Harmonic Coordinates for Character Articulation Pushkar Joshi Mark Meyer Tony DeRose Brian Green Tom Sanocki Pixar Technical Memo #06-02b Pixar Animation Studios (a) (b) (c) Figure 1: A character posed using using harmonic coordinates. (a) The character and cage (shown in black) at bind-time; (b) and (c) are two poses from an animated clip. All images c Disney/Pixar. Abstract Character articulation, sometimes called rigging, is an important component of high-end animation systems of the kind used in fea- In this paper we consider the problem of creating and controlling ture film production. Modern high-end systems, most notably Sof- volume deformations used to articulate characters for use in high- tImage XSI R and Maya R , offer a variety of articulation methods end applications such as computer generated feature films. We in- such as enveloping [Lewis et al. 2000], blend shapes [Joshi et al. troduce a method we call harmonic coordinates that significantly 2006], and chains of arbitrary deformations. In the realm of de- improves upon existing volume deformation techniques. Our de- formations, free-form deformations as introduced by Sederberg and formations are controlled using a topologically flexible structure, Parry [1986] are particularly popular for a number of reasons. First, called a cage, that consists of a closed three dimensional mesh. The they offer smooth and intuitive control over the motion of the char- cage can optionally be augmented with additional interior vertices, acter using only a few parameters, namely, the locations of the free- edges, and faces to more precisely control the interior behavior of form lattice control points. Second, there are virtually no restric- the deformation. We show that harmonic coordinates are general- tions on the three-dimensional model of the character — the only ized barycentric coordinates that can be extended to any dimension. requirement is that the character model is completely enclosed by Moreover, they are the first system of generalized barycentric coor- the control lattice. dinates that are non-negative even in strongly concave situations, However, free-form deformation has some drawbacks. Articulat- and their magnitude falls off with distance as measured within the ing a multi-limbed character is best accomplished using a lattice cage. that conforms to the geometry of the character. However, given the topological rigidity of a lattice, it is often necessary to combine CR Categories: I.3.5 [Computational Geometry and Object Mod- several overlapping lattices, and each of the lattices possess interior eling]: Geometric algorithms, languages, and systems. points that can be difficult and annoying to articulate. The prob- lem of multiple overlapping lattices was addressed by MacCracken and Joy [1996] where lattices were generalized to arbitrary volume Keywords: Barycentric coordinates, mean value coordinates, free meshes, but their method still requires the introduction and articu- form deformations, rigging. lation of numerous interior control points. Ju et al [2005] introduced a promising new approach that is even 1 Introduction more topologically flexible, wherein the character to be deformed (henceforth called the object) is positioned relative to a coarse closed triangular surface mesh (henceforth called the cage). The object is then “bound” to the cage by computing a weight gi(p) of each cage vertex Ci evaluated at the position of every object point p. 0 As the cage vertices are moved to new locations Ci, the deformed points p0 are computed from 0 0 p = ∑gi(p)Ci: (1) i An example is shown in Figure 2(b). The weight functions gi(p) used by Ju et al. are known as mean value coordinates [Floater 2003; Floater et al. 2005; Ju et al. 2005]. Mean value coordinates Bind Mean Value Harmonic (a) (b) (c) (d) (e) (f) Figure 2: A comparison between deformations based on mean value and harmonic coordinates. The first row shows a torso bend and the second row shows a leg bend for a typical character. The left column shows the cage (shown in black) and object (shown in beige). The middle column shows modified cages and the corresponding deformed objects using mean value coordinates. The right column shows modified cages and deformed objects using harmonic coordinates. Notice that the two methods perform similarly for the torso bend. However, in strongly concave situations such as the legs, harmonic coordinates produced significantly more pleasing results. are a form of generalized barycentric coordinates that have a num- characters in feature film production. ber of uses, but they are particularly interesting in the context of character articulation because: The undesirable behavior illustrated above occurs because mean value coordinates lack two properties that are essential for high-end • The cage that controls the deformation can be any closed tri- character articulation; namely: angular surface mesh, so there is a great deal of topological and geometric flexibility when designing