DOI: 10.1111/cgf.12246 Pacfc Graphcs 2013 B. Levy, X. Tong, and K. Yn (Guest Edtors) Volume 32 (2013), Number 7 As-Rgd-As-Possble Dstance Feld Metamorphoss Yanln Weng 1 Mengle Cha 1 Wewe Xu 2 Yyng Tong 3 Kun Zhou 1 1 State Key Lab of CAD&CG, Zheang Unversty, Chna 2 Hangzhou Normal Unversty, Chna 3 Mchgan State Unversty, USA Abstract Wdely used for morphng between obects wth arbtrary topology, dstance feld nterpolaton (DFI) handles topologcal transton naturally wthout the need for correspondence or remeshng, unlke surface-based nterpolaton approaches. However, lack of correspondence n DFI also leads to neffectve control over the morphng process. In partcular, unless the user specfes a dense set of landmarks, t s not even possble to measure the dstorton of ntermedate shapes durng nterpolaton, let alone control t. To remedy such ssues, we ntroduce an approach for establshng correspondence between the nteror of two arbtrary obects, formulated as an optmal mass transport problem wth a sparse set of landmarks. Ths correspondence enables us to compute non-rgd warpng functons that better algn the source and target obects as well as to ncorporate local rgdty constrants to perform as-rgd-as-possble DFI. We demonstrate how our approach helps acheve flexble morphng results wth a small number of landmarks. Categores and Subect Descrptors (accordng to ACM CCS): I.3.5 [Computer Graphcs]: Computatonal Geometry and Obect Modelng Geometrc algorthms, languages, and systems 1. Introducton Shape nterpolaton/morphng between 2D or 3D obects wth arbtrary topology s of great nterest for varous applcatons. As a useful tool for smooth transton from a source obect to a target obect, t has long been used n anmaton and specal effects n the flm ndustry. There are two man approaches for the morphng problem, surface morphng and dstance feld nterpolaton (DFI). Surface morphng usually establshes one-to-one correspondence between the source and target surfaces va cross-parameterzaton and remeshng, and acheves as-rgd-as-possble nterpolaton results. It, however, has dffculty n handlng obects wth dfferent topologes. DFI approaches, on the other hand, represent the shape as an sosurface, whch does not requre dense surface correspondences and allows topologcal change wthout specal treatment. However, wthout dense correspondences, DFI approaches also lack effectve control over the morphng process. One soluton [COSL98] s to specfy sparse anchor ponts n the source and target obects, compute a warpng functon assocated wth the anchors, and then blend the two warped dstance felds to create ntermedate obects. In ths manner, the morphng process s roughly controlled by the sparse anchor ponts. Another soluton [TPG01] approxmates the volume enclosed by the surface wth spheres, usng correspondences between source and target spheres to provde control. Unfortunately, the sparse correspondences used n these approaches are not enough to precsely algn the features durng the morphng process. There s no way to perform as-rgd-as-possble nterpolaton, whch needs neghborhood correspondences. In order to mprove the controllablty and qualty of the DFI process, we propose a novel approach to as-rgd-aspossble dstance feld morphng. We frst construct shape nteror correspondence, a per-voxel fuzzy correspondence between two dstance felds. Such a correspondence allows mantanng shape algnment throughout the morphng process, wthout an excessve number of landmarks. Makng use of ths correspondence, we then warp the source and target dstance felds to an ntermedate tme and blend them to generate the ntermedate dstance feld, takng nto account the local rgdty constrants mpled n the correspondence. The result s an as-rgd-as-possble dstance feld metamorphoss algorthm that can generate superor morphng sequences to those produced by prevous DFI methods. Our shape nteror correspondence s computed through Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd. Publshed by John Wley & Sons Ltd.
