Modeling n -Symmetry Vector Fields using Higher-Order Energies

Abstract

We introduce a variational approach for modeling n-symmetry vector and direction fields on surfaces that supports interpolation and alignment constraints, placing singularities and local editing, while providing real-time responses. The approach is based on novel biharmonic and m-harmonic energies for n-fields on surface meshes and the integration of hard constraints to the resulting optimization problems. Real-time computation rates are achieved by a model reduction approach employing a Fourier-like n-vector field decomposition, which associates frequencies and modes to n-vector fields on surfaces. To demonstrate the benefits of the proposed n-field modeling approach, we use it for controlling stroke directions in line-art drawings of surfaces and for the modeling of anisotropic BRDFs which define the reflection behavior of surfaces.

Octahedral Frames for Feature-Aligned Cross Fields

“…Their results on wavey-box, and wedge are successfully aligned to the creases. ; • fields from [Brandt et al 2018] are challenging to tune with λ representing alignment to a guiding extrinsic curvature field. We show their method for the biharmonic energy (m = 2) as a point of contrast to Dirichlet energy.…”

“…The generation of tangential n-RoSy fields over surfaces has many applications in computer graphics ranging from surface BRDF modification [Brandt et al 2018] to meshing [Bommes et al 2009;Jakob et al 2015] to texture synthesis [Knöppel et al 2015] and sketchbased modeling [Bessmeltsev and Solomon 2019;Iarussi et al 2015]. Surveys of n-RoSy field design methods are provided in [Vaxman et al 2016] and[de Goes et al 2015].…”

“…We observe generally better alignment in the quad meshes generated from our method. By placing singularities on the cylindrical region of the anchor, our quad meshing manages to align better We test on the three-cylinder-intersection, wavey-box, wedge, and fandisk meshes and compare against the following works: [Brandt et al 2018;Jakob et al 2015;Knöppel et al 2013] with various parameters. We use normal aligned octahedral fields generated by minimizing E 2 .…”

“…An alternative strategy is to include energy terms that explicitly enforce alignment to an input guiding field of principal curvature directions during cross-field design [Brandt et al 2018;Knöppel et al 2013]. Drawbacks include the difficulty of robustly computing principal curvature directions on noisy meshes, the fact that forcing alignment to a guiding field based on local geometry may exclude cross-field designs that are globally more optimal, and more critically the fact that principal curvature directions are often different from features e.g.…”

Fast Approximation of Laplace‐Beltrami Eigenproblems

“…Song et al [SLMR14] defined a saliency measure on surfaces that combined spectral and spatial information and takes advantage of the global nature of information embedded in the low‐frequency eigenfunction. Spectral methods have also been used for tangential vector and n‐vector field processing on surfaces [ABCCO13, AOCBC15,BSEH17,BSEH18].…”