Featurepreserving variational smoothingof terrain data

TolgaTasdizen RossWhitaker

Schoolof Computing Schoolof ComputingUniversityof Utah Universityof Utah

SaltLakeCity, UT 84112 SaltLakeCity, UT 84112e-mail: tolga@sci.utah.edu e-mail: whitaker@cs.utah.edu

Abstract

In this paper, we presenta novel two-step,variationalandfeaturepreservingsmoothingmethodfor terraindata. Thefirst step computesthe field of 3D normal vectors fromthe heightmapandsmoothesthemby minimizing a robustpenaltyfunction of curvature. This penaltyfunction favorspiecewise planarsurfaces;therefore,it is bettersuitedforprocessingterraindatathenpreviousmethodswhichoperateon intensity images. We formulatethe total curvatureof aheightmapasa functionof its normals.Then,the gradientdescentminimization is implementedwith a second-orderpartial differentialequation(PDE) on the field of normals.For thesecondstep,we defineanotherpenaltyfunctionthatmeasuresthe mismatchbetweenthe the 3D normalsof aheightmapmodelandthe field of smoothednormalsfromthefirst step.Then,startingwith theoriginal heightmapastheinitialization,wefit anon-parametricterrainmodelto thesmoothednormalsminimizing this penaltyfunction. Thisgradientdescentminimization is also implementedwith asecond-orderPDE.We demonstratetheeffectivenessof ourapproachwith a ridge/gullydetectionapplication.

1 Intr oduction

Terraindatacontainsinformationthatis pertinentto avarietyof applications.For instance,scientistsestimateslopegradi-entandaspectfor analysisof hydrologicalflow over terrain.Theseestimatesare later usedin flood pathpredictionsforsafetyplanning.Anotherapplicationcouldbetheanalysisoftheterrainof otherplanetsto plana routethatwill be takenby anunmanned,explorationvehicle.However, thelevel au-tomationin thisanalysisis currentlylow andtherefore,thesearemostly time consuming,manualtasks. Standardimageprocessingtechniquesarenot optimal whenappliedto ter-rain data. Therefore,the first stepin a pushtowardsmoreautomatedanalysisshouldbethedevelopmentof processingtechniquesspecificto terraindata.

Terrainsare often representedas a collection of heightmeasurements,i.e. a digital elevation model (DEM). Ele-vationvaluescanbe representedaspoints,contourlinesortriangulatedirregularnetworks. In this paper, we areinter-

estedin terraindatathat is representedon a 2D regular, rec-tilinear grid, i.e. a height image. SuchDEMs areusuallycreatedby stereophotogrammetryfrom aerialphotographs,field surveys,ormostcommonlyby manuallydigitizing con-tourmaps.

DEMs areavailablein many differentforms that vary inaccuracy, andhorizontalandverticalresolution.ThehighestresolutionDEMsdistributedby theUnitedStatesGeologicalSurvey (USGS)correspondto 7.5 minutequadranglemaps(1:24000scale)with horizontalresolutionof 10 or 30 me-ters[20]. Vertical accuracy of theseDEMs varieswith thedesiredmeanerrorof 7 meters,but it is not unusualto haveerrorsof up to 15 meters. The 7.5-MinuteDEMs arecre-atedby optically scanningcontourmapsandthenfitting anapproximatingsurface. Errors can be producedwhen ele-vationsareinterpolatedfrom digitized contours.Therearemany potentialsourcesof errorsthateffectaccuracy andun-certaintyof terrainfeatureanalysisusingDEM data[21].

Someof the necessarytools for analysisof terrain dataareridge detection,segmentationandcompression.All ofthesetools require that we are able to extract featuresofthe terrain such as ridges and gullies in the presenceofnoise. In image processing,Perona& Malik (P&M) in-troducedan anisotropicdiffusion processthat canpreserveedgesbetweendistinct regions while smoothingthe noisewithin the regions[12]. Unfortunately, a direct applicationof this methodto heightmapsyields unsatisfactoryresults,seeSection3. In this paper, we formulatea correctgener-alizationof this methodto heightmapsby posingit asanenergy minimizationproblemon curvature.Thevariationalmethodwe proposeis alsogeometric,i.e. it is independentof theparameterization.

