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Inputs to this method are a static triangle mesh M and affine transformations (rotation and scale/shear components), to be applied to nc selected triangles of the input model. The Poisson-based editing scheme manipulates the mesh gradient field instead of directly deforming the spatial coordinates of a triangle mesh. By expressing the mesh in terms of the gradient operators Gj , for each triangle t j , Poisson-based methods are able to derive a novel surface mesh M that matches the deformed gradient field subject to the user’s constraints. Gradient operators Gj contain the gradients of the triangle’s shape functions. Here pj are the three vertices of the triangle t j and n is its unit normal. The matrices Gj can be combined into a large 3mn gradient operatormatrix G, and the gradients of the entire input triangle mesh then can be represented. The same holds true for the other two coordinate functions (gy and gz). By multiplying with GTM an both sides where the 3m3m weight matrix M contains the areas of the triangles. The matrix GTMG is the cotangent discretization of the Laplace-Beltrami operator Ls and GTMgx represents the differential coordinates of M. This construction allows us to manipulate M by applying the user constraints as separate transformations.
At the end, we can reconstruct M in its new target configuration by computing the new vertex positions p such that the resulting mesh complies with the new, rotated gradients. This can be computed by solving the Poisson system Lsp, which is formulated as a least-squares system for each x, y and z-coordinate separately. Unfortunately, this formulation is only able to correctly reconstruct M if constraints are given for all triangles, i.e. such that we can transform all gradients. Alternatively, if only a sparse set of constraints is giving, the idea proposed can be used to propagate the rotations over the wholemodel based on harmonic field interpolation. After converting the input transformations Rj to unit quaternions, we regard each component of the quaternion q as a scalar field defined over the entire mesh. A smooth interpolation is generated by regarding these scalar fields as harmonic fields defined over Mtri, and can be computed efficiently by solving the Laplace equation (Lsq = 0) with constraints at the selected vertices. Once the rotational components (qx, qy, qz and qw) are computed for all vertices, we average the quaternion rotations of the vertices to obtain a quaternion rotation for each triangle. This way, we establish a geometric transformation Rj for each triangle t j ofM. After estimating the rotations for all triangles, we perform the procedure described above to transform all gradients and obtain a realistic reconstruction of the model in a new pose. During the interactive editing process, the differential operator matrix Ls does not change. Furthermore, since it is symmetric positive definite, we can perform a sparse Cholesky decomposition as a preprocessing step and use back-substitution for each new set of input constraints R.
The input to this approach is a static triangle mesh Mtri and positional constraints vj for selected nc vertices of Mtri. The Laplacian-based editing scheme represents the surface by the differential coordinates the goal, which is to reconstruct the vertex positions of M such that the mesh approximates the initial differential coordinates and the positional constraints given by the user. Differential coordinates for M are computed by solving a linear system of the form LsVtri, where Ls is the discrete Laplace operator based on the cotangentweights. Thereafter, themodel M can be reconstructed in a new pose subject to the positional constraints pc by solving the following least-squares system. Pc is the vector of positional constraints specified by the user and the matrix A is a diagonal matrix containing non-zero weights Ai j = wj for constrained vertices vj . The weights wj indicate the influence of the corresponding positional constraint pcj on the final deformation result. Unfortunately, if the mesh undergoes large rotations, this scheme will reconstruct the mesh with an unnatural look, since most of the triangles will be oriented according to the original differential coordinates of M. However, the quality of the deformation result can be improved by carefully handling the local transformations of the differential coordinates. As before after converting the input rotations R to quaternions, we interpolate the transformations q over M.
Each component of the quaternion q is regarded as a scalar field defined on the entire mesh. A smooth interpolation is guaranteed by regarding these scalar fields as harmonic fields. The interpolation is performed efficiently by solving the Laplace equation Lsq = 0 over the entire mesh with constraints at the selected vertices. Thereafter, we use the interpolated local transformations to rotate the differential coordinates. At the end, the vertex positions V tri of M are reconstructed such that the mesh approximates the rotated differential coordinates as well as the positional constraints pc. During the mesh editing process, the Laplacian matrix Ls does not change. Therefore, we are able to perform a sparse matrix decomposition and execute only backsubstitution for each new set of input constraints.