Layered Animation using Displacement Maps

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1 Layeed Animation using Displacement Maps Raymond Smith, Wei Sun, Adian Hilton and John Illingwoth Cente fo Vision, Speech and Signal Pocessing Univesity of Suey, Guildfod GU25XH, UK Abstact This pape pesents a layeed animation famewok which uses displacement maps fo efficient epesentation and animation of highly detailed sufaces. The model consists of thee layes: a skeleton; low-esolution contol model; and a displacement map image. The novel aspects of this appoach ae an automatic closed-fom solution fo displacement map geneation and animation of the layeed displacement map model. This appoach povides an efficient epesentation of complex geomety which allows ealistic defomable animation with multiple levels-of-detail. The epesentation enables compession, efficient tansmission and level-of-detail contol fo animated models. 1 Intoduction Animation of 3D models with highly detailed sufaces is a common equiement of compute geneated imagey fo film and boadcast poduction. Chaacte models ae often sculpted fom clay to achieve the desied shape and fine suface detail. Lase-scanning devices ae used to digitise the suface. Advances in scanning technology togethe with development of suface econstuction algoithms [3, 7] have enabled automatic captue of complex 3D models with highly detailed sufaces. Cuently, animation of captued models equies a skilled animato seveal weeks to manually estuctue the model geomety and topology. In this pape we intoduce a layeed animation famewok which addess the following issues: 1. Rapid econstuction fom captued data. 2. Efficient epesentation of suface detail. 3. Model geneation at multiple levels-of-detail. 4. Seamless defomable animation of the suface detail. Displacement maps ae used to epesent the highesolution suface detail in an efficient image-based fom. A new appoach is intoduced fo automatically mapping between a captued high-esolution mesh model and a lowesolution contol model. This mapping is used to geneate a displacement map image epesentation of the highesolution suface detail based on 2D textue coodinates fo the low-esolution model. The displacement map epesents the distance between the low and high esolution model sufaces as a scala image. The low-esolution contol model epesents the object shape and topology and can be manipulated efficiently using an undelying skeleton. The esulting layeed model povides an efficient epesentation. Real-time seamless defomable animation of the contol model enables efficient seamless animation of the high-esolution suface detail. Automatic mesh simplification algoithms have been widely developed fo educing the size of econstucted 3D models fom suface measuements [10, 8, 13, 16, 2]. Typically, toleances ae imposed to geneate a epesentation which appoximates the oiginal suface within a specified geometic eo. Howeve, automatic simplification algoithms do not esult in meshes which ae diectly suitable fo efficient and ealistic animation. As the captue pocess is pefomed on a static object in a single pose thee is no captued infomation on the undelying aticulation stuctue. Theefoe a manual pocess is equied to position a subset of the mesh vetices to coincide with the equied aticulation stuctue. Pehaps the most closely elated wok to ou appoach is the wok of Kishnamuphy and Levoy [14] whee the boundaies of a low-esolution B-spline patch model ae manually identified on a high-esolution captued model. Automatic fitting and mapping is then pefomed to obtain a B-spline epesentation of smooth suface egions and displacement map images to epesent fine suface detail. The displacement maps povides an efficient epesentation of complex sufaces without loss of fine detail. Howeve, the esulting

2 model can not be diectly animated. In this pape we pesent a geneal appoach to obtaining a displacement map epesentation based on an low-esolution contol model. Animation of the contol model [12] enables seamless defomable animation of the high-esolution suface data. Pevious wok has addessed automatic econstuction of suface models with subdivision connectivity based on polygonal meshes [5] and highe ode suface [9, 6] with piecewise smooth continuity. Subdivision models povide an efficient mechanism fo geneating models at multiple levels-of-detail. In this pape we use subdivision of the contol model togethe with esampling of the displacement map image to geneate models at multiple levels-of-detail. The combination of displacement maps with mesh subdivision povides a mechanism fo epesenting highly complex sufaces. Adaptive mesh subdivision geneates epesentations with vaiable level-of-detail o a specified geometic accuacy. Unlike smooth subdivision suface epesentations this appoach enables accuate and efficient epesentation of sufaces with complex geomety including bump pattens and discontinuities. 