The restof this paperis organizedasfollows. Section2presentsa brief overview of relatedwork in the literature.Section3 discussesedgepreservingsmoothingmethodsinimageprocessing.Section4 formulatesa robust curvatureenergy for

���� D height surfaces. Section5 introducesoursplittingstrategy thatgivesanefficientandstableminimiza-tion procedurefor thisenergy. Section6 illustratesresultsoftheproposedapproachanddemonstratesits advantageswithin the context of a ridge/gully detectionapplication. Sec-tion 7 summarizesthecontributionsof thispaper.

2 RelatedWork

Perona& Malik introducedthe anisotropicdiffusion PDEfor intensity imagesin their pioneeringwork [12]. Nord-strom[11] andBlack et al. [1] have shown that P&M’sdiffusionis thegradientdescentprocessfor a robustenergyfunction. Otherresearchershave proposeddifferentenergymetrics,suchastotal variation,which yield variationsof theanisotropicdiffusion PDE [14]. Theseapproachesare allsecond-orderPDEs. Tumblin and Turk proposea fourth-orderPDEasa detail preservingcontrastreductionmethodfor depictinghigh contrastimageson low contrastdisplaydevices[19]. TheirLCIS(low curvatureimagesimplifier)al-gorithmis relatedto therobustcurvatureminimizationstrat-egy proposedin this paper. However, their approachis for2D images,whereas,we take into accountthe geometryof���� D heightsurfaces.Furthermore,we provide a variationalgeneralizationof anisotropicdiffusionto suchsurfaces.

A problemrelatedto oursis thesmoothingof 3D surfaces.In the context of level set representations,meancurvatureflow (MCF), a geometricPDE that minimizessurfaceareahasbeena popularchoicefor enforcingsmoothnessof themodel [9, 24]. MCF is not a featurepreservingprocess.Furthermore,it suffersfrom severalproblemsincludingvol-umeshrinkage,pinchingof thin structures.In thecomputergraphicsliterature,smoothingsurfacemesheshasbeenap-proachedasanenergy minimizationproblem[10, 8, 22] andasa filtering problem [17, 4, 7]. More recently, anisotropicdiffusion processeshave beenproposedfor surfacemeshesandlevel sets[3, 13, 16]. Noneof thesemethodsmake useof thespecificpropertiesof

���� D heightsurfaces.Depth reconstructionhas beena major focus in earlier

computervision research[6, 18, 2]. More recently, in thecontext of height maps,a

������ versionof MCF hasbeenderived [23]. Desbrunet al. proposea featurepreservingdenoisingprocessfor heightmaps[5]. Their approachis tousea second-orderflow that is a modificationof MCF thatdoesnot take into accountthegeometryof heightmaps.

3 Edge preserving smoothing for in-tensity images

An establishedsolutionto thesmoothingproblemis to poseit asthegradientdescentof anenergy function,andto imple-mentthegradientdescentasa non-linearpartialdifferentialequation(PDE).Thechoiceof theenergy functiondependson the application,andit determineswhat part of the inputsignalis preservedandwhatpart is eliminatedasa resultofthegradientdescentPDE.In imageprocessing, it is typicalto usean energy that favors smoothness.For instance,theheatequationimplementsthegradientdescentfor themini-mizationof the integral of a quadraticpenaltyon the imagegradientmagnitude���� ���������� ������������ � ��� � �"!#�$�&% (1)

0 0.5 1 1.5 20

1

2

3

4

|x|

Quadraticpenalty

Robustpenalty

Figure1: Comparisonof quadraticandrobustpenaltyfunc-tions.

where ' is the domainof the image. Processingan imagewith the heatequationis mathematicallyequivalentto con-volving it with a Gaussiansmoothingkernel. It is a wellknown fact thatsuchsmoothingblursboundariesin the im-age,andwill goto aconstantintensitylevel in thelimit. Thisresultis alsoexpectedfrom thevariationalpointof view, be-causethe quadraticpenaltyfunction, seeFigure1(a), doesnot permit thepresenceof high gradientmagnitudeoutliers(edges).Figure2(a) illustratesa surfacedescribedby a partof the Mt.Hood DEM. Figure2(b) illustratesthe resultsoftreating this data as an intensity image and smoothingitwith a Gaussiankernel. The ridgesin the original dataareroundedanddistorted.

