This is the peer reviewed version of the following article: Susín, A. and Ramírez, J.E. (2015). Segmentation-based skinning. Computer animation and

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1 This is the eer reviewed version of the following article: Susín, A. and Ramírez, J.E. (). Segmentation-based skinning. Comuter animation and virtual worlds, (), 1 which has been ublished in final form at [doi:./cav.]. This article may be used for non-commercial uroses in accordance with Wiley Terms and Conditions for Self-Archiving."

2 Journal Code: Article ID Disatch:.. CE: Jan Aril Same C A V No. of Pages: ME: Q1 Q2 RESEARCH ARTICLE Segmentation-based skinning Jorge Eduardo Ramírez Flores* and Antonio Susin Sánchez COMPUTER ANIMATION AND VIRTUAL WORLDS Com. Anim. Virtual Worlds () Published online in Wiley Online Library (wileyonlinelibrary.com). DOI:./cav. MOVING Universitat Politecnica de Catalunya (UPC), c/ Jordi Girona, 1-3, 4 Barcelona, Sain ABSTRACT Skeleton-driven animation is oular by its simlicity and intuitive control of the limbs of a character. Linear blend skinning (LBS) is u to date the most efficient and simle deformation method; however, ainting influence skinning weights is not intuitive, and it suffers the candy-wraer artifact. In this aer, we roose an aroach based on mesh segmentation for skinning and skeleton-driven comuter animation. We roose a novel and fast method, based in watershed segmentation to deal with characters in T-Pose and arbitrary oses, a simle weight assign algorithm based in the rigid skinning obtained with the segmentation algorithm for the LBS deformation method, and finally, a modified version of the LBS that avoids the loss of volume in twist rotations using the segmentation stage outut values. Coyright John Wiley & Sons, Ltd. KEYWORDS mesh segmentation; skinning; weight assignment algorithm; rigging; comuter animation *Corresondence Jorge Eduardo Ramírez Flores, MOVING Universitat Politecnica de Catalunya (UPC), c/ Jordi Girona, 1-3, 4 Barcelona, Sain. jramirez@lsi.uc.edu 1. INTRODUCTION Skeleton-driven animation is one of the most common 3D animation techniques used nowadays with alications in video games and film industry. The common ieline begins with an artist sculting a 3D character, creating the rig of the character mesh, and deending on the selected deformation method, a set of weights are associated from every joints of the skeleton to a secific art of the character s body. The described rocess is time-consuming and usually is hand made by the artist itself. The most difficult art in this ieline is the weight creation, a roer result deends on a recise weight assignment. Another well-known roblem related with the most oular deformation methods are artifacts generated at joint rotation. In linear blend skinning (LBS) we have the candy-wraer artifact (a loss of volume associated with a twist rotation), dual quaternion skinning (DQS) avoidsthe candy-wraer artifact of LBS but introduces its own artifact: the joint-bulging artifact (an artifact roduced by the sheric nature of the quaternion interolation). The main focus in this aer is mesh segmentation alied to skinning: we roose a novel and fast segmentation algorithm, that is going to be alied over a reviously rigged mesh to organize a set of vertices by its satial distribution. Each vertex in the inut rigged mesh is assigned to a secific link of the underlying skeleton (segmentation stage), creating what is known as rigid skinning. A roer rigid skinning is a good starting oint for a weight distribution algorithm, in [1,2] and [3] is used as starting oint. We also roose a weight assignment algorithm based in the segmentation information of each vertex to create a simle and fast algorithm. Finally, we use the segmentation in the limbs of the character to create an algorithm based on LBS, but without volume loss on twist rotations, link-oriented twist scheme, and fast enough to be used in real time animations. 2. PREVIOUS WORKS The main classification for segmentation algorithms according to Shamir [4] are art-tye segmentation and surface-tye segmentation. Part-tye segmentation is oriented in artitioning the object into semantic comonents; surface tye uses geometric roerties of the mesh to create surface atches. The most common alication of mesh segmentation is skeleton extraction: an inut mesh is taken and artitioned in segments that will reresent a region that belongs to a skeleton bone; examles are found in [5,6] and [7]; in [5] is roosed a segmentation method that takes as inut a set of meshes that reresent an animated mesh sequence through time. The inut mesh is segmented in atches that undergo aroximately the same rigid transformation over time. In [6], the segmentation is based in a maing function that creates level lines around concave areas that define area of the surface mesh. A region merge algorithm Coyright John Wiley & Sons, Ltd. 1

