Improved Laplacian Smoothing of Noisy Surface Meshes

EUROGRAPHICS’99 / P. BrunetandR. Scopigno(GuestEditors)

Volume18 (1999), Number3

Improved Laplacian Smoothing of Noisy Surface Meshes

J.Vollmer, R. Mencl,andH. Müller

Informatik VII (ComputerGraphics),Universityof Dortmund,Germany

AbstractThispaperpresentsa techniquefor smoothingpolygonalsurfacemeshesthat avoidsthewell-knownproblemofdeformationandshrinkage causedbymanysmoothingmethods,like e.g. theLaplacianalgorithm.Thebasicideais to pushtheverticesof thesmoothedmeshback towardstheir previouslocations.Thistechniquecanbealsousedin order to smoothunstructuredpoint sets,by reconstructinga surfacemeshto which thesmoothingtechniqueisapplied.Thekey observationis that a surfacemeshwhich is not necessarilytopologically correct,but which canefficientlybereconstructed,is sufficientfor thatpurpose.

1. Introduction

A usefulapproachto acquirecomplex geometricmodelsincomputergraphicsis digitization.From thecloudof pointsscannedby digitizing devices like laserscannersor tactilescanners,a surfacedescriptionmaybeobtainedby connect-ing thepointsin anappropriatemannerinto a surfacemesh,e.g.a meshof triangles.

Figure 1: Fromleft to right: origin, noisymesh,threeitera-tionsof theLaplacianalgorithm.

Unfortunately, thedigitizedpointsoftendonot reflectthecorrectlocationontherealsurface,becauseof physicalnoiseaddedby the technicalscanningdevice. The effect is thatthereconstructedsurfacesoftendonotlook satisfactory. The

consequenceis that the meshhasto be smoothed,with thegoalto remove thenoise.

Figure 2: Fromleft to right: origin, noisymesh,threeitera-tionsof our algorithm.

One approachto surfacesmoothingis the Laplacianal-gorithm.Versionsof Laplaciansmoothingaree.g.known inimageprocessingandfinite-element-meshing.1 � 2� 3� 4 Theba-sic ideain imageprocessingis to replacethegrey valueofa pixel with anaverageof thegrey valuesof pixels in someneighborhood.Similarly, the locationof a vertex in a finite-elementmeshis correctedby calculatinga new locationastheaverageof the locationsof verticesin theneighborhood

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J. Vollmer, R.Mencl,H. Müller / ImprovedLaplacianSmoothingof NoisySurfaceMeshes

on themesh.Thisprocessof averagingcanbeapplieditera-tively, until theresultis satisfiable.

Averagingcausesa smoothingeffect which is similar tothat which we desire in our applicationof surface meshsmoothing.This observation can be transferredto surfacemeshesby calculatinga new locationof a vertex on a sur-facemeshby averagingthelocationsof theneighboringver-tices.Figure1 shows theresultof aniterative applicationofthisapproachto a dataset.For demonstration,theoriginallysmoothverticesweredisarrangedby noise.

However, the exampleof figure 1 shows an undesirableeffectof thatapproach.Theessentialdifficulty is thattheap-plicationof theLaplacianalgorithmshrinksthemesh.Evi-dentlytheresultingsmoothskull is significantlysmallerthantheoriginalone.

In thiscontribution,wepresentageneralapproach,calledHC-algorithm,which preventstheeffect of shrinking,whilepreservingtheeffect of smoothing.Thekey ideais to pushthe verticesobtainedin eachstepof iterationof the Lapla-cianalgorithmbackinto thedirectionof theoriginalvertices.Figure2 shows theresultof anapplicationon thedatasetoffigure1. Thesmoothedmeshis quitesimilar to theoriginal,alsoin size.

