We describe and demonstrate an algorithm that takes as input an unorganized set of points fx1�:::�xng IR 3 on or near an unknown manifold M, and produces as output a simplicial surface that approximates M. Neither the topology, the presence of boundaries, nor the geometry of M are assumed to be known in advance — all are inferred automatically from the data. This problem naturally arises in a variety of practical situations such as range scanning an object from multiple view points, recovery of biological shapes from two-dimensional slices, and interactive surface sketching.

In this paper we describe a new tool for interactive free-form fair surface design. By generalizing classical discrete Fourier analysis to two-dimensional discrete surface signals -- functions defined on polyhedral surfaces of arbitrary topology --, we reduce the problem of surface smoothing, or fairing, to low-pass filtering. We describe a very simple surface signal low-pass filter algorithm that applies to surfaces of arbitrary topology. As opposed to other existing optimization-based fairing methods, which are computationally more expensive, this is a linear time and space complexity algorithm. With this algorithm, fairing very large surfaces, such as those obtained from volumetric medical data, becomes affordable. By combining this algorithm with surface subdivision methods we obtain a very effective fair surface design technique. We then extend the analysis, and modify the algorithm accordingly, to accommodate different types of constraints. Some constraints can be imposed without any modification of the algorithm, while others require the solution of a small associated linear system of equations. In particular, vertex location constraints, vertex normal constraints, and surface normal discontinuities across curves embedded in the surface, can be imposed with this technique. CR Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/image generation - display algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, solid, and object representations;J.6[Com- puter Applications]: Computer-Aided Engineering - computeraided design General Terms: Algorithms, Graphics. 1

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.

We present a general method for automatic reconstruction of accurate, concise, piecewise smooth surface models from scattered range data. The method can be used in a variety of applications such as reverse engineering — the automatic generation of CAD models from physical objects. Novel aspects of the method are its ability to model surfaces of arbitrary topological type and to recover sharp features such as creases and corners. The method has proven to be effective, as demonstrated by a number of examples using both simulated and real data. A key ingredient in the method, and a principal contribution of this paper, is the introduction of a new class of piecewise smooth surface representations based on subdivision. These surfaces have a number of properties that make them ideal for use in surface reconstruction: they are simple to implement, they can model sharp features concisely, and they can be fit to scattered range data using an unconstrained optimization procedure.

Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)-regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semi-regular meshes. The original mesh is typically decomposed into a set of disk-like charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a geometry image. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization — texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders.

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 an efficient method for constructing a smooth surface that interpolates the vertices of a mesh of arbitrary topological type. Normal vectors can also be interpolated at an arbitrary subset of the vertices. The method improves on existing interpolation techniques in that it is fast, robust and general. Our approach is to compute a control mesh whose Catmull-Clark subdivision surface interpolates the given data and minimizes a smoothness or "fairness" measure of the surface. Following Celniker and Gossard, the norm we use is based on a linear combination of thin-plate and membrane energies. Even though Catmull-Clark surfaces do not possess closed-form parametrizations, we show that the relevant properties of the surfaces can be computed efficiently and without approximation. In particular, we show that (1) simple, exact interpolation conditions can be derived, and (2) the fairness norm and its derivatives can be computed exactly, without resort to numerical integration.

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.

In this paper we disprove the belief widespread within the computer graphics community that Catmull-Clark subdivision surfaces cannot be evaluated directly without explicitly subdividing. We show that the surface and all its derivatives can be evaluated in terms of a set of eigenbasis functions which depend only on the subdivision scheme and we derive analytical expressions for these basis functions. In particular, on the regular part of the control mesh where Catmull-Clark surfaces are bi-cubic B-splines, the eigenbasis is equal to the power basis. Also, our technique is both efficient and easy to implement. We have used our implementation to compute high quality curvature plots of subdivision surfaces. The cost of our evaluation scheme is comparable to that of a bi-cubic spline. Therefore, our method allows many algorithms developed for parametric surfaces to be applied to Catmull-Clark subdivision surfaces. This makes subdivision surfaces an even more attractive tool for free-form surface modeling. 1

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 . . . . . . . . . . .