382 Fgure 1: Top row: a morphng sequence generated by our algorthm. Bottom row: the correspondng sequence generated usng [COSL98]. The user-specfed anchor ponts are also shown n the top row. mnmzng earth mover s dstance [Kan06], based on sparse correspondng anchor ponts. Thus, our ntal correspondence s a fuzzy many-to-many correspondence, represented by a functon storng the porton of a source voxel transported to each target voxel. Wth a smple weghted average, each target voxel center corresponds to a pont n the source doman. Ths procedure avods the dffculty n drectly establshng a one-to-one correspondence between voxels, whle reducng the shape dstorton n the morphng as well as provdng flexble control. We tested our approach on several 2D and 3D obects wth complcated shapes and topologes. In these tests, our voxel-level as-rgd-as possble nterpolaton scheme produced smooth morphng results wth proper feature algnment, as shown n Fg. 1, Fg. 5 and Fg. 7. 1.1. Related Work We brefly revew both the surface-based and the volumetrc approaches for morphng. We also dscuss the mass transport problem, the technque used n our fuzzy correspondence algorthm, n the context of related areas. Surface morphng: Surface morphng technques usually start wth buldng dense vertex-to-vertex correspondences and then compute vertex paths to create ntermedate shapes. Snce the source and target shapes are usually of dfferent mesh connectvty, a cross parameterzaton method s requred to remesh the shapes and buld correspondence [KS04, SAPH04]. The vertex correspondences can also be establshed through functonal correspondences on the mesh, such as harmonc coordnates [ZRKS05] and functonal maps [OBCS 12]. The challenge n vertex path computaton s how to reduce the shape dstorton n the morphng. Sederberg et al. [SGWM93] developed an ntrnsc coordnate for 2D shape blendng, usng edge lengths and nteror angles. Ths method has been extended to polyhedral meshes n [SWC97]. To reduce the non-rgd dstorton n surface shape blendng, the as-rgd-as possble nterpolaton technque was ntroduced n [ACOL00, IMH05]. The technque proposed by Alexa et al. [ACOL00] can handle volumetrc obect nterpolaton but requres that the source and target obects have the same number of tetrahedra and the same connectvty. For 3D meshes, low-dstorton surface morphng results can be acheved by changng the underlyng surface representaton. Laplacan coordnates [SCOL 04], rotaton-nvarant coordnates [LSLCO05] and Posson shape representaton [XZWB05] are popular technques to enforce the local rgdty constrant durng morphng. A recent contrbuton [KMP07] realzed as-sometrc-as-possble shape morphng results va fndng a geodesc path n shape space wth the desgnated metrc. DFI morphng: Sgned dstance feld s an mplct representaton of a surface. One of ts advantages n shape morphng s that t can handle the topologcal change durng the morphng between two surfaces wth dfferent topology wthout any addtonal processng, whle a trangle-based representaton requres specal topologcal modfcatons for surface splt and merge operatons and may have to remesh some regons to mantan reasonable trangle shapes. The early work n DFI morphng smply cross-dssolves two dstance felds and reconstructs the ntermedate surface from the blended dstance feld [PT92]. However, ths smple scheme cannot preserve the essental features n the source and target obects. To mprove feature preservaton, a wavelet volumetrc morphng technque s developed n [HWK94]. Fne control of feature algnment for two dstance felds s acheved n [LGL95] by usng varous types of feature handles. Sphere correspondences are adopted n [TPG01] to control the DFI morphng process. The most related DFI method to our work s the 3D dstance feld metamorphoss by Cohen-Or et al. [COSL98]. It separates the transformaton between source and target obects nto rgd and non-rgd parts accordng to the sparse correspondence ponts on the surface, and the ntermedate c 2013 The Author(s) Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
