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.

We show how spectral methods may be applied to 3D mesh data to obtain compact representations. This is achieved by projecting the mesh geometry onto an orthonormal basis derived from the mesh topology. To reduce complexity, the mesh is partitioned into a number of balanced submeshes with minimal interaction, each of which are compressed independently. Our methods may be used for compression and progressive transmission of 3D content, and are shown to be vastly superior to existing methods using spatial techniques, if slight loss can be tolerated.

A basic element of Digital Geometry Processing algorithms is the establishment of a smooth parameterization for a given model. In this paper we propose an algorithm which establishes parameterizations for a set of models. The parameterizations are called consistent because they share the same base domain and respect features. They give immediate correspondences between models and allow remeshes with the same connectivity. Such remeshes form the basis for a large class of algorithms, including principal component analysis, wavelet transforms, detail and texture transfer between models, and n-way shape blending. We demonstrate the versatility of our algorithm with a number of examples.

Normal meshes are new fundamental surface descriptions inspired by differential geometry. A normal mesh is a multiresolution mesh where each level can be written as a normal offset from a coarser version. Hence the mesh can be stored with a single float per vertex. We present an algorithm to approximate any surface arbitrarily closely with a normal semi-regular mesh. Normal meshes can be useful in numerous applications such as compression, filtering, rendering, texturing, and modeling.

We introduce multi-chart geometry images, a new representation for arbitrary surfaces. It is created by resampling a surface onto a regular 2D grid. Whereas the original scheme of Gu et al. maps the entire surface onto a single square, we use an atlas construction to map the surface piecewise onto charts of arbitrary shape. We demonstrate that this added flexibility reduces parametrization distortion and thus provides greater geometric fidelity, particularly for shapes with long extremities, high genus, or disconnected components. Traditional atlas constructions suffer from discontinuous reconstruction across chart boundaries, which in our context create unacceptable surface cracks. Our solution is a novel zippering algorithm that creates a watertight surface. In addition, we present a new atlas chartification scheme based on clustering optimization.

Lossless transmission of 3D meshes is a very challenging and timely problem for many applications, ranging from collaborative design to engineering. Additionally, frequent delays in transmissions call for progressive transmission in order for the end user to receive useful successive refinements of the final mesh. In this paper, we present a novel, fully progressive encoding approach for lossless transmission of triangle meshes with a very fine granularity. A new valence-driven decimating conquest, combined with patch tiling and an original strategic retriangulation is used to maintain the regularity of valence. We demonstrate that this technique leads to good mesh quality, near-optimal connectivity encoding, and therefore a good rate-distortion ratio throughout the transmission. We also improve upon previous lossless geometry encoding by decorrelating the normal and tangential components of the surface. For typical meshes, our method compresses connectivity down to less than 3.7 bits per vertex, 40% better in average than the best methods previously reported [5, 18]; we further reduce the usual geometry bit rates by 20% in average by exploiting the smoothness of meshes. Concretely, our technique can reduce an ascii VRML 3D model down to 1.7% of its size for a 10-bit quantization (2.3% for a 12-bit quantization) while providing a very progressive reconstruction.

We present a novel method to extract iso-surfaces from distance volumes. It generates high quality semi-regular multiresolution meshes of arbitrary topology. Our technique proceeds in two stages. First, a very coarse mesh with guaranteed topology is extracted. Subsequently an iterative multi-scale force-based solver refines the initial mesh into a semi-regular mesh with geometrically adaptive sampling rate and good aspect ratio triangles. The coarse mesh extraction is performed using a new approach we call surface wavefront propagation. A set of discrete iso-distance ribbons are rapidly built and connected while respecting the topology of the iso-surface implied by the data. Subsequent multi-scale refinement is driven by a simple force-based solver designed to combine good iso-surface fit and high quality sampling through reparameterization. In contrast to the Marching Cubes technique our output meshes adapt gracefully to the iso-surface geometry, have a natural multiresolution structure and good aspect ratio triangles, as demonstrated with a number of examples.

Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains, i.e., how rapidly the derivatives of the parameterization change. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and parametric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures; the rate distortion behavior of semi-regular remeshes produced with these parameterizations; and a comparison with globally smooth subdivision methods. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.

Polygonal models acquired with emerging 3D scanning technology or from large scale CAD applications easily reach sizes of several gigabytes and do not fit in the address space of common 32-bit desktop PCs. In this paper we propose an out-of-core mesh compression technique that converts such gigantic meshes into a streamable, highly compressed representation. During decompression only a small portion of the mesh needs to be kept in memory at any time. As full connectivity information is available along the decompression boundaries, this provides seamless mesh access for incremental in-core processing on gigantic meshes. Decompression speeds are CPU-limited and exceed one million vertices and two million triangles per second on a 1.8 GHz Athlon processor.

Many high-resolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axis-aligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate out-of-core execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting “mesh surgery. ” Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.