2 Layeed animation famewok In this pape we popose a layeed animation famewok using displacement maps to efficiently epesent and animate high-esolution suface detail. The model consists of thee layes: skeleton, contol model and displacement map. Skeleton stuctues ae widely used as the basis fo manipulating animated models eithe inteactively o fom motioncaptue data. The contol model is a low-esolution polygon mesh,, epesenting a coase appoximation of the chaacte shape and topology. A contol model can be deived eithe by simplification of a high-esolution model o fom a libay of geneic objects models which have been stuctued fo efficient animation such as those available fom ViewPoint DataLabs [4]. The contol model is animated based on the undelying skeleton stuctue and enables ealtime visualisation. As in pevious wok [17] the vetices of the low-esolution model ae mapped to the skeleton stuctue. Real-time seamless animation is achieved using a geometic tansfom [1, 12]. Moe ealistic defomation coesponding to a paticula suface type could also be implemented using a physics-based appoach to defom the contol model such as FFD [11] and DFFD [15]. In this wok we intoduce the use of a displacement map epesentation of the high-esolution suface detail. The displacement map is an image-based epesentation analogous to a textue map which encodes the distance along the suface nomal between the low-esolution contol model and a high-esolution polygonal mesh model,. The high-esolution model is automatically econstucted fom suface measuements captued using a ange-senso system [7]. An automatic mapping technique is intoduced in section 2.2 to establish a continuous mapping between the high-esolution captued suface model and the lowesolution contol model. This mapping is used to geneate an efficient displacement map image epesentation of the high-esolution suface detail. Animation is achieved by manipulation of the skeleton based on eithe keyfame o motion captue data. Movement of the skeleton is used to seamlessly defom the lowesolution contol model. Recusive subdivision of the animated low-esolution contol model is used to efficiently econstuct models at multiple levels-of-detail. This layeed animation famewok, combining displacement maps with mesh subdivision, enables efficient seamless animation of the high esolution suface detail. 2.1 Nomal volume mapping In pevious wok [17] we intoduced the nomal-volume mapping to automatically paameteise an abitay highesolution mesh model,, with espect to a low- esolution contol model,. It was shown that this paameteisation could be used to seamlessly animate the high-esolution model based on defomation of the undelying contol model. In this section we summaise the key details of the nomal-volume mapping which ae used fo displacement map geneation. Fo each tiangle in an abitay mesh we define a nomal-volume,, by displacing the tiangle vetices,, along the vetex nomals,, as illustated in Figue 1(a). The union of the nomal-volumes fo all tiangles in mesh enclose a continuous volumetic envelope which can be used to define a continuous mapping between points in 3-space and the mesh suface. To obtain a continuous mapping we poject points in 3- space along the coesponding intepolated tiangle nomal. A point on the suface of tiangle and its unit nomal can be defined by bi-linea intepolation using baycentic coodinates as: #%$ '&)( ( $ %$ '&*( ( $ (1) whee fo a point inside the tiangle the baycentic coodinates, $ and '&)( ( $ ae all positive scala vaiables. Bilinea intepolation of the nomal gives a continuous vaiation in the tiangle nomal,, acoss the plana suface and between adjacent tiangles. The esulting nomal field is continuous such that fo evey point inside the nomalvolume thee is a coesponding nomal which passes though that points. Figue 1(b) illustates the nomal passing though a point. Thus any point inside the nomal volume,- can be expessed as:

3 v v t v n v - v s n s vs v - s n t vt v- t (a)tiangle nomal volume p 1 v v t dn v n v - v s n s p dns 2 vs v - s x dn j p j nt (b) Point mapping Figue 1. Nomal volume mapping. $ '& ( ( $ p 3 dn t v t v - t (2) whee is the Euclidean distance of the points along the nomal fom. The above equation can be viewed geometically as an offset suface at a distance which defines a plane passing though point as illustated in Figue 1(b). As shown in pevious wok [17] fo an abitay point we can obtain the paametes $ fo tiangle by solving fo the plane in the nomal-volume which passes though the point. If, $ and & ( ( $, ae in the ange then the point,, maps to a point on the tiangle suface. Paameteisation of a high-esolution model,, with espect to a low-esolution model,, is achieved by evaluation of the nomal-volume mapping to the high-esolution model vetices. Each model vetex has an associated set of fou mapping paametes $ whee is the low-esolution tiangle index, $ ae the baycentic coodinates of the coesponding point on the low-esolution model tiangle and is the distance along the intepolated tiangle nomal. Having computed this mapping the model can be exactly epoduced and animated fom the low-esolution model,, togethe with the highesolution vetex paametes and mesh topology. 