383 dstance feld s generated by blendng the warped dstance felds. As descrbed earler, the advantage of our algorthm s to compute a dense shape nteror correspondence, whch allows the shape algnment to be consstent wth the 3D obects enclosed by the surfaces. To our knowledge, ours s the frst algorthm that ncorporates local rgdty constrants nto DFI morhpng, leadng to reduced dstorton. Mass transport: We model the morphng problem as a mass transport problem, a shape transformaton wth a mass preservaton property, where the weghted 2D or 3D volume s treated as the mass to be transported. The optmal mass transport of ths sort was frst consdered n 1781 by Gaspard Monge, who referred to t as the Earth Mover s Dstance (EMD). Its modern formulaton frst appeared n [Kan06]. Optmal mass transport has been appled to mage regstraton [HZTA04], content-based mage retreval [RTG00] and feature smlarty measure [GD05]. In [MY11], EMD s also appled to topology free 2D mage morphng. However, explct non-rgd dstorton control s not drectly avalable n ths method, whch hnders ts applcaton to shape morphng wth the feature preservaton requrement. In [LPD11, SNB 12], EMD s used to compute soft correspondences between surfaces. 1.2. Overvew We present the overall framework here, before elaboratng on the algorthm n the followng sectons. The two gven 3D sold obects, a source Ω 0 and a target Ω 1, are represented as sgned dstance felds to ther respectve boundary surfaces, D 0 and D 1, stored on a volumetrc grd. More precsely, for any pont q n the doman, the value of D 0 (q) (or D 1 (q)) s defned as the sgned shortest Eucldean dstance between q and the boundary of Ω 0 (or Ω 1 ), negatve for ponts nsde the obects and postve for the outsde ponts. Our goal s to contnuously deform Ω 0 to Ω 1 and produce the n-between obects{ω t,0<t < 1} (.e., {D t, 0<t < 1}). The user can specfy a set of anchor pont pars {(p 0, p 1 ), 1 K}, such that pont p 0 n the source doman corresponds to pont p 1 n the target doman. Our algorthm frst establshes a fuzzy correspondence between the nterors of Ω 0 and Ω 1 (Sec. 2). Treatng morphng as a process of transportng the mass of Ω 0 to Ω 1, we formulate the correspondence as the soluton to an optmal mass transport problem from the nteror voxels of Ω 0 to those of Ω 1. The underlyng physcal optmalty of ths correspondence makes the morphng results reasonable, n the sense that t moves the voxels wth the mnmum total transport cost. To produce correspondences consstent wth the userspecfed anchor pars, we defne a mass transportaton cost nduced by the anchor ponts. Makng use of the above correspondence, we use thn plate splne nterpolaton to compute warpng functons W 0 1 and W 1 0 that can algn the shapes of Ω 0 and Ω 1 as well as possble (Sec. 3.1),.e., W 0 1 (Ω 0 ) Ω 1, W 1 0 (Ω 1 ) Ω 0. (1) Next, gven a morphng parameter t [0,1], we fnd warpng functons W 0 t (W 1 t ) that can transform Ω 0 (Ω 1 ) to tme t (Sec. 3.2) as an as-rgd-as-possble nterpolaton between the dentty transformaton and W 0 1 (W 1 0 ). We then compute ther nverses, the backward mappng functons B t 0 and B t 1. Fnally, the dstance feld D t at t can be evaluated as the nterpolaton of D 0 and D 1 guded by the backward mappngs: D t(v)=(1 t)d 0 (B t 0 (v))+td 1 (B t 1 (v)), (2) where v s an arbtrary voxel n the doman. Once we have the dstance feld, we can use the marchng cubes method [LC87] to extract the zero set surface, whch s represented as a trangular mesh. Note that the extracted meshes at contguous tme frames may have dfferent number of vertces/trangles and connectvty, and t s dffcult to enforce any temporal coherence between the two meshes. Fortunately, n practce, we found that the generated morphng sequences exhbt smooth transton as demonstrated n our vdeo demo. It s also possble to use the recent surface trackng technque [BHLW12] to track the correspondence for the mesh sequence and mprove the mesh coherence. 2. Shape Interor Correspondence We formulate the desred shape nteror correspondence between Ω 0 and Ω 1 as the optmal mass transport. Suppose that the total mass of a sold obect s a constant value 1, evenly dstrbuted among ts nteror voxels. The problem s then to fnd a mappng f that transports all the mass from the nteror voxels of Ω 0 to those of Ω 1 wth the mnmal transportaton cost. Specfcally, let Ω 0 have N 0 nteror voxels and Ω 1 have N 1 nteror voxels. We construct a complete bpartte graph, wth N 0 source nodes each of whch corresponds to an nteror voxel of Ω 0 and N 1 target nodes each of whch corresponds to an nteror voxel of Ω 1. We can then regard the mappng f as an assgnment of edge weghts, wth f(, ) representng the amount of mass transported from source voxel to target voxel (see Fg. 2). The goal s thus to solve for the mappng f that mnmzes the total transportaton cost, arg mn f d(, ) f(, ), (3), subect to f(, ) 0, f(, ) = 1/N 1 and f(, ) = 1/N 0, where 0 < N 0 and 0 < N 1. Here d(, ) s some dstance between the -th source node and the -th target node. We solve Eq. (3) usng the network smplex algorthm proposed by Bonneel et al. [BPPH11]. Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
