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Volumexx(200y),Numberz, pp.1–11 Interactive Character Posing by Sparse Coding RanchY.Q.Lai1andPongC.Yuen1andKelvinK.W.Lee2andJHLai3 1DepartmentofComputerScience,HongKongBaptistUniversity,HongKong,China 2DepartmentofCommunicationstudies,HongKongBaptistUniversity,HongKong,China 3SchoolofInformationScienceandTechnology,SunYat-senUniversity,GuangZhou,China 2 1 0 2 Abstract n a J Characterposingisofinterestincomputeranimation.Itisdifficultduetoitsdependenceoninversekinematics (IK) techniques and articulate property of human characters . To solve the IK problem, classical methods that 6 relyonnumericalsolutionsoftensufferfromtheunder-determinationproblemandcannotguaranteenaturalness. ] Existingdata-drivenmethodsaddressthisproblembylearningfrommotioncapturedata.Whenfacingalarge R variety of poses however, these methods may not be able to capture the pose styles or be applicable in real- G timeenvironment.Inspiredfromthelow-rankmotionde-noisingandcompletionmodelin[LYL11],wepropose anovelmodelforcharacterposingbasedonsparsecoding.Unlikeconventionalapproaches,ourmodeldirectly . s capturestheposestylesinEuclideanspacetoprovideintuitivetrainingerrormeasurementsandfacilitatepose c synthesis.Aposedictionaryislearnedintrainingstageandbasedonitnaturalposesaresynthesizedtosatisfy [ users’ constraints . We compare our model with existing models for tasks of pose de-noising and completion. 1 Experimentsshowourmodelobtainslowerde-noisingandcompletionerror.WealsoprovideUserInterface(UI) v examplesillustratingthatourmodeliseffectiveforinteractivecharacterposing. 9 0 Categories and Subject Descriptors (according to ACM CCS): I.7 [Computer Graphics]: Computer Graphics— 4 Animation 1 . 1 0 2 1. Introduction To solve the character posing problem, inverse kinematics 1 isoftennecessarytofindtheskeletoninanglespacerepre- : Characterposingisanimportantstepforkey-frameanima- sentation.Theclassicalinversekinematicssolvesanunder- v i tion.Itisdifficultfornovicesandevenskilledartistsdueto determined non-linear system to find the joint angles. One X thearticulatepropertyofhumanmotion.Withthemostpre- popular method is to exploit the gradient information–that r vailing input device still being the mouse, the users’ input is,toconstructtheJacobianmatrixandthensolvethesys- a canonlyprovidebasicinformationsuchas2Dscreencoor- temiterativelystartingfromarandominitialpoint.However, dinates.Basedonthislimitedinformation,itischallenging themappingfrom3DEuclideanspacetojointanglespaceis to generates satisfactory character poses efficiently. More- one-to-manyiftheusers’constraintsareinsufficient.Forex- over, it requires considerable information to determine the ample,givenasetofincompletejointconstraintssuchasthe character’salldegreesoffreedom(DOF’s).Givensome3D 3Dpositionsofsomejoints,thesolutionobtainedfromJaco- positionalinformationofoneorsomejoints,itisusefulto bianmethodwillnotbeuniquebutdependsontheinitiation, re-position the rest of joints or even the whole pose if the nottomentionthatallposesresultingfromthepossibleso- informationprovidedbyusersisnotaccurate.Forexample, lutionsareunlikelytobenatural.Onenotonlyhastonarrow a novice animator is likely to pose an unnatural character downthesolutionset,butmustalsorefinethesolutionsso withinashorttimelimit.Itisthenuptothealgorithmtoex- thattheresultingposeisnatural. tractusefulinformationsuchasposestylefromtheunnatural characterandcreateanewnaturalone.Thisshallcarryout Onewaythatmayhelpisbylearningfrommotioncapture interactivelyforsynthesizingnaturalposes,andtheprocess data.Eventhoughthespaceofjointconfigurationislarge, isreferredtoascharacterposing. thedesirableposesonlyspanamuchsmallerspace.Forex- submittedtoCOMPUTERGRAPHICSForum(1/2012). 2 RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding f is the forward-kinematic function which involves a set oftranslationsandrotations,usuallyimplementedprocedu- Dictionary DLiecatitronninagry Mocap database rally in some programming language; J(q) is the Jacobian matrix defined as J(q) = ∂f(q) . The Jacobian matrix is Training ∂q usually not full-rank and the update ∆q is given by ∆q= J(q)†(p− f(q)), where J(q)† =(J(q)J(q)T +ηI)−1J(q) Input pose &rSoptaartisoen c loedarinngin g NormPaolsizeation isthepseudo-inverseoftheJacobianmatrixandηisasmall positivenumber.Toaccommodateconstraintssuchasangle limitorspatialrelations,Zhaoetal[ZB94]minimizedthe(cid:96) 2 Constraints Output pose JMaceothboiadn distance between input pose and forward-kinematics func- tionsubjecttotheconstraintsusingnon-linearprogramming Interactive Character Posing technique.Roseetal[RRP97]solvedtheIKproblemusing BFGS optimization method [GMW81], which is an quasi- NewtonalgorithmthatdoesnotrequirethecomplicatedHes- sianmatrix.Theirworkaimedatbuildingthefinalmotionin Figure1:Frameworkofproposedapproach anglerepresentationusingthesensordata,whichwereob- tainedfrommotioncaptureprocessinEuclideanspace. Data-drivenInverseKinematicsIngeneral,data-drivenIK ample,humanbeingscanmakealargenumberposes,butthe leverages the mocap data and models the IK problems as spaceofnaturalposesissmaller.Byrecordingtheseposesin follows motioncapturedataandlearningfromthem,wecanprovide heuristicsforsolvingtheIKproblem.Thisistherecentap- min. EPrior+λEIK (1) proachtakenbyresearchersandisreferredtoasdata-driven