3 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez Q3 is used to revent over segmentation; this algorithm creates a fine abstraction of hierarchy levels to merge segmented areas; the lower the hierarchy used, the lower the number of surface atches the object will have. In [8], a segmentation method is used over rigged meshes alying Euler distances and a normal test over the surface of the inut mesh; however, their segmentation method had the same goal as our method: create a weight assignment method for the LBS algorithm. They also use their segmentation algorithm to correct and reserve volume during rotations, but they are limited to twists rotations under 0 ı, because they are using LBS as deformation scheme, and their reservation method is alied after a joint rotation is erformed with its consequent deformation; therefore, they cannot eliminate the candy-wraer artifact. An evolution of this method is found in [3]; they imrove the original method using a more robust segmentation algorithm based in voxelizing a closed mesh; then, a segmentation algorithm is erformed using geodesic distances and adding deformation effects to their revious framework, but their twist rotation constraint is still resent. A similar method is roosed in [9]; they also achieve mesh segmentation using a voxelization scheme to create a weight assignment algorithm for the LBS algorithm. Their voxelization algorithm is a novel method based in slicing a 3D mesh in the canonical axis directions to create a set of images that allows their algorithm to create a solid voxelized version of the inut mesh. One of their most interesting features is that their algorithm can work with multile meshes, and they are not limited to inut closed meshes, but all the LBS deformation roblems still remains. Our aroach is different than the classic segmentation methods that need inut arameters to create a segmentation; our method is oriented to create a segmentation based on an underlying skeleton reviously created, in a similar way as in [3,8] and [9], but our method works on the vertex ositions as inut to a region growing algorithm instead of voxelizing the target mesh or simly using Euler distances over models with ideal oses. As mentioned earlier, LBS is one of the most oular skinning algorithms; it has been used widely in video games and film industry since []. In this method, the deformation for each vertex is the roduct of the sum of each joint in the skeleton multilied by a weight to comute the final osition of the comuted vertex in a 3D mesh. This kind of comutation is used extensively in methods such as skeleton subsace deformation [], enveloing or vertex blending []. The main advantages of the LBS algorithm are its simlicity (based in a linear combination) and, as consequence, efficiency to comute (which leads to low rocessing times). This algorithm is used natively in rofessional animation software (such as AUTODESK MAYA, where it is called smooth skinning), but as is known, suffers from artifacts when some rotations are made by the influence joints, leading to the collasing elbow [] and the candy-wraer [] artifacts. These well-known artifacts cannot be revented by any user ainting the weights of a target mesh (usually the weight distribution are ainted or assigned by an artist to achieve the desired effects; the automatic software made an initial aroximation) because the artifacts are inherent to the LBS deformation scheme. Large number of works has been ublished about LBS; based in the way these methods comute the set of weights w ik, they fall in one of the next categories: Examle based: The number of aers develoed within this category make it a very oulated one [7,, ]; all these methods main idea is comute the weights of a 3D mesh by using a set of examles (a set of 3D meshes in different oses). The new oses will be the result of an interolation scheme to comute the vertices osition on the target mesh. In the method known as multi-weight enveloing [], an extension of the LBS is made by assigning more than one weight value er rigid transformation in the skinning main equation; a weight value is assigned to each element of the rigid transformation matrix. A more recent aroach is the work described in [7]; the main idea of this work is using the roxy bones to comute the weights of the target mesh. In our develoed method, we use the segmentation as base to generate a weight distribution algorithm using the neighborhood information of the skeleton s joints for each vertex. Function based: These methods have two modalities: (1) Comute automatically the weights of the LBS. (2) Relace artially or totally the blending method (substituting the rigid transformation matrix by a different rigid transformation tool) or adding a correcting method to the deformation achieved with the LBS. In the first category, we find []; its main aroach uses a nonlinear model to comute the weights of the LBS. The weights are comuted using a olynomial function that is based in a quantity called influence ratio r. In [1], a function based in heat diffusion is used to comute the weights of the LBS algorithm, where the Lalacian of a discrete surface is alied over a vector, i is a vector using the initialization j j D 1 (rigid skinning) if the nearest bone to vertex j is i and j j D 0 otherwise. Finally, an H diagonal matrix is comuted, which will have in H jj the closest weight contribution to the vertex j. One of the latest methods to comute weights automatically is [] where the weights are comuted using a Lalacian energy function subject to an uer and lower bound constraint minimizer. This work is interesting in articular because of its general treatment of the roblem; the weights are values that deend on elements called handlers; the handlers can be the elements of a cage (in 2D or 3D) or the joints of a Q4 2 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

4 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning skeleton. Therefore, it can be alied to images or skinning in 3D cage based or skeleton driven. In the second category, we find aroaches like [], where we find a change in the interolation method from LBS to sherical blend skinning (SBS); SBS uses quaternions to blend the final osition of a deformed vertex; all the matrix in the LBS are changed to its equivalent in quaternions. A more sohisticated aroach introduced by the authors of SBS is DQS []; in this work, dual quaternions are used to solve in an efficient way the well-known roblems of the LBS. Dual quaternions are quaternions whose elements are dual numbers (Oq D Ow C iox C joy C koz). Because of the equivalence of oerations between quaternions and dual quaternions, a version of QLERP for dual quaternions is made (called dual quaternion linear blending; lately, it will be known as DQS). Interolation with SBS and DQS eliminates the candy-wraer artifact and roduces better results than the LBS algorithm but roduces what is known as the bulging joint artifact. The bulging joint is an artifact roduced by the sherical interolation that is caused by nature of quaternions; an interesting work that corrects this articular roblem is [], using rigid skinning to comute the vertex length to its main joint and correcting it when a rotation is erformed. In [], the candy-wraer artifact is corrected by comuting an additional weight ı i er vertex based on the angles at animation and binding time. ı i is used to comute a rotation matrix (restricted to the X canonical axis in the aer) that is multilied rior to M ık in the LBS equation; also, an oeration over the vertices to comensate the collasing joint is introduced. This oeration consists in choosing a collasing joint; then the vertices affected by this articular joint will be recomuted by a stretch oeration that is basically a vector length comensation from the chosen joint to the affected vertex. One of the methods that adds a ost-rocessing to the LBS is [3]; the method adds a volume correction stage after the skinning deformation of a target mesh. The volume correction is treated as minimization roblem of a correction vector u that is comuted using Lagrange multiliers. A similar method can be found in []; this method is also a ost-rocessing to correct the deformed volume obtained with LBS. The change of volume is comuted by a dislacement vector field and a scale factor alied to volume V. By solving the vector field, the correction of volume is erformed over the deformed mesh; this method uses a set of new weights S to control the correction in a localized level; however, this method cannot solve the candy-wraer artifact in LBS. An otimization method based on level of details can be found in [], using LBS skinning a model s matrix transformation are recomuted based in its joints hierarchy. If more detail in the animation is required, the method adjusts it rogressively alying the matrix oerations for the desired level of detail with no noticeable errors in the final animation. An extension of the LBS algorithm is found in []; the method uses two meshes to achieve advanced deformations. One with low level of detail that will control the final deformation of the detailed one in the control mesh, the LBS is alied and combined with bar-net deformers to achieve some degree of realism (hysically based) in some selected vertices of the control mesh; the fine mesh is deformed using the wraing method and can be directly maniulated or thought the bar-net if the user needs and advanced deformation effect such as wrinkles. Two of the latest methods are [] and []. In [], blend bones are used to aroximate nonlinear skinning with a set of weights comuted secifically to work with the extra blend bones. Kavan and Sorkine [] take an interesting aroach by using two deformers deending on the kind of rotation. For twist rotations, they use a quaternion deformer, and linear blend deformation is used for any other kind of rotation. Each deformer had its own set of weights that are comuted and otimized using examles for some reresentatives oses, using biharmonic weights as base. To hel to understand the main features of some of the main skinning algorithms, we have created Table I. Our roosed method relays on the segmentation to create a weight distribution algorithm and a deformation scheme without the candy-wraer artifact. We use the outut of the delta value (ı) as a normalized distribution value to comute the rogressive change in the twist angle for each segmented limb (a link between the joints of the skeleton). Our segmentation algorithm is based in art-tye segmentation, using a reviously rigged mesh that can be created by an artist or by an automatic method such as []. In our articular case, the semantic (the elements that reresents a limb) arts of the art-tye segmentation are already created: the underlying logic skeleton. Therefore, our algorithm is not a full automatic segmentation algorithm, and a benchmark such as the one described in [] is not viable because of our deendency in a re-defined skeleton and our lack of exlicit control arameters. Our method is created with the urose of detecting vertices that belong to each semantic art, in this case: the link between a joint and its child; therefore, we do not have a set of control arameters; our method is imlicit, and our only control arameter is the skeleton that has been created reviously. The number of segments of the segmented mesh will have a direct relationshi with the number of bones or links that our skeleton rig had; the same model will have different segmentations if our method is alied to different skeletons bounded each time to the same mesh.although our algorithm works for non-closed T1 Q5 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. 3 DOI:./cav

