No abstract

Discrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so-called cotan Laplacian is widely used since it preserves several important properties of its smooth counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two-dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles. We provide explicit formulas for the lesser-known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two-dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi-definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes. CCS Concepts • Mathematics of computing → Mesh generation; Discrete optimization; • Computing methodologies → Mesh geometry models; • Theory of computation → Computational geometry;

Extending the ARAP energy with a term that depends on the face normal, energy minimization becomes an effective stylization tool for shapes represented as meshes. Our approach generalizes the possibilities of Cubic Stylization: the set of preferred normals can be chosen arbitrarily from the Gauss sphere, including semi‐discrete sets to model preference for cylinder‐ or cone‐like shapes. The optimization is designed to retain, similar to ARAP, the constant linear system in the global optimization. This leads to convergence behavior that enables interactive control over the parameters of the optimization. We provide various examples demonstrating the simplicity and versatility of the approach.

Within the last decade, neural network based predictors have demonstrated impressive -and at times super-human -capabilities. This performance is often paid for with an intransparent prediction process and thus has sparked numerous contributions in the novel field of explainable artificial intelligence (XAI). In this paper, we focus on a popular and widely used method of XAI, the Layer-wise Relevance Propagation (LRP). Since its initial proposition LRP has evolved as a method, and a best practice for applying the method has tacitly emerged, based on humanly observed evidence. We investigateand for the first time quantify -the effect of this current best practice on feedforward neural networks in a visual object detection setting. The results verify that the current, layer-dependent approach to LRP applied in recent literature better represents the model's reasoning, and at the same time increases the object localization and class discriminativity of LRP.