Konrad Polthier

Freie Universität Berlin | FUB · Institute of Mathematics

Point clouds arise from acquisition processes applied in various scenarios, such as reverse engineering, rapid prototyping, or cultural preservation. To run various simulations via, e.g., finite element methods, on the derived data, a mesh has to be created from it. In this paper, a meshing algorithm for point clouds is presented, which is based on...

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 subdivi...

The gyroid is a triply periodic minimal surface that belongs to the associate family of the Schwarz P and D surfaces, and has several point reflection, rotational and translational symmetries. A discrete gyroid can be built from triangles—it is a discrete surface with the same symmetries as the smooth gyroid surface, and it is discrete minimal. Eac...

Surface representations play a major role in a variety of applications throughout a diverse collection of fields, such as biology, chemistry, physics, or architecture. From a simulation point of view, it is important to simulate the surface as good as possible, including the usage of a wide range of different approximating elements. However, when i...

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 positio...

Point set denoising is a vital pre-processing step in several computer graphics and medical imaging applications. This article introduces an effective point set denoising software, which is able to remove noise components from the input point set without blurring important sharp features. This software is capable of producing a high fidelity point...

In this work, we present a translation of the complete pipeline for variational shape approximation (VSA) to the setting of point sets. First, we describe an explicit example for the theoretically known non-convergence of the currently available VSA approaches. The example motivates us to introduce an alternate version of VSA based on a switch oper...

In this work, we present a translation of the complete pipeline for variational shape approximation (VSA) to the setting of point sets. First, we describe an explicit example for the theoretically known non-convergence of the currently available VSA approaches. The example motivates us to introduce an alternate version of VSA based on a switch oper...

Learning mathematics can be supported by technology in various creative ways. Appliances for 3D digital setups are widely available. They have transcended their intended use as simple in- or output devices and nowadays play a part in many artistic setups. Thus, they change the way we both perceive and create (digital) models. In this paper, we exam...

Point Sets are acquired as representations of surfaces in R 3 e.g. via laser-scanning or LiDaR. At first, they are unstructured and neighborhood relations have to be established. Then, noise added during generation has to be removed robustly also on non-uniform input while retaining features. The obtained cleaned point set can then be used for e.g....

We present a discrete gyroid surface. The gyroid is a triply periodic minimal surface, our discrete version has the same symmetries as the smooth gyroid and can be constructed from simple translational units.

This work proposes an algorithm for point set segmentation based on the concept of Variational Shape Approximation (VSA), which uses the k-means approach. It iteratively selects seeds, grows at planar proxy regions according to normal similarity, and updates the proxies. It is known that this algorithm does not converge in general. We provide a con...

We present a method for optic nerve head (ONH) 3-D shape analysis from retinal optical coherence tomography (OCT). The possibility to noninvasively acquire in vivo high-resolution 3-D volumes of the ONH using spectral domain OCT drives the need to develop tools that quantify the shape of this structure and extract information for clinical applicati...

The artist Piet Mondrian (1872-1944) is most famous for his abstract works utilizing primary colors and axes-parallel black lines. A similar structure can be found in visualizations of the KdTree data structure used in computational geometry for range searches and neighborhood queries. In this paper, we systematically explore these visualizations a...

Our point set denoising is an iterative, 3-phase algorithm for noisy point sets. Its parameters offer a variety of tuning opportunities. Used models are the gargoyle (real, noisy, irregular), the Chinese ball and rabbit (real, noisy, many features), the fan disk (sharp features, near-flat areas), the sphere and the cube (sharp features), with the l...

In many applications, point set surfaces are acquired by 3D scanners. During this acquisition process, noise and outliers are inevitable. For a high fidelity surface reconstruction from a noisy point set, a feature preserving point set denoising operation has to be performed to remove noise and outliers from the input point set. To suppress these u...

This paper presents a simple and effective two-stage mesh denoising algorithm, where in the first stage, the face normal filtering is done by using the bilateral normal filtering in the robust statistics framework. \textit{Tukey's bi-weight function} is used as similarity function in the bilateral weighting, which is a robust estimator and stops th...

The Neighborhood Grid approximates neighborhood information. A (quadratic) matrix contains the coordinates of the points such that in each row the x-values are increasing while in each column the y-values are increasing. For the algorithm, the order of the points suffices, the exact coordinates are irrelevant. If the above ordering is given, we cal...

The Neighborhood Grid approximates neighborhood information. A (quadratic) matrix contains the coordinates of the points such that in each row the x-values are increasing while in each column the y-values are increasing. For the algorithm, the order of the points suffices, the exact coordinates are irrelevant. If the above ordering is given, we cal...

