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In contrast to the two previous methods, the iterative volumetric Laplacian method works on a tetrahedral mesh Ttet and mt tetrahedra Ttet.. A tetrahedralmesh can, for instance, be created from a triangle mesh Mtri by performing a quadric error decimation on M and then building a face-constrained Delaunay tetrahedralization. The input to this approach isTtet and positional constraints pcj for nc selected vertices. This method infers rotational constraints from the given positional constraints and also improves the overall deformation performance by implicitly encoding stronger prior on the shape properties that should be preserved after the deformation, such as local cross-sectional areas. It is our goal to deform the tetrahedral mesh Ttet as naturally as possible under the influence of a set of positional constraints vt j. To this end, we iterate a linear Laplacian deformation step and a subsequent update step, which compensates the (mainly rotational) errors introduced by the nature of the linear deformation.
This algorithm is related to. However, here a tetrahedral construction is used rather than a triangle mesh, as this enables the implicit preservation of certain shape properties, such as cross-sectional areas, after deformation. The approach starts by constructing the tetrahedral Laplacian system where G is the discrete gradient operator matrix for the volumetric model, D is a 4mt 4mt diagonal matrix containing the tetrahedra's volumes, g is the set of tetrahedron gradients, each being calculated as gj = Gj pj, and pj is a matrix containing the vertex coordinates of tetrahedron tt j . The constraints pcj can be factorized into the matrix Ls by eliminating the corresponding rows and columns in the matrix and incorporating the values into the right-hand side. By solving the previous tetrahedral Laplacian system, we obtain a set of new vertex positions V. After calculating a transformation matrix Ti which brings tti into configuration tti, the matrix Ti is split into a rigid part Ri and a non-rigid part Si using an iterative polar decomposition method. Thereafter, only the rigid transformations are applied to the gradients of all respective tetrahedra and we rebuild the right-hand side of the linear system using these rotated gradients g. It is possible to pre-calculate a factorization of the left-hand side matrix once (since it never changes) and only perform an efficient back-substitution in each iteration. During the iteration process we search for a new configuration of the input shape that minimizes the amount of non-rigid deformation Si remaining in each tetrahedron.
We refer to this deformation energy as ED. In comparison with simulation methods this technique has the advantages of being extremely fast, of being very easy to implement, and of producing plausible results even if material properties are unknown. Propagating the deformation from Ttet to M After deforming Ttet , we can transfer the pose from T to the input triangle mesh. Initially, we represent the vertices of M as linear combinations of tetrahedra in the local neighborhood. To this end, for each vertex vi in M, we find the subset T of all tetrahedra from T that lie within a local spherical neighborhood of radius r and contain a boundary face with a face normal similar to that of vi.
Subsequently, we calculate the barycentric coordinate coefficients ci( j) of the vertex with respect to all tt and compute the combined coefficient vector as a compactly supported radial basis function with respect to the distance of vi to the barycenter of tetrahedron. The coefficients for all vertices of M are combined into a matrix B. Thanks to the smooth partition of unity definition and the local support of our parameterization, we can quickly compute a smooth and natural looking deformed pose M by calculating the new vertex positions.