Konrad Polthier

Computing discrete minimal surfaces and their conjugates

We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations. Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.

QuadCover - Surface Parameterization using Branched Coverings

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 converts a given frame field into a single vector field on a branched covering of the 2‐manifold and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.

Mesh parameterization: theory and practice

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

Visualization and Mathematics III

A collection of state-of-the-art presentations on visualization problems in mathematics, fundamental mathematical research in computer graphics, and software frameworks for the application of visualization to real-world problems. Contributions have been written by leading experts and peer-refereed by an international editorial team. The book grew out of the third international workshop Visualization and Mathematics, May 22-25, 2002 in Berlin. The variety of topics covered makes the book ideal for researcher, lecturers, and practitioners.

Combinatorial Image Analysis

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), Madras (India 1999), Philadelphia (USA 2001), Palermo (Italy 2003) and Auckland (New Zealand 2004). For this workshop we received 59 papers from all over the world. Each paper was assigned to three independent referees and carefully revised. Finally, we selected 34 papers for the conference based on content, significance, relevance, and presentation. Conference papers are presented in this volume in the order they were presented at the conference. The topics of the conference covered combinatorial image analysis, grammars and models for analysis and recognition of scenes or images, combinatorial topology and geometry for images, digital geometry of curves or surfaces, algebraic approaches to image processing, image, point-clouds or surface registration as well as fuzzy and probabilistic image analysis. The programm followed a single-track format with presentations of all published conference papers. Non-overlapping oral and poster sessions ensured that all attendees had opportunities to interact personny with presenters. Among the highlights of the meeting were the talks of our two invited speakers, renowned experts in the field of descrete geometry, digital topology, and image analysis. - David Coeurjolly (University of Lyon, France): Computational Aspects of Digital Plane and Hyperplane Recognition - Longin Jan Latecki (Temple University, Philadelphia, USA): Polygonal Approximation of Point Sets.

Anisotropic Filtering of Non-Linear Surface Features

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 efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well‐known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non‐linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments.

Straightest geodesics on polyhedral surfaces

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 a unique movement in a given tangential direction, a property not available in the well-known concept of locally shortest geodesics.An immediate application is the definition of parallel translation of vectors and a discrete Runge-Kutta method for the integration of vector fields on polyhedral surfaces. Our definitions only use intrinsic geometric properties of the polyhedral surface without reference to the underlying discrete triangulation of the surface or to an ambient space.

Identifying Vector Field Singularities Using a Discrete Hodge Decomposition

We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient are rotation-free respectively divergence-free components of the vector field. The sinks and sources respectively vortices are then automatically identified as the critical points of the corresponding scalar-valued potentials. The global nature of the decomposition avoids the approximation problem of the Jacobian and higher order tensors used in local methods, while the two potentials plus a harmonic flow component are an exact decomposition of the vector field containing all information.

Anisotropic smoothing of point sets

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 the surface such as edges and corners. We derive a shape operator, principal curvature properties of a point set, and an anisotropic Laplacian of the surface. This anisotropic Laplacian reflects curvature properties which can be understood as the point set analogue of Taubins curvature-tensor for polyhedral surfaces. We combine these discrete tools with techniques from geometric diffusion and image processing. Several applications demonstrate the efficiency and accuracy of our method.

Smooth feature lines on surface meshes

Feature lines are salient surface characteristics. Their definition involves third and fourth order surface derivatives. 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 algorithm producing visually more pleasing feature lines: First, a new computation scheme based on discrete differential geometry is presented, avoiding costly computations of higher order approximating surfaces. Secondly, this scheme is augmented by a filtering method for higher order surface derivatives to improve both the stability of the extraction of feature lines and the smoothness of their appearance.