Abstract Signed distance fields are often represented by discrete samples (e.g., on a grid). Recovering the contour implicitly represented by the distance samples requires an approximation algorithm. Several recent approaches have shown that exploiting the information carried in each distance sample by explicitly constructing a surface point gives better results than classical contouring algorithms. We explore the idea of generating surface points by simply approximating the gradient of the signed distance function from a tesselation of the sample locations. The distance value together with gradient yields a potential surface point. To avoid problems resulting from bad approximation, surface points are removed if they are too close to any of the distance samples. Using the regular triangulation as tesselation facilitates this filtering. The resulting approximation algorithm is conceptually simple, easy to implement, and significantly faster than existing alternatives, yielding reconstructions that are on par.

Abstract Unsigned distance fields (UDF) are a versatile, implicit representation of geometry. They can represent surfaces that are not bounding a solid or contain points or curves that are not manifold, for example several sheets meeting along a common curve. Contouring the implicit representation, i.e. turning it into an explicit one, requires finding the zero level set. This is challenging because of the lacking sign information. We present an adaptive re‐sampling approach based on regular triangulations that allows efficiently querying the UDF function at the most important locations. Using a dual contouring approach, information on the topology of the reconstructed surface patches is available during refinement, enabling to increase the resolution in the more critical non‐manifold regions.

We investigate the construction of visually smooth spline surfaces that interpolate the vertices of triangulations by blending local patches. Each triangle star carries a locally interpolating surface patch. The patches are only required to interpolate the vertex, whereas in previous methods the patches are often defined per edge, imposing multiple constraints on local approximations. We adopt simple rational blend functions for the triangular domains, that are constructed so that they retain the interpolation and tangent behavior on the patch boundaries. Decoupling local approximation from blending facilitates the exploration of visually pleasing constructions, while controlling the complexity.

Abstract Contouring an implicit function typically considers function values in the vicinity of the desired level set, only. In a recent string of works, Sellán at al. have demonstrated that signed distance values contain useful information also if they are further away from the surface. This can be exploited to increase the resolution and amount of detail in surface reconstruction from signed distance values. We argue that the right tool for this analysis is a regular triangulation of the distance samples, with the weights chosen based on the distance values. The resulting triangulation is better suited for reconstructing the surface than a standard Delaunay triangulation of the samples. Moreover, the dual power diagram encodes the envelope enclosing the surface, consisting of spherical caps. We discuss how this information can be exploited for reconstructing the surface. In particular, the approach based on regular triangulations lends itself well to refining the sample set. Refining the sample set based on the power diagram outperforms other reconstruction methods relative to the sample count.

Abstract Many common approaches for reconstructing surfaces from point clouds leverage normal information to fit an implicit function to the points. Normals typically play two roles: the direction provides a planar approximation to the surface and the sign distinguishes inside from outside. When the sign is missing, reconstructing a surface with globally consistent sidedness is challenging. In this work, we investigate the idea of squaring the Poisson Surface Reconstruction, replacing the normals with their outer products, making the approach agnostic to the signs of the input/estimated normals. Squaring results in a quartic optimization problem, for which we develop an iterative and hierarchical solver, based on setting the cubic partial derivatives to zero. We show that this technique significantly outperforms standard L‐BFGS solver and demonstrate reconstruction of surfaces from unoriented noisy input in linear time.

Abstract Screened Poisson Surface Reconstruction creates 2D surfaces from sets of oriented points in 3D (and can be extended to co‐dimension one surfaces in arbitrary dimensions). In this work we generalize the technique to manifolds of co‐dimension larger than one. The reconstruction problem consists of finding a vector‐valued function whose zero set approximates the input points. We argue that the right extension of screened Poisson Surface Reconstruction is based on exterior products: the orientation of the point samples is encoded as the exterior product of the local normal frame. The goal is to find a set of scalar functions such that the exterior product of their gradients matches the exterior products prescribed by the input points. We show that this setup reduces to the standard formulation for co‐dimension 1, and leads to more challenging multi‐quadratic optimization problems in higher co‐dimension. We explicitly treat the case of co‐dimension 2, i.e., curves in 3D and 2D surfaces in 4D. We show that the resulting bi‐quadratic problem can be relaxed to a set of quadratic problems in two variables and that the solution can be made effective and efficient by leveraging a hierarchical approach.

K-surfaces are an interactive modeling technique for Bézier-spline surfaces. Inspired by k -curves by [Yan et al. 2017], each patch provides a single control point that is being interpolated at a local extremum of Gaussian curvature. The challenge is to solve the inverse problem of finding the center control point of a Bézier patch given the boundary control points and the handle. Unlike the situation in 2D, bi-quadratic Bézier patches may exhibit none, one, or several extrema, and finding them is non-trivial. We solve the difficult inverse problem, including the possible selection among several extrema, by learning the desired function from samples, generated by computing Gaussian curvature of random patches. This approximation provides a stable solution to the ill-defined inverse problem and is much more efficient than direct numerical optimization, facilitating the interactive modeling framework. The local solution is used in an iterative optimization incorporating continuity constraints across patches. We demonstrate that the surface varies smoothly with the handle location and that the resulting modeling system provides local and generally intuitive control. The idea of learning the inverse mapping from handles to patches may be applicable to other parametric surfaces.

Abstract 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 superhuman - 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 however on humanly observed evidence alone. In this paper we investigate - and 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 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.

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.