After fixing therequirednotationin section2, in section3we introducein somemoredetail the Laplacianalgorithmfor surfacemeshes,including several useful variants.Sec-tion 4 describestheHC-algorithm.In section5, thebehaviorof convergenceof the iterationof theoriginal Laplacianal-gorithmandtheHC-algorithmis given.Section7 discussesthe moregeneralcasethat only a cloud of pointsnot con-nectedby a meshhasto besmoothed.Thebasicideaof oursolutionto thatproblemis to constructa surfacemeshcon-nectingthe given pointswhich, whensmoothed,yields thedesiredsmoothedpointset.Theinterestingobservationwiththis approachis that it is not necessaryto constructa topo-logically correctmeshwhichmakesthetaskmuchmoreeasythantherathercomplex surfacereconstructionproblem.

2. Formal conventions

In thefollowing a meshM with vertex setV �� 1 �� n isgivenby a tupel K � p � , whereK � 2V is a simplicial com-plex andp : V IR3 is a function,whichmapsevery vertexi to its position pi . In generaldifferent verticesi �� j canhave the samepositionpi � p j . The "set" of pointsp willbe representedin the following by the vertical point vectort p1 �� pn � . The setof verticesV is split up into two dis-joint setsof fixed verticesVf ix and movable verticesVvar.The set of adjacentverticesof one vertex i is denotedbyadj i � . Let E : �� e � K �� e�� 2 be the setof edgesandF : �� f � K �� f �� 3 bethesetof facesof M.

It is characteristicallyfor smoothingalgorithms in con-trast to swapping-(seesect.7) andsubdivision algorithms

(seesect.6) that they modify the positionspi � i � Vvar ofthe(moveable)verticeswhile keepingthetopologyK of themeshunchanged. We follow the convention,that oi aretheoriginal givenpoints,qi thecurrentpoints(beforetheappli-cationof the algorithm)andpi the new, modifiedpositions(afteroneiterationof thealgorithm),e.g.a smoothingalgo-rithm startswith vertex positionso � : q andmapsthecurrentpositionsq ontop by onestep.

Usually, thenew positionpi of any vertex i dependssolelyon thepositionsof its adjacentverticesj � adj i � . To avoidproblematiccases,westatetheconventionthatif adj i ��theni � Vf ix.

3. Laplacian smoothing

In this sectiontheoriginal versionandtwo modificationsoftheLaplacianalgorithmwill be introduced.They belongtotheclassof smoothingalgorithmsdefinedasabove.

3.1. The original version

The Laplacianalgorithmis quite simple:the positionpi ofvertex i is replacedwith theaverageof thepositionsof adja-centvertices(figure3). Wehave

pi : � �1�

adj � i � � ∑ j � adj � i � q j � i � Vvar �qi � i � Vf ix �

qj6

qj5

qi

qj4qj2

qj1

qj3

qj1

qj2

qj3

qj4

qj6

qj5

ip

Figure 3: TheLaplacianalgorithm.

Thereare two techniquesto calculatenew positionspi .Thefirstmethodis tomodifyall positionsq p byonestep.So every new positionpi , i � 1 �� n, dependscompletelyonthesamesetof positions,namelyq. Thismethodis calledthesimultaneousversion. Thesecondvariantis to updatethenew positionspi immediately. This variantis calledthe se-quential version. In this casea position pi may not solelydependon the “set” of old positionsq but can dependona previously calculatednew positionp j , too. Hencethe re-sult of onesmoothingpassthroughall verticesi � Vvar willdependon theorderhow theverticesareconsidered.

The simultaneousversionneedsmore storagespaceforall old positionsq until the new arecalculatedcompletely.However, the resultsof this techniquesarebetter. Surpris-ingly, amongall othersmoothingalgorithmsthe Laplacian

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algorithmsolely hasthe propertythat the limits of the twotechniquesarethesameif they exist (see3.3and5).

Figure 4: Laplacianalgorithmappliedto a noisytorus.

Figure4 showsmesheswith Vf ix �!� after1,2,and3 itera-tionssmoothedwith theLaplacianalgorithm.Wecanrecog-nizeastrongdegreeof shrinkage.If thenumberof iterationsk goesto " themeshwill beshrinkingtowardsa point.