384 f (, ) Fgure 2: Illustraton of shape nteror correspondence. For every source node and target node, f(, ) represents the amount of mass transported from source voxel to target voxel. The dstance d(, ) between source nodes and target nodes should be defned wth the correspondence of userspecfed anchor pont pars taken nto account. Intutvely, f (, ) s an anchor pont par, d(, ) should be a mnmal value to favor the mass transportaton between the two nodes. Inspred by Zayer et al. s work [ZRKS05], we compute a K-dmensonal vector feld n ether of the source or target domans wth respect to the K anchor pont pars. Specfcally, for each source anchor pont, we compute a harmonc feld h 0 n the source doman wth Drchlet boundary condtons by settng ts value to 1 at ths anchor pont and to 0 at all other source anchor ponts. Ths s equvalent to solvng the followng lnear system n the volumetrc grd h 0 = 0, wth h 0(p 0)=1, h 0(p 0 )=0,, (4) where = 2 + 2 + 2 s the Laplacan operator. In our x 2 y 2 z 2 current mplementaton, the Laplacan at voxel v, h 0 (v), s calculated as h 0 (v) 1 6 u Ψ(v) h 0 (u), where Ψ(v) represents v s 6-connected neghborng voxels. Solvng the above equaton for all K source anchor ponts gves us a K-dmensonal vector feld h 0 =(h 1 0,..., hk 0 ). Smlarly we can compute a K-dmensonal vector feld h 1 = (h 1 1,..., hk 1 ) n the target doman. We then defne the dstance between source node and target node as d(, )= h 0 (v 0) h 1 (v 1 ), (5) where v 0 and v 1 are the postons of the -th source node and the -th target node respectvely. Note that unlke [ZRKS05], we cannot drectly use the harmonc felds to construct the correspondence as ths may make exteror regons of the source correspond to nteror regons of the target or vce versa. We only use harmonc felds to calculate the dstance between source and target nodes. The fnal correspondence s computed by mnmzng the total mass transportaton cost. The mnmzer f of the total transportaton cost gves a many-to-many correspondence: for each target node, any source node that has a nonzero f(, ) could partally correspond to. To perform DFI, we need to compute a unque correspondng poston n the source doman for each target node. For ths purpose, we frst fnd the source node wth the largest f(, ) wth gven,.e., l = argmax f(, ). The source correspondng poston of the -th target node, q 0, s then computed through q 0 = k ϒ(l) f(k, )v k 0 k ϒ(l) f(k, ), (6) where ϒ(l) s the 3 3 3 volume grd centered at source node l. We can compute the target correspondng poston q 1 of every source node n a smlar way. Although we compute a unque correspondng poston n the source for each target node, such a correspondence s not a one-to-one mappng between the source and target domans. Our ntuton s that although the correspondng poston of each ndvdual node may not be optmal, the correspondng postons of all nodes can be used as a good gudance to algnng the source and target shapes. Note that as we only need a fuzzy correspondence, we can also compute q 0 as the the average poston of all source nodes havng nonzero f(, ) values. Accordng to our experments, ths hardly affects the fnal morphng results. 3. As-Rgd-As-Possble Dstance Feld Interpolaton We frst descrbe how to compute warpng functons that algn the source and target shapes as well as possble, then ntroduce an algorthm to perform as-rgd-as-possble nterpolaton for these warpng functons and generate the ntermedate shape. 3.1. Computng Warpng Functons We employ thn plate splne (TPS) nterpolaton [Boo89] to compute the warpng functons from the dense correspondence computed n the prevous secton. Specfcally, to compute the warpng functon W 1 0 from the target shape to the source shape, we solve the followng problem W 1 0 = argmn W(v W 1 ) q 0 2 + λ D 2 W 2, (7) where D 2 W s the matrx of second-order partal dervatves of W and the matrx norm used n D 2 W 2 s the Frobenus norm. The frst term preserves the nteror correspondence, and the second term ensures the smoothness of the warpng. The weght λ s set to 0.5 n all of our tests. The above problem has a closed form soluton when restrcted to the followng form: W 1 0 (p)=a 1 + a 2 p x+ a 3 p y+ a 4 p z+ c ϕ( p v 1 ), (8) where p x, p y and p z ndcate coordnate components of p, ϕ(r) = r 2 logr s the kernel functon, and a 1, a 2, a 3, a 4 and c are mappng coeffcents ( R 3 ) solved from Eq. (7), Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