5 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez Q7 Q6 F1 Table I. Weight distribution and skinning methods features comarison. Features [] [1] [] [] [] [] [] [3] [] [] Nonlinear olynomial weight function Heat-diffusion weight function Rigid skinning re-roc. Quaternion based Linear blend skinning matrix modification Post-roc. volume correction Nonlinear arox., using additional bones Two deformers combination Methods: realistic skeleton driven skin deformation [], Pinocchio [1], bounded biharmonic weights [], dual quaternion skinning/dual quaternion iterative blending [], sherical blend skinning [], strech-it [], volume-reserving mesh skinning [], exact volume reserving skinning with shae control [3], automatic linearization of nonlinear skinning [], and elasticity-insired deformers for character articulation []. meshes and characters that are defined in multile meshes, we will exlain the method with the assumtion that we have as inut a single closed mesh. We have chosen a region growing method, because it is fast and deends directly on the number of vertices in a 3D mesh. Works like in [3] and [9] are similar to our method but relays in a voxelization of the inut model. A voxelization rocess can be very time-consuming deending in the voxel size, and as is exlained in [9], the main roblem is to know which voxels are internal voxels to roduce a solid model. One of our objectives is to roose a novel and fast method that can be used in a model with an arbitrary ose (not constrained to an ideal T-Pose). In our roosed weight assignment method, inut mesh segmentation is the base, but we use it at a high level, because we store for each vertex their articular segmentation information to create a fast weight assignment, our aroach is to create the segmentation and the weight assignment as two searated methods, because we also use the segmentation to create a new set of weights to solve the candy-wraer artifact effectively. Our algorithm is comosed of three stages: (1) Region growing. In this stage, we assign to each vertex a set of segments that can be the segment where it belongs. (2) Belonging test. For each candidate segment in a vertex, defined in the revious ste, we aly a set of rules to discriminate which is the most suitable segment to be assigned. (3) Region merge. If false ositives exist, we merge them with one of their surrounding neighbor regions. In the following section, we are going to describe in detail each stage. 3. METHODS Before we start exlaining our segmentation algorithm, we will exlain a key feature of our method: The maing of the skeleton to a tree data structure (Figure 1). Skeleton maing allows us to traverse the skeleton hierarchically, the maing is not erformed by joints but by joint airs, that is, a joint jn a and its child jn b define a node in our tree data structure (a segment that we will reference as s j )that has end joints as secial cases. Therefore, a skeleton will have m number of segments for a skeleton with n joints, with m > n (m is greater than n because the end nodes are counted as segments of its own) and will be related directly by their hierarchy deending their osition within the tree data structure Region Growing Our algorithm starts using one of the root node related segments as initial growing region. We test if a vertex v i belongs to a segment defined by the joints jn a and jn b, using their coordinates to comute the orthogonal rojection value ı as defined in []: ı D.v i jn a /.jn b jn a / kjn b jn a k 2 (1) The ı value classifies the relative osition of the oint and the segment: ı < 0 if the oint rojection is before the segment, 0 <ı<1 if the oint rojection is inside the segment, and ı > 1 if the oint rojection is after the segment. We aly our test for each segment in the skeleton hierarchically, using the delta function combined with region grown as tool to check if this vertex is candidate to being inside a segment for each vertex traversed. Region growing needs a seed to begin with,; for the first segment, the seed can be a manually chosen vertex, or can be the closest vertex to the root node that belongs to the initial segment measured in Euler distance. In our case, we use a vertex that had a delta value between 0 and 1 as seed (the vertex is in the influence sace of the segment). If the value of delta is greater than 1, we comute the delta value outut for the child segment (next segment in hierarchy). If its value is not in the child segment influence sace, we mark it as candidate for being art of the current Q8 Q Q Q9 Q Q 4 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