In theirseminalwork, Perona& Malik (P&M) introduceda non-linear, anisotropicdiffusion PDE that can preserveedgeswhile smoothingnoise[12]. This methodhasgainedpopularityasa successfuledgedetectionmethodin imageprocessing.Nordstrom[11] andBlack et al. [1] show thatP&M anisotropicdiffusionPDEis thegradientdescentfor arobustenergy function:���� �)(+*-,�.0/21435/768 6 ��������� � ��� � �"!�9 ,�.0/21&35/768 6 �$��:<;

(2)This robust penalty function, seeFigure 1(b), is boundedfor high valuesof its argument; therefore, it allows thepresenceof a limited number of outliers. Indeed, theP&M anisotropicdiffusion hasbeenshown to converge toa piecewise-constantsolution, i.e. a solution that haszerogradientat mostlocationsanda limited numberof locationswith highgradientmagnitude.Shockformationatedgesandsmoothingof detailsat otherlocationsis themechanismbywhich the solution is achieved. The parameter= controlsthe degreeof edgepreservation; however, it is not a sim-ple threshold.Pointswith gradientmagnitudeslessthan =mayform shocksandbeincorporatedinto edgesin thepres-enceof many pointswith stronggradientsnearby. On theother hand,isolatedpoints with large gradientmagnitudeswill besmoothed.Motivatedby this successfulmethod,oneapproachto smoothinga heightmap is to treat it asan in-tensityimageandapplyP&M anisotropicdiffusiondirectly.

Figure2(c)illustratestheproblemswith thisapproach;P&Manisotropicdiffusioncausesa “staircasing”effect. This ef-fecthasbeendocumentedin [25, 19] andis causedby shocksforming at seeminglyrandomlocationsin regionsof high,uniformgradientmagnitude,e.g.theslopesof amountaininterraindata.

4 Feature preserving smoothing forheight maps

The original P&M anisotropicdiffusion fails whenappliedto terraindataasshown in Figure2(c), becauseit wasde-signedto preserve andcreate>0? discontinuitiesin intensityimagesto detectedges.However, >0? discontinuitiesareveryrarein terraindata,they occuronly on verticalclif fs. In ter-rain data,one is insteadinterestedin ridgeswhich are > �discontinuities.Wewould like to find aprocessthatrespects> � continuousregionssuchastheslopesof a mountainandpreservesandcreates> � discontinuities.

Let usconsiderthe parametricsurface @ definedby theheightmap @BA � % ��CD� A � % � %FE A � % ��CGC ; (3)

Theareaof this surfaceover thedomain ' is computedasH A E CI� ��J (LK ���E A � % ��C � � ������ ; (4)

The gradientdescentPDE on the first variationof this areais� ENM �� $� �O! A �$ENM J (LK ���$EP� � C , which wasusedfor

smoothing� �� D surfaces[23]. This PDE is the MCF for

heightmaps.It hastheappearanceof beinganisotropicdueto the division by

J (LK �+�EQ� �beforethe divergenceop-

eration.However, this is nota featurepreservinganisotropicprocess.The rescalingis the projectionof the 3D motionontotheimageplanewhich is aresultof representingthe3Dsurfaceasa2D graph.

Explicitly preserving> � discontinuitiesrequiresapenaltytermonthesecond-orderstructureof surfaces,i.e. curvature.We proposea family of energies that are surfaceintegralsof penaltyfunctionson total curvature, R �S A � % ��C , which isthe sumof the squaresof the two principal curvatures.LetTVU�WYXZW

beanarbitrarypenaltyfunction,thenwe definethefollowing family of energy functions[ A E CD� � T A\R S A � % ��CGC J (]K ���E A � % ��C � � ������� ; (5)

Notice that ifT A\R S C�� (

, this energy reducesto the areaof @ . We can penalizethe integral of total curvaturebychoosing

T A\R S C^� R �S . However, similar to heatflow inimageprocessing,this energy doesnot preserve discontinu-ities becausethequadraticpenaltyfunctiondoesnot permitany outliers. Instead,we usethepenaltyfunction thatgivesriseto P&M diffusionin (1), but replacethe imagegradientmagnitudewith total curvature:T A\R S CD� (+*-, .�_ 6`8 6 ; (6)

Theparameter= determinestherangeof curvaturesthatarepreserved. Figure 2(d) illustratesthe resultsof smoothingtheMt. Hooddatasetusingthis energy definition.