Optical coherence tomography (OCT) allows three-dimensional (3D) imaging of the retina, and is commonly used for assessing pathological changes of fovea and macula in many diseases. Many neuroinflammatory conditions are known to cause modifications to the fovea shape. In this paper, we propose a method for parametric modeling of the foveal shape. O...

Subdivision surfaces are a common tool in geometric modeling, especially in computer graphics and computer animation. Nowadays, this concept has become established in engineering too. The focus here is on quadrilateral control grids and generalized B-spline surfaces of Catmull–Clark subdivision type. In the classical theory, a subdivision surface i...

In high accuracy numerical simulations and optimal control of time-dependent processes, often both many timesteps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refine...

We present an interactive modeling framework for 3D shapes and for texture maps. The technique combines a differential-based deformation method with the idea of geometry brushes that allow to interactively apply modifications by painting on the geometry. Whereas most other deformation techniques demand the designer to define and move hard constrain...

This paper proposes a generalization of the ordinary de Casteljau algorithm to manifold-valued data including an important special case which uses the exponential map of a symmetric space or Riemannian manifold. We investigate some basic properties of the corresponding Bézier curves and present applications to curve design on polyhedra and implicit...

This article gives a short overview of domain coloring for complex functions that have four-dimensional function graphs and therefore can't be visualized traditionally. The authors discuss several color schemes, focus on various aspects of complex functions, and provide Java-like pseudocode examples explaining the crucial ideas of the coloring algo...

Creating motions of objects or characters that are physically plausible and follow an animator's intent is a key task in computer animation. The spacetime constraints paradigm is a valuable approach to this problem, but it suffers from high computational costs. Based on spacetime constraints, we propose a framework for controlling the motion of def...

We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT (centroidal Voronoi tessellation) optimization framework. We devise a CVT-CMC energy function defined as a co...

In recent years, substantial progress in shape analysis has been achieved through methods that use the spectra and eigenfunctions of discrete Laplace operators. In this work, we study spectra and eigenfunctions of discrete differential operators that can serve as an alternative to the discrete Laplacians for applications in shape analysis. We const...

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). Cross-sectional pilot trial comparing 19 IIH patients and controls matched for gender,...

Multiresolution meshes provide an efficient and structured representation of geometric objects. To increase the mesh resolution only at vital parts of the object, adaptive refinement is widely used. We propose a lossless compression scheme for these adaptive structures that exploits the parent-child relationships inherent to the mesh hierarchy. We...

We propose a framework for deformation-based surface modeling that is interactive, robust, and intuitive to use. The deformations are described by a nonlinear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a...

Discrete Laplace–Beltrami operators on polyhedral surfaces play an important role for various applications in geometry processing and related areas like physical simulation or computer graphics. While discretizations of the weak Laplace–Beltrami operator are well‐studied, less is known about the strong form. We present a principle for constructing...

Despite the success of quad-based 2D surface parameterization methods, effective parameterization algorithms for 3D volumes with cubes, i.e. hexahedral elements, are still missing. CubeCover is a first approach for generating a hexahedral tessellation of a given volume with boundary aligned cubes which are guided by a frame field. The input of Cube...

We introduce hexagonal global parameterization, a new type of surface parameterization in which parameter lines respect sixfold rotational symmetries (6-RoSy). Such parameterizations enable the tiling of surfaces with nearly regular hexagonal or triangular patterns, and can be used for triangular remeshing. Our framework to construct a hexagonal pa...

This work concerns the approximation of the shape operator of smooth surfaces in R3R3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximat...

We present a discretization of Koiter's model of elastic thin shells based on a finite element that employs limit surfaces of Catmull-Clark's subdivision scheme. The discretization can directly be applied to control grids of Catmull-Clark subdivision surfaces, and, therefore, integrates modeling of Catmull-Clark subdivision surfaces with analysis a...

We present a novel algorithm for automatic parameterization oftube-like surfaces of arbitrarygenus such as the surfaces of knots, trees, blood vessels, neurons, or any tubular graph with a globally consistentstripe texture. Mathematically these surfaces can be described as thickened graphs, and the calculatedparameterizationstripe will follow eithe...

The QuadCover algorithm is a well received and flexible algorithm for surface parameterization. For demonstrating QuadCover, we originally used principal curvature lines to guide the parameterization. The current work focuses on fine‐tuning the vector fields used in QuadCover to yield state‐of‐the‐art surface parameterizations. We propose new techn...