3.2. Inclusion of the central point

It is noticeablethatonly thepositionsof theadjacentverticesandnot thecurrentpositionqi is includedin thecalculationof thenew positionpi . This couldbedoneby extendingtheLaplacianrule to

pi �$#% & αqi ' 1 ( α�adj � i � � ∑ j � adj � i � qi � i � Vvar �

qi � i � Vf ix �It turnsout thatthis modificationis not essential.We obtainneithera bettersmoothingquality nor betterpropertiescon-cerningthe shrinkingeffect. This modificationonly delaysthesmoothingprocessin comparisonwith theoriginal ver-sion.

3.3. Inclusion of the original point

In general,theLaplacianalgorithmmight not converge.Forthis,considerameshconsistingof only two verticesi � j jointby oneedge � i � j . Theoriginal simultaneousLaplacianal-gorithmletsthepositionspi ) p j exchangealternately. Butin mostcasesthemeshconvergestowardsonepoint.

Figure 5: ExtendedLaplacianalgorithmwith α � 0 � 2.

Figure 6: ExtendedLaplacianalgorithmwith α � 0 � 4.

An ideato avoid thesedisadvantagesandto forceconver-genceagainsta non-trivial meshis to include the originalpointsoi α-weightedin thecalculation:

pi : � �αoi ' 1 ( α�

adj � i � � ∑ j � adj � i � q j � i � Vvar �qi � i � Vf ix �

Dependingon the location of α nearat 1 or nearat 0,we canchoosebetweena strongor a weakbinding of i tothe original position oi . It will be shown in section5 thatnon-trivial convergenceis guaranteedif α * 0. An interest-ing questionis whetherwe can solve the deformationandshrinkageproblem by this idea. Obviously, the last torusin figure 5 is quite better in comparisonwith the resultproducedby the original Laplacianalgorithmshowed in 4.However, it is not free from shrinkage.Concerningshrink-agewe ratherwould like to obtaina meshshown in figure6. Unfortunately, thismeshis notsmoothenough.All of ourexampleshave shown that theresultsof this techniqueonlyform a badcompromisebetweensmoothingquality andde-greeof deformationor shrinkage,respectively.

4. The HC-algorithm

The ideaof the HC-algorithm(HC standsfor Humphrey’sClassesandhasno deepermeaning)is to pushthemodifiedpoints pi (producedby the Laplacianalgorithm e.g.) backtowardsthepreviouspointsqi and(or) theoriginalpointsoi

by theaverageof thedifferences

bi : � pi + αoi ' 1 + α � qi �,� i.e. by

di : � + 1� adj i �-� ∑j � adj � i � b j �

p j2

jq1

p i

p j3

q i

q j3

q j4

q j2

b j2

d i b i

b j3

d ip i+

Figure 7: The definition of di in the case of the HC-modificationbasedontheLaplacianalgorithm(for simplifi-cationwith α � 0, i.e. withoutinfluenceof oi).

It turnsout (seesect.5) that in this case(in contrasttosect.3.2) thedifferencebi at thecentervertex i mustbe in-cluded,weightedby a scalarβ �/. 0 � 10 sothat

di : � βbi ' 1 + β� adj i �� ∑j � adj � i � b j �

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Figure 8: Fromleft to right: noisymesh,four stepsof Lapla-ciansmoothingonly. Wenoticea highdegreeof deformationandshrinkage.

Algorithmus 1 HC-algorithm(simultaneously)p : � o; // initialize points with original locations

repeatq : � p;for all i � Vvar do

n : �1� adj i �� ;if n �� 0 then

pi : � 1n ∑ j � adj � i � q j ; // Laplacian operation

end ifbi : � pi + αoi ' 1 + α � qi � ;

end forfor all i � Vvar do

n : �1� adj i �� ;if n �� 0 then

pi : � pi + βbi ' 1 ( βn ∑ j � adj � i � b j � ;

end ifend for

until 2 condition* // e.g. smooth enough

Figure 9: Fromleft to right: origin, noisymesh,alternatelya Laplacianoperation anda HC-modification.