385 k l (a) k l (b) k l (c) k l (d) Fgure 3: Illustraton of the eght trrectangular tetrahedra surroundng voxel. For each tetrahedron, an affne transformaton can be computed from ts node postons n the source (a) and target (d). Interpolatng the transformatons of the eght tetrahedra ndependently wll dsconnect them (b). Our global optmzaton sttches them together agan (c). whch s smply a quadratc energy mnmzaton wth respect to these coeffcents. See [Boo89] for detals on computng the mappng coeffcents. The warpng functon from the source to the target, W 0 1, can be computed n a smlar way. 3.2. As-Rgd-As-Possble Interpolaton Gven the warpng functons W 0 1 and W 1 0 computed above, we can now warp both the source Ω 0 and the target Ω 1 to tme t and compute the ntermedate shape Ω t. Frst, we need to compute the warpng functons W 0 t and W 1 t, and ther backward mappngs B t 0 and B t 1. Then, Ω t (represented by D t) s generated accordng to Eq. (2). Note that W 0 t should be smoothly changng wth t, and be the dentty transformaton at t = 0 and W 0 1 at t = 1. At tme t, each nteror voxel v n the source doman wll be transformed to v t = W 0 t (v ). In the followng we descrbe an approach for computng W 0 t as an as-rgd-as-possble nterpolaton between the dentty transformaton and W 0 1. We frst explan how to solve for the optmal postons of all nteror voxels v t =(v t 1,..., vt N 0 ) at tme t. For each nteror voxel v and ts 6-connected neghbors n the source doman (see Fg. 3), we can construct eght trrectangular tetrahedra wth voxel center v as ther rght angle. Let (v, v, v k, v l ) be one of the tetrahedra, we can fnd ts correspondng tetrahedra (u, u, u k, u l ) n the target doman through W 0 1, u m = W 0 1 (v m), m {,,k, l}. (9) An affne transformaton defned by a 3 3 matrx M and a dsplacement vector b can transform the source tetrahedron to the target tetrahedron u m = Mv m+ b, m {,, k, l}. (10) M can be easly solved through M =[u u u k u u l u ][v v v k v v l v ] 1. (11) Fgure 4: A horse s warped to algn wth a mechancal-lke horse. Left: our algnment result. Rght: the algnment result by [COSL98]. We only show the slhouette of the horse for better llustraton. Followng [ACOL00,SZGP05], we decompose M (deformaton gradent) through polar decomposton nto a sngle rotaton and a symmetrc matrx (rght stretch tensor): M = R rs. Based on ths decomposton, the transformaton for the tetrahedron at tme t can be constructed by lnearly nterpolatng the free parameters n the factorzaton where I s an dentty matrx. M(t)=R t((1 t)i+ts), (12) Under ths transformaton and gnorng the translaton, the postons of the tetrahedron at tme t should be g t m = M(t)v m, m {,,k,l}. (13) Now we solve for the optmal v t that can match the shape of tetrahedron(v,v,v k,v l ) at tme t to the shape of the tetrahedron computed wth Eq. (13) as much as possble. Specfcally, we measure the smlarty usng the relatve postons of v, v k and v l to v, and propose the followng energy E a(v t,v t,v t k,v t l)= v t m v t (g t m g t ) 2. (14) m {,k,l} The optmal postons of all nteror voxels can be calculated by summng the energes for all possble tetrahedra together arg mn v t E a(v t,v t,v t k,v t l), (15) where the summaton s over all tetrahedra whose vertces are all nteror voxels. As E a s a quadratc functon, the above equaton can be solved va least-squares optmzaton (usng a conugate gradent solver n our mplementaton). To set the boundary condton, we ask the user to specfy an anchor pont whose poston s drectly computed va lnear nterpolaton. Alternatvely, the center of mass can be forced to be statc. Ths forms the boundary condton for Eq. (15). Havng solved v t, we drectly apply the TPS nterpolaton descrbed n Sec. 3.1 to calculate W 0 t and ts backward mappng B t 0. W 1 t and B t 1 can be computed n a smlar way. Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
