The ability to position a small subset of mesh vertices and produce a meaningful overall deformation of the entire mesh is a fundamental task in mesh editing and animation. However, the class of meaningful deformations varies from mesh to mesh and depends on mesh kinematics, which prescribes valid mesh configurations, and a selection mechanism for choosing among them. Drawing an analogy to the traditional use of skeleton-based inverse kinematics for posing skeletons, we define mesh-based inverse kinematics as the problem of finding meaningful mesh deformations that meet specified vertex constraints.
Our solution relies on example meshes to indicate the class of meaningful deformations. Each example is represented with a feature vector of deformation gradients that capture the affine transformations which individual triangles undergo relative to a reference pose. To pose a mesh, our algorithm efficiently searches among all meshes with specified vertex positions to find the one that is closest to some pose in a nonlinear span of the example feature vectors. Since the search is not restricted to the span of example shapes, this produces compelling deformations even when the constraints require poses that are different from those observed in the examples. Furthermore, because the span is formed by a nonlinear blend of the example feature vectors, the blending component of our system may also be used independently to pose meshes by specifying blending weights or to compute multi-way morph sequences.
Top row: Ten lion example poses. Bottom row: A sequence of posing operations. (A) Two handle vertices are chosen. (B) The front leg is pulled forward and the lion continuously deforms as the constraint is moved. (C) The red region is selected and frozen so that the front leg can be edited in isolation. (D) A similar operation is performed to adjust the tail. The final pose is different from any individual example.
Articulated shapes are aptly described by reduced deformable models that express required shape deformations using a compact set of control parameters. Although sufficient to describe any shape deformation, the control parameters can be ill-suited for animation tasks, particularly when reduced deformable models are inferred automatically from example shapes. Our algorithm provides intuitive and direct control of reduced deformable models similar to a conventional inverse-kinematics algorithm for jointed rigid structures. With only a few manipulations, an animator can automatically and interactively pose detailed shapes at rates independent of their geometric complexity. Our resolution-independent metric ensures that even a few vertex constraints generate example-like meshes.
 Robert W. Sumner, and Jovan Popović. Deformation Transfer for Triangle Meshes. In ACM Transactions on Graphics 23(3), pp. 399--405, 2004.
 Robert W. Sumner, Matthias Zwicker, Craig Gotsman, and Jovan Popović. Mesh-Based Inverse Kinematics. In ACM Transactions on Graphics 24(3), pp. 488--495, 2005.
 Kevin G. Der, Robert W. Sumner, and Jovan Popović. Inverse Kinematics for Reduced Deformable Models. In ACM Transactions on Graphics 25(3), pp. 1174--1179, 2006.
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