Hence,there are two steps:the given smoothingalgo-rithm, thatmapsq to p, andtheHC-modification,thatmapsp to p ' d as a kind of correction.It is remarkablethatthis calculationis similar to the Laplacianoperation(fromsecti.3.2) too, not relatedto thepointspi but to the differ-encesbi .

Four iterationsof thetwo stepsareap-pliedto anoisycubein figure9. Wemakeout that the size of the cubede- and in-creasesalternately. Thelastcubeseemstobe shrunka little. The right figure showsthatthisisnotthecase.Thereasonfor thisopticalillusion arethecutedges.

Obviouslythecubessmoothedby theLaplacianalgorithm

only (figure 9) areshrinkingmuchmore.In both casesthesimultaneousversionof thealgorithmshasbeenperformed.

5. Mathematical treatment

Thegoalof thissectionis to investigatetheconvergencebe-havior of the Laplacian-and of the HC-algorithm.In sec-tion 3.3wasmentionedthattheversiondescribedtherecon-vergesin every case.It becameclear too that convergenceonly doesnot guaranteea goodsmoothingalgorithm.How-ever, it shouldbetheminimumrequirementwe expectfroman algorithmthat works iteratively. While the convergenceof theextendedLaplacianalgorithmdescribedin 3.3is quiteevident, this is not the casefor the HC-algorithm.We for-mulatetheiterationstepof thesimultaneousLaplacianalgo-rithm in matrix notation,sinceit is themoreimportantcasein practice.The treatmentof thesequentialversionis moreextensive. Both versionscanbe interpretedastwo kinds ofnumericalalgorithmsfor solving the samematrix equationAx � b. Additionally, this implies that the two limits areequalas mentionedabove. Furtherdetailson this are notconsideredhere.

Let M �3 K � p � be a meshwith vertex setV andsetofedgesE. Without lossof generalitywe requirethatVvar �� 1 �� m andVf ix �� m ' 1 �� n . Sothepointsp1 �� pmarethe interestingmoveablepoints.Onestepof the Lapla-cianalgorithmcanberepresentedin matrix notation:456

p1...

pm

798: � T

456q1...

qm

798: ' 456r1...

rm

798: �It is easyto verify thatthematrixT �; ti j � i < j = 1 < > > > <m is definedby

ti j : �@? 1AB� adj i �-� if � i � j C� E0 else

�andthevectorr : � t r1 ��D� rm � by

ri : � 1� adj i �� n

∑l = mE 1

? pl if � i � l F� E0 else

�The iterationsof the Laplacianalgorithm provide a se-

quenceof vectorst p1 �� pm � thatwill bedenotedby x � k �wherek is theindex of theiterationnumber.

If we initialize x � 0� : � t o1 �� om � thesequencesof thepointsproducedby thefollowing equations

x � k E 1� : � Tx � k � ' r (1)

x � k E 1� : �G.αE ' 1 + α � T0 x � k � ' 1 + α � r (2)

x � k E 1� : �G.H 1 + α � T0 x � k � ' αx � 0� ' 1 + α � r (3)

representthe iteratedpoint vectorsof the threeversionsofthe Laplacianalgorithmdescribedin sections3.1, 3.2, and3.3,respectively.

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It follows from Banach’s fixpoint theoremthat thosese-quencesconvergeif thematrixstandingin front of theitera-tion variablex � k � hasonly eigenvaluesλ with � λ ��2 1. It caneasilybe seenthat the sum∑m

j = 1 ti j of valuesof every ma-trix row is I 1.Equalityhappensif thecorrespondingvertexis joinedwith no fixedvertex. Sincethis is thegeneralcasewe only can conclude � λ �JI 1 for the eigenvaluesλ of T.The sameholds for the matrix αE ' 1 + α � T. But in thethird casewe canensureconvergencebecause 1 + α � T haseigenvalues � λ �K2 1 if α * 0. The following theoremsum-marizestheseobservations.