IK.Ourapproachfallsinthiscategory. whereE istheenergytermthatmeasuresthe(negative Prior log) observation likelihood. E is the energy term that The framework of our model is showed in figure 1. In our Prior measuresthe(negativelog)probabilityofcurrentposeun- proposed model, each pose is assumed to have sparse rep- der some prior distribution. This prior distribution is what resentation given a pose dictionary. The pose dictionary is makesthemodeldistinctivefromotherones. learnedfrommotioncapturedatainEuclideanspace.These data cover a large number of different motion styles, such A straightforward model for modelling the prior is to use aswalking,runningandothersportsactivities.Intheinter- theGaussiandistribution.Duetotheconnectionintheco- activecharacterposingstage,ourmodelcanrespondtothe variancematrix,thisapproachisrelatedtoPrincipalCom- users’inputsandconstraintsinreal-timeandconstructnat- ponentanalysis(PCA)whichrestrictsthesolutiontoliein uralposesthatmeettheusers’intentions.Wesolvethepose thesubspacespanbytheprincipalcomponents.Byimpos- synthesisproblembybreakingtheoptimizationprobleminto ing the Gaussian prior, we force the solution to approach threecomponents:1)findingthesparsecoefficientandrota- themeanfromthedirectionofoneprincipalcomponentor tion parameters; 2) normalizing the pose to determine the alinearcombinationofthem.InsteadofusingtheGaussian scaling parameter and 3) building the output pose in angle modeldirectly,wecanalsofirstpartitionthemotiondataby spacebyJacobianmethod.Detailsonourmodelandhowto clusteringalgorithmandthenbuildaGaussianpriorforeach solvetheoptimizationproblemarepresentedinsection3. cluster.Thisissimilartothemixtureoflocallinearmodels whichhavebeenusedasbaselinemodelsin[CH05]. Therestofthispaperisorganizedasfollows.Wereviewthe relatedworkintheremainingofthissection.Startingfrom Wei et al [WC11] modelled the prior using the mixture of thesubspacemodels,wepresenttheinspirationforthiswork factor analyzers (MFA) [GH96]. MFA is similar to Mix- andderiveourmodelinsection2.Insection3weintroduce tureofGaussian,butincludesadimensionreductioncom- ourproposedmodelandthealgorithmforsolvingthemodel ponent and avoids the ill-condition problem of covariance indetail,followedbyapplicationsandexperimentalresults matrix.Kallmannetal[Kal08]introducedanalyticalIKfor insection4.Wepresentsomediscussionsandconcludeour the arms instead of the whole body. To construct a natural workinsection5. whole-body,asetofpre-designedkeybodyposesareused forposeblending(interpolation).Grochowetal[GMHP04] proposedstyle-basedIK(SIK)formodellinghumanmotion. 1.1. Relatedwork TheirmodelisbasedonthescaledGaussianProcessesLa- ClassicalinversekinematicsTheuseofJacobianmatrixfor tent variable Models (GPLVM) [Law04]. Specifically, the inverse kinematics can be at least traced back to [GM85], training samples and the target poses are mapped to low- in which Girard et al linearised the equation p= f(q) at dimensionallatentvariablesusingGPLVM.Thesevariables current estimate q, yielding p = f(q)+J(q)T∆q, where are connected by a kernel function. The information then submittedtoCOMPUTERGRAPHICSForum(1/2012). RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding 3 passes from the training set through the correlation of la- cationisoncharacterposinginsteadofactionretrievaland tent variables to the target pose. Their model can general- classification. izetounseenposesthankstothegoodgeneralizationabil- ity of Gaussian processes. However, due to the limitation SummaryAmongtheexistingmodels,thenumericalIKal- of Gaussian processes, the complexity of their approach is gorithms do not have access to motion capture data, thus asymptotically cubic to the size of training set. To reduce cannotguaranteethenaturalnessofsynthesizedposes.For complexity,theymaintainedanactivesetduringthetraining data-drivenmodels,Gaussianmodelwillintroducelargeer- and synthesis. Despite the improved efficiency brought by rorforlargedatasetasittriestoapproximatetheunderlying theactiveset,itisstillprohibitivetolearnfromlargescale complicateddistributionbyGaussian.Imposingaclustering posedataforreal-timeapplications.Tofurtherimproveef- stepbeforeapplyingGaussianisanimprovementbutleaves ficiency, Wu et al [WTR11] considered using different ap- ustheproblemofchoosingtheclusternumber.ForLPCA, proximationstothespeedupGaussianprocessesandapply searchingtheposeinon-linemodeistooslowforlearning them to solve the IK problem. Other than GPLVM and its fromlargetrainingset.Themodelaccuracyalsodependson variants, modelling the motion based on dimension reduc- both the searching result and the neighbourhood size. For tionisalsoverypopular.Examplesarethosemethodsthat SIK, the complexity problem is prohibitive for moderate- based on state model [BH00,LWS02] and PCA [SHP04], scaletrainingset,andtheintroducedactive-setapproxima- etc. tionisdifficulttocapturethediversityofmotionstyles.For MFA, the introduction of a diagonal matrix in the covari- Givenincompletemeasurementsofmotioncapturesensors, anceavoidstheill-conditionproblemwhentheclusternum- Chaietal[CH05]constructafull-bodyhumanmotionuse ber is large (and thus the pose number in each cluster is LocalPCA(LPCA)model.Thisbasicideaistoincremen- small). However, being a variant of Gaussian model, it is tally estimate the current pose based on the previous esti- stillattheriskofunder-fittingthecomplicateddata.Apart