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Spectral Geometry Processing with Manifold Harmonics

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Spectral Geometry Processing with Manifold Harmonics Author manuscript, published in "Computer Graphics Forum 27, 2 (2008) 251-260" DOI : 10.1111/j.1467-8659.2008.01122.x EUROGRAPHICS 2008 / G. Drettakis and R. Scopigno Volume 27 (2008), Number 2 (Guest Editors) Spectral Geometry Processing with Manifold Harmonics B. Vallet1 and B. Lévy1 1ALICE team, INRIA Abstract We present an explicit method to compute a generalization of the Fourier Transform on a mesh. It is well known that the eigenfunctions of the Laplace Beltrami operator (Manifold Harmonics) define a function basis allowing for such a transform. However, computing even just a few eigenvectors is out of reach for meshes with more than a few thousand vertices, and storing these eigenvectors is prohibitive for large meshes. To overcome these limitations, 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. We also propose a limited-memory filtering algorithm, that does not need to store the eigenvectors. Using this latter algorithm, specific frequency bands can be filtered, without needing to compute the entire spectrum. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering. These technical achievements are supported by a solid yet simple theoretic framework based on Discrete Exterior Calculus (DEC). In particular, the issues of symmetry and discretization of the operator are considered with great care. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling, Hierarchy and geometric transformations 1. Introduction the one described in [PG01] for point sets. The main dif- ference is that no resampling nor segmentation is needed: 3D scanning technology easily produces computer represen- the Fourier transform is computed directly on the mesh. The tations from real objects. However, the acquired geometry pipeline is outlined in Figure 1: often presents some noise that needs to be filtered out. More • A: given a triangulated mesh with n vertices, compute a generally, it may be suitable to enhance some details while function basis Hk,k = 1...m that we call the Manifold removing other ones, depending on their sizes (spatial fre- th inria-00331894, version 1 - 20 Oct 2008 Harmonics Basis (MHB). The k element of the MHB quencies). In his seminal paper, Taubin [Tau95] showed that is a piecewise linear function given by its values Hk at the formalism of signal processing can be successfully ap- i vertices i of the surface; plied to geometry processing. His approach is based on the similarity between the eigenvectors of the graph Laplacian • B: once the MHB is computed, transform the geometry and the basis functions used in the discrete Fourier trans- into frequency space by computing the Manifold Har- form. This Fourier function basis enables a given signal to be monic Transform (MHT) of the geometry, that is to say decomposed into a sum of sine waves of increasing frequen- three vectors of coefficients [x˜1,x˜2,...x˜m], [y˜1,y˜2,...y˜m], cies. He used this analogy as a theoretical tool to design and and [z˜1,z˜2,...z˜m]. analyse approximations of low-pass filters. Several variants • C: apply a frequency space filter F(ω) by multiplying of this approach were then suggested, as discussed below. each (x˜k,y˜k,z˜k) by F(ωk), where ωk denotes the fre- k In this paper, instead of only using Fourier analysis as a the- quency associated with H ; oretical tool to analyse approximations of filters, our idea is • D: finally, transform the object back into geometric space to explicitly implement its generalization to surfaces of ar- by applying the inverse MHT. bitrary topology, and use this to achieve interactive general convolution filtering. Our processing pipeline is similar to c 2008 The Author(s) Journal compilation c 2008 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. B. Vallet & B. Lévy / Spectral Geometry Processing with Manifold Harmonics Figure 1: Processing pipeline of our method: A: Compute the Manifold Harmonic Basis (MHB) of the input triangulated mesh. B: Transform the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. Note that this approach shares some similarities with sig- ied, motivated by the large number of its applications, that nal processing with spherical harmonics or with the Dis- include parameterization, remeshing, compression, recon- crete Cosine Transform. The main difference is that our ap- struction and minimal surfaces. The eigenfunctions of the proach does not require any resampling of the original sur- Laplacian are known to define a Fourier-like function ba- face. However, our MHB depends on the surface, and there- sis [ZvKD07], that can be used to define spectral analysis on fore requires to be precomputed (step A). Once the MHB