Theorem 5.1 Thesequenceof pointsmaintainedby thethreeversionsin sections3.1,3.2,3.3 of theLaplacianalgorithmaregivenby thethreeequations(1), (2), (3), resp.Let ρ de-notethe radiusof the spectrum.Convergenceis ensuredinall casesif ρ T �L2 1.Sincethisdoesnotholdin general,con-vergencecanbeguaranteedonly in thethird caseif α * 0.

A more preciseanalysisshows that convergencein thefirst two casescanbeconcludedif we only requirethat themeshis connectedandhasat leastonefixedpoint.However,in the usualcaseVf ix �M� for surfacemeshesnot even thiscanbeassumed.

Now, we will considerthe HC-algorithm.Let T be thematrix asdefinedabove. We canexpressthe HC-algorithmin termsof T, namelyby

x � k E 1� � Tx � k � ' r + βE ' 1 + β � T �JNOTx � k � ' r + O αx � 0� ' 1 + α � x � k �DPLP �

Theorem 5.2 TheHC-algorithmconvergesif α * 0 andβ *0 � 5.

Proof Transformingthe describingequationof the HC-algorithmsyields

x � k E 1� � Tx � k � ' r + βE ' 1 + β � T �JNOTx � k � ' r + O αx � 0� ' 1 + α � x � k � PQP�O + 1 + β � T2 ' 2 + α �- 1 + β � T ' 1 + α � βE P x � k �' αβE ' α 1 + β � T � x � 0� ' 1 + β �, E + T � r �

Responsiblefor theconvergenceis thematrixin frontof x � k � ,i.e.

H : � + 1 + β � T2 ' 2 + α �- 1 + β � T ' 1 + α � βE �Theeigenvaluesof H areobtainedby applyingthepolyno-mial

p X � : � + 1 + β � X2 ' 2 + α �, 1 + β � X ' 1 + α � βto theeigenvaluesλ of T whichareknown to satisfy � λ �RI 1.Calculatingthe amountof p X � for X � CI �,�X �KI 1 is noteasy. Fortunately, all eigenvaluesof T arereal.To seethis,

we consider

D : �455556 � adj 1�� 0 �� 0

0 � adj 2�� ......

. . . 00 �� 0 � adj m��

7 8888:andnoticethatDT is symmetric.Matricesremainsymmet-ric if they are multiplied from the left and the right withthesamediagonalmatrix. Now it is � adj i ��B* 0 for all i �� 1 �� mS� Vvar becauseof theconventionthatadj i �T�;�implies i � Vf ix. Therefore,it exists an inverseelementand

asquareroot.Hence,D ( 1U 2 exists(notunambiguously)andis a diagonalmatrix. Multiplying DT from the left andtheright with D ( 1U 2 resultsin D1U 2TD ( 1U 2 whichis symmetricandsimilar to T. It follows that all eigenvaluesof T areinIR.

This result makes it much more easy to show that� p X ��V2 1 if X I 1, α * 0, andβ * 0 � 5. This technicaltaskwill notbeshown herein detailbecauseof thelackof space.A proof canbefoundin 5. W

Theconclusionλ � IR is importantsincetheupperbound1 for p X � will beexceededif X � CI and �X �XI 1. For β �0 � 6 andα � 0 � 1 is p X �Y� + 0 � 4X2 ' 0 � 76X ' 0 � 54 andso� p i ��Z�1� 0 � 94 ' 0 � 76i �Z[ 1 � 2.