matedposesandamotioncapturedatabase.Thispriorterm fromthelimitationsmentionedabove,mostofthesemodels measuresthedeviationofreconstructedposesfromthemo- areprobabilisticandthetrainingerrorismeasuredbylikeli- tion capture database. Since the database can be large and hood,whichisnotintuitive:givensuchameasurement,itis heterogeneous,theyintroduceaLPCAmodel:givenanin- notstraight-forwardtodeterminewhetherthemodelisade- completepose,theyfirstsearchthedatabasetofindthenear- quateforfittingthedata.Consequently,itishardtochoose estposeandbuildaGaussianmotionprioraroundtheneigh- the model parameters (e. g. , the cluster number). In con- bourhood.Theprioristhenusedforposesynthesis. trast,ourmodelmeasuresthetrainingerrordirectlybythe mean square error, which is very intuitive for determining Motion data de-noising and completion was considered by modelparameters.Besides,ourmodelcanlearnfromlarge Louetal[LC10].Theideaistofirstconstructasetoffil- datasets (up to millions of poses) with an arbitrarily small ter bases from the motion capture data and use them for trainingerror(although)attheexpenseoftheincreasingthe motion completion or de-noising. The resulting motion is sizeofdictionary.Wefoundthatthisincreasedoesnotcause the solution to a cost function that consists of the bases- theover-fittingproblemandthede-noisingandcompletion representationerrorandtheobservationlikelihood.Thefil- algorithmstillmaintainsefficiency,asthecomplexityofour terbasescapturespatial-temporalpatternsofthehumanmo- synthesisalgorithmislineartothesizeofposedictionary. tion,andtheden-noisingprocessalsoreliesonthespatial- We present some real-time applications demonstrating that temporal patterns, which in our case do not exist. Another our model is effective for interactive character posing. We workonthisdirectionwasintroducedbyLaietal[LYL11], alsocompareourmodelwiththeexistingmodelstotestthe inwhichlow-rankmatrixcompletionalgorithmwasusedfor performanceofposecompletionwhenalargeproportionof unsupervisedmotionde-noisingandcompletion.Themajor jointsaremissing,andposede-noisingwhentheposeiscor- difference of our working environment from both of these rupted by dense and sparse Gaussian noise. Experimental isthatwedonothavetemporalinformationavailablewhen results show that our model has lower completion and de- synthesizingnewposes. noisingerror. SparsecodingOntheotherside,sparserepresentationhas beenwidelyappliedtoimageprocessingandpatternrecog- Contributions Our contributions are two-fold: 1) starting nition. Examples include face recognition [WYG∗09], im- from the prevailing subspace models in modelling human agesuper-resolution[YWHM08],etc.Formodellinghuman motion,weproposesparserepresentationofposesforchar- motion, [LFAJ10] considered each joint’s movement as a acterposing;toourknowledge,wearethefirsttopropose signal that admits sparse representation over a set of basis sparsecodingforcharacterposing;2)differentfromprevi- functions.Thesebasisfunctionsarelearnedfromthemotion ousapproaches,weproposetolearnfromthemotioncapture capturedata.Theydemonstratedthattheproposedmodelis data in Euclidean space, which not only provides intuitive usefulforactionretrievalandclassification.Ourworkisdif- measurementsintrainingerror,butfacilitatesparsecoding ferentaswemodeleachposeseparatelyandourtargetappli- andposesynthesis. submittedtoCOMPUTERGRAPHICSForum(1/2012). 4 RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding 2. Overviewofproposedmodel fromSVDisasolutiontothefollowingoptimizationprob- lemwithvariablesU∈RD×kandX∈Rk×m NotationsettingInthispaper,matrices,vectorsandscalers are denoted in bold face upper-case, bold face lower-case min. ||Y−UX||2F (2) andnon-boldlower-caselettersrespectively.||·||2,||·||0and s.t. UTU=I ||·||F denote the vector (cid:96)2-norm, vector (cid:96)0-(pseudo)norm andmatrixFrobeniusnormrespectively.||·|| measuresthe Asarelaxation,theorthonormalconstraintonUischanged 0 to unit ball constraint on its columns and the size of U is numberofnon-zerocomponentsinavector. extendedtobeD×nposedictionarywithn>D.However, Fromlow-rankapproximationofmotiontosparserepre- eachcolumnofXshallbesparsetoreflectthesparseprop- sentationofposesAsthemovementsofthebodypartsare ertyofposes.Thesparsityconstraintismeasuredbythe(cid:96) - 0 correlated,whenwerepresentthehumanmotionasamatrix, normofeachcolumnofX.WerefertothematrixUinthis itwillbeapproximatelylow-rank,endowingafast-decaying caseasposedictionaryanddenoteitasAtobeconsistent spectrum[LYL11].Low-rankapproximationisthereforeef- withthesparsecodingliterature.Wepresentthisposedic- fectiveformodellinghumanmotion.Ourworkonthispaper tionarylearningprocessinnextsection. is inspired from the low-rank motion completion approach ModellingtheposesinEuclideanspaceIntheanalysispre- proposedbyLaietal[LYL11],inwhichtherankofamotion sentedabove,wehavenoassumptionontheambientspace isminimizedforcompletingandde-noisinghumanmotion. ofposes.Althoughthemotionmatrixislow-rankinbothEu- Theconnectionbetweenrankfunctionandthe(cid:96) -normmin- 0 clideanspaceandanglespace(whenpreprocessedproperly) imizationisclearifweobservethattherankofadiagonal ,wechoosetheformerforsparserepresentation.Thisisdif- matrixisequivalenttothe(cid:96) -normofthediagonalvector. 