2.2 Displacement map geneation In this section we intoduce a pocess fo computing a displacement map epesentation of the high-esolution suface model. The displacement map is an efficient image based epesentation of the high-esolution suface detail. An appoximation of the high-esolution model can be apidly econstucted fom the displacement map image togethe with the low-esolution contol model. Animation of the low-esolution contol model can be used to achieve efficient animation of the high-esolution suface detail. Figue 2 illustates the pocess of displacement map geneation fo a single tiangle in the low-esolution contol & model,. Initially a mapping is established between the high-esolution mesh,, and the contol model,, in 3D space using the tiangle nomal-volume as shown in Figue 2(a). The high-esolution model is then mapped to a 2D image plane using a set of textue coodinates fo each tiangle in the low-esolution model, Figue 2(b). The distance between the high and low esolution model sufaces along the intepolated suface nomal is then sampled on a egula gid to obtain the displacement map image, Figue 2(c). The nomal-volume mapping defines a paameteisation fo all tiangle vetices in a high-esolution model,, in tems of the neaest tiangle on a low-esolution con- tol model. The mapping of a vetex,, to a point on tiangle is defined by equation 2 as a point on the tiangle suface in baycentic coodinates $ and the distance along the intepolated tiangle nomal. Thus, fo evey vetex in the high-esolution model we obtain a mapping specified by fou paametes $ whee is the low-esolution tiangle index. Let us define a set of textue coodinates,, fo the low-esolution model which uniquely map each tiangle to a two-dimensional textue map plane,. Fo each tiangle,, thee is a continuous mapping to a coesponding tiangle in the textue image domain. We can obtain a mapping of the high-esolution model to the textue map plane,, by combining the low-esolution model textue mapping with the nomal-volume mapping of the high-esolution model onto the low-esolution model. Given the nomal-volume mapping $ fo vetex in the high-esolution model onto tiangle in the low-esolution model the mapping to textue coodinates is: $ &)( ( $ (3) This enables us to map any tiangle # in the high-esolution model into a tiangle the in the textue image plane,. Thus any point on the high-esolution model suface can be mapped to a point in the textue image plane. It should be noted that this mapping is not injective, multiple points on the high-esolution model may map to the same point on the low-esolution model due to ovefolding of the suface. This many-to-one pojection is a known limitation of displacement maps. In pactice this poblem can be avoided by eithe adding additional tiangles to the low-esolution model to ensue a one-to-one mapping fo the entie suface o by appoximating the suface geomety as discussed in section 3.2. Given the mapping of the high esolution model,, to the textue plane,, we can obtain a sample of the high-

4 (a) Mapping of to (b) Mapping to 2D plane (c) Sampling distance to Figue 2. Displacement map geneation fo high-esolution model,. esolution model suface,, fo any point in the textue plane which is inside the egion to which the high-esolution model maps. Fo a point we can find the tiangle fom the high-esolution model which it is inside such that: # &*( ( (4) whee & ( ( & ae baycentic coodinates. Fom the nomal-volume mapping we know the distance of each vetex along the intepolated low-esolution tiangle nomal. The distance fo any point inside the tiangle is: &)( ( (5) The mapping defined above enables the distance between the low and high esolution models to be sampled at any point in the image plane. This mapping is used to geneate a displacement map image whee the distance is sampled fo a set of discete points in the image plane. Geneation of the displacement map image is discussed futhe in section Multiple Level-of-Detail Reconstuction In this section we pesent a subdivision scheme fo the low-esolution model which enables us to efficiently econstuct high-esolution models at multiple levels-of-detail (LOD). An appoximation of the high-esolution model can be econstucted fom the low-esolution contol model togethe with the set of vetex textue coodinates and the displacement map image. The location of points on the high-esolution model can be econstucted by sampling the displacement image fo any point on the lowesolution model using equation 5. Figue 3 illustates the econstuction of a high-esolution model fo a single tiangle in the low-esolution contol model. Initially the contol model,, is subdivided accoding to the equied level-of-detail to geneate a model, as shown in Figue 3(a). The displacement map image is then sampled at the vetices of the subdivided mesh,, to obtain a distance value. Finally, a new 3D mesh model,, is econstucted by displacing the vetices of the subdivided mesh,, along the intepolated contol model tiangle nomal by the sampled distance, as shown in Figue 3(c). Given