Mathieu Desbrun

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

Implicit fairing of irregular meshes using diffusion and curvature flow

In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface. The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features. The problem arises mainly when creating high-fidelity computer graphics objects using imperfectly-measured data from the real world. Our approach contains three novel features: an implicit integration method to achieve efficiency, stability, and large time-steps; a scale-dependent Laplacian operator to improve the diffusion process; and finally, a robust curvature flow operator that achieves a smoothing of the shape itself, distinct from any parameterization. Additional features of the algorithm include automatic exact volume preservation, and hard and soft constraints on the positions of the points in the mesh. We compare our method to previous operators and related algorithms, and prove that our curvature and Laplacian operators have several mathematically-desirable qualities that improve the appearance of the resulting surface. In consequence, the user can easily select the appropriate operator according to the desired type of fairing. Finally, we provide a series of examples to graphically and numerically demonstrate the quality of our results.

Variational shape approximation

A method for concise, faithful approximation of complex 3D datasets is key to reducing the computational cost of graphics applications. Despite numerous applications ranging from geometry compression to reverse engineering, efficiently capturing the geometry of a surface remains a tedious task. In this paper, we present both theoretical and practical contributions that result in a novel and versatile framework for geometric approximation of surfaces. We depart from the usual strategy by casting shape approximation as a variational geometric partitioning problem. Using the concept of geometric proxies, we drive the distortion error down through repeated clustering of faces into best-fitting regions. Our approach is entirely discrete and error-driven, and does not require parameterization or local estimations of differential quantities. We also introduce a new metric based on normal deviation, and demonstrate its superior behavior at capturing anisotropy.

Intrinsic Parameterizations of Surface Meshes

Parameterization of discrete surfaces is a fundamental and widely‐used operation in graphics, required, for instance, for texture mapping or remeshing. As 3D data becomes more and more detailed, there is an increased need for fast and robust techniques to automatically compute least‐distorted parameterizations of large meshes. In this paper, we present new theoretical and practical results on the parameterization of triangulated surface patches. Given a few desirable properties such as rotation and translation invariance, we show that the only admissible parameterizations form a two‐dimensional set and each parameterization in this set can be computed using a simple, sparse, linear system. Since these parameterizations minimize the distortion of different intrinsic measures of the original mesh, we call them Intrinsic Parameterizations. In addition to this partial theoretical analysis, we propose robust, efficient and tunable tools to obtain least‐distorted parameterizations automatically. In particular, we give details on a novel, fast technique to provide an optimal mapping without fixing the boundary positions, thus providing a unique Natural Intrinsic Parameterization. Other techniques based on this parameterization family, designed to ease the rapid design of parameterizations, are also proposed.

Dynamic real-time deformations using space & time adaptive sampling

This paper presents a robust, adaptive method for animating dynamic visco-elastic deformable objects that provides a guaranteed frame rate. Our approach uses a novel automatic space and time adaptive level of detail technique, in combination with a large-displacement (Green) strain tensor formulation. The body is partitioned in a non-nested multiresolution hierarchy of tetrahedral meshes. The local resolution is determined by a quality condition that indicates where and when the resolution is too coarse. As the object moves and deforms, the sampling is refined to concentrate the computational load into the regions that deform the most. Our model consists of a continuous differential equation that is solved using a local explicit finite element method. We demonstrate that our adaptive Green strain tensor formulation suppresses unwanted artifacts in the dynamic behavior, compared to adaptive mass-spring and other adaptive approaches. In particular, damped elastic vibration modes are shown to be nearly unchanged for several levels of refinement. Results are presented in the context of a virtual reality system. The user interacts in real-time with the dynamic object through the control of a rigid tool, attached to a haptic device driven with forces derived from the method.

Smoothed particles: a new paradigm for animating highly deformable bodies

This paper presents a new formalism for simulating highly deformable bodies with a particle system. Smoothed particles represent sample points that enable the approximation of the values and derivatives of local physical quantities inside a medium. They ensure valid and stable simulation of state equations that describe the physical behavior of the material.

Variational tetrahedral meshing

In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. As this energy is known to be the ∠1 distance between an isotropic quadratic function and its linear interpolation on the mesh, our minimization procedure generates well-shaped tetrahedra. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes through a series of examples.

Interactive geometry remeshing

We present a novel technique, both flexible and efficient, for interactive remeshing of irregular geometry. First, the original (arbitrary genus) mesh is substituted by a series of 2D maps in parameter space. Using these maps, our algorithm is then able to take advantage of established signal processing and halftoning tools that offer real-time interaction and intricate control. The user can easily combine these maps to create a control map --- a map which controls the sampling density over the surface patch. This map is then sampled at interactive rates allowing the user to easily design a tailored resampling. Once this sampling is complete, a Delaunay triangulation and fast optimization are performed to perfect the final mesh.As a result, our remeshing technique is extremely versatile and general, being able to produce arbitrarily complex meshes with a variety of properties including: uniformity, regularity, semi-regularity, curvature sensitive resampling, and feature preservation. We provide a high level of control over the sampling distribution allowing the user to interactively custom design the mesh based on their requirements thereby increasing their productivity in creating a wide variety of meshes.

Progressive compression for lossless transmission of triangle meshes

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.

Generalized barycentric coordinates on irregular polygons

Abstract In this paper we present an easy computation of a generalized form of barycentric coordinates for irregular, convex n-sided polygons. Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to ray tracing. Our new equations preserve many of the familiar properties of the triangular barycentric coordinates with an equally simple calculation, contrary to previous formulations. We illustrate the propert ies and behavior of these new generalized barycentric coordinates through several example applications.