0 ferentfrompreviousdata-drivenapproaches,whichdirectly Let {yi ∈ RD}i=1,...,m denote a set of poses and modelthemotioninanglespace.Wedosofortworeasons. Y be the motion matrix: Y = [y1,...,ym]. The Sin- Onereasonistoavoidtheperiodicityofangles,whichpoten- gular Value Decomposition (SVD) of Y gives Y = tiallycorruptsthesparserepresentation:giventwoidentical [u1,..uD]diag([s1,...,sD])[v1,..vD]T, where ui ∈ RD,vi ∈ poses and add 2π to the one pose vector while leaving the Rm.Bysettingsk+1...sDtozeros,wecanapproximateeach otherthesame,thentheresultingtwoposesarestillidenti- pose yi by the first k singular vectors: yî = ∑kj=1ujα(ji), cal, but the 2π-shifted one is unlikely to have same sparse whereα(ji)=sjv(ji) andv(ji) istheithcomponentofvj.The representationastheotherunderthesamedictionary.This number k can be small and we still have a good approxi- will be a problem especially in pose synthesis, due to the mation for human motions. Another words, each pose has non-smoothnessof(cid:96)0-norm.Thesamewillnothappenfor asparserepresentationgiventhesetofbases{uj}j=1,...,D, poses represented in Euclidean space. Other parametriza- andthesupportsinthiscaselieinthefirstkbases. tionssuchasquaternionandexponentialmapare(also)non- linear, and thus inconvenient for sparse coding which in- Following this idea, to learn from a large motion capture volvessolvingalinearsystem. dataset,onepossiblewayistofirstpartitionthewholetrain- Theotherreasonisthatbydoingso,wecandirectlymea- ingsetintoK clustersandfindasetofbasesUi forcluster suretherepresentationerror,whichprovidesusanintuitive i . If we then collect all the bases into a matrix, i.e., U= measurementsintraining.Specifically,weareoptimizingdi- [U1,...,UK], each pose in the training set will be sparsely rectly the (mean) square error of the sparse representation representedundersuchamatrix. Thisapproachisgeneral, ofthetrainingsetwithoutinvokingtheforward-kinematics as we can set K to 1 to get back to original low-rank ap- mapping.Moreover,sincetheinputobservationssuchasan proximation of the whole dataset, and set K to the size of editedposesand2D/3DcoordinatesareinEuclideanspace, thedatasettousethewholedatasetasbases.However,this theoptimizationforposesynthesiswillbemoreefficientbe- leavesustheproblemtofindawaytoproperlypartitionthe cause we can defer the demand of Jacobian matrix till we dataintoclusters.Iftheposesinaclusteraretoolinearly- findaposerepresentedbyafullsetofjointcoordinates.And uncorrelated because of improper partition(e.g., too many theneedforconvertingtheposeintoanglespaceinthelast diversified poses in a cluster), then the sparse approxima- stage(seefigure1)isonlynecessarywhenwewanttofur- tion error will tend to be large. On the other hand, if each therprocesstheposesuchaschangingtheskeletonconfigu- clusteronlyconsistsofafewposes,thesparseapproxima- ration(e.g.jointanglelimits,bonelength). tionissmallbutthenumberofbaseswillbeverylarge,with thelimitbeingthesizeoftrainingset. 3. Learningsparserepresentationofposesforcharacter InsteadofdeterminingthematrixUintheabovemeans,we posing takeanotherwayaround:welearnthematrixfromthetrain- ingsetwithoutbeingworriedaboutthepartitioning.Tobe- Theideaofmodellingtheposesbasedonsparsecodingis ginwith,wereturntoone-clustercaseandnotethatfroman similartothesubspaceapproachessuchasPCA,exceptthat optimizationperspective,thebasesU=[u ,...,u ]obtained the’subspace’isgeneralizedtothespanofactiveatomsin 1 k submittedtoCOMPUTERGRAPHICSForum(1/2012). RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding 5 the pose dictionary and the ’bases’ are no longer assumed Thesecondtermmeasuresthesparsecodingerrorofthein- to be orthonormal or even independent. Given a set of ob- put under a rigid-body rotation. τ∈R3 is the rotation pa- servationsandconstraints,thereconstructedposeshallbea rameter. The notation τ◦(t) denotes a 3D rotation of the trade-off between having a sparse supports under the pose vectortwhichisconcatenationofasetof3Dpoints.y is 0 dictionary and being consistent with the observations and theinputposeandy¯ isroot-shiftedversionofy .Pisthe 0 0 constraints. diagonalmatrixanditsdiagonalentriesareeither1or0,in- dicatingwhetherthecorrespondingentryoftheinputpose isavailableornot.ThroughthisintroductionofP,weallow 3.1. Learningtheposedictionary theinputobservationtobeincompletewhilemaintainingthe formulaintegrityforcompleteobservationbysettingPtobe Before applying sparse representation, we need to first de- theidentitymatrix.Thiscanconvenientlymodeltheusers’ terminetheunderlyingdictionary,whichshouldbeableto constraintsonspecifyingthefixedandmoving(ormissing) capturetheposevariationsandbeinsensitivetoglobalorien- tationandtranslation.Givenatrainingset{yi∈RD}i=1,...m, joints. theposedictionaryA∈RD×nislearnedsuchthattheposes Thefinaltermprovidesapriorconstraintontherotationpa- inthetrainingsetaresparselyrepresentedunderthisdictio- rameters, where the diagonal matrix W gives a weight for nary.Specifically,thelearningproblemismodelledas eachofthe3rotationparameters.Usually,theweightonthe min. ||Y¯ −AX||2 (3) second rotation parameter τy (rotation around y-axis) shall F belargerthantheothertwo,asthisrotationisusuallymore s.t. ||xj||0<κ,j=1,...,m common. ||a || =1,k=1,...,n k 2 By solving this problem, we find a pose that on one hand whereX=[x1,...,xm],Y¯ =[y¯1,...,y¯m]andy¯iistheithpose staysclosetotheinputposesubjecttoasimilaritytransform, intrainingset,withglobalorientationandtranslationsetto