6 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning F2 Figure 1. Equivalence between a logic skeleton used in animation and a n-ary hierarchy tree. segment; therefore, only vertices with delta value greater than 0 can be candidates if all the conditions are accomlished, and as consequence, only a ortion of the vertices of the mesh are checked er segment. The region-growing method marks as candidate a vertex for a secific segment; therefore, when all the segments are comuted, we will have a list of segments for each vertex v i,beingv i a candidate vertex to be art of the influence sace of a segment s j Belonging Test Because of the nature of the delta value, some vertices that are not art of a articular segment are marked as candidates (Figure 2). Therefore, a discrimination of segments in a candidate vertex v i is needed. We aly the following test to select the segment s j in the segment list denoted by Ls for each vertex v i : For each segment s j in the segments list Ls, we comute the angle ij between the weighted normal n i (the mean value of the sum of the normals of the vertex and its 1 connected neighbors) and the vector v i Es j (the vector that had the shortest distance d ij from a segment s j to the vertex v i ). The segments with an angle ij > are discarded. End nodes cannot be discarded by the anterior rule. We assign the vertex v i to the segment s j with the lowest distance d ij from the segment to the vertex. This simle set of rules allows us to aly our algorithm in meshes that are not in the ideal T-Pose;ascanbe seen in Figure 4, it can be alied to rigged meshes with arbitrary oses Region Merging Figure 3(a) shows an examle of vertex assignation with our region-growing algorithm and the belonging test in an arbitrary ose that results in false ositives. This roblem is caused because of the orientation of the ondered normal of some vertices with the vector v i Es j, and it deends basically on the face orientation in some vertices; we solve this roblem using region merging. Our region-merging method creates for each region s j (corresonding to a segment) a list with subsets of vertices connected; we basically create subsets of vertices interconnected in a segment region. The largest subset in the list will be the definitive set for the comuted segment region s j ; the remaining subsets will be merged, each one with its largest neighbor region (the region that had the highest number of vertices connected with the analyzed subset). The comlexity of our segmentation method is O.Sn 2 /, being n the number of vertices in a 3D mesh, and S the number of segments created from a skeleton, the O comlexity analysis of our method is included in Aendix A. 4. SEGMENTATION-BASED SKINNING In this section, we will exlain how our roosed segmentation algorithm can be used for automatically generating vertex weight information for skinning. The main advantage of generating weights based on our segmentation algorithm is that we have identified already the main influence joint for each vertex in a mesh, which also alies for the case of meshes with arbitraryoses F3 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. 5 DOI:./cav

7 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez Q F4 F5 Figure 2. Candidate vertices for the right-hi segment selected by our region-growing algorithm before the belonging test. Figure 3. Comarative between the same mesh before and after alying the merge region rocedure; (a) false ositives in a segmented mesh and (b) segmented mesh after merge region (some ondered normals in yellow). (Figure 4). Then, the weight generation is comuted hierarchically involving only the joints that had a direct relation with the main joint, instead of distributing the weights by a geometrical method, where the influence joints are comuted by its distance to a vertex [1]. As an examle, in the automatic weight algorithm used by AUTODESK MAYA artifacts are created (Figure 5); aarently the weights are calculated using a shere with its center in the current vertex using Euclidean distances in the weight comutation for each influence joint. Using the segmentation algorithm described in the revious section for each vertex v i, we had stored in a data structure the main influence joint jn k. Then, as we will show next, we use a distance metric to comute the weight for the main joints an its siblings. We exress the distribution function for each weight as w ik D F.d ik /; in our secific case, we use the normalized rojection of a vertex v i over the skeleton s links Selection of the Distribution Function and Distance Function. All the algorithms that calculate automatically the weights for the LBS have exlicitly or imlicitly two comonents: (1) Distance function.dstf/. A function that calculates a number. This number can be a direct or indirect relation with a kind of distance from a vertex to the main influence joint( and consequently the main influence link). The distance function is imortant because its outut will be used directly by the distribution function; therefore, a function that calculates the Euclidean distance will give us a different outut that another one uses a geodesic distance. (2) Distribution function.dtbf/. This function takes a set of numbers and mas it to a set of values between 0 and 1. Its outut will be the weights assigned to a vertex for each joint of the character. If the Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

8 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning Figure 4. Segmentation in meshes with different oses, our segmentation algorithm is not restricted to a secific ose to roduce an adequate mesh segmentation. Figure 5. Artifact generated in an animation frame of a mesh (a) because of an imroer weight assignation by the AUTODESK MAYA automatic weight assignation algorithm, corrected by our segmentation-based weight assign algorithm (b). sum of the weight values is different from 1, the roduced deformation will have artifacts deending on the influence of each joint. The distribution function is the most imortant art in the weight comutation for the LBS algorithm; the deformation behavior of a mesh deends on the distribution function. Any function can be used as distribution function, but the quality of the outut deformation can change according to the chosen function. In [1], a heat equilibrium equation is used as DstF Used Distance and Distribution Functions. In our articular case, we use the ı value described in Section 3.1 as DstF (a consequence of the segmentation), because its outut is a normalized measure based in distance of the rojection of the vertex over the link instead of the Euclidean distance from the vertex to a joint that is more deendent of the shae of the inut mesh. As distribution function (DtbF), we use a Gaussian function, defining the center of the function in the center of the main influence link to create a skinning behavior mostly rigid in the center of the links and smooth in the joint areas. Our DstF is defined as f.x/ D ae.x 0.5/2 2c 2 (2) where a is the maximum value of f.x/ and x is the outut value from the distance function. The inflection oint is controlled by the value of the constant c; c is imortant because if we choose the incorrect value (an extremely low or a big value), we might have rotational continuity artifacts. The values comuted by the function f.x/ are the weights for each influence joint in a articular vertex v i, after assigning the weight values, we normalize it; if the weights are not normalized, artifacts are roduced because of the sum greater than 1 in the influence joints weight values Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. 7 DOI:./cav

