-byteintegers.TheotherthreeterraindatasetsaretheNeedles,Yakima,andDalles(seeFig. 14) datasets,all of 1201-by-1201resolutionwith single-byte integer heightvalues.We have alsousedsimulationdataof the autoignitionof a spatiallynon-homogeneoushydrogen-air mixture, courtesyof EchekkiandChen[35], at resolution512-by-512with temperaturevaluesrepresentedby single-byteunsignedinteger values.Additionally, we applied our tech-niquesto threescientificdatasetswith multiple scalarfields.In particular, we useasmanifold domain � an isosurfaceofonescalarfield andasfunction f thevaluesof thesecondfieldon � . The first data set, seeFig. 20, representsa methanemolecule with � being an isosurface of the electrostaticpotentialand f the correspondingvan der Waalsenergy. Theseconddataset,seeFig. 21, describesthe interactionenergybetweena ligand (glucose)anda receptor(ethane)underthethreetranslationaldegreesof freedom.The domain � is alsoan isosurface of the electrostaticpotential with f being thevan der Waalsenergy. The third example,seeFig. 22, showsa groundremediationprocessafter an oil spill contamination.The domain � is an isosurface of the oil concentrationinthe soil (groundlevel at the top). The superimposedpseudo-coloredfunctionshows theconcentrationof microbesconsum-ing the oil andperformingthe remediationprocess.

The most basic application of our algorithm is removalof topological noise without smoothing. This functionalitydoes not dependon the hierarchy and is implementedbyrepeatedcancellationof critical pointswith lowestpersistence.Our experiencesuggeststhat this noise removal step shouldalways be appliedat least to remove the artifactscausedbysymbolicperturbation.We defineall featureswith persistence

1http://www.webgis.com

9

below 0.1%of the total datarangeasnoise.Fig. 14 illustratesthis procedurefor The Dalles data set. Removing the noisereducesthe numberof critical points from 24,617to 2,144.As oneof themainproblemin topologicaldataanalysisis thelarge numberof spurioustopologicalfeaturesthis (symbolic)clean-upis a valuablepre-processingstepfor many techniquesproposedin recentyears.

Fig. 14. (Left) Original TheDallesdatasetcontaining24,617critical points.(Right) Samedata with 2,144 critical points after removing all topologicalfeatureswith persistencelessthan0 9 1% of height range.

We have testedseveral strategiesfor creatingthe hierarchy.Thefirst strategy is basedon performingcancellationsin orderof increasingpersistence.The secondstrategy is basedonperforming simplification in “batches” of maximal indepen-dentsetsof cancellations.Eachbatchis createdby cancelingiteratively the critical point pair with smallestpersistenceandmarking any dependentcancellationas “not-allowed.” Onceno more cancellationscan be performedall cancellationsareresetto be “permitted” and the processis repeated.

Thethird strategy is a hybrid methodof thefirst two, whichbuilds maximal independentsets with a restrictedrange ofpersistencevalues.In particular, we constructthe independentsetsas usualbut stop if the next cancellationhaspersistencelarger than twice of the smallestin the sameset. In all threemethodswe never remove topology with persistencelargerthan20% of the initial height range.

TableI summarizesstatisticsfor thehierarchiesconstructedfor our terrain data.As expected,a hierarchybecomesmoreshallow when cancelingcritical points in independentsets.One problem is the presenceof nodeswith a high numberof parentsand children; this problem is the result of high-valencevertices in the MS complex. Fig. 15(a) shows thehighest-resolutionMS complex of theNeedlesdataset,whereeachpath is representedby a straight line from the saddleto the extremumto betterhighlight the problem.As the dataset containsfew minima eachof them is incident to a verylarge numberof edges.Any cancellationinvolving one suchminimum hasa large numberof parentsand children.High-valenceverticesare the causefor the uneven distribution ofnodesover the levels.

In Fig. 15(b), we show the numberof nodesper level foreachcancellationstrategy for the PugetSounddataset. Themaximumindependentset strategy clearly producessuperiorresults in terms of overall shapeof the hierarchy. However,this observation does not necessarilyimply better practicalbehavior. Fig. 15(c)shows thenumberof critical pointsin theMS complex dependingon a uniform error. Equivalentgraphs

TABLE I

HIERARCHY STATISTICS FOR DIFFERENT CANCELLATION STRATEGIES.

MAXIMAL AND AVERAGE DEPTH (DISTANCE FROM THE ROOT) ARE

SHOWN IN THE FIRST TWO COLUMNS. THE LAST THREE COLUMNS LIST

THE MAXIMUM NUMBER OF PARENTS AND CHILDREN AND THE AVERAGE

DEGREE FOR A NODE.

depth avg dep max #p max #c avg degPuget Sound org. no. of CPs49185,no. of significantCPs17470

purepersistence 381 128 148 110 3.80max. indep.set 157 118 131 112 4.28

indep.set-cut-off 238 105 147 106 3.94Yakima org. no. of CPs21275,no. of significantCPs6082


indep.set-cut-off 96 52 74 37 3.82Dalles org. no. of CPs24617,no. of significantCPs2144


indep.set-cut-off 73 33 63 57 3.52Needles org. no. of CPs17375,no. of significantCPs3772


indep.set-cut-off 149 62 124 101 3.68

for the other data setsare shown in Figs. 16 and 17. Eventhough the hierarchy createdby maximal independentsetsis the most shallow it producesa significantly densermesh.The hybrid methodproducesnearly identical resultsduringtraversalas the one basedon pure persistence.However, thehybrid methodcreatesa moreshallow hierarchyanda higherbranchingfactorand it is thereforethe methodof choice.