andontheotherhandadmitsasparserepresentationgiven zerossincetheyareusuallyirrelevantinaffectingthepose thelearneddictionary.Theinputposecanbeincompleteor style.Notethesimilaritybetweenproblem(3)and(2). corruptedbynoise,anditcanalsoconsistof2Dpointclouds obtained from an image, as showed in our experiments in To solve the above learning problem, we use the K-SVD nextsection. algorithm proposed by Aharon et al [AEB06]. K-SVD al- ternates between sparse coding and dictionary updating in Theoptimizationvariablesintheproblem(P0)aretheout- everyiterate.Specifically,inthesparsecodingstage,theOr- putposeq,sparsecoefficientsx,rotationparametersτand thogonal matching pursuit (OMP) [PRK93] is used to find positivescalings.Tosolvetheproblem,wefirstfindxand asparserepresentationofthetrainingset,whileinthedic- τalternatively:ineachiteratewefirstfixτandfindxusing tionaryupdatingstage,thecolumnsofthedictionaryareup- OMPalgorithm,andthenfixxtofindτbygradientdescend. datedsequentiallybycomputingthesingularvaluedecom- Basedonthexandτfound,wethencalculatethescalingsby positionofthesparsecodingresidualmatrix.Itisreported analgorithmreferredtoasPoseNormalization,afterwhich that this method is better than the naive method of simply wefinallydeterminetheoutputposeJacobianmethod.The computingtheleastsquaresolutionbyfixingXtoupdateA. frameworkforsolvingproblem(P0)isshowedinfigure1, Wealsoreferreadersto[AEB06]fordetails. andtheoptimizationdetailsarepresentedinthenextsubsec- tion. 3.2. Posesynthesisproblem 3.3. Solvingtheposesynthesisproblem(P ) 0 Now by assuming that the pose dictionary A is given, we proposethefollowingmodelforposereconstruction: To efficiently solve the problem (P0), we first assume that given x, we can find a positive s and a q such that Ax= (P0) min. γ||Ax−s·f(q)||22+||P{τ◦(s·f(q))−y¯0}||22 s·f(q)(approximately)holds.Bysubstitutingitto(4),we +||W(τ−τ )||2 (4) arriveatthefollowing, 0 2 s.t. ||x||0≤κ (5) (P1) min. ||P{τ◦(Ax)−y¯0}||22+||W(τ−τ0)||22 (7) s>0 (6) s.t. ||x|| ≤κ (8) 0 In the objective (4), the first term measures the difference We use the alternating minimization framework to solve between sparse representation and the forward-kinematics problem (P ). More specifically, we first solve the sparse 1 function f scaledbyapositivefactors.Thescalesisapplied codingproblembyfixingτ, toall3dimensionsinEuclideanspacetomaintaintheskele- tonscalingratio.Theconstraint(5)guaranteesthesparsity min. ||P{Ax−τ−1◦y¯0}||22 (9) ofxinthesolution. s.t. ||x|| ≤κ, (10) 0 submittedtoCOMPUTERGRAPHICSForum(1/2012). 6 RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding whereτ−1 denotestheinverseofrigid-bodyrotationτ.Let Algorithm1Posenormalization v=τ−1◦y¯0,thentheaboveproblemisequivalenttosolving Input:AposeyinEuclideanspace,standardbonelength (P1a) min. ||Apx−vp||22 (11) matrixL. Output:Anewposey˜ s.t. ||x|| ≤κ 0 Dividetheskeletonintofivechains(seefigure2)asfol- whereApisextractionofrowsofAwhichcorrespondtothe lows: non-zerosdiagonalentriesofP,andthesamegoesforvp. c =(1,2,3,4,5) 1 Theproblem(P1a)issolvedbyOMP. c =(1,6,7,8,9) 2 Wethenfindtherotationparametersτbyfixingthesparse c =(1,10,11,12,13,14,15) 3 coefficientx.Thatis,wesolvethefollowingunconstrained c =(12,16,17,18,19) sub-problem: 4 c =(12,20,21,22,23) (P ) min. ||P{τ◦(Ax)−y¯ }||2+||W(τ−τ )||2(12) 5 1b 0 2 0 2 The gradient information of τ can be used to solved this Initialize∆y(k)=0fork=1,...,23andy˜(1)=y(1) problem.Noteagainthatthenotationτ◦ydenotestheop- fori=1→5do erationthatsubsequentlyrotatestheposebythreerotation for j=2→length(ci)do anglesaroundx,yandzaxis.lett1=(π2,0,0),t2=(0,π2,0), Letk=ci[j],k−1=ci[j−1],apply: t =(0,0,π),thenthegradientgoftheaboveobjectivefunc- ti3onisgive2nby y˜(k)=∆y(k−1)+ y(k)−y(k−1) L ||y(k)−y(k−1)|| k,k−1 2 g(i)=<P{τ◦y−y¯0},P{(τ+ti)◦y}>+W(ii)(τ(i)−τ(0i)) ∆y(k)=y˜(k)−y(k) (13) where<,>denotesinnerproduct. endfor endfor Wealternativelysolvetheabovetwosub-problems(P1a)and wherey(k)isthe3Dcoordinateofjointk.Jointk−1isthe (P )untilconvergenceisreached.Oncewehavefoundthe 1b parentofthejointk,asalreadyarrangedsointhechain. finalsparsecoefficientsxandrotationparametersτ,wecan L is the standard bone length between joint k and determinethejointanglesdenotedasvectorqbysolvingthe k,k−1 k−1.y˜(k)isthenewpositionofjointk. IKproblem: (P ) min. ||Ax−s·f(q)||2 (14) 2 2 s.t. s>0 Because of the involvement of Jacobian matrix, the Hes- sianforproblem(P )isdifficulttofind.Moreover,Jacobian 2 method,gradientdescendorquasi-Newtonmethodsseemto belessefficientforthisproblembecauseoftheunknownar- bitrary scaling s. Since we already know the lengths of all bones in our case, we can leverage this knowledge to determine the normalized pose y˜ =def 1Ax. This process is re- s ferredtoasPoseNormalizationandispresentedinalgorithm 1.Notethenormalizationschemeisnotsimplymakingthe posevectornormalizedin(cid:96) -normsense. 