A Smoothness Energy without Boundary Distortion for Curved Surfaces

Abstract

Laplacian energy (zero Neumann boundary conditions) planar Hessian energyStein et al. [2018] our Hessian energy Fig. 1. Solving an interpolation problem on an airplane. Using the Laplacian energy with zero Neumann boundary conditions (left) distorts the result near the windows and the cockpit of the plane: the isolines bend so they can be perpendicular to the boundary. The planar Hessian energy of Stein et al. [2018] (center)is unaffected by the holes, but does not account for curvature correctly, leading to unnatural spacing of isolines at the front and back of the fuselage. Our Hessian energy (right) produces a natural-looking result with more regularly spread isolines, unaffected by the boundary.Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: one can either have natural behavior in the interior, or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.

Surface multigrid via intrinsic prolongation

“…We also pick the applications that involve different system matrices with different sparsity patterns. This includes the cotangent Laplacian (1-ring sparsity), the Bilaplacian (2-ring sparsity), the squared Hessian (2-ring sparsity) [Stein et al 2020], a system matrix derived from Lagrange multipliers [Azencot et al 2015], and also the Hessian matrices from shell simulation which has 3|𝑉 |-by-3|𝑉 | dimensionality. Fig.…”

“…We compare the runtime of our multigrid solver against the Cholesky solver on smoothing the noisy data on a sphere cap at different resolutions until reaching a sufficiently small mean squared error (visually indistinguishable). We evaluate the smoothing with the Dirichlet energy 𝐸 Δ (1-ring sparsity) and with the squared Hessian energy 𝐸 𝐻 2 [Stein et al 2020] (2-ring sparsity). Our method is asymptotically faster than the direct solver.…”

“…19), the squared Laplacian energy 𝐸 Δ 2 (Fig. 20 top), and the squared Hessian energy 𝐸 𝐻 2 [Stein et al 2020] Fig. 22.…”

Frame Field Operators

“…It is used for applications in surface fairing [DMSB99], surface deformation [SCL∗04], data interpolation [JWS12], data smoothing [WGS10], the computation of smooth distances [LRF10], skinning and character animation [JBPS11], physical simulation [BWH∗06], and more [SKČ∗14; AJC11]. The Bilaplacian with zero Neumann boundary is often discretized using mixed finite elements for the Laplacian [JTSZ10], although other approaches are also popular for different boundary conditions [BWH∗06; SGWJ18; SJWG20].…”

A Simple Discretization of the Vector Dirichlet Energy

“…by minimizing the vector Dirichlet energy. In Figure 3, this approach for a linear function u using Raviart-Thomas and Nédélec basis functions fails, while the Crouzeix-Raviart approach recovers the function exactly (up to numerical error To extend the Crouzeix-Raviart element to vectors, we multiply the scalar basis functions be with appropriate vectors [SJWG20]. At the midpoint of each edge e, we represent all tangent vectors as a linear combination of the following two vectors,…”