a low-esolution tiangula mesh model we use unifom quateney subdivision to geneate a mesh to any specified level of detail. A ecusive subdivision scheme is used to subdivide each tiangle in the mesh into fou subtiangles. New vetices ae placed at the midpoints of the tiangle edges. An abitay tiangle in the given mesh is split into fou tiangles afte one level subdivision. Caying on the pocess each of the fou new tiangles is futhe subdivided into fou new tiangles. Figue 4(a) shows a low-esolution cube model subdi- & ). Fig- vided at thee diffeent esolution levels ( ue 4(b) shows the econstuction of a head model at multiple levels-of-detail. The new high-esolution mesh models ae geneated by subdivision of the low-esolution cube and esampling of the displacement map image, shown in Figue 6(g). The accuacy of the econstucted model depends on the sampling esolution of the displacement image. If this esolution coesponds to the smallest tiangle edge size,, on the oiginal high-esolution model then the maximum e- o will be. Results indicate that with this sampling esolution thee is no visible loss of accuacy in the econstucted models. 2.4 Animation The layeed model stuctue enables us to efficiently animate the high-esolution suface based on manipulation of the undelying skeleton joint positions o angles. The model

5 (a) Subdivision of mesh to obtain map fo (c) Geneation of new model (b) Resampling of displacement Figue 3. Subdivision egeneation of mesh with vaiable level-of-detail is animated in the following stages: 1. Manipulate the skeleton (manual o motion captue). 2. Seamless defomation of low-esolution model,, by computation of new vetex positions. 3. Subdivision of low-esolution model to obtain,. 4. Sampling of displacement map image fo each vetex in,. 5. Geneation of a new model,, by displacement of vetices in. Seamless defomation of the low-esolution contol mesh,, is implemented using a geometically based appoach. Initially the skeleton model is manually located inside the low-esolution contol model. The mapping between the vetices of the contol model and the skeleton is automatically established using a point-to-line mapping algoithm [17]. This mapping egistes each contol model vetex,, in a local-coodinate system with espect to a skeleton segment,, whee is the position of the skeleton joint. The mapping is defined in tems of thee paametes whee its position is given by: '&)( &)( (6) whee is a scala epesenting the distance along the segment and epesents the distance fom the segment along an intepolated nomal. Vecto is a unit vecto defined by the intesection of the plane defined by: ( and the joint plane. Whee the joint plane fo the joint is defined automatically by the position of its adjacent joints and such that: with unit vecto ( (. As the joint planes ae dependent on the joint positions animation of the joint positions will esult in seamless defomation of the vetex positions as defined by equation 6, fo futhe details see [17]. To avoid ceases in the mesh fo acute joint angles we intoduce a scaling facto to ensue that the distance of the vetex fom the segment is maintained at a constant distance. This scaling intoduces ounding fo vetices nea the joint esulting in a smooth defomation of the mesh aound the joint. Animation of a new high-esolution model can be efficiently implemented fom the animation of the low-esolution model,. Having, animated the low-esolution model, to obtain the new highesolution model is econstucted accoding to the algoithm intoduced in the pevious section. Initially the low-esolution model, is subdivided to obtain based on the animated low-esolution vetex position. The displacement image is then esampled based on the textue coodinates fo the low-esolution model. The new highesolution model,, is then geneated by displacing the vetices of the mesh,, along the nomal. The nomals fo the animated low-esolution model ae ecomputed esulting in defomation of the nomal-volume fo the low-esolution mesh. The high-esolution vetex positions fo the animated high-esolution model ae then ecomputed using equation 2 based on the modified lowesolution vetex positions and nomals. The cost of animating the high-esolution model is theefoe the combined cost of seamless defomation of the lowesolution model (which is eal-time) and the cost of geneation of the new mesh. The esulting animation of the high-esolution model gives a smooth seamless defomation based on the defomation of the undelying low-esolution model. 3 Implementation In this section we pesent details of the implementation of the algoithms used fo mesh subdivision and displacement map geneation. The implementation of these algoithms is citical to the computational efficiency of geneat-