9 Segmentation-based skinning RR EC TE D In LBS, one of the main roblems of this widely used deformation scheme is the candy-wraer artifact. The candy wraer in lain words is a loss of volume over the mesh caused mainly because of the abrut change of osition between two sets of vertices that is generated when a link rotates more than ı and is at its maximum when the rotation reaches over 0ı in the axis that is aligned with the link direction (Figure 6(c)). The LBS is basically a weighted sum of the set of vectors for each vertex of a olygonal mesh. A more detailed exlanation can be found in [] and [] To eliminate the loss of volume in a rotation over the link vector, we will avoid the abrut change of angle in a twist rotation, which is the main reason of volume loss in LBS. Our aroach is based on keeing the same rotation angle over all influence joints in a vertex when a twist rotation is alied; although the rotation angle will be the same for all the joints in a vertex, this angle will be assigned rogressively deending on its rojection on the link segment (the closest link) using ı. In our modification over the LBS deformation scheme, we use the segmentation algorithm. A segment sj is defined by two joints as main comonents: jna and jnb. The vertex with lower hierarchy will be the main joint; therefore, twist rotations over jna in a segment will be alied normally. When a twist rotation is made over the joint jnb, we comute the rotation angle rogressively for each vertex on that secific segment. As can be seen in Figure 7, when we made a 0ı rotation over its link axis, the LBS alies the deformation in two segments (Figure 7(a)), but our deformation scheme is alied only in one segment F 5. SEGMENTATION-BASED LINEAR BLEND SKINNING 5.1. Our Aroach. PR OO In our test over multile meshes, we had used a hierarchy value of 1; for most of the vertices, this roduces three influence joints for each vertex, which is the number of influence joints that is usually used for the vertices of a rigged character. The values used for the Gaussian function are a D 1.3 and c D 0.; these arameter values generate the best results in our tests. CO Figure 6. Linear blend skinning deformation method alied to a cylinder (a), a roer deformation in the x -axis (b), and the candy-wraer artifact roduced by a twist rotation (c). UN F6 J. E. Ramírez Flores and A. Susin Sánchez Figure 7. Deformation of a bar (a) with linear blend skinning (b), the deformation is alied in two segments with its resective loss of volume roduced by a twist rotation greater than 0ı. In our aroach, we aly the deformation solely in one segment, reventing the loss of volume and self intersection (c) and (d). 8 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav F

10 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning (Figure 7(b)); we believe that this is a more natural behavior if we take the way of how a human limb deforms itself Simlest Case. To exlain our deform scheme, we start with the simlest case: a link with the same direction of the canonical axis. A twist rotation over a link axis can be classified by the hierarchy that has the rotating joint jn k in the segment s j of a articular vertex v i. (1) Our segmentation algorithm is alied to the target mesh obtaining an additional weight that will be the value ı.v i / for each vertex and its assigned segment s j. This value will be stored to be used in ste 4. (2) For a vertex v i in the target mesh, all the joints that influence v i are stored and sorted in a list by its hierarchy. Inside our list of influence joints, every time a child joint is added, its hierarchy will be increased by one of its father hierarchy value. (3) If a rotation with angle i is erformed over the joint jn b of the segment assigned to v i,then i is stored for comutation. As we had mentioned, a segment made by the joints, jn a and jn b (! ba), is arallel to the canonical axis in this case. (4) For a joint jn k, which is also the joint jn a in the segment s j assigned of a vertex v i, we will comute the rotation angle as i 0 D i ı.v i /. The rotation matrix Mjn ık is comuted with i 0 ; in the chain of rotations, Mjn ık will be multilied by Mjn k : Mjn 0 ık D! ky Mjn i Mjn ık (3) (5) The joints with different hierarchy than jn k need to be rotated with the same angle of the segmented link axis; then, the exression for any joint with hierarchy lower than jn k will be i 0 1 k n M 0 Y ky ık n Mjn i Mjn 0 ha Mjn ık (4) 0 k nc1 kq where Mjn 0 h is an iterative roduct of rotation k nc1 matrices that will have only rotations over the link axis of every joint with higher hierarchy between the joints jn k n and jn k. (6) Joints with higher hierarchy than jn k will have the same rotation of the assigned link axis. In a similar way as in the revious oint, any joint with higher hierarchy than jn k (jn kcn ) needs to be multilied by the negative angle of the link axis of each of the revious joints. The exression for a joint with higher hierarchy than jn k will be Figure 8. Candy-wraer artifact in the uer art of a segment roduced by not alying our method to joints with hierarchy greater than j n. 0 1 kcn M 0 Y kcn Y ıkcn Mjn i Mjn 0 ha Mjn ık (5) 0 kc1 where kcn Q Mjn 0 h is an iterative roduct of rotation kc1 matrices that will have negative angle rotation over the link axis of every joint with lower hierarchy between joints jn kc1 and jn kcn. (7) If a rotation over the arent segment of v i assigned segment is erformed, a rotation over the joint jn a of the segment has been made. This rotation needs to be corrected; otherwise, the candy-wraer artifact will be resent again (Figure 8). To revent this situation, the chain of rotations for influence joints with hierarchy lower than jn a must be multilied by the rotation chain described in Equation (4) but with the rotation matrix Mjn ık as identity ( D 0). As has been mentioned reviously, the idea to avoid the loss of volume is based on having the same rotation of the segmented link in all the influence joints when a rotation is erformed but only if that rotation is over the segmented link of a vertex. As can be seen in Equation (4), the rotationexressedby Q k nc1 k Mjn 0 h roduct has to be alied before any rotation from the skeleton sace to the world coordinates has been erformed. If the roduct is alied Qk n after the original set of rotation 0 Mjn i, volume loss artifacts are roduced. Putting all the revious cases and information in one exression, we obtain 0 i D X w m M 0 ım M Lm i (6) where M 0 ım will be Mjn 0 ık,m 0 ık n or M 0 ıkcn deending on the hierarchy of the joint F8 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. 9 DOI:./cav

