115 research outputs found
Regular map smoothing
A regular map is a family of equivalent polygons, glued together to form a closed surface without boundaries which is vertex, edge and face transitive. The commonly known regular maps are derived from the Platonic solids and some tessellations of the torus. There are also regular maps of genus greater than 1 which are traditionally viewed as finitely generated groups. RMS (Regular Map Smoothing) is a tool for visualizing a geometrical realization of such a group either as a cut-out in the hyperbolic space or as a compact surface in 3−space. It provides also a tool to make the resulting regular map more appealing than before. RMS achieves that by the use of a coloring scheme based on coset enumeration, a Catmull-Clark smoothing scheme and a force-directed algorithm with topology preservation
Surface Denoising based on Normal Filtering in a Robust Statistics Framework
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 points-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 afterwards. 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 non-linear 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
Weaving patterns inspired by the pentagon snub subdivision scheme
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. They create subdivisions that are (mainly)
comprised of triangles, quadrilaterals, or hexagons. Furthermore, they ensure
that (almost) all vertices have the same number of neighboring vertices. This
paper illustrates a subdivision scheme that splits the input domain into
pentagons. Repeated application of the scheme 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.Comment: Submitted for publication to the Journal of Mathematics and the Arts
(2022
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Hexagonal global parameterization of arbitrary surfaces
In this paper we introduce hexagonal global parameterizations, a new type of parameterization in which parameter lines respect six-fold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with regular hexagonal texture and geometry patterns and can be used to generate high-quality triangular remeshing. To construct a hexagonal parameterization given a surface, we provide an automatic technique to generate a 6-RoSy field that respects directional and singularity features in the surface. This is achieved by applying the trace-and-deviator decomposition to the curvature tensor, which allows us to identify regions of appropriate directional constraints. We also introduce a technique for automatically merging and cancelling singularities. This field will then be used to generate a hexagonal global parameterization by adapting the framework of QuadCover parameterization. In particular, we formulate the energy terms needed to solve for a hexagonal parameterization. We demonstrate the usefulness of our geometry-aware global parameterization with applications such as surface tiling with regular textures and geometry patterns and triangular remeshing.Keywords: Surface parameterization, Pattern synthesis, Rotational symmetry, Triangular remeshing, Hexagonal tilin
Uniform convergence of discrete curvatures from nets of curvature lines
We study discrete curvatures computed from nets of curvature lines on a given
smooth surface, and prove their uniform convergence to smooth principal
curvatures. We provide explicit error bounds, with constants depending only on
properties of the smooth limit surface and the shape regularity of the discrete
net.Comment: 21 pages, 8 figure
Optic Nerve Head Quantification in Idiopathic Intracranial Hypertension by Spectral Domain OCT
Objective: To evaluate 3D spectral domain optical coherence tomography (SDOCT) volume scans as a tool for quantification of optic nerve head (ONH) volume as a potential marker for treatment effectiveness and disease progression in idiopathic intracranial hypertension (IIH). Design and Patients: Cross-sectional pilot trial comparing 19 IIH patients and controls matched for gender, age and body mass index. Each participant underwent SDOCT. A custom segmentation algorithm was developed to quantify ONH volume (ONHV) and height (ONHH) in 3D volume scans. Results:Whereas peripapillary retinal nerve fiber layer thickness did not show differences between controls and IIH patients, the newly developed 3D parameters ONHV and ONHH were able to discriminate between controls, treated and untreated patients. Both ONHV and ONHH measures were related to levels of intracranial pressure (ICP). Conclusion: Our findings suggest 3D ONH measures as assessed by SDOCT as potential diagnostic and progression markers in IIH and other disorders with increased ICP. SDOCT may promise a fast and easy diagnostic alternative to repeated lumba
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