5 A splitting strategy

A direct minimizationof the energy definedin (5) requiressolving a fourth-orderPDE. This is a computationallyex-pensive andunstabletask. In this section,we introduceastrategy thatsplitsthesolutioninto apairof coupledsecond-orderPDEsthatcanbesolvedefficiently.Step1a: Total curvature fr om normal vector variationsThefirst stepin our splitting strategy is to expresstotal cur-vature R �S in termsof thenormalvectorsto theparametricsurface@ . Thenormalvectorto @ canbefoundasa � (b (�K E �c K E �dfeg E cE d*h(jik %

(7)

whereE c

andE d

denotethex andy partialderivativesof theheightmap

E. Letsdefinethe l^m � matrixof thegradientof

thenormalvectorsin theimageplane� a �On a c % a dpo ;(8)

Also definethe� ml matrix thatprojects

� ainto 3D spaceq � (n (�K E �c K E �d osr5t � 9 (�K E �d * E c E d E c* E c E d (�K E �c E d : ; (9)

The variation of the normal vectorsintrinsic to the heightmapsurfaceis

�u a � � avq. Thentotal curvaturecanbe

foundas R �S � ��� u a � � %(10)

wherethe norm denotesthe Frobenius norm of the lwmxlmatrix,which is thesquareroot of thesumof thesquaresofall of thematrix elements.Step1b: Decouplingthe normals fr om the height mapThe secondstepin our strategy is to decouplethe normalvectorsfrom thesurfacedefinedby theheightmap. Substi-tuting (10) in (5) gives[ A a CI� �� T A �+� u a � Czy A � % ��C{������� % (11)

wherey A � % ��CI� J (LK �+�$E A � % ��C � � . We initialize thenor-

mal vectorsaccordingto (7). Thenwe fixE

andprocessthenormalvectorsto minimize the energy (11). Sincenormalvectorshaveto remainunit length,this is aconstrainedmin-imizationproblem,which is accomplishedby thefollowingsecond-orderPDEon thenormalvectors:| a| � * A2} * a�~xa C ��! A y A � % ��C Th� A �+�$u a � C �u a C %

(12)where

~denotesthetensorproduct,and

T �is thederivative

ofT

. Theoperator} * a�~-aprojectsthechangevectorto

(a) (b) (c) (d)

Figure2: (a)A portionof the� �� D surfacedefinedby theMt. HoodDEM, (b) Gaussiansmoothingof theheightimage,(c)

P&M diffusionon theheightimage,and(d) ourgeneralizationof P&M diffusionto� �� D surfaces.

(a)

(b)

Figure 3: (a) An imageof noisy unit vectors,and (b) theresultof minimizing therobustcurvatureenergy.a

ontotheplaneperpendiculartoa

, thusenforcingthatthenormalvectorsremainunit length.

This PDE is the generalizationof P&M anisotropicdif-fusion for a field of unit vectorsdefinedon a

� �� surface.Figure3(a)illustratesanartificial imageof noisyunit lengthvectors. The vectorswerechosento have a differentmeaneachquadrantof the image. Figure 3(b) illustratesthe re-sultsof smoothingthis vectorimagewith thePDE(12) andwith

Tasdefinedin (6). Thediscontinuitiesin thenormals

betweenthe quadrantsarepreservedwhile the noisewithinthequadrantsis smoothed.Step2: Refitting the height map to the normal vectorsSinceourgoalis to denoisetheheightmap,wehaveto relatetheprocessingof thenormalvectorsbackto thesurface.Wedo this with a refitting stepthat minimizesan energy thatreflectsthediscrepancy betweentheheightmapsurfaceand

Noisy�height�map

Compute�N from�h Process�N Process�h

Done?No

Smoothed�height�map

Figure4: Flow chart.

theprocessednormalvectors.Thisenergy is� A E C�� �� a u ! a uh* a u ! a)� ������� % (13)

wherea u

anda)�

denotethenormalvectorscomputedfromEaccordingto (7) and the processednormal vectorsob-

tainedfrom iteratingthePDE(12), respectively. Leta � �n�� � � � � r o#� , then� A E CI� �� J (]K ���EP� � * �E!�9 � �� � : K � r ������� ;