the cage. • Interior locality: Informally, the coordinates should fall off as a function of the distance between cage vertices and object • The coordinates are smooth, so the deformation is smooth. points, where distance is measured within the cage. • The coordinates reproduce linear functions, so the object • Non-negativity: As illustrated in Figures 2(e) and 4(b), if an doesn’t “pop” when it is bound. That is, the coordinates are 0 0 object point whose coordinate relative to a cage vertex is neg- such that setting Ci to Ci in Equation 1 results in p reducing ative, the object point and cage vertex will move in opposite to p. directions. To prevent this unintuitive behavior, we seek coor- However, mean value coordinates have a drawback for use in char- dinates that are guaranteed to be non-negative on the interior acter articulation, as illustrated by the bipedal character shown in of the cage, even in strongly concave situations. Figure 2. Notice how the modified cage vertices for the leg on In this paper, we show that coordinates possessing the these two the left in Figure 2(e) significantly influence the position of object critical properties can be produced as solutions to Laplace’s equa- points in the leg on the right. This occurs because mean value co- tion. Since solutions to Laplace’s equation are generically referred ordinates are based on Euclidean (straight-line) distances between to as harmonic functions, we therefore call these coordinates har- cage vertices and object points. Since the distance between the monic coordinates, and the deformations they generate harmonic modified cage vertices and the object points in the leg on the right deformations.1 are relatively small in the bind pose, the influence is relatively large. Notice too that the displacement of those object points is in a direc- Unlike mean value coordinates, harmonic coordinates do not, in tion opposite to the displacement of the cage vertices. This occurs because the mean value coordinates are negative, as shown in Fig- 1Since each component of a harmonic deformation is a harmonic func- ure 4(b). This undesirable movement is particularly striking in in- tion, many texts refer to such deformations as harmonic maps. We prefer teractive use, as demonstrated in accompanying video (see [Joshi the term harmonic deformation because of the context in which they’re used et al. 2007]). Such behavior is unacceptable for the articulation of in this paper. general, possess a closed form expression. Instead, they must be of a character. Use of subdivision is crucial for their purposes approximated using a numerical solver. There are two potential since refinement is used to create the hierarchical basis they use drawbacks of a numerical solution: time and accuracy. For charac- to accelerate the simulation of elastic deformation. However, we ter articulation, as the animator moves the cage vertices, the object wish to avoid subdivision-based methods because, like the origi- must deform in real-time. For such fully interactive deformations nal free-form deformation, they require the interior of the cage to the coordinates should be computed as a pre-process, prior to any be meshed, and they contain interior vertices that must be articu- user interaction. This precomputation is necessary even if a closed lated. This makes authoring of cage articulation more difficult than form exists, as noted by Ju et al. [2005]. Given that coordinates cage-based schemes like ours that do not require interior points. A must be precomputed whether or not a closed form exists, the time direct manipulation volume deformation method was introduced by required to run the harmonic coordinate solver is acceptable (see Igarashi et al. [2005]. The deformations they produce are intuitive, Sections 6 and 7). The key issue regarding accuracy is whether it but their method does not employ a cage, so the reuse advantages is possible to reproduce linear functions using reasonable time and mentioned above are not realized. space. For harmonic coordinates this is in fact the case, again as shown in Section 6. In the area of barycentric coordinates, Ju et al. [2005] used gener- alized barycentric coordinates, specifically, mean value coordinates To put our analysis of previous work and contributions in context, (see also Floater et al.[2005]) to define deformations controlled by we reiterate the problem we are trying to solve and why. Our goal is a surface mesh rather than a three dimensional lattice. The prob- to develop a topologically flexible “cage-based” method of control- lem of generalizing barycentric coordinates is rich and has received ling volume deformations. We specifically do not seek a direct ma- considerable attention in recent years ([Wachpress 1975], [R.Sibson nipulation method, such as methods based on local differential co- 1981], [Loop and DeRose 1989], [Warren 1996], [Meyer et al. ordinates (c.f. Sorkine [2006] and Sumner et al. [2005]), though we 2002], [Floater 2003]).