Curvature-Aware Regularization on Riemannian Submanifolds

Anisotropic Laplace-Beltrami Operators for Shape Analysis

“…Essentially, ∆ Dα conserves the strong analytical properties of the conventional Laplace-Beltrami operator ∆. This is in contrast with the previous proposal of [11]; in the following, we will also provide a brief comparison of the two variants.…”

“…This differentiates our approach from the anisotropic LB operator ∆D introduced in [11], where the anisotropic factorD only modifies the norm of a given input vector, but not its direction (see Eq. (5) below).…”

“…It should be noted that the operator∆D as defined in [11] is also not invariant to isometries, furthermore it loses other key properties of ∆. In particular, the anisotropic factorD has the form:…”

“…Recently, Kim et al [11] have extended these methods by approximating an anisotropic LB operator in point clouds of high dimension using graph reweighting.…”

“…In the case of a diffusion process, D controls both the direction and the magnitude of the diffusion on the surface S, as shown in Figure 3. Inspired by successful applications of curvature-aware anisotropic diffusion to denoising of images and surfaces [16,2,8,6,25], and by recent advances in the analysis of high-dimensional data [11], we attempt to incorporate extrinsic information into the anisotropic tensor D for shape analysis purposes. We do so by introducing the following generic linear operator, defined in the orthonormal basis (v m , v M ):…”