manifolds. We will now review some of the most significant is known, the subsequent stages of the pipeline can be very definitions and applications of discrete Laplacians, focusing efficiently computed. This allows the solution to be interac- mainly on the works related with spectral analysis and geo- tively updated when the F(ω) filter is modified by the user. metric filtering. Combinatorial Laplacians: A combinatorial Laplacian Contributions solely depends on the connectivity of the mesh. Spectral The main contribution of this paper is an efficient numeri- analysis of the combinatorial graph Laplacian was first used cal mechanism to compute the eigenfunctions of the Lapla- by Taubin [Tau95] to approximate low pass filters. A basis cian that overcomes the current limits (thousands vertices) of eigenfunctions of the graph Laplacian was used by Karni by several orders of magnitude (up to a million vertices). The et al. [KG00] for geometry compression. Zhang [Zha04] eigenfunctions are computed band by band based on spectral studies several variants of combinatorial laplacians and their transforms and an efficient eigensolver, and an out-of-core properties for spectral geometry processing and JPEG-like inria-00331894, version 1 - 20 Oct 2008 implementation that can compute thousands of eigenvectors mesh compression. However, as pointed-out in [MDSB03], for meshes with up to a million vertices. We also propose the analogy between the graph Laplacian and the discrete a limited-memory filtering algorithm, that does not need to cosine transform supposes a uniform sampling of the mesh. store the eigenvectors. Once computed, the MHB allows for Moreover, different embeddings of the same graph yield the interactive filtering on meshes which we demonstrate ap- same eigenfunctions, and two different meshings of the same plied to geometry. These computations use a clean, rigorous object yield different eigenfunctions which is problematic as yet simple DEC-based framework which clarifies the issues shown in Figure 2. of weighting and symmetry of the discrete cotangent Lapla- cian. In particular, the symmetry guaranties that the eigen- Geometric Laplacians and applications: More geome- functions form an orthonormal "Manifold Harmonics Basis" try can be injected into the definition of a discrete Lapla- (MHB) with positive eigenvalues. cian through the ubiquitous cotan weights [PP93,MDSB03]. These weights can also be derived from Finite Element Mod- eling (FEM) such as done in [WBH∗07], and they converge Previous Work to the continuous Laplacian under certain conditions as ex- The discretization of the Laplacian operator plays a central plained in [HPW06] and [AFW06]. Reuter et al. [RWP05] role in geometry processing and has been extensively stud- also use FEM to compute the spectrum (i.e. the eigenval- c 2008 The Author(s) Journal compilation c 2008 The Eurographics Association and Blackwell Publishing Ltd. B. Vallet & B. Lévy / Spectral Geometry Processing with Manifold Harmonics Figure 2: Filtering an irregularly sampled surface (twice denser on the right half) with different discrete laplacians. Combina- 0 0 torial Laplacian (A), unweighted cotan cot(βi j)+cot(βi j) (B) and symmetrized weighted cotan (cot(βi j)+cot(βi j))/(Ai +A j) 0 (D) produce deformed results (right leg is bigger). Weighted cotan (cot(βi j) + cot(βi j))/Ai are not symmetric which does not 0 p allow for correct reconstruction (C). Only our symmetric weights (cot(βi j)+cot(βi j))/ AiA j are fully mesh-independent (E). ues) of a mesh, which provides a signature for shape clas- angle shapes) and outer fairness (surface smoothness), by sification. The cotan weights were also used in [DBG∗06] using a combination of the combinatorial Laplacian and the to compute an eigenfunction to steer a quad-remeshing pro- discrete Laplace-Beltrami operator. cess. The cotan weights alone are still dependent on the sam- pling as shown in Figure 2-B, so they are usually weighted Spectral transform: To directly implement the spectral by the one ring or dual cell area of each vertex, which makes transform on manifolds, several methods consist in putting them loose their symmetry. As a consequence, they are im- the input surface in one-to-one correspondence with a sim- proper for spectral analysis (2-C). An empirical symmetriza- pler domain [ZBS04], or to partition it into a set of sim- tion was proposed in [Lev06] (see Figure 2-D). We clarify pler domains [LSS∗98, PG01] on which it is easier to de- these issues based on a rigorous DEC formulation, and re- fine a frequency space. Note that these methods generally cover symmetry by expressing the operator in a proper basis. need to resample the geometry, with the exception of Mousa This ensures that its eigenfunctions are both geometry aware et al.
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