Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online .

During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as point set or triangulated mesh). The noise removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterward. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method.

In this paper, we introduce such a framework to establish relations between a number of widely used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and nonlinear normal filtering. Although these methods originate in different mathematical theories—which include diffusion-, bilateral-, and directional curvature-based algorithms—we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods.

Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.

The approximation of a golden logarithmic spiral by quarter circles is well known. Starting from this, we show that any logarithmic spiral can be approximated by quarter circles in a similar way. Using our construction on a rectangle with aspect ratio √휙 and performing a coordinate reparametrization, we obtain an aesthetic partition of the plane as our main artwork.