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Princeton COS 598B - Spectral Processing of Point-Sampled Geometry

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Spectral Processing of Point-Sampled GeometryMark Pauly Markus GrossAbstractWe present a new framework for processing point-sampled objectsusing spectral methods. By establishing a concept of local frequen-cies on geometry, we introduce a versatile spectral representationthat provides a rich repository of signal processing algorithms.Based on an adaptive tesselation of the model surface into regu-larly resampled displacement fields, our method computes a set ofwindowed Fourier transforms creating a spectral decomposition ofthe model. Direct analysis and manipulation of the spectral coeffi-cients supports effective filtering, resampling, power spectrumanalysis and local error control. Our algorithms operate directly onpoints and normals, requiring no vertex connectivity information.They are computationally efficient, robust and amenable to hard-ware acceleration. We demonstrate the performance of our frame-work on a selection of example applications including noiseremoval, enhancement, restoration and subsampling.Keywords: Signal processing, spectral filtering, subsampling,Fourier transform, point-based representations1 IntroductionToday’s range sensing devices are capable of producing highlydetailed surface models that contain hundreds of millions ofsample points. Due to a variety of physical effects and limitationsof the model acquisition procedure, raw range datasets are prone tovarious kinds of