Controlled-distortion constrained global parametrization

Abstract

The quality of a global parametrization is determined by a number of factors, including amount of distortion, number of singularities (cones), and alignment with features and boundaries. Placement of cones plays a decisive role in determining the overall distortion of the parametrization; at the same time, feature and boundary alignment also affect the cone placement. A number of methods were proposed for automatic choice of cone positions, either based on singularities of cross-fields and emphasizing alignment, or based on distortion optimization. In this paper we describe a method for placing cones for seamless global parametrizations with alignment constraints. We use a close relation between variation-minimizing cross-fields and related 1-forms and conformal maps, and demonstrate how it leads to a constrained optimization problem formulation. We show for boundaryaligned parametrizations metric distortion may be reduced by cone chains, sometimes to an arbitrarily small value, and the trade-off between the distortion and the number of cones can be controlled by a regularization term. Constrained parametrizations computed using our method have significantly lower distortion compared to the state-of-the art field-based method, yet maintain feature and boundary alignment. In the most extreme cases, parametrization collapse due to alignment constraints is eliminated.

Robust field-aligned global parametrization

“…These properties ensure compatibility with well-developed techniques for field generation [Bommes et al 2009;Myles and Zorin 2012;Myles and Zorin 2013], which generate singularities at vertices, and place no restrictions on the structure of the resulting parametrization.…”

“…Our algorithm is agnostic about the origin of the fields: in particular, it works with any algorithm generating piecewise constant fields on faces ( [Bommes et al 2009;Myles and Zorin 2012;Myles and Zorin 2013]); vertex-based fields can also be adapted (e.g., [Knöppel et al 2013]). All of these are converted to our representation identical to the one used in concurrent work [Ray and Sokolov 2013].…”

Discrete Derivatives of Vector Fields on Surfaces -- An Operator Approach

“…Such vector fields occur often in applications. For example, scalar functions are often discretized as piecewise linear on the vertices of the mesh, and their gradients [Kälberer et al 2007;Bommes et al 2009Bommes et al , 2013Campen et al 2012;Myles and Zorin 2013;Myles et al 2014;Panozzo et al 2014] piecewise constant vector fields are often given as constraints for controlling the alignment of the result. Hence, as we work directly with piecewise constant vector fields without requiring additional conversions to 1-forms or atlas-based representations, our approach is simpler, more intuitive, and easier to implement.…”

Seamless surface mappings

“…Bommes et al, [2009] apply a mixed integer solver to ensure the grid connects across the seam while minimizing a smoothness energy defined via a cross-field. [Myles and Zorin 2012;Myles and Zorin 2013] compute a seamless metric while placing cone singularities to reduce curvature at other locations. Our method shares quite a few similar components with quadrangulation methods, in particular the fact that a transformation group relates copies of the cuts in the plane.…”