11 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez T General Case. The general case of our method takes into account that a limb of a virtual character is not always aligned with the canonical axis; therefore, when a rotation over a limb is made, the rotation had to be over the axis made by the link of the joints jn a and jn b. In Equation (6), we made the assumtion that this link is aligned with one of the axis; in the general case, we will take the axis made by the link and we will rotate around it the segmented vertices of the mesh. The rotation over an arbitrary link can be carried out with quaternions or with its equivalent in matrix rotation comuted by Rodrigues rotation formula using the following exression: v R D v cos C.u v/ sin C u.u v/.1 cos / (7) if we exress the cross roduct.uv/ as the rotation matrix M u v, the Rodrigues formula can be exressed as R u D I C M 2 u.1 cos /C M u sin therefore, v R D R u v. When a rotation is alied in a link l k with an arbitrary osition, we use the next rocedure to comute the rotation over a vertex. (1) We identify the nearest axis ax k in orientation with l k. (2) The angle is extracted if a rotation over the ax k exists. (3) If the angle between ax k and l k is greater than ı, we take the negative value of as the rotation angle using the negative of M u. (4) We aly Equations (3), (4), or (5) deending on the case, but M 0 ım is relaced with R u with 0 i D i ı.v i / as rotation angle and kl k k as u, wherel k will be the link of the segment where the influence joint jn k is assigned. Our aroach is similar to [] but with substantial differences: we use the ı value to obtain an additional weight that had the urose of being the rotation amount of the twist rotation; our rotation scheme alies twist rotations over segments instead of the classic way that is over the influenced joint vertices. We aly our deformation correction in more than one influence joint down and u in hierarchy, and finally, we are not restricted to canonical axis only. 6. RESULTS As can be seen in Table II, the times of our segmentation algorithm deends on the number of vertices of the inut model mesh. The weight assignment algorithm uses the segmented vertices; therefore, the segmentation method used on the inut mesh does not have an imact in the weight assignment rocessing times, but their outut Table II. Segmentation rocessing times. Model Num. Vert. Seg. (sec.) Weight Assg. (sec.) Low res. 0.. Mid res High res Table III. Comarison between deformation methods (rocessing times). Rotation # DualQuat (ms) LBS (ms) SLBS (ms) LBS, linear blend skinning; SLBS, segmentation-based linear blend skinning. values deend on the segmentation outut. All the rogramming and test of our algorithms where erformed in an Intel Core i3 at 2.1 GHz, 6 GB in RAM, Windows 7 with Visual Studio with a bits C++ comiler and tested over AUTODESK MAYA. In the skinning world, it is customary to comare the erformance of a new roosed skinning algorithm with the most oular algorithm because of its simlicity and its linear nature: LBS, robably LBS has the best erformance of all the skinning algorithms used u to date. We also comare the erformance of our algorithm with another oular skinning solution: DQS. DQS has a lower erformance than LBS, but it solves one of its main roblems: the well-known candy-wraer artifact. Therefore, these are the two main algorithms to comare a new roosed method in the field. As we have carried out in Section 6.0.3, a sequence of six deformations is alied to a test model (a bar) and are reorted in Table III. The imlementation of our segmentation-based LBS for general urose can be almost six times slower than LBS; in our test, we use an otimized version of segmentation-based LBS for three segments (the main one, their father, and child) alying it to a total of four joints. All the remaining influence joints with or without twist rotation will be solved with LBS; four joints are the usual number of influence joints for almost all vertices in a rigged mesh; even if that is not the case, the influence weight is commonly retty low for joints related in second degree to the main segment s joints that can be solved with LBS without affecting the outut (as can be seen in Figure 9). With the otimization, our algorithm is very fast having a difference of 0. milliseconds with LBS, and it is faster than DQS; in terms of quality, our algorithm shows the roer results without the candy-wraer artifact of the LBS or the artifacts caused by the weight distribution showed in the DQS algorithm that are solved roerly with dual quaternion iterative blending (DIB) T3 F9 Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

12 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning Figure 9. Segmentation-based linear blend skinning outut alied over an animated character in different frames over time (character hitting a soccer ball) Volume Preservation. To know how much volume is lost when a rotation is made with our method, we have made a set of rotations over a mesh. The volume is comuted using a tetrahedron reresentation with negative volumes (a negative volume will be comuted for each face in the inut mesh with negative direction in its surface normal); the result is in the next table. In all cases, the set of weights for the deformation methods are the same. We aly six rotations over Table IV. Comarison between outut volumes from deformation methods (error ercentage). Rotation # DualQuat (%) LBS (%) SLBS (%) LBS, linear blend skinning; SLBS, segmentation-based linear blend skinning. the joints jn 1 to jn 3 of the five joints in the bar mesh with an initial volume of units, leaving left of the test the end joints (jn 0 and jn 4 ). The set of rotations are lanned to show the behavior of every deformation scheme; the results that are shown in Table IV are error ercentages, where e i D V 0 V i. The rotations in sequence are V 0 (1) 0 ı in the y axis, joint jn 1. (2) 0 ı in the y axis, joint jn 2. (3) 0 ı in the y axis, joint jn 3. (4) ı in the x axis, joint jn 1. (5) ı in the z axis, joint jn 2. (6) ı in the z axis, joint jn 3. In Table IV, as exected, the method that had lost volume the most is LBS, followed by our method with DQS with the best erformance of the three methods. Figures and show the surface areas were DQS and LBS oerates, one with the test erformed and a new test of a 0 twist rotation over the arm of two characters; the surface areas are lower in each one than the area were our method oerates because the main weight distribution is the same for all the methods. In our method, we have two set of weights: Figure. Outut volumes for different deformation methods. From (b) to (d), rotation 3 and from (e) to (g), rotation 6. LBS, linear blend skinning; DQS, dual quaternion skinning; SLBS, segmentation-based linear blend skinning T4 F F Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