6 (a) Low-esolution cube contol mesh subdivision ( & ) 0 V Edge 1 Edge V s Vt Figue 5. Subdivision indices (b) Reconstucted head models at thee LOD ( ) Figue 4. Reconstucted subdivision head model at thee levels-of-detail ing the layeed model and econstuction of new models at multiple levels-of-detail. The esulting layeed model enables eal-time geneation and animation of high-esolution suface models. 3.1 Mesh subdivision Subdivision of the low-esolution tiangle mesh is used to geneate new high-esolution suface models at multiple levels-of-detail as pesented in section 2.2. The econstuction time fo geneating new models depends on the time taken to subdivide the low-esolution model. Theefoe, a highly efficient scheme has been developed fo geneating the set of vetices and the tiangle topology fo a new mesh,, at a specified subdivision level,. A scheme has been developed fo ecusively geneating a set of vetex indices fo subdividing a tiangle to a given subdivision level. The indices ae numbeed such that the set of vetex positions and moe impotantly the set of tiangles fo the subdivision mesh can be geneated automatically. Subdivision tiangles ae defined with vetices in counte-clockwise ode. Figue 5 shows the vetex index scheme fo subdivision of a tiangle in the low-esolution model with. The new tiangles ae divided into two categoies shown as shadowed and unshadowed. The vetex connections fo the shadowed tiangle ae defined as: & & & ( & (7) whee, & $#%#%#, '& )(*. As shown in Figue 5, &,, & and.-. Unshadowed tiangle vetex connections, & ( & & ( &, ae defined as: & whee 0 & ( & / & / /01 2& 3(4. As shown in Figue 5, 65 and.76-. Each new vetex in tiangle of the thee tiangle vetices, on the tiangle edge & / & (8), & $#%#%# &, &,, is expessed in baycentic coodinates, $, & ( ( $ in tems and. New vetices ae given by: $ (9) whee, &8&9:(. & (.;, $ 6;. Similaly new vetices on the tiangle edge ae given by: whee < $. $ ( &, &=&>:( (10). &)( ;, Based on the vetices on the edge vetices given by equations 9 and 10 the est of the new vetices in can be computed as: '&)( whee &)(, $ &)( (11). &

7 The above scheme defines the subdivision fo each individual tiangle in the low-esolution model,. In pactice adjacent tiangles will shae vetices in the subdivision mesh. To avoid edundancy edge-shaing infomation is used in the algoithm to avoid epetition of vetices. When a tiangle is subdivided, the new vetices on its edges will be shaed automatically by its neighbou tiangles. The output of the subdivision algoithm is a new mesh composed of a set of vetices and tiangles, togethe with the baycentic coodinates of each new vetex with espect to a low-esolution tiangle. The baycentic coodinates of each new vetex give its elative position inside a tiangle of the oiginal mesh. As the baycentic coodinates do not change unde affine tansfomations they can be used to map the vetices of the subdivided model into the textue image plane. Resampling of the displacement image fo each vetex on the subdivision model enables geneation of a new high-esolution model. The baycentic coodinates also emain constant when the low-esolution contol mesh is defomed duing animation and can theefoe be used to econstuct and animate new high-esolution models. This subdivision scheme gives a closed fom algebaic solution fo geneating a new mesh at subdivision level in eal-time. Geneation of the new high-esolution model is based on a simple lookup in the displacement image using baycentic coodinates. Theefoe, the new model can be geneated in close to eal-time. 3.2 Displacement map geneation This section explains the pactical details of displacement map geneation. Thee ae a numbe of issues which must be consideed in implementing the displacement map geneation. The pincipal consideation is to ensue a continuous mapping (without gaps) of the high-esolution model suface to the displacement image. Secondly, implementation should conside the computational efficiency of the displacement map geneation. Finally, implementation should deal with the possibility of many-to-one mapping between the high-esolution model suface and displacement map. Figue 6 illustates the displacement map geneation fo a high-esolution 3D head model on to a simple cube model. Figue 6(a-c) shows the 3D model and model mapping, (d) shows the mapping of the contol model to the textue image plane, (e-f) show images of intemediate stages and (g) shows the esulting displacement map image. The scala displacement maps values ae shown in pseudocolou fom ed to blue with inceasing magnitude of the displacement. Two basic stategies to the displacement map geneation could be adopted: Fowad mapping takes each tiangle,, on the high-esolution model and maps it via the lowesolution contol model tiangle,, into the textue map plane,. Fo each displacement image pixel,, we then sample the distance between the low and high esolution model along the nomal using equation 5. Backwad mapping takes each point in the displacement map image,, and find the tiangles on the low, #, and high,, esolution models which the 2D point is inside. The distance between the low and high esolution models is then computed fom equation 5. Backwad mapping equies a seach to find the coesponding tiangles fo each point. This seach even in the 2D textue plane is potentially computationally expensive fo high-esolution models