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We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices.In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

We present a new globally smooth parameterization method for triangulated surfaces of arbitrary topology. Given two orthogonal piecewise linear vector elds dened over the input mesh (typically the estimated principal curvature directions), our method computes two piecewise linear periodic functions, aligned with the input vector elds, by minimizing an objective function. The bivariate function they dene is a smooth parameterization almost everywhere on the surface, except in the vicinity of singular vertices, edges and triangles, where the derivatives of the parameterization vanish. We extract a quadrilateral chart layout from the parameterization function and propose an automatic procedure to detect the singularities, and x them by splitting and re-parameterizing the containing charts. Our method can construct both quasi-conformal (angle preserving) and quasi-isometric (angle and area preserving) parameterizations. The more restrictive class of quasi-isometric parameterizations is constructed at the expense of introducing more singularities. The constructed parameterizations can be used for a variety of geometry processing applications. Since we can align the parameterization with the principal curvature directions, our result is particularly suitable for surface tting and remeshing.

One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces.In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object.Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing. MotivationsThese last few years, mesh parameterization is a topic for which much time and effort has been devoted to. Floater in his recent survey [10] reviews a large number of methods for objects with disc topology. More recently, the geometry processing community started to study methods that could be applied to objects of arbitrary topology, based for instance on holomorphic one-forms [14]. Such an interest for parameterization methods is justified by the fact that it facilitates attaching attributes to surfaces. For instance, such attributes may be physical properties simulated by a PDE solver. Adapting a finite-element method to this representation of surfaces is easy, and is often referred to as the master element method (see e.g. [20]). However, recent advances in PDE solving and in calculus suggest that a parameterization may not be the best representation of geometry and attributes attached to the geometry. For instance, to construct a hierarchical representation, the CHARMS method presented in [13] suggests to use function basis refinement instead of finite element refinement. This function-centric view leads to simpler computations. The external calculus and its discrete counterparts are also leading towards this direction (see e.g. Desbrun and Schroeder's course at Siggraph 2005).For this reason, rather than seeking for a global parameterization of the objects, we will investigate algorithms that create a function basis over an object of arbitrary topology. Surprisingly, as shown further, the underlying basic principles (Laplace operator) are present in a wide variety of disciplines of computer graphics and geometry processing.The remainder of this paper is organized as follows. Section 2 reviews spectral analysis used in the discrete setting. Section 3 explains the links with the continuous setting and the Laplace-Beltrami operator. Section 4 presents some possible applications in geometry processing.