Fig. 18 shows thePugetSounddataset.Theoriginal topol-ogy has49,185critical pointsand is too denseto be printed.Theupper-left pictureshows thedatawith 4,045critical pointsobtainedafter removing the topologicalnoiseusing a persis-tencethresholdof 0 � 5% of the elevation range.The upper-right pictureshows the approximationof the datawith 2,025critical pointsobtainedby increasingthepersistencethresholdto 1 � 2%. The lower-left picture shows a significantly coarserMS complex containing289 critical points that remainafterincreasingthe persistencethresholdto 20%. The differencesbetweenthe resolutionsare most notable along the front-facingboundary. The pre-processingof the PugetSounddatawas done in about three and a half hours, due to the slowconvergenceof the geometricfitting procedure.The traversalof the hierarchyandrenderingare fully interactive.

Thehierarchyalsosupportsadaptiverefinements,two exam-plesareshown in Fig. 19. Fig. 19(a)shows a view-dependentrefinementof the Puget Sound data. The full resolution ispreserved inside the view frustum yielding a total of 1,070critical points.Outsidethe view frustum,we have simplifiedthe data to the extent possible.One can observe a quickdrop in resolution away from the view frustum. Reducingthetopologyoutsidethe frustumnaturallyreducesthenumberof quadrangularregions that mustbe rendered(and thereforethe numberof triangles)and frustum culling can be performdirectly on the regionsculling largepartsof themeshwithouttraversal. Fig. 19(c) shows the combustion data adaptivelyrefinedto preserve only maximawith a high function value.Only maximaabove 90% of the maximal function valueandtheir ancestorsin the hierarchy are preserved. Notice thatseveral lower maximahave completelydisappeared.

10

0 100 200 300 400level

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Fig. 15. (left) HighestresolutionMS complex of Needlesdataset.(middle)Nodedistribution over the levels for different cancellationstrategies for PugetSounddataset. (right) Numberof critical points in MS complex during uniform refinementof PugetSounddataset.

0 50 100level

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0 50 100 150 200level

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Fig. 16. Nodesdistribution over the levels for differentcancellationstrategies for the Yakima(left) , The Dalles(middle) , andthe Needles(right) datasets.

0 0.025 0.05 0.075 0.1 0.125 0.15uniform persistence

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0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2uniform persistence

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Fig. 17. Numberof critical points in MS complex for the Yakima(left), The Dalles (middle), and the Needles(right) datasets.

Fig. 18. (Left) PugetSounddataafter topologicalnoiseremoval. (Middle) Dataat persistenceof 1 9 2% of the maximumheight.(Right) Dataat persistence20% of maximumheight.

11

Fig. 19. (Left) View-dependentrefinementof PugetSounddata(purple: view frustum).(Middle) Combustiondataafter topologicalnoiseremoval. (Right)Adaptive refinementof combustiondatabasedon function value.All maximaabove 90% of the maximal function value arepreserved.

Fig. 20. Electrostaticandvander Waalspotentialsfor a methanemolecule.(Left) Isosurfaceof the electrostaticpotentialpseudo-coloredwith van derWaalspotential.(Middle) Full MS complex with 30 critical points.(Right) Simplified MS complex with 14 critical pointshighlighting the hydrogenatomsnearthemaximaandthe carbonatomnearthe minium at the center.

VII I . CONCLUSIONS

We have describeda new topology-basedmulti-resolutiondatastructurefor functionsover a planardomainanddemon-stratedits usefor two-dimensionalheightfields.Thehierarchyallows one to extract geometryadaptively for a given topo-logical error. Due to its robustnessin the presenceof noiseandits well-definedsimplificationprocedures,theapproachisappealingfor applicationsbasedon topologicalanalysis,forexample,datasegmentationandfeaturedetectionandtrackingin medical imaging or simulatedflow field datasets.Futurework will be concernedwith fitting the completegeometrywithin a given error bound and the extensionto volumetricdata.

ACKNOWLEDGMENTS

This work was performedunder the auspicesof the U. S. Department

of Energy by University of California LawrenceLivermoreNational Labo-

ratory undercontractNo. W-7405-Eng-48.HerbertEdelsbrunneris partially

supportedby the National ScienceFoundation(NFS) undergrantsEIA-99-

72879 and CCR-00-86013.Bernd Hamannis supportedby the NSF under

contractACI 9624034,through the LSSDSV programunder contractACI

9982251,and through the NPACI; the National Institute of Mental Health

and the NSF under contractNIMH 2 P20 MH60975-06A2; the Lawrence

Livermore National Laboratory under ASCI ASAP Level-2 Memorandum

AgreementB347878andunderMemorandumAgreementB503159.

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