with a lage numbe of tiangles. A second poblem with the backwad mapping is that due to the many-to-one mapping fom the high-esolution model to the textue domain we must seach fo the set of coesponding tiangles which map each point in the textue image. Theefoe, in this wok we have used fowad mapping to implement the displacement map geneation. Fowad mapping gives a closed fom solution to finding the set of points in the textue map fo each tiangle on the high-esolution model. This povides an efficient method fo geneating the displacement map image povided that the high-esolution tiangle,, only maps to a single tiangle on the low-esolution model,. This is the case fo the majoity of tiangles whee all thee vetices of map to a single tiangle. As both the low and high esolution tiangles ae by definition convex it is guaanteed that all points inside map to points inside. Howeve, if the vetices of map to multiple tiangles in the low-esolution model then we must conside the mapping of each pat of the high-esolution tiangle to the low-esolution model sepaately. In this case the mapping of the high-esolution tiangle is only piecewise linea fo each low-esolution tiangle, this is illustated in Figue 7(a). In addition, the high-esolution tiangle may map to disconnected egions of the displacement map if the lowesolution tiangles ae not adjacent in the textue plane. In this section we intoduce a closed fom solution which enables efficient implementation of the mapping fo highesolution tiangles which map to multiple low-esolution tiangles Vetex Mapping The fist stage in geneating a displacement map is to map the low-esolution contol model into the 2D textue image plane. Each tiangle in the contol model will have textue coodinates fo each vetex as discussed in section 2.1. This enables each low-esolution model tiangle,, to be

8 (a) High-esolution Model (b) Low-Resolution Contol Model (c) Nomal-Volume Mapping (d) High-esolution Vetex Mapping (e) Patial displacement map (f) Many-to-one mapping (g) Final displacement Map (h) Reconstucted head animation by moving one vetex of cube Figue 6. Displacement map geneation fo head model with simple cube contol model

9 A B (a) Bode textue mapping A B (c) Left-hand tiangle B B (b) Bode nomal-volume A B A (d) Right-hand tiangle Figue 7. Nomal-volume mapping of a highesolution bode tiangle to multiple lowesolution tiangles uniquely mapped to non-ovelapping tiangles in the textue image plane,. The mapping pesented in section 2.1 enables us to map each vetex fom the high-esolution model to a point in the textue image plane,. Mapping of vetices in the high-esolution model is illustated in Figue 6(d). The distance fo each vetex between the high-esolution and low-esolution suface is given by the nomal-volume mapping distance paamete, Filling Tiangles Having mapped the individual high-esolution vetices the next stage is to evaluate the distance fo points in the displacement map which coespond to points on the highesolution suface. The fowad mapping of the highesolution tiangle vetices to the textue image plane defines textue coodinates fo the high-esolution tiangle vetices. As discussed peviously thee ae two cases of tiangle mapping to be consideed: 1. Tiangles whose vetices map to a single lowesolution contol tiangle. 2. Bode tiangles whose vetices map to multiple low-esolution contol tiangles. A In the fist case, we simply evaluate a displacement value using equation 5 fom the peviously computed highesolution tiangle vetex distances fo each point # which is inside the tiangle,. Computation of the distance is a linea intepolation using baycentic coodinates and is theefoe highly efficient. This case coves the majoity of tiangles in the high-esolution model. The esulting patial displacement map fo a typical model is shown in Figue 6(e). The emaining unmapped bode tiangles have vetices which map to moe than one low-esolution tiangle. The esulting mapping of the high-esolution tiangle suface will intesect the set of edges between adjacent tiangles to which the vetices map. The mapping acoss multiple tiangles is illustated in Figue 7(b). As the nomal-volume mapping is in geneal diffeent fo adjacent tiangles the continuous mapping of the suface of the high-esolution tiangle to the low-esolution model is only piecewise linea fo each tiangle in the low-esolution model. This esults in the high-esolution tiangle mapping to a non-tiangula polygon # in the displacement map image plane, as illustated in Figue 7(a). Each intesection of the highesolution tiangle edge with a low-esolution tiangle edge will esult in one additional vetex in the polygon #. Geneation of the displacement map fo bode tiangles equies evaluation of the mapping fo points whee the tiangle edge intesect the bounday. This can be achieved by evaluating the nomal-volume mapping fo each of the high-esolution tiangle vetices in the plane of each lowesolution tiangle which it intesects. The key diffeence hee is that one o moe of the tiangle vetices may map outside the low-esolution tiangle esulting