386 Fgure 5: Top row: a morphng sequence generated by our algorthm. Bottom row: the correspondng sequence generated usng [COSL98]. The user-specfed anchor ponts are also shown n the top row. Fgure 6: Top row: a morphng sequence generated by our as-rgd-as-possble DFI. Bottom row: the correspondng sequence generated usng lnearly nterpolated warpng functons. We note that many tetrahedra constructed n our approach overlap wth each other and do not form a tetrahedralzaton of the obect. They are only used to mpose local transformaton constrants and fnally produce global as-rgd-aspossble nterpolaton. Ths dstngushes our approach from [ACOL00], where an somorphc tetrahedralzaton needs to be constructed for two obects wth the correspondence on the boundares already establshed. 4. Expermental Results We have mplemented the correspondence and DFI morphng algorthms on an Intel Xeon E5620 workstaton. We provde statstcs for the models presented n ths paper n Table 1, ncludng the tmngs and the numbers of anchor ponts used. We selected several examples to demonstrate the effects of the resultng algorthm. See also the accompanyng vdeo for morphng anmatons. As shown n Table 1, among the stages of our approach, constructng correspondence takes the least amount of tme as t s performed on grds of smaller szes. TPS nterpolaton and as-rgd-as-possble DFI spend comparable tme. Example Grd Sze #Achors Corresp. TPS Interp. Fg. 1 180 3 /40 3 12 10 58 24 Fg. 5 500 2 /50 2 12 1 14 1 Fg. 7 180 3 /30 3 5 9 20 22 Fg. 9 180 3 /40 3 13 10 62 18 Fg. 10 180 3 /40 3 14 9 21 33 Fg. 11 180 3 /40 3 12 10 19 27 Table 1: The volume grd szes for dstance felds/mass transport, the number of anchor ponts and the tmng results (n seconds) for buldng correspondence, TPS nterpolaton, and as-rgd-as-possble DFI for a sngle frame. The zero set surface extracton (.e., marchng cube) takes less than two seconds for a sngle frame n all examples. Overall, we can generate an anmaton frame n less than two mnutes, wth much room for acceleraton as the current mplementaton s not well optmzed. Fg. 1 demonstrates the advantages of our dense nteror correspondence. As only twelve anchors are specfed by the user, some shape features of the source and target are not well algned by the warpng functon computed n [COSL98] (see Fg. 4, rght), producng artfacts (e.g., small blobs) durc 2013 The Author(s) Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
387 Fgure 7: Morphng a smple obect of genus 2 to the Bunny model. Top row: a morphng sequence generated by our as-rgdas-possble DFI. Mddle row: the correspondng sequence generated usng lnearly nterpolated warpng functons wth our nteror correspondence. Bottom row: the correspondng sequence generated usng [COSL98]. ng morphng as observed n the bottom row of Fg. 1. Whle ncreasng the number of anchor ponts can help allevate ths problem, t becomes a tedous tral-and-error process. In contrast, usng the same sparse anchor ponts, our algorthm can compute a dense nteror correspondence that better algns shape features such as the horse leg (see Fg. 4, left), generatng superor results to those produced by [COSL98]. Fg. 5 compares our algorthm wth [COSL98] n a 2D example. Whle udgng the qualty of the morphng results n ths example could be subectve and dfferent people may have dfferent opnons, we would lke to pont out that our nteror correspondence helps to avod creatng unpleasant ghostng features durng morphng that are not exhbted n the source and target, as hghlghted n the the bottom row of Fg. 5. Our as-rgd-as-possble DFI can greatly mprove the qualty of the morphng results. Fg. 6 shows a 2D example. The results n both the frst and second rows are generated usng our dense nteror correspondence, whch s employed to compute the warpng functons between the source and target. In the frst row, we perform as-rgd-as-possble DFI, whle n the second row we lnearly nterpolate the warpng functons. A shown, as-rgd-as-possble DFI can generc 2013 The Author(s) Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd ate more natural ntermedate obects wth less dstorton lnear nterpolaton makes the dnosaur s tal shrnk n the morphng process. In Fg. 7, an obect of genus two s changng smoothly to the Bunny model. Agan, the sparse correspondence of fve landmarks cannot algn the source and target well, resultng n dsconnected components n the morphng results produced by [COSL98] (see the bottom row). Whle lnear nterpolaton usng our computed dense nteror correspondence greatly mproves the results (the