6. Discussion

We could see that the HC-algorithm yields quite satis-fying results.However, a smoothingalgorithm with HC-modificationis not completelyfree from shrinkage,too. Sofurthermodificationsarepossible:

Smoothingalgorithmsmap qi to pi for eachvertex i �V, accordingto our definition. Immediatelyafter a localLaplaciansmoothingoperationat vertex i it is µ i � : � pi +∑ j � adj � i � q j � 0. Clearly, after a completetreatmentof allverticesµ i �\� 0 cannotbe expected.Rather, µ i � is a dis-cretenormalvectorandcanberegardedasa measureof thecurvature at vertex i (it even is a discreteapproximationoftheLaplacianoperator∆, which is for its partanapproxima-tion of themeancurvature.6) A meshis planar if µ i � tendstowardsthezerovectorfor all i. On theotherhand,a meshis smoothif neighboringnormalvectorsµ j �,� j � adj i � , aresimilar. This is an important difference.Hence,for meshsmoothing,the µ i � neednot necessarilytend towardsthezerovectorlike for theLaplacianalgorithm.Rather, it is ap-propriateto filter the high frequency partsfrom µ. For thispurpose,a spectraldecompositionor Fourier analysis,re-spectively, of µ would benecessaryto geta filteredfunctionµ] .

TheHC-algorithmis a stepinto this direction.α � 0 andβ � 0 � 5 imply µ] � 0 � 5µ ^ µ, i.e. theHC-algorithmfilters thedifferencesof secondorder. Probably, therecanbe founda

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betterchoicefor µ] which perhapscan avoid deformationandshrinkagecompletely.

TheHC-modificationcanalsobeappliedin combinationwith subdivisionalgorithmsliketheDoo-Sabin-or Catmull-Clark-algorithm.7� 8 Mostof thesubdivision algorithmscon-tain rulesthatcalculateaveragesof adjacentpoints.Hence,they haveacomponentof shrinkage,too.It is possibleto ex-pandsuchrulesby theHC-modificationin thesameway aswe did with theLaplacianalgorithm.

7. Smoothing of noisy sample points

Mesh smoothingtechniquesmay also be usedin order tosmoothanoriginal cloudof pointswithout surfaceinforma-tion. The immediateidea is to reconstructa surfacemeshfrom the cloud of points.Smoothingof the resultingmeshyields vertices in new locations which are taken as thesmoothedcloud.Thedifficulty with thisapproachis thatsur-facereconstructionin generalis not an easyproblem,cf. 9

However, it will turn out in the following that, for the pur-poseof smoothing,it is not necessaryto reconstructa per-fect mesh.The only requirementis that the neighborhoodson thereconstructedmeshareaboutthesameasthey areontheunderlyingrealsurface.Whenever a reconstructedmeshthroughthe noisy point set is not available,the RFS-mesh(replacementfor surface)canbetaken in orderto eliminatethenoise.This RFS-meshcontainsnearlythesamenumberof facesasa valid surfacemesh,but about50 percentmoreedges.Facescancrossandoverlapin aRFS-mesh.

Figure 10: Left: thenoisypoint set.Right: theEMST.

TheRFS-meshis basedon theSDG(surfacedescriptiongraph)of Mencletal. 10� 11. Thekernelof theSDG,in turn,istheEMST of the given point cloud.TheEMST (Euclideanminimum spanningtree) is a tree connectingall points ofP with line segmentsso that the sumof its edgelengthsisminimized(seefigure 10, right). The EMST (asinitial sur-facedescriptiongraph)is thenextendedto the surfacede-scription graph(SDG) of higherorder by addingedgesatappropriateregions(cf. figure11, left for anexampleof theSDG).This graphconsistsof significantlymoreedgesthantheEMST andcanserve asa basefor gettingneighborhoodinformationwhengeneratingtheRFS-mesh.

Independentlyfrom the degreeof noise,the SDG-meshis alwaysdefined.It containslessedgesthana surfacemesh

Figure 11: Left: theSDG.Right: theRFS-mesh.

but "good"edges,i.e. it containsedges,that"lie" closeto theassumed(but actuallyunknown) surfacethatis describedbythenoisypoint set.

In the following we outline thecomputationof theRFS-meshoutof theSDG.