2 Afterfindingthenormalizedposey˜,itisreadytoapplyJa- cobian method [GM85] to find q by solving the following non-linearsystem f(q)=y˜ (15) Convergence and complexity By breaking the pose syn- Figure 2: Skeleton configuration. Each number is a joint thesis problem (P0) in to sub-problems (P1) and (P2), we IDlabellingthecorrespondingjoint.Theskeletoncanbedi- greatlyreducetheproblemcomplexity.Theassumptionfor videdintofivechains:thetorso,botharmsandbothlegs. thisbreak-downisthattheresidualofthetermin(4)dimin- ishes,whichcorrespondstosettingγtoalargevalue.Thus thisassumptionholdsandthebreak-downmakessense.The convergenceofproblem(P )isguaranteedasineachiterate 1 submittedtoCOMPUTERGRAPHICSForum(1/2012). RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding 7 bothOMPandgradientdescenddecreasetheobjectivevalue Task Ourmodel MFA Gaussian LPCA CG PCA andtheobjective(7)isboundedbelow.Theconvergenceof densenoise 0.12 0.26 0.25 0.20 0.95 3.34 Sparsenoise 0.05 0.15 0.14 1.02 0.17 1.36 (P )isalsoguaranteedastheposenormalizationalgorithm 2 completion 0.01 0.24 0.30 0.80 0.20 4.11 is deterministic with constant complexity and the Jacobian Table 1: Average MSE of Large-scale comparison for de- method with complete and normalized target pose usually noising and completion. The total testing poses for all the convergewithin20iteratesinourexperiments. tasks are 2075280. As we see, our model performs better thanothermodelsforallthethreetasks.PCAobtainscon- 4. ApplicationsandExperiments siderably large errors for all tasks because the principal Componentscannotexplainsuchalargedataset. 4.1. Experimentalsetting ThetrainingsamplesweusedareobtainedfromCMUmo- tion capture website†. We manually trim the toes and fin- Subject No.Frames Ourmodel MFA Gaussian LPCA CG SIK PCA 07 2161 0.02 0.13 0.08 0.07 0.02 0.70 0.07 gersfromtheposedata.Thesedataareoriginallyin62di- 09 769 0.03 0.08 0.10 0.06 0.06 1.26 0.09 63 7529 0.07 0.41 2.44 0.35 0.34 3.90 0.41 mensionalanglespace,andtheyaretrimmedto46dimen- 102 4252 0.10 1.39 0.17 0.08 0.11 1.37 1.28 sionalsothattheresultingskeletonscontainonlysignificant Table 2: Average MSE of small-scale comparison for de- DOF’s,asin[WC11].WhenconvertedtoEuclideanspace, noising when the noise is dense standard Gaussian noise. theskeletonmodelhas23jointswitheachinthreedimen- Forsubject07,whichcontainssimilarstylesofwalkingmo- sional.ThetotaldimensionforaposeisD=69.Wepre- tions,allthemodelsperformverywell.However,whenthe shiftallthetrainingsamplestoberootedattheoriginand motionsbecomemorecomplex,forexample,subject63,our set the global orientation to zeros. The corresponds to set- modeloutperformsallothermodels. tingthefirstsixcomponentsintheanglevectortozeros. We determine the size of pose dictionary n by the follow- ingprocedure:givenatargetlearningerroret,werandomly samplen posefortheposedictionaryanduseittotestthe should be trusted more than the likelihood, thus λ should 0 sparsecodingerrores ofthetrainingset.Ifes≤δet,setn bedecreased.Thesamegoesforκinourmodel.Therefore, ton andusethecurrentsampledposesasinitializationfor thevalueofλforallothermodelsandκforoursynthesis 0 dictionarylearningalgorithm;otherwise,setn to1.5n and modelarechosenusingbrute-forcesearchforafaircompar- 0 0 continuethesearching.Weusees≤δet asacriterionwithδ ison.Specifically, tofind theapproximatelybest value,we usuallysetto2becausethedictionarylearningcanusually firstrandomlysample0.1%poses(about2000)posesfrom decreasetheerrorby50%,aswefoundinourexperiments. thetrainingsetandusethemtoselectthebestvaluewithin aproperinterval.Thenthebestvalueforeachmodelisused fortestingthewholetestingset. 4.2. Large-scaleComparison For MFA, we use the same setting as stated in the paper Inlarge-scalecomparison,weusethewholedatabasefrom [WC11], except for the λ, which was not given originally CMU,whichsumsupto4150384poses,coveringavarsity andisfoundbybrute-forcesearch.wesettheclusternum- ofmotionstylesrangingfrombasictypessuchaswalking, berto20forCG,neighbourhoodsizeto100forLPCA.For runningtomorecomplextypessuchasbasketballandgolf. PCA,weusethefirstseveralprincipalcomponentssuchthat Werandomlysample50%ofposesfortrainingallmodels 90%energyofthecorrespondingeigenvaluesispreserved. and the rest for testing. In training, we set the cardinality SincetheSIKistoocomputation-demanding,wehaveomit- upper-bound κ to 3. The resulting pose dictionary consists teditfromthislarge-scaledcomparison. of262759atoms. Wetesttheperformanceofde-noisingandposecompletion. We test our model and other models using the testing set Forde-noising,wetesttwotypesofnoise:denseandsparse which consists of 2075280 poses. These models are MFA, Gaussiannoise.FordenseGaussiannoise,wegeneratestan- themodelwithaGaussianprior,themodelwithclustering dard Gaussian noise and add to the testing set. For sparse andthenbuildaGaussianpriorforeachcluster(CG),LPCA Gaussian noise, we generate standard Gaussian noise and andPCA.Thetestingschemeisasfollows. randomlycorrupt20%ofthejoints.Themeansquareerror As we mentioned, the existing models can be generalized (MSE) for each recovered pose is calculated and the aver- to (1) (except for PCA which will be discussed soon), in ageMSEforthewholetestingsetisshowedinfigure1.We which the parameter λ is important in determining the re- alsotesttheperformancethecompletionwhenonlyasmall sultingpose,anditsvalueprovidesatrade-offbetweenthe portionofjointsareobserved.Specially,theinputsarethe priorandthelikelihood.Ifthenoiselevelishigh,theprior 3DcoordinatesofjointID16,20,19,23,5and9(seefig- ure2forjointIDmap).Thismissingpatternisthesameas thatin[WC11].Thecomparisonresultisshowedin1.Fora † http://mocap.cs.cmu.edu:8080/allasfamc.zip visualinstanceofthelarge-scalecomprison,seefigure3. submittedtoCOMPUTERGRAPHICSForum(1/2012). 