Design of tangent vector fields
(From Hugues Hoppe's publications)
ACM Transactions on Graphics (Proc. SIGGRAPH 2007), 26(3), 56.
paper movie - interactive design movie - no extraneous singularities
Abstract: Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.
Hindsight: It is interesting to generalize to N-symmetry fields as in Palacios and Zhang 2007 and Ray et al 2008. One drawback in all approaches (including ours) based on linear systems is that the objective functional under-penalizes smoothness of the vector field in regions where the vectors have small magnitude. In contrast, a nonlinear approach such as [Hertzmann and Zorin 2000] is able to directly measure angle variation of (unit-norm) direction vectors. However, it is more expensive to solve.
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