(14)Thegradientdescentfor thefirst variationof thisenergywithrespectto

Eis� E�� � * ��!�� �EJ (LK �+�EQ� � *�9 � �� � :�� ; (15)

5.1 The algorithm

Theflow chartfor the algorithmis shown in Figure4. Thenormalsprocessingstageof thealgorithmcomputesthegra-dientdescentfor the normals(12) for a fixed numberof it-erations(25 for the experimentsin this paper). Hence,weavoid evolving evolving thenormalstoo faraway from theirinitializationfrom

E. Theheightmapfitting to theprocessed

normalvectorsis given asa gradientdescentPDE in (15).Thisstageof thealgorithmis rununtil thediscrepancy mea-sure(13) betweenthe new normalsandthe normalsof the

height mapceasesto decrease,which signalsthe needforanotherroundof processingthenormalvectors.Theoverallalgorithmrepeatsthesetwo stepsto denoisetheinputheightmap. This algorithmconsistsof solving two second-orderPDEsin seriesinsteadof a direct fourth-orderPDE,whichmakesit computationallytractable.

5.2 The parameters

Therearetwo free parametersin our algorithm: = andthenumberof iterationsof themain loop in Figure4. Thecon-ductanceparameter= determinestherangeof curvaturesthatis smoothedandtherangethatis preserved.As in P&M im-agediffusion,it is notasimplethreshold.In wasfixedat � ; (for all of the resultsshown in this paper. Unlike, in P&Mimagediffusion,this parameterdoesnot needto bechangedfor differentsurfacemodels. In the context of P&M imagediffusion,theunitsof = arein graylevels;consequently, theoptimal choiceof = is imagedependent.However, for sur-faces,theunitsarein curvature,which is dataindependent.This makesit possibleto choosea = valuethatgivesconsis-tentresultsoverabroadrangeof surfaces.

The numberof times we repeatthe main loop (process-ing thenormalvectorsfollowedby refitting) determinestheamountof smoothingappliedto thedata.This is thesecondfreeparameterin thesystem.In Section6, wepresentexper-imentalresultsthatillustratevariousamountsof smoothing.Thisparametercouldbeexchangedfor adatatermweightbyposingthesmoothingproblemasa reconstructionproblem.In a reconstructionproblemthe energy will be a weightedsum of a smoothnessterm and a data term. A PDE thatminimizesthis typeof energy is run until convergence,andthefreeparameteris therelative weightingbetweenthetwoterms.

6 Experiments

In this section,we presentexperimentswith two differentDEM datasets.We demonstratetheeffectivenessof our ap-proachas a pre-processingstep for ridge/gully detection.Therearemultiple definitionsof a ridge. For terraindata,weconsideraridgeto bepointsof localmaximumcurvatureon the isocurvesof the heightfunction

E A � % ��C . The curva-tureof theisocurvesof

E A � % ��C canbefoundasR � E �d E c�c * � E c E d E c#d K E �c E djdn E �c K E �d osr5t � ;(16)

Let � bethetangentvectorto theisocurvesofE A � % ��C . Then

point of maximalcurvatureof theisocurvesarefoundasthezerocrossingsof thefollowing directionalderivative:� R� � � � ; (17)

Among points that satisfy this equation,thosewith R��O�areridgesandthosewith R���� aregullies.

The main point of this discussionis not the ridge detec-tion itself, but therole of our smoothingalgorithmasa pre-processingstep. Equation(17) involvesthird derivativesofE A � % ��C ; therefore,it is hard to computestably. Typically,thesethird derivativesof Gaussiankernelsareusedfor thiscomputation.However, theGaussiankernelseliminatesanddislocatesridgelines.WeproposethatlikeP&M diffusionisa betterchoicethanGaussiansmoothingfor edgedetectionin intensityimages,our anisotropicsmoothingalgorithmisa better approachfor smoothingthe data for purposesofridge/gullydetection.