In this work, we study the spectra and eigenmodes of the Hessian of various discrete surface energies and discuss applications to shape analysis. In particular, we consider a physical model that describes the vibration modes and frequencies of a surface through the eigenfunctions and eigenvalues of the Hessian of a deformation energy, and we derive...

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in \({\mathbb R}^3\) from explicitly given branch points with corresponding branch...

The structuring of surface meshes is a labor intensive task in reverse engineering. For example, in CAD, scanned triangle meshes must be divided into characteristic/uniform patches to enable conversion into high-level spline surfaces. Typical industrial techniques, like rolling ball blends, are very labor intensive.We provide a novel, robust and qu...

We present a novel coder for lossless compression of adaptive multiresolution meshes that exploits their special hierarchical structure. The heart of our method is a new progressive connectivity coder that can be combined with leading geometry encoding techniques. The compressor uses the parent/child relationships inherent to the hierarchical mesh....

We present an automatic method for computing an accurate parametric model of a piecewise defined pipe surface, consisting of cylinder and torus segments, from an unorganized point set. Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G1 continuous circular a...

Classical surface parameterization algorithms often place singularities in order to enhance the quality of the resulting parameter map. Unfortunately, singularities of positive integral index (as the north pole of a sphere) were not handled since they cannot be described with piecewise linear parameter functions on a triangle mesh. Preprocessing is...

Abstract Complex-valued functions are fundamental objects in complex analysis, algebra, differential geometry and in many other areas such as numerical mathematics and physics. Visualizing complex functions is a non-trivial task since maps between two-dimensional spaces are involved whose graph would be an unhandy submanifold in four-dimensional sp...

Die euklidische Geometrie von Eiflächen und Eikörpern bietet eine gute Gelegenheit, geometrische Begriffsbildungen und Sachverhalte aus der Flächentheorie einem breiten Publikum anschaulich zu machen. In dieser Arbeit erörtern wir die erstaunliche Stabilität von Eierschalen und dazu verwandte Probleme unter heuristisch-mathematischen Gesichtspunkte...

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.

We describe an approach to visually analyzing the dynamic behavior of 3D time-dependent flow fields by considering the behavior of the path lines. At selected positions in the 4D space-time domain, we compute a number of local and global properties of path lines describing relevant features of them. The resulting multivariate data set is analyzed b...

Visualization research aims at providing insights into large, complex bodies of data. Topological methods are distinguished by their solid mathematical foundation, guiding the algorithmic analysis and its presentation among the various visualization techniques. This book contains 13 peer-reviewed papers resulting from the second workshop on "Topol...

We propose a method to identify planar regions in volume data using a specialized version of the discrete Radon transform operating on a structured or unstructured grid. The algorithm uses an efficient discretization scheme for the parameter space to obtain a running time of O(N (TlogT))\mathcal O(N (T\log T)) , where T is the number of cells an...

We present a method for parametric reconstruction of a piecewise defined pipe surface, consisting of cylinder and torus segments, from an unorganized point set. Our main contributions are reconstruction of the spine curve of a pipe surface from surface samples, and approximation of the spine curve by G^1 continuous circular arcs and line segments....

The extraction of curvature information for surfaces is a basic problem of Geometry Processing. Recently an integral invariant solution of this problem was presented, which is based on principal component analysis of local neighborhoods defined by kernel ...

Abstract We introduce an algorithm for the automatic computation of global parameterizations on arbitrary simplicial 2-manifolds, whose parameter lines are guided by a given frame field, for example, by principal curvature frames. The parameter lines are globally continuous and allow a remeshing of the surface into quadrilaterals. The algorithm con...

We propose a constraint-based method for the fairing of surface meshes. The main feature of our approach is that the resulting smoothed surface remains within a prescribed distance to the input mesh. For example, specifying the maximum distance in the order of the measuring precision of a laser scanner allows noise to be removed while preserving th...

We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidean 3-space. Under the assumption of convergence of surfaces in Hausdorff distance, we show that convergence of the following properties are equivalent: surface normals, surface area, metric tensors, and Laplace–...

Geodesic curves are the fundamental concept in geometry to generalize the idea of straight lines to curved surfaces and arbitrary manifolds. On polyhedral surfaces we introduce the notion of discrete geodesic curvature of curves and define straightest geodesics. This allows a unique solution of the initial value problem for geodesics, and therefore...