If required,theSDGcanbefilled with additionalnearest-neighbor-edges.To respectvariationsconcerningthedensityof the noisy points we usean adaptive numberof nearestneighbors(estimatedby usingtheSDG-meshasreference)for differentvertices.

Around eachvertex a coronaof faceswill be createdonbaseof incidentedgeslike aspanningumbrella.By this, foreachfaceonenew edgewill becreated,if it doesnotalreadyexist. Theseedgesareimportantfor theconnectivity of thenew mesh.

Facesthatdonot fulfill agivenmeshqualitymeasurewillbedeleted.Weusea quality measureintroducedby Bank&Smith.12

To make edgeswapping(betweenneighboringtriangles)applicable,incidentfacesto anedgearedeletedby compar-ing their encloseddihedralanglesuntil thereremainat mosttwo at anedge.If therearemorethantwo facesat an edgewe keepthatpair of edgesthat form themaximumdihedralangle.

All edgesthat do not have at leastoneincident facearedeleted.

The resultingmeshis called the RFS-meshand can betakenasinput for theHC-algorithmin orderto eliminatethenoiseoutof thepoint set.In figure11 (right) anexampleforanRFS-meshis depicted.If wewould take theRFS-meshofthis figureasinput for thesmoothingalgorithm,a point setof similarquality (with respectto thenoisereductionfactor)to theoneof figure2 in section1 will beobtained.

During thesmoothingprocessit is advantageousto applyedgeswappingandonestepof theHC-algorithmalternately.Edgeswappingbetweentwo adjacentfacesis usedto in-creasethesumof dihedralanglesatall five edgesof thetwofacesin thetriangularmesh.Freitagetal.13 foundoutthatthecombinationof edgeswappingandsmoothingyields better

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resultsthanoneof thetwo techniquesonly. This wasexam-ined in the field of finite elementmethods,i.e. for meshesin IR2 andtetrahedronmeshesin IR3. We foundout thatthisis alsovalid for surfacemeshes.Thereasonis, thatsmooth-ing techniquesdonotchangetheconnectivity formedby theedgesandfaces(seesect.2). All examplesshow thatwecanimprove thedegreeof smoothnessif we permitedgesto ad-just themselvesin a new manner. Note, thatedgeswappingdoesnotchangeany vertex positionandhencethemeshdoesnotshrinkthereby.

Figure 12: Noisysetof points.

Figure 13: Temporary RFS-mesh.

An examplefor the smoothingprocessis shown with alargepointsetin figures12–16.Thefirst figure12shows the

noisy input point setandfigure 13 its correspondingRFS-mesh.In figure14 theRFS-meshis smoothed.Thecompu-tation time for the whole smoothingprocesswasabout10seconds.It canbeseenthattheRFS-meshis not a valid sur-facemesh.Thesmoothedpoint setof the RFS-meshis de-pictedin figure15.Thissmoothpoint setwastakenasinputfor thesurfacereconstructionalgorithmof Mencl et al. 10� 11

andtheresultis shown in figure16.

Figure 14: RFS-meshsmoothedby the HC-algorithm incombinationwith edge swapping.

Figure 15: Noisereducedset of points maintainedby thesmoothedRFS-mesh.

8. Conclusions

A techniquefor the eliminationof noisein setsof samplepointsandsurfacemesheshasbeendevelopedandapplied

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Figure 16: Reconstructionbasedonthenoisereducedsetofpoints.

successfully. The proposedtechniquepresentscertain ad-vantages:_ it canproducemesheswith thesamesmoothingdegreeas

theLaplacianalgorithm,_ in contrastto the Laplacianalgorithmit preservesshapeandsizeof thesampledobjectfar better,_ it preservesthepointdensityof themesh,_ the algorithmworks very fastandrequiresonly calcula-tionsof simplevectorarithmetic,_ it is easyto implement.

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Improved Laplacian Smoothing of Noisy Surface Meshes

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