Design of tangent vector fields

Design of tangent vector fields
Matthew Fisher, Peter Schröder, Mathieu Desbrun, Hugues Hoppe.
ACM Trans. Graphics (SIGGRAPH), 26(3), 2007.
Interactive control of direction fields for real-time surface texture synthesis.
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.
Hindsights:

One generalization is N-symmetry fields as in Palacios and Zhang 2007 and Ray et al 2008.

A common drawback in most linear approaches (including ours) is that the objective functional under-penalizes smoothness of the vector field in regions where the vectors have small magnitude. The work of Crane et al. 2010 overcomes this by optimizing over direction angles instead of vectors.

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