During the last years the concept of multi-resolution modeling has gained special attention in many fields of computer graphics and geometric modeling. In this paper we generalize powerful multiresolution techniques to arbitrary triangle meshes without requiring subdivision connectivity. Our major observation is that the hierarchy of nested spaces which is the structural core element of most multi-resolution algorithms can be replaced by the sequence of intermediate meshes emerging from the application of incremental mesh decimation. Performing such schemes with local frame coding of the detail coefficients already provides effective and efficient algorithms to extract multi-resolution information from unstructured meshes. In combination with discrete fairing techniques, i.e., the constrained minimization of discrete energy functionals, we obtain very fast mesh smoothing algorithms which are able to reduce noise from a geometrically specified frequency band in a multiresolution decomposition. Putting mesh hierarchies, local frame coding and multi-level smoothing together allows us to propose a flexible and intuitive paradigm for interactive detail-preserving mesh modification. We show examples generated by our mesh modeling tool implementation to demonstrate its functionality.

Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the original data exactly, and play an important role in fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme, which yields C 1 surfaces in the topologically regular setting. Unfortunately it exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.

We describe a multiresolution representation for meshes based on subdivision. Subdivision is a natural extension of the existing patch-based surface representations. At the same time subdivision algorithms can be viewed as operating directly on polygonal meshes, which makes them a useful tool for mesh manipulation. Combination of subdivision and smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex meshes of arbitrary topology. Simplicity of the essential algorithms for re nement and coarsi cation allows to make them local and adaptive, considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.

sis Denis Zorin and Jorg Peters Afternoon Session: Applications and Algorithms The afternoon session will focus on applications of subdivision and the algorithmic issues practitioners need to address to build efficient, well behaving systems for modeling and animation with subdivision surfaces. Interactive Multiresolution Mesh Editing Denis Zorin Subdivision Surfaces and Wavelets Michael Lounsbery A Variational Approach to Subdivision Leif Kobbelt Exploiting Subdivision in Modeling and Animation David R. Forsey Subdivision Surfaces in the Making of Geri's Game Tony DeRose 6 1 Introduction 13 2 Foundations I: Basic Ideas 17 2.1 The Idea of Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Review of Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Piecewise Polynomial Curves . . . . . . . . . . .

We propose a new progressive compression scheme for arbitrary topology, highly detailed and densely sampled meshes arising from geometry scanning. We observe that meshes consist of three distinct components: geometry, parameter, and connectivity information. The latter two do not contribute to the reduction of error in a compression setting. Using semi-regular meshes, parameter and connectivity information can be virtually eliminated. Coupled with semi-regular wavelet transforms, zerotree coding, and subdivision based reconstruction we see improvements in error by a factor four (12dB) compared to other progressive coding schemes. CR Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - hierarchy and geometric transformations; G.1.2 [Numerical Analysis]: Approximation - approximation of surfaces and contours, wavelets and fractals; I.4.2 [Image Processing and Computer Vision]: Compression (Coding) - Approximate methods Additional K...

A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented which generates C 1 surfaces in the limit. The scheme satisfies important requirements for practical applications in computer graphics and engineering. These requirements include the necessity to generate smooth surfaces with local creases and cusps. The scheme can be applied to open nets in which case it generates boundary curves that allow a C 0 -join of several subdivision patches. Due to the local support of the scheme, adaptive refinement strategies can be applied. We present a simple device to preserve the consistency of such adaptively refined nets. Keywords: Curve and surface modeling, Interpolatory subdivision, Adaptive mesh-refinement 1 Introduction The problem we address in this paper is the generation of smooth interpolating surfaces of arbitrary topological type in the context of practical applications. Such applications range from the design of free-form surfaces an...

Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas. The main

Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

Most systems that support the visual interaction with 3D models use shape representations based on triangle meshes. The size of these representations imposes limits on applications, where complex 3D models must be accessed remotely. Techniques for simplifying and compressing 3D models reduce the transmission time. Multi-resolution formats provide quick access to a crude model and then refine it progressively. Unfortunately, compared to the best non-progressive compression methods, previously proposed progressive refinement techniques impose a significant overhead when the full resolution model must be downloaded. The CPM (Compressed Progressive Meshes) approach proposed here eliminates this overhead. It uses a new “patching ” technique, which refines the topology of the mesh in batches, which each increase the number of vertices by up to 50%. Less than 4 bits per triangle encode where and how the topological refinements should be applied. We estimate the position of new vertices from the positions of their topological neighbors in the less refined mesh using a new estimator that leads to representations of vertex coordinates that are 50 % more compact than previously reported progressive geometry compression techniques. 1.