This volume presents the proccedings of the 11th International Workshop on Combinatorial Image Analysis. IWCIA 2006 was the 11th in a series of international workshopfs devoted to combinatorial image analysis. Prior meetings took place in Paris (France 1991), Ube (Japan 1992), Wahington DC (USA 1994), Lyon (France 1995), Hiroshima (Japan 1997), Mad...

We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2D. This pap...

The use of point sets instead of meshes became more popular during the last years. We present a new method for anisotropic fairing of a point sampled surface using an anisotropic geometric mean curvature flow. The main advantage of our approach is that the evolution removes noise from a point set while it detects and enhances geometric features of...

Abstract We introduce FreeLence, a novel and simple single-rate compression coder for triangle manifold meshes. Our method uses free valences and exploits geometric information for connectivity encoding. Furthermore, we introduce a novel linear prediction scheme for geometry compression of 3D meshes. Together, these approaches yield a significant e...

Feature lines are salient surface characteristics. Their definition involves third and fourth order surface deriva- tives. This often yields to unpleasantly rough and squiggly feature lines since third order derivatives are highly sensitive against unwanted surface noise. The present work proposes two novel concepts for a more stable algo- rithm pr...

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A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and e...

The archive Electronic Geometry Models is a new electronic journal for the publication of digital geometry models from a broad range of mathematical topics. The geometry models are distinguished constructions, counter examples, or results from elaborate computer experiments. Each submitted model has a self-contained textual description and is peer...

this paper we define the new alignment energy for non-conforming triangle meshes, and describes its use to compute unstable conforming discrete minimal surfaces. Our algorithm makes use of the duality between conforming and non-conforming discrete minimal surfaces which was observed earlier. In first experiments the new algorithm allows us the comp...

ids. ii Printer: Opaque this Introduction to Polyhedral Meshes Polyhedral meshes belong to the most basic structures for the representation of geometric shapes not only in numerics and computer graphics. Especially the niteness of the set of vertices and of their combinatorial relation makes them an ideal tool to reduce innite dimensional problems...

Contents Preface vii 1 Introduction to Polyhedral Meshes 1 1.1 SimplicialComplexes................... 2 1.2 TopologicalProperties.................. 6 1.3 DistanceandMetric ................... 8 1.4 DiscreteGauCurvature................. 9 1.5 GridsinNumericsandGraphics ............ 15 1.5.1 DelaunayTriangulation ............. 18 1.5.2 VoronoiDiagram...

On a curved surface the front of a point wave evolves in concentric circles which start to overlap and branch after a certain time. This evolution is described by the geodesic flow and helps us to understand the geometry of surfaces. In this paper we compute the evolution of distance circles on polyhedral surfaces and develop a method to visualize...

Modern mathematical visualization has always been related with special graphics workstation although visualization was always part of mathematics.

In this paper we present the implementation of a partial knot recognition algorithm as a mathematical web service on the internet. Knots may interactively be loaded and edited, and then checked for being unknottet.

this paper we use a slightly more general definition of the spaces S h respectively S # h , namely we include functions which are only defined at vertices respectively at edge midpoints. For example, the (total) Gau curvature is defined solely at vertices. Here for a given vector field # we will have div h # S h (respectively div # h # S # h ) to b...

JavaViewLib is a new Maple package combined with the JavaView vi- sualization toolkit that adds new interactivity to Maple plots in both web pages and worksheets. It provides a superior viewing environment to enhance plots in Maple by adding several features to plots' interactivity, such as mouse-controlled scaling, translation, rotation in 2d, 3d,...

The future of mathematical communication is strongly related with the internet. On a number of examples, the present paper gives a futuristic outlook how mathematical visualization imbedded in the internet will provide new insight into complex phenomena, influence the international cooperation of researchers, and allow to create online hyperbooks c...

JavaView is a 3D geometry viewer and a numerical software li- brary written in Java which allows one to publish interactive geometries and mathematical experiments in online web pages. Its numerical software library provides solutions and tools for problems in difierential geometry and math- ematical visualization. This allows the creation of one's...

dle the geometric objects they thought about. In particular, Felix Klein and Hermann Amandus Schwarz in Gottingen built many models of curves, surfaces and mechanical devices for teaching and other educational purposes. What are the main reasons for today's mathematicians to construct digital models of geometric shapes and make them available via t...

. We define triangulated piecewise linear constant mean curvature surfaces using a variational characterization. These surfaces are critical for area amongst continuous piecewise linear variations which preserve the boundary conditions, the simplicial structures, and (in the nonminimal case) the volume to one side of the surfaces. We then find expl...