in negative baycentic coodinates. Figue 7(c,d) shows the nomalvolume mapping of the high-esolution bode tiangle to two adjacent low esolution tiangles. Having evaluated the mapping fo each vetex of the high-esolution tiangle,, with espect to a paticula low-esolution tiangle,, we can map it to a tiangle in # the textue image plane,. Whee one o moe of the vetices of is outside #. The pat of which is inside gives the coect mapping of the potion of which maps onto though the nomal-volume mapping. This is the coect mapping because the intesection of the mapping of with the edge of coesponds to the bounday suface between the nomal-volume of and the adjacent tiangle. This bounday suface is continuous and intesects the high-esolution along a line, as shown in Figue 7(b). Theefoe pojection of tiangle to the plane of gives the coect points of intesection with the tiangle edge. Bode tiangles which ovelap multiple low-esolution tiangles can theefoe be coectly mapped to the textue image in the following steps. Fo each low-esolution tian-

10 ( ( gle to which the high-esolution tiangle 1. Map onto the plane of maps: using the nomal-volume. 2. Map into the 2D textue image plane using the textue coodinates fo # to obtain. 3. Fo each point which is inside both the highesolution tiangle and inside the low-esolution # tiangle evaluate the distance between the low and high esolution tiangle fom equation 5. Repeating the above algoithm fo each tiangle to which the high-esolution tiangle is mapped is implemented by steping between adjacent low-esolution tiangles which the edges of the high-esolution tiangle intesect. This ensues that the high-esolution tiangle is mapped continuously into the textue image domain. Figue 6(g) shows the esulting pseudo-colou displacement map image with all tiangles in the high-esolution model mapped Resolving many-to-one mapping As discussed peviously, ovefolding of the suface may esult in multiple pats of the high-esolution model being mapped to the same location on the low-esolution model o textue map plane. This is illustated in Figue 6(f) whee image points ae coloued accoding to the numbe of highesolution tiangles which map to that point. The majoity of the image is dak blue indicating that one tiangle maps pe point, geen indicates two tiangles and ed thee. It should be noted that in pactice even in the case of a complex shape such as the head mapped to a simple cube contol model only a few points in the displacement map image have coespond to moe than one high-esolution tiangle. Multiple tiangle mapping occu aound the eas, nose and hailine whee the model ovefolds elative to the cube. In egions of the textue map plane whee multiple maps occu we implement the displacement map geneation by taking the lagest distance to any point on the suface:. This esults in the econstucted model suface being the convex hull of the oiginal mesh. Fo meshes such as the head model this does not esult in significant visual degadation of the econstucted model as the unmodelled egions such as behind the ea ae only visible fom a small numbe of views. Howeve, fo moe complex models, such as a hand mapped to a cube, taking the lagest distance would esult in incoect econstuction of the finges. In this situation it is necessay to have a moe sophisticated contol model which appoximates the object shape. In the case of the hand this could be achieved by using a contol model with finges Image Sampling, quantisation and scaling The mapping defined in section 2.1 enables us to sample the distance function fo points on a discete gid,, in the textue map plane to fom a displacement map image fo ( & and ( &. To obtain an efficient displacement image epesentation we quantise the displacement into levels. Distance values ae scaled using the ange of values in the entie image and quantised into discete levels as: (12) The esulting quantised displacement image epesentation of the high-esolution model gives an efficient appoximation which can be stoed using standad image epesentations. Typically the displacement at each pixel can be epesented by a single 8 o 16 bit numbe wheeas each vetex in the oiginal high-esolution model equied thee 32 bit floating point numbes. A quantitative compaison of epesentation costs is given in the esults section. The esults pesented in this wok use single byte values with 7 giving 256 distance levels. Images ae stoed in the standad gey scale image fomat with the ange stoed in the file heade fo econstuction. 4 Results This section pesents initial esults of fo the geneation of layeed models using displacement maps. Layeed displacement map models have been geneated fom highesolution data of a human head and monste head. Results fo these models ae shown in Figues 6 and 8. The top-line of each figue shows the oiginal high-esolution model, the low-esolution model and the nomal volume mapping. Geneation of models at multiple levels-of-detail is shown in Figue 8(d). Animation of the econstucted models based on defomation of the low-esolution contol model is shown in Figue 6(h) and 8(e). Results show that defomation of the contol model enables efficient defomable animation of the high-esolution suface detail. Table 1 shows the epesentation and computation costs fo the layeed animation model. The epesentation cost fo the layeed model is the combined size of the low-esolution contol model and the displacement map image. Results show that the epesentation cost fo the layeed model is an ode of magnitude smalle than fo the oiginal model. Results fo the computation show the combined time fo mesh subdivision and esampling the displacement map to geneate the high-esolution model on a Intel PIII450MHz. Initial esults indicate that even fo the monste head model with subdivision level - the computation is pefomed

