New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples

Glyphs for space-time Jacobians of time-dependent vector fields

Glyphs have proven to be a powerful visualization technique for general tensor fields modeling physical phenomena such as diffusion or the derivative of flow fields. Most glyph constructions, however, do not provide a way of considering the temporal derivative, which is generally nonzero in non-stationary vector fields. This derivative offers a deeper understanding of features in time-dependent vector fields. We introduce an extension to 2D and 3D tensor glyph design that additionally encodes the temporal information of velocities, and thus makes it possible to represent time-dependent Jacobians. At the same time, a certain set of requirements for general tensor glyphs is fulfilled, such that the new method provides a visualization of the steadiness or unsteadiness of a vector field at a given instance of time

Line-art rendering of 3D-models

We present an interactive system for computer aided generation of line art drawings to illustrate 3D models that are given as triangulated surfaces. In a preprocessing step an enhanced 2D view of the scene is computed by sampling for every pixel the shading, the normal vectors and the principal directions obtained from discrete curvature analysis. Then streamlines are traced in the 2D direction fields and are used to define line strokes. In order to reduce noise artifacts the user may interactively select sparse reference lines and the system will automatically fill in additional strokes. By exploiting the special structure of the streamlines an intuitive and simple tone mapping algorithm can be derived to generate the final rendering

Convex Boundary Angle

Angle Based Flattening is a robust parameterization method that finds a quasi-conformal mapping by solving a non-linear optimization problem. We take advantage of a characterization of convex planar drawings of triconnected graphs to introduce new boundary constraints. This prevents boundary intersections and avoids post-processing of the parameterized mesh. We present a simple transformation to effectively relax the constrained minimization problem, which improves the convergence of the optimization method. As a natural extension, we discuss the construction of Delaunay flat meshes. This may further enhance the quality of the resulting parameterization

Discrete tensorial quasiharmonic maps

We introduce new linear operators for surface parameterization. Given an initial mapping from the parametric plane onto a surface mesh, we establish a secondary map from the plane onto itself that mimics the initial one. The resulting low-distortion parameterization is smooth as it stems from solving a quasi-harmonic equation. Our parameterization method is robust and independent of (the quality of) the initial map. 1

Feature Sensitive Sampling for Interactive Remeshing

We present a technique for remeshing irregular triangles meshes where the distribution and alignment can be adapted to the underlying geometry. Following the interactive virtual range scanner approach we overcome aliasing problems by introducing a special sampling technique. A sampling grid that can be aligned to the local features of the mesh is constructed interactively in an intuitive way and without adding reasonable overhead to the virtual scanning process.