8 RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding Subject No.Frames Ourmodel MFA Gaussian LPCA CG SIK PCA We choose four subjects from CMU mocap database web- 07 2161 0.03 0.08 0.05 0.07 0.19 0.26 0.08 09 769 0.04 0.05 0.06 0.06 0.05 0.70 0.05 site: 07 (walking), 09 (running), 63 (golf) and 102 (bas- 63 7529 0.03 0.12 0.47 0.09 0.10 0.99 0.36 ketball). These subjects are representative as they are dif- 102 4252 0.05 0.08 0.09 0.07 0.08 1.34 1.09 ferent styles of motion and their size varies from 1538 to Table 3: Average MSE of small-scale comparison for de- 15079 poses. For each subject, we randomly sample 50% noisingwhenthenoiseissparsestandardGaussiannoise.In fortrainingandtherestfortesting.Thetrainingandtesting thistest,MFA,LPCAandourmodelareallrobusttochange schemesarethesameasthatinlargescalecomparison.In of dataset size and motion styles, and our model performs trainingstage,thesettingforSIKisthesameasmentioned slightlybetteramongthetree. in[GMHP04];forothermodels,thesettingissimilartothe above.Wealsotesttheperformanceofcompletionandde- Subject No.Frames Ourmodel MFA Gaussian LPCA CG SIK PCA noisingundertwotypesofnoises.Theresultsforthesethree 07 2161 0.07 0.11 0.23 0.25 0.95 0.31 0.07 tasksareshowedthetable2,3and4respectively.Aswesee, 09 769 0.09 0.09 0.26 0.26 0.25 0.74 0.05 63 7529 0.07 0.18 0.26 0.18 0.21 3.90 0.53 ourmodeloutperformstheothermodelsforthreetaskseven 102 4252 0.09 0.24 0.28 0.25 0.27 1.48 1.30 forsmalldatasets. Table4:AverageMSEofsmall-scalecomparisonforcom- pletion.Inthistest,ourmodelistheonlyonethathasob- tainedMSEsmallerthan0.1forallfoursubjects. 4.4. Interactivecharacterposing We provide an real-time application of our model in interactive character posing. The user interface is implemented inC++andweusetheposedictionarylearnedinthelarge- Eventhoughallmodelsconsideredinthispaperdonottake scalecomparisonforposesynthesis.Weconsidertwokinds intoaccountthemotiondynamicsintrainingstage,wecan of input here. Free-dragging interface provides a freely- stillcomparetheirperformanceonmotioncompletionbyap- edited complete pose as an input. Pose completion takes a plying completion algorithm to each (incomplete) pose in setof2Dor3Dpointsasinputsandreconstructsthewhole themotion,asthisprovidesagoodreflectionontheperfor- pose.Otherinputsarepossible,aslongastheycanfitinto manceofposecomplete.Thiscomparisonisdonetoarun- themodel,perhapsaftersomenecessarypreprocessing. ningmotionandtheresultisshowedinfigure6.Ourmodel outperforms other models in that it preserves a better pose Free-dragging A common scenario is that when the user structureoftheupper-body(seethefigurecaptionformore drags one or multiple joints of the skeleton, the computer details). is required to respond to this drag and create a new pose. Chancesarethattheeditedposelookslikebeingcorrupted bynoiseiftheuserisnovice.Tosynthesizeanewposebased 4.3. Small-scaleComparison onthecorruptedposeandtheposedictionary,(P )issolved 0 Similartotheabovelargescalecomparison,wealsotestthe withPsettotheidentitymatrix.Thiscorrespondstosetting performanceofeachmodelforlearningfromsmalldatasets. alljointsoftheinputposey assoft-constraints. 0 As our model is trained from the pose data in Euclidean space, it is well-suited for interactive character posing in whichtheusercanarbitrarilymodifytheposewithoutbe- 15 10 10 ingworriedaboutbone-lengthconstraintsandanglelimits. 10 This provides a great continence for the user because s/he 5 5 cannowmovethejointstowherevers/hewants.Afterthe 5 0 userfinishesthemodification,ourmodelsynthesizesnatu- 0 y0 y ral poses that satisfies the user’s intention on the style and y −5 correctsalltheviolations.Seefigure4forexamples. −5 −5 −10 PosecompletionInposecompletionproblem,weinferthe −10 −10 wholeposewhileonlyaportionofthejointsareobserved. −15 4 2 0x−2−4−62z0−2 2 0 −2 −4 8 6 4 2 0 −1520−2 8 6 4 2 0 Istoltvuirnngs othuitstphraotboleumr meovdenelwcahnenbethceomnvoedneilenistlytraaidnaepdtewdittoh z z Figure3:Avisualsnapsxhotofthelarge-scxalecomparison. the full-body pose data. To do this, we simply set the en- Fromlefttoright:de-noisingresultwhenthenoiseisdense, triesofPcorrespondingtothejointsthatwewanttosetas sparseandcompletionresultwhenonlyfivejoints(ID:16, observed(fixed)toonesandtheresttozeros.Withthisin- 20,19,23,5and9)areobserved.green:ground-truth,red: troductionofP,wecanconvenientlyincorporate2Dinputs: corrupted, cyan: our model, black: MFA, magenta: Gaus- givenapicturewhichcontainsapose,theusercanlabelthe sian,yellow:LPCA,blue:CG,gray:PCA jointswiththe2Dcoordinateoftheposeinthepictureand reconstructa3Dpose.Wegivetwoexamplesinfigure5. submittedtoCOMPUTERGRAPHICSForum(1/2012). RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding 9 25 20 y15 10 5 −−42024 −30 −20 −10 0 10 20 30 40 x 25 z 20 Figure4:Someexamplesoffree-draggingandposerecon- y15 structionresults.Firstandthird:modifiedposes;Secondand 10 fourth:thecorrespondingreconstructedposes 5 −−4202 −30 −20 −10 0 10 20 30 40 z x 25 20 y15 10 5 −−4202 −30 −20 −10 0 10 20 30 40 z x 25 Figure5:Reconstructing3Dposesgiven2Dinputsshowed 20 asreddots.The2Dcoordinatesoftheinputsareassigned y15 tothecorrespondingjoints(chosenbyusers)forposerecon- 10 struction. 5 −−4202 −30 −20 −10 0 10 20 30 40 z x 25 5. Discussionsandconclusions 20 De-nosing and completion We have compared our model y15 withtheexistingonesfortheperformanceofde-noisingand 10 completion. One may think that de-noising is irrelevant as 5 the motion capture data are usually ’clean’. This perhaps −−4202 is true for the already-available databases. In the process −30 −20 −10 0 10 20 30 40 ofmotioncapturehowever,de-noisingisnecessarybecause 30 z x of measurement error and sensor failures [RRP97,LC10]. 