11 Model Size Geneation Time - Oiginal Layeed Human Head 127Kb 11Kb 34ms 61ms Monste Head 935Kb 34Kb 167ms 332ms Table 1. Size and time costs in 0.3 seconds fo 60K vetices. This indicates that with the use of moe efficient adaptive subdivision new models could be geneated and animated in eal-time using the layeed displacement map model. 5 Conclusions In this pape we have intoduced a layeed animation famewok which uses a displacement map image to epesent the high-esolution suface detail. Automatic geneation of a displacement map image is achieved by mapping the high-esolution suface detail onto a low-esolution contol model using the nomal-volume mapping. The pincipal contibutions of this appoach ae: 1. Automatic displacement map geneation fo a lowesolution model. 2. Seamless defomable animation of displacement. 3. Efficient epesentation of highly detailed sufaces. 4. Rapid geneation of suface models at multiple levelsof-detail. 5. Efficient animation of high-esolution suface based on a low-esolution contol model. The esults pesented in this pape demonstate the potential of animated displacement maps fo epesenting complex sufaces such as those obtained using 3D suface measuement devices. To ou knowledge, this is the fist wok to pesent both automatic geneation and animation of displacement maps. Results demonstate that the displacement map epesentation gives an ode of magnitude eduction in model size and an ode of magnitude incease in computational efficiency fo animation. Futhe wok is equied to addess: adaptive subdivision; many-to-one mapping; and geneation of models fo a fixed geometic o appeaance toleance. Refeences [1] C. Babski and D. Thalmann. A seamless shape fo hanim compliant bodies. In Poc. VRML 99, pages 21 28, [2] P. Cignoni, C. Montani, and R. Scopigno. A compaison of mesh simplification algothms. Compute and Gaphics, 22(1), [3] B. Culess and M. Levoy. A Volumetic Method fo Building Complex Models fom Range Images. In ACM Compute Gaphics Poceedings, SIGGRAPH, NewOleans,USA, pages , [4] Viewpoint Datalabs. Viewpoint Catalog. ( [5] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbey, and W. Stuetzle. Multiesolution analysis of abitay meshes. In SIGGRAPH, pages , [6] M. Eck and H. Hoppe. Automatic econstuction of b-spline sufaces of abitay topological type. In Poc. ACM SIG- GRAPH, pages , [7] A. Hilton, A.J. Stoddat, J. Illingwoth, and T. Windeatt. Implicit suface based geometic fusion. Intenational Jounal of Compute Vision and Image Undestanding, Special Issue on CAD Based Vision, 69(3): , Mach [8] H. Hoppe. Pogessive meshes. In Poc. ACM SIGGRAPH, pages , [9] H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, J. Hin, H. McDonald, J. Schweitze, and W. Stuetzle. Piecewise smooth suface econstuction. In Compute Gaphics Poceedings, SIGGRAPH, pages , [10] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Mesh optimization. In Compute Gaphics Poceedings, SIGGRAPH, pages 19 26, [11] John E.Chadwick, David R.Haumann and Richad E.Paent. Layeed Constuction fo Defomable Animated Chaactes. In Poceedings of SIGGRAPH 89, Boston, 31 July-4 August, 1989, pages , [12] N. Kala, N. Magnenat-Thalmann, L. Moccozet, G. Sannie, A. Aubel, and D. Thalmann. Real-time animation of vitual humans. IEEE Compute Gaphics and Applications, 18(5):42 56, [13] A. Kalvin and R. Taylo. Supefaces: polygonal mesh simplification with bounded eo. IEEE Compute Gaphics and Applications, 16(3):64 77, [14] V. Kishnamuthy and M. Levoy. Fitting smooth sufaces to dense polygon meshes. In In ACM Compute Gaphics Poc. SIGGRAPH, NewOleans, USA, [15] L. Mocozzet and N. Mangnenat Thalmann. Diichlet feefom defomations and thei applications to hand simulation. In IEEE Int.Conf. on Compute Animation, [16] W.J. Schoede, J.A. Zage, and W.E. Loensen. Decimation of tiangula meshes. In SIGGRAPH 26(2), pages 65 70, [17] W. Sun, A. Hilton, R. Smith, and J. Illingwoth. Building layeed animation models fom captued data. In Euogaphics Wokshop on Compute Animation, pages , Septembe 1999.

12 (a) High-esolution monste head model (b) Low-esolution contol model (c) Mapping of head to cube (d) Reconstucted monste head at multiple levels-of-detail & 6- (e) Monste head mouth animation using layeed displacement map model - Figue 8. Layeed displacement map model and animation fo monste head