mddle row), t ntroduces sgnfcant volume shrnkage n the bunny s body part, leadng to more dstorton than our as-rgd-as-possble DFI results (the top row). Lmtatons and Dscusson. Although our technque can produce vsually plausble morphng anmatons wth a small number of landmarks, we note that n order to get satsfactory results these landmarks need to be carefully placed n meanngful postons that correspond to geometrc features of the source and target obects. And f the number of landmarks s too low, our approach may generate unwanted results (see Fg. 8, top row). Addng more landmarks can certanly mprove the results (see Fg. 8, bottom row). One problem wth our current correspondence s that the
388 Fgure 9: Top row: a morphng sequence generated usng harmonc felds computed n the whole volume. Bottom row: the correspondng sequence generated usng harmonc felds computed n the shape nteror. Fgure 8: Morphng results wth dfferent numbers of landmarks. Top row: the results of three landmarks. Bottom row: the results of fve landmarks. computaton of harmonc felds s performed n the ambent space and does not consder the shapes of the source and target obects. Ths may cause correspondences not n accordance wth the user s percepton of the shapes and generates unsatsfactory morphng results (see Fg. 9, top row). To remedy ths problem, we can restrct the harmonc feld computaton n Sec. 2 to be wthn the shape nteror. Specfcally, we only compute the harmonc feld values for nteror voxels. And when calculatng the Laplacan at a voxel, we only consder those nteror voxels n ts 6-connected neghbors. The harmonc felds computed ths way lead to a better correspondence and sgnfcant mprove the morphng results (see Fg. 9, bottom row). Fg. 10 and Fg. 11 show two more morphng results generated usng ths approach. Note that ths approach works only for shapes contanng a sngle connected component (.e., all nteror voxels of the shape are connected). For shapes wth multple dsconnected components such as those shown n Fg. 1 and Fg. 5, we stll have to compute harmonc felds n the ambent space. Another problem of our work s that the computed correspondence s not a one-to-one correspondence between shapes. It s thus dffcult to evaluate the qualty of the correspondence as n surface parameterzaton where the dstorton of the correspondence can be rgorously defned. Whle the morphng results based on ths correspondence are better than prevous DFI methods, we found t hard to further mprove the results wthout usng more landmarks. Furthermore, lack of one-to-one correspondence also makes our method problematc when handlng colored or textured shapes. The correspondence of each ndvdual voxel s fuzzy, whch may cause unpleasant ghostng artfacts f t s drectly used for color nterpolaton. A possble soluton s to frstly generate the morphng surfaces usng our algorthm, track the correspondence nformaton through tme for the surfaces usng the recent method ntroduced n [BHLW12], and use the tracked correspondence for color nterpolaton. Fnally, f an obect contans multple sematc components, we have to treat the obect as a sngle dstance feld and gnore the semantc nformaton. Ths may generate morphng results that do not capture structure-aware effects, whch are crtcal for some manmade obects (e.g., furnture). We would lke to explore these problems n the future. 5. Concluson We have ntroduced a novel approach to performng as-rgdas-possble nterpolaton between two dstance felds. It establshes dense correspondence between the nteror of two arbtrary obects based on the theory of optmal mass transportaton. The resultng correspondence s consstent wth the user-specfed anchor pars, enablng us to compute nonrgd warpng functons that better algn the source and target obects, and thus makes t possble to ncorporate local rgdty constrants to perform as-rgd-as-possble DFI. We found n our tests that our approach greatly mproves the qualty and flexblty of DFI morphng technques. Acknowledgements: We would lke to thank Stanford Unversty and AIM@SHAPE for sharng ther 3D models, and the anonymous revewers for ther constructve comments. The work s partally supported by NSF of Chna (No. 61003145, 61272305 and 61272392) and the Open Proect of State Key Lab of CAD&CG. References [ACOL00] ALEXA M., COHEN-OR D., LEVIN D.: As-rgd-aspossble shape nterpolaton. In ACM SIGGRAPH 00 (2000), pp. 157 164. 2, 5, 6 [BHLW12] BOJSEN-HANSEN M., LI H., WOJTAN C.: Trackng surfaces wth evolvng topology. ACM Trans. Graph. 31, 4 (July 2012). 3, 8 [Boo89] BOOKSTEIN F. L.: Prncpal warps: Thn-plate splnes and the decomposton of deformatons. IEEE Trans. Pattern Anal. Mach. Intell. 11, 6 (1989), 567 585. 4, 5 [BPPH11] BONNEEL N., PANNE M. V. D., PARIS S., HEIDRICH W.: Dsplacement nterpolaton usng Lagrangan mass transport. ACM Trans. Graph. 30, 6 (Dec. 2011). 3 [COSL98] COHEN-OR D., SOLOMOVIC A., LEVIN D.: Threedmensonal dstance feld metamorphoss. ACM Trans. Graph. 17, 2 (Apr. 1998), 116 141. 1, 2, 5, 6, 7 Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd
389 Fgure 10: Morphng a ar of genus one to another ar of genus fve. Fgure 11: Morphng a hand model of genus zero to a statue model of genus fve. [GD05] G RAUMAN K., D ARRELL T.: The pyramd match kernel: dscrmnatve classfcaton wth sets of mage features. In ICCV 05 (2005), vol. 2, pp. 1458 1465 Vol. 2. 3 [HWK94] H E T., WANG S., K AUFMAN A.: Wavelet-based volume morphng. In IEEE Vsualzaton 94 (1994), pp. 85 92, CP8. 2 [HZTA04] H AKER S., Z HU L., TANNENBAUM A., A NGENENT S.: Optmal mass transport for regstraton and warpng. Internatonal Journal on Computer Vson 60 (2004), 225 240. 3 [IMH05] I GARASHI T., M OSCOVICH T., H UGHES J. F.: Asrgd-as-possble shape manpulaton. ACM Trans. Graph. 24, 3 (July 2005), 1134 1141. 2 [Kan06] K ANTOROVICH L.: On a problem of Monge. Journal of Mathematcal Scences 133, 4 (2006), 1383 1383, translated from Uspekh Mat. Nauk, 3(2), 225l C226 (1948). 2, 3 [KMP07] K ILIAN M., M ITRA N. J., P OTTMANN H.: Geometrc modelng n shape space. ACM Trans. Graph. 26, 3 (July 2007). 2 [KS04] K RAEVOY V., S HEFFER A.: Cross-parameterzaton and compatble remeshng of 3d models. ACM Trans. Graph. 23, 3 (Aug. 2004), 861 869. 2 [LC87] L ORENSEN W. E., C LINE H. E.: Marchng cubes: A hgh resoluton 3d surface constructon algorthm. In ACM SIGGRAPH 87 (1987), pp. 163 169. 3 [LGL95] L ERIOS A., G ARFINKLE C. D., L EVOY M.: Featurebased volume metamorphoss. In ACM SIGGRAPH 95 (1995), pp. 449 456. 2 [LPD11] L IPMAN Y., P UENTE J., D AUBECHIES I.: Conformal wassersten dstance: I. computatonal aspects and extensons. arxv preprnt arxv:1103.4681 (2011). 3 [LSLCO05] L IPMAN Y., S ORKINE O., L EVIN D., C OHEN -O R D.: Lnear rotaton-nvarant coordnates for meshes. ACM Trans. Graph. 24, 3 (July 2005), 479 487. 2 [MY11] M AKIHARA Y., YAGI Y.: Earth mover s morphng: topology-free shape morphng usng cluster-based emd flows. In ACCV 10, Volume Part IV (2011), pp. 202 215. 3 c 2013 The Author(s) Computer Graphcs Forum c 2013 The Eurographcs Assocaton and John Wley & Sons Ltd [OBCS 12] OVSJANIKOV M., B EN -C HEN M., S OLOMON J., B UTSCHER A., G UIBAS L.: Functonal maps: a flexble representaton of maps between shapes. ACM Trans. Graph. 31, 4 (July 2012), 30:1 30:11. 2 [PT92] PAYNE B. A., T OGA A. W.: Dstance feld manpulaton of surface models. IEEE Comput. Graph. Appl. 12, 1 (Jan. 1992), 65 71. 2 [RTG00] RUBNER Y., T OMASI C., G UIBAS L.: The earth mover s dstance as a metrc for mage retreval. Internatonal Journal of Computer Vson 40, 2 (2000), 99 121. 3 [SAPH04] S CHREINER J., A SIRVATHAM A., P RAUN E., H OPPE H.: Inter-surface mappng. ACM Trans. Graph. 23, 3 (Aug. 2004), 870 877. 2 [SCOL 04] S ORKINE O., C OHEN -O R D., L IPMAN Y., A LEXA M., RÖSSL C., S EIDEL H.-P.: Laplacan surface edtng. In SGP 04 (2004), pp. 175 184. 2 [SGWM93] S EDERBERG T. W., G AO P., WANG G., M U H.: 2-D shape blendng: an ntrnsc soluton to the vertex path problem. In ACM SIGGRAPH 93 (1993), pp. 15 18. 2 [SNB 12] S OLOMON J., N GUYEN A., B UTSCHER A., B EN C HEN M., G UIBAS L.: Soft maps between surfaces. Comp. Graph. Forum 31, 5 (Aug. 2012), 1617 1626. 3 [SWC97] SUN Y. M., WANG W., CHIN F. Y. L.: Interpolatng polyhedral models usng ntrnsc shape parameters. The Journal of Vsualzaton and Computer Anmaton 8, 2 (1997), 81 96. 2 [SZGP05] S UMNER R. W., Z WICKER M., G OTSMAN C., P OPOVI C J.: Mesh-based nverse knematcs. ACM Trans. Graph. 24, 3 (Aug. 2005), 488 495. 5 [TPG01] T REECE G., P RAGER R., G EE A.: Volume-based threedmensonal metamorphoss usng regon correspondence. Vsual Computer 17 (2001), 397 414. 1, 2 [XZWB05] X U D., Z HANG H., WANG Q., BAO H.: Posson shape nterpolaton. In SPM 05 (2005), pp. 267 274. 2 [ZRKS05] Z AYER R., R ÖSSL C., K ARNI Z., S EIDEL H.-P.: Harmonc gudance for surface deformaton. Comput. Graph. Forum 24, 3 (2005), 601 609. 2, 4