25 Moreover, for interactive character posing, the pose edited 20 byusersisusuallynoisyinthesensethatitisinconsistent y15 withthetrainingset.Wecanseetheprocessofposeediting 10 as a measurement of users’ intentions, which will always 5 introduce measurement noise. Apart from dense noise, we −5 0 have also considered sparse noise. This is meaningful be- 5 causeinmotioncaptureprocess,noisecanbesparsedueto −30 −20 −10 0 10 20 30 40 x z errorintroducedinafewsensors.Similarly,incharacterpos- Figure6:Comparisonofallmodelsforcompletingthefirst ing,theusermayonlyeditafewjoints,makingthemeasure- 128 poses of a running sequence in subject 09 of CMU mentnoisesparse.Posecompletionalsoisuseful,notonly database.Allmodelsaretrainedasinthelarge-scalecom- indealingwithmotioncapturedatawhenthemeasurements parison.Foraclearview,theposesareshowedatevery20 are incomplete, but also in character posing to account for frames. The ground-truth is showed in green together with users’constraints. theresultsobtainedfromallmodels(inblue)whicharere- spectively from top to bottom: our model, MFA, Gaussian, Sparsity The choice of κ in pose synthesis stage depends CG, LPCA and PCA. Our model performs best as it pre- on the noise level. It reflects our initial knowledge on the serves a better pose structure. For the rest models, Gaus- propertyofnoise.Ifwebelievethatthenoiselevelishigh, sian, CG and MFA fail to capture the running style of the wecanreducetheκ;otherwise,wecanincreaseκsuchthat upper-body(fromback-bonetohead)anddefectscanalso bespottedintheheadandfeet.NeitherLPCAnorPCAob- submittedtoCOMPUTERGRAPHICSForum(1/2012). tainsanacceptableresultinthiscomparison. 10 RanchY.Q.Lai&PongC.Yuen&KelvinK.W.Lee&JHLai/InteractiveCharacterPosingbySparseCoding itcanbetterapproximatetheinput.Thisthenoffersatrade- Conclusion In this paper, we have proposed a model for off between stratifying the users’ exact constraints (which articulatecharacterposing.Wehaveshownthatourmodel may result in an invalid pose) and synthesizing a realistic canbetrainedtolearntheposedictionaryfromalarge-scale andnaturalpose. trainingset.Wealsodemonstratedhowtoapplyourmodelin de-noisingandcompletionproblem.Wehavealsoprovided CombiningdictionariesOursynthesismodelisflexiblein UI examples showing how to use our model for character that it can combine dictionaries that are learned separately posing.Experimentshaveshownthatourmodeloutperforms bysimplyConcatenatingallsub-dictionaries..Thisprovides theexistingmodelsinposede-noisingandcompletion. a friendly solution for accepting large scale training set in thetrainingstage.Intermsofcomplexity,K-SVDisO(m2), Onelimitationofourmodelisthattoachieveasmalllearn- wheremisthesizeoftrainingset.Byusingfastalgorithms ingerror,theposedictionarysizecouldbelargeforlearn- forclusteringsuchasK-means,wecandividethetraining ingfromalargedataset.Thiscouldbeaproblemforappli- setintosmallersubsetswithcomplexityO(m)beforeapply- cationsindevicesthathavelimitedmemory.Nevertheless, ingK-SVDtoeachofthem.Inthisway,theoveralltraining ourmodeliscurrentlydesignedforapplicationsinpersonal complexityisreduced.SinceK-SVDisageneralizationof computers. k-means,thisapproximationisanalogicaltothehierarchical versionofk-means.Weusethisapproachforthelargescale Acknowledgement training. ThisprojectissupportedbytheFacultyResearchGrantof PhysicalconstraintsTheonlyphysicalconstraintswecon- DepartmentofComputerScience,HongKongBaptistUni- sider in this paper is the bone length. Angle limits are not versity. considered here as we found that the solving the problem (P )usuallywillnotviolatetheanglelimitconstraints.How- 0 ever,theycanbeinteroperatedintosub-problem(P )ifnec- References 2 essary,andtheproblemcanbesolvedforexamplesimilarly [AEB06] AHARONM.,ELADM.,BRUCKSTEINA.: k-svd:An to[ZB94]. algorithmfordesigningovercompletedictionariesforsparserep- resentation. Signal Processing, IEEE Transactions on 54, 11 Connection to subspace models By setting κ to a large (nov.2006),4311–4322.5 numberandimposingextraorthogonalconstraintsonA,the [BH00] BRANDM.,HERTZMANNA.: Stylemachines. 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Then this mixturesoffactoranalyzers.UniversityofTorontoTechnicalRe- port(1996).2 sensing matrix will have no effect as we can take inverse andremoveitfromthemodel.Thatis,Wedonotperform [GM85] GIRARD M., MACIEJEWSKI A.: Computationalmod- elingforthecomputeranimationofleggedfigures. ACMSIG- anyreducedmeasurementonthepose.Thisisbecausethe GRAPHComputerGraphics19,3(1985),263–270.2,6 inputposemightbeincompletealready,asindicatedbyP. Introducing a (fat) sensing matrix will complicate the (in- [GMHP04] GROCHOW K., MARTIN S. L., HERTZMANN A., POPOVIC´ Z.: Style-based inverse kinematics. 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Advancesinneural thesparsecodingproblem.Otheralgorithmssuchaslinear informationprocessingsystems16(2004),329–336.2 programming,shrinkageandinteriorpointmethodcanalso [LC10] LOUH.,CHAIJ.:Example-basedhumanmotiondenois- beused.However,wefindthatOMPismoreefficientinour ing. VisualizationandComputerGraphics,IEEETransactions case. on16,5(sept.-oct.2010),870–879.3,9 submittedtoCOMPUTERGRAPHICSForum(1/2012).