Figure5(a) illustratesthe surfacedefinedby the original( � �p� m ( � �]� Mt. Hoodheightdata. Figure5(b) and(c) il-lustratetheresultsof smoothingwith theproposedapproachafter2 and10 iterationsof our algorithm,respectively. Oneiterationtakesapproximately20 minutesfor this datasetona Intel 1.7 Ghz processor. For denoisingpurposesa cou-ple of iterationsaresufficient. More iterationsstartforcingthe surfacetowardsbeingpiecewiseplanar. The prominentstructureof the mountainis preserved as the smallerscaledetailandnoiseis eliminated.Thiscanbeperformedaspre-processingfor ridgedetectionor compression.

Figure6 illustratesthe ridge/gully detectionexperiment.Theblueandtheredcurveson thesurfacedepictridgesandgullies, respectively. For the resultsshown in the top row,we useda Gaussiansmoothingkernelwith a low anda highstandarddeviation. The low standarddeviation did not de-noisethedataenough;hence,therearealot of falsepositivesin thedetectionresults.Ontheotherhand,thehighstandarddeviation Gaussianresultedin too many falsenegatives,i.e.missedridges. Figures6(c) and(d) illustratedetectionre-sultsobtainedby usingtheproposedanisotropicdiffusionasthesmoothingstep(after10 and20 iterations,respectively).The resultsaremuchbetterthanwith Gaussiansmoothing.For instance,the prominentridge runningdown the centerright sideof themountainwasmissedin both low andhighstandarddeviation Gaussiansmoothingresults. In contrast,this ridge was able to self-organizeand strengthenduringanisotropicdiffusionandwassuccessfullydetected.In sum-mary, the detectionresultswith anisotropicdiffusion havefewer falsepositivesand fewer falsenegatives.

Figure 7 illustratesa different type of terrain: the tran-sition from a flat valley to rolling hills. Two shallow riverbedscan be observed as depressionsin the valley. Fig-ure 7 (a) shows the original height surface,while (b) and(c) show the surfaceafter 1 and10 iterationsof our algo-rithm. Figures7(d)-(f) illustrate the correspondingdetec-tion results.Although,only aminimalamountof smoothingcanbeobservedin Figure7(b), thecorrespondingdetectionresult shown in Figure 7(e) is much betterthan the detec-tion from the original datain Figure7(d). Notice thatbothof the shallow river bedsweredetectedasgullies. Furtheranisotropicsmoothingresultsin acleanerdetection,but alsolossesthe weaker river bed. The processingtimes for thissmallerdatasetwasapproximately3 minutesper iterationona Intel 1.7 Ghz processor.

(a) (b) (c)

Figure5: (a)Mt. Hood,(b) after2 iterations,and(c) 10 iterationsof themainprocessingloop.

(a) (b)

(c) (d)

Figure6: Ridgeandgulliesaredepictedby blueandredcurves,respectively. ResultsusingGaussiankernelswith standarddeviation (a)1 pixel and(b) 3 pixels.Resultsusingour anisotropicsmoothingwith (c) 10 iterations,and(d) 20 iterations.

(a) (b) (c)

(d) (e) (f)

Figure7: (a) Original data,(b) after 1 iterations,and(c) 10 iterationsof our algorithm. Ridges(blue) andgullies (red)detectedfrom therespectivedatasetson thefirst row.

7 Conclusions

We derive a variationalgeneralizationof P&M anisotropicdiffusion for featurepreservingsmoothingof terrain data.Theproposedmethodis derivedfrom thegeometryof

� �� Dsurfaces.It preservesandenhancesdiscontinuitiesin thesur-facenormalvectors;hence,forcing surfacestowardspiece-wise smoothness.This type of processingis bettersuitedto terraindatathandirect applicationsof imageprocessingtechniquesandtheirheuristicmodifications.

Measureson surface normal variations require solvingfourth-orderPDEson level sets. However, by processingthenormalsseparatelyfrom thesurface,we cansolve a pairof second-orderequationsinsteadof a fourth-orderequa-tion. This methodis numericallymore stableandcompu-tationally lessexpensive thansolving the fourth-orderPDEdirectly. The shortcomingof this methodis the computa-tion time; however, the processlendsitself to parallelism,and therefore,the useof multi-threading. Also, recentde-velopmentsin solvingnonlinearimagePDEon commoditygraphicshardwarepromisesignificantspeed-upsfor our al-gorithm[15].

Acknowledgements

This work is supportedin part by ONR (#N00014-01-10033), NSF (#CCR0092065)and ARO (DAAD19-01-1-0013).

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