13 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez T5 Figure. Deformation over an arm for two meshes erformed by different deformation methods, with a 0 ı rotation. SLBS, segmentation-based linear blend skinning; LBS, linear blend skinning; DQS, dual quaternion skinning. Table V. Comarison between outut errors from two models with different volume magnitude. Rotation #.8 units model (%) units model (%) the main (taken directly from LBS) will have effect over all the rotations that are not aligned with the segmented link axis (twist rotations); the second set is obtained from the ı value. If a behavior different than the lineal one obtained through the ı value is desired, an outut function must be alied over the results ı. Our method can manage degenerated cases such as rotations equal and greater than 0 ı because of its rogressive nature. As an examle of this feature, in LBS, if a rotation angle is greater than 0 ı, will be equivalent to the difference between and 0 ı ; in general, the rotation angle in LBS will behave by the relation 0 D 0 j 0 j j 0 ; j in DQS, the rotation about 0 ı roduces serious artifacts as is showed in []; only DIB roduces a correct outut. With our method, this degenerated case is roerly solved, because is changing smoothly between vertices by the ı value, instead of changing deending on the weights values. To test how stable is our deform method, we have modified the bar model; we have made two modifications: one varying tuning down the total volume of our bar and other increasing the volume. The same set of rotations had been alied to these modified models; the results are shown in Table V. As seen in Table V, the variation between the two models are indicative of a stable method. When the results of the set of rotations of the original model (Table IV) and the result of the outut errors on the modified volume models are comared, the outut errors are similar Discussion The segmentation in the automatic rigging algorithms [2] and [] shares one main feature in their segmentation: the segmentation is art of the skeleton extraction rocess; therefore, their segmentation will fit erfectly with the segments of their outut skeleton. Our case is different; we are not working directly with the vertices of the target mesh to roduce a skeleton; we instead take a rigged mesh and roduce a segmentation deending of the bound skeleton to the target mesh. Our automatic weight assign algorithm was develoed with the same objective as the one showed in [1]: create a set of weights having only as inut a 3D rigged mesh. Other algorithms such as [,,,,,] had a set of examles to comute (or extend in some cases) the weights of each vertex in a character 3D mesh. Works like [3,8] and [] reserve the volume of a mesh after its deformation; however, they comensate the loss of volume as a ost-rocess; their results are notable, but it adds comutation time to the animation ieline, and they can only solve roerly twist rotations with an angle 0 ı < < 0 ı, because if 0 ı, a self intersection artifact is roduced, and it cannot be corrected by any volume reservation ost-rocess; our roosed method solves correctly twist rotation angles greater than 0 ı with low loss of volume. A main oint of comarison in the case of the skinning algorithm is the work develoed by Kavan in []; DQS uses the same weight influence base of LBS and also corrects the candy wraer, but it also introduces a bulging artifact in rotation over limbs such as elbows or knees by the nonlinear behavior (similar to a shere) of DQS; our method does not have that kind of roblem because it uses fundamentally LBS with the excetion of twist rotations. Another roblem addressed in [] are the artifacts that DQS Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

14 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning Table VI. Skinning algorithms main advantages comarison. Features DQS [] Segmentation-based skinning LBS DIB [] ALNS [] Uses LBS weights Solves candy-wraer artifact Solves or not roduces bulging artifact Produces correct results with rotations over 0 ı Short rocessing times DQS, dual quaternion skinning; LBS, linear blend skinning; DIB, dual quaternion iterative blending; ALNS, automatic linearization of nonlinear skinning. Figure. Segmentation algorithm alied over meshes with different shaes and number of joints in their skeletons. creates when the rotation over the link axis is close to 0 ı, artifacts that our solution does not have with the trade off of having longer comuting times than LBS and DQS in its unotimized version but similar in quality to DIB, which is more than five times slower than DQS []; in its otimized imlementation, our method is aroximately as fast as LBS. The method used in [] is similar to our aroach in the sense that they use as base LBS; for rotations in local coordinates XY lane (swing) and for rotations over Figure. Segmentation alied over a multi-mesh character. local z axis (twist), they change to a nonlinear interolation method (an aroximation to DQS). They also aly the twist rotation in the middle of the link segment and not over the target joint in a similar way we aly it in our extension of LBS. When a rotation is made over a segment link axis, their erformance is not clear because of its lack of rocessing times on the skinning stage, but their main overhead is resent when the two deformers are evaluated; because of its close relation to DQS, we believe that Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

15 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez T6 F F Q deformations closer or higher than 0 ı lead to artifacts resent also [] that are roerly solved by DIB or our method; we had created Table VI that shows the features that our method share with other geometry-based skinning methods. 7. CONCLUSIONS AND FUTURE WORK We have develoed a novel segmentation algorithm that works over the vertices of an inut rigged mesh only; their main features are that it is not restricted to a secific ose (can deal with rigged meshes in arbitrary oses), arbitrary number of joints in the rig, or to a secific kind or shae (Figure shows the segmentation algorithm alied to some non-anthroomorhic meshes), and it is low in comutation times. Because it works solely with the vertices of a rigged mesh, its adatation to rigs with multile meshes is easy (Figure ). Segmentation is imortant because it is a concet that simlifies and makes easier the weight assign rocess, while other algorithms deend on minimization algorithms or uses the weights assigned by other methods as a starting oint. A good segmentation can be useful even for digital 3D artist as base to aint influence weights on a desired model. The widely used LBS algorithm had the well-known candy-wraer artifact; to overcome this roblem, we had develoed a skinning algorithm based in LBS. Our algorithm can handle advance deformations (twists over a link greater than 0 ı ), without volume loss or unrealistic artifacts, is not deendent on examles, uses weights generated for an LBS deformation scheme, and generates automatically the extra weights needed. Our method was develoed entirely in AUTODESK MAYA as a lug-in in ANSI C++; this had the objective of making easier the diffusion around the animation community and being indeendent of the hardware used. Although the roject was made using the Visual C++ comiler of Microsoft, with some changes, a ort to Mac, Linux, or another oerating system that suorts Maya and a C++ comiler will be ossible. Our segmentation algorithm can be imroved in the region-growing stage; we are using Euclidean distances and rojections to discriminate the candidates, but in inut meshes with arbitrary oses, the task can be difficult, leading to false ositives that must be refined by hand. To solve this roblem, we want to exlore an algorithm that uses geodesic distances (such as []) to unfold the arbitrary ose to something closest to T-Pose; therefore, we can use geodesic distances directly in the region-growing stage and use a simler rule set to discriminate candidates. To imrove our skinning weight assign algorithm, we want to exlore an algorithm based on examles, such as [], that allow us to aly the information obtained by the examles in models with similar shae. We are using a Gaussian function as weight distribution function, but we want to test which are the results with different kind of functions such as Bezier curves or a function of high order to roduce smoother transitions between two connected segments to avoid weight-based artifacts. The algorithms deicted in this aer are sequential because of the nature of our imlementation as a Maya lug-in; an interesting alternative will be a arallelized version in CUDA to imrove its erformance. The distribution function of the extra weight in our skinning algorithm is linear; a different distribution function may lead to different twist behavior. A toic to exlore in our skinning method as future work will be volume reservation; to achieve this goal, we can lug an algorithm to the outut of our skinning method to achieve a volume loss of 0% or closer; although the volume loss of our deformation algorithm is low, we want to test their imrovement with a volume-correction algorithm such as []. 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16 J. E. Ramírez Flores and A. Susin Sánchez Segmentation-based skinning Q. Lewis JP, Cordner M, Fong N. Pose sace deformation: a unified aroach to shae interolation and skeleton-driven deformation. In Proceedings of the th Annual Conference on Comuter Grahics and Interactive Techniques, SIGGRAPH 00, New York, NY, USA, 00; 5 2, ACM Press/Addison-Wesley Publishing Co.. Jacka D, Reid A, Merry B. A comarison of linear skinning techniques for character animation. In Afrigrah, ; 7 6, ACM.. Wang XC, Phillis C. Multi-weight enveloing: least-squares aroximation techniques for skin animation. In Proceedings of the ACM SIG- GRAPH/EUROGRAPHICS Symosium on Comuter Animation, SCA, New York, NY, USA, ; 9 8, ACM.. Sloan P-PJ, Rose CF, III, Cohen MF. Shae by examle. In Proceedings of the Symosium on Interactive 3D Grahics, I3D, New York, NY, USA, ; 5 3, ACM.. Mohr A, Gleicher M. Building efficient, accurate character skins from examles. In ACM SIGGRAPH Paers, SIGGRAPH, New York, NY, USA, ; 2 8, ACM.. Merry B, Marais P, Gain J. Animation sace: a truly linear framework for character animation. ACM Transactions on Grahics ; (4): 00.. Mohr A, Gleicher M. Deformation sensitive decimation. Technical Reort,.. Yang XS, Zhang JJ. Realistic skeleton driven skin deformation. In Proceedings of the International Conference on Comutational Science and Its Alications Volume Part III, ICCSA, Berlin, Heidelberg, ;, Sringer-Verlag.. Jacobson A, Baran I, Poovic J, Sorkine O. Bounded biharmonic weights for real-time deformation. ACM Transactions on Grahics ; (4):.. Kavan L, Collins S, Žára J, O Sullivan C. Geometric skinning with aroximate dual quaternion blending. ACM Transactions on Grahics ; (4): 1:1 1:.. Kim YB, Han JH. Bulging-free dual quaternion skinning. Comuter Animation and Virtual Worlds ; (3-4): Yang X, Zhang JJ. Stretch it realistic smooth skinning. In Proceedings of the International Conference on Comuter Grahics, Imaging and Visualisation, CGIV, Washington, DC, USA, ; 3 3, IEEE Comuter Society.. von Funck W, Theisel H, Seidel HP. Volume-reserving mesh skinning. Proceedings of Vision, Modeling, and Visualization : 4.. Pilgrim S, Steed A, Aguado A. Progressive skinning for character animation. Comuter Animation and Virtual Worlds ; (4-5): Zhang JJ, Yang X, Zhao Y. Bar-net driven skinning for character animation. Comuter Animation and Virtual Worlds ; (4-5): Kavan L, Collins S, O Sullivan C. Automatic linearization of nonlinear skinning. In Proceedings of the Symosium on Interactive 3D Grahics and Games, ;, ACM.. Kavan L, Sorkine O. Elasticity-insired deformers for character articulation. ACM Transactions on Grahics (TOG) ; (6): 6.. Ramirez JE, Lligadas X, Susin A. Adjusting animation rigs to human-like 3D models. In AMDO : Proceedings of the 6th International Conference on Articulated Motion and Deformable Objects, Berlin, Heidelberg, ; 0 3, Sringer-Verlag.. Chen X, Golovinskiy A, Funkhouser T. A benchmark for 3D mesh segmentation. ACM Transactions on Grahics (Proc. SIGGRAPH) ; (3).. Pan JJ, Yang X, Xie X, Willis P, Zhang JJ. Automatic rigging for animation characters with 3D silhouette. Comuter Animation and Virtual Worlds ; (2/3): Weber O, Sorkine O, Liman Y, Gotsman C. Context-aware skeletal shae deformation. Comuter Grahics Forum ; : 5 4. Wiley Online Library.. Jacobson A, Baran I, Kavan L, Poović J, Sorkine O. Fast automatic skinning transformations. ACM Transactions on Grahics (TOG) ; (4):.. Katz S, Leifman G, Tal A. Mesh segmentation using feature oint and core extraction. The Visual Comuter ; (8-): 6 6. APPENDIX A: COMPLEXITY ANALYSIS OF THE SEGMENTATION ALGORITHM The calculation of the comlexity O of the segmentation algorithm based in our code imlementation is P SiD1 a 1i n v a 2i n v C Region grow. P njd1.a 3j v j s C a 4j v j s/ C n v SC P mkd1 Vertex belong test..v k a 5k n v / C a 6 n v SC n v C P s ld1.n v a 7l a 8l n v C a 9l n v / Region merge. where 0 5 a 1 :::a are constants, n v is the total number of vertex in a 3D rigged mesh, and S is the total number of segments roduced by the skeleton bounded to the mesh. The simlification of this formula taking some of the constants a1 :::a9 as1is Q Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

17 Segmentation-based skinning J. E. Ramírez Flores and A. Susin Sánchez n 2 v SC 3n v S C mn 2 v C a 6n v SC n v C n 2 v S C n vs D 2n 2 v S C n2 v C Sn v.4 C a 6 / C n v D 2.S C 1/n 2 v C n v.s.4 C a 6 / C 1/ D n 2 v S C n vs b C 1 s D n 2 v S Region grow. Vertex belong test. Region merge. therefore, the comlexity of our segmentation algorithm is O Sn 2. AUTHORS BIOGRAPHIES Jorge Eduardo Ramírez Flores Antonio Susin Sánchez Q Com. Anim. Virtual Worlds () John Wiley & Sons, Ltd. DOI:./cav

18 Research Article Segmentation-based skinning Jorge Eduardo Ramírez Flores and Antonio Susin Sánchez In this aer, we roose an aroach based on mesh segmentation for skinning and skeleton-driven animation. Our method is based in watershed segmentation to deal with characters in T-Pose and arbitrary oses; the segmentation is the core algorithm of our method; we use it to develo a simle weight that assigns the LBS deformation method and a modified version of the LBS that avoids the loss of volume (candy-wraer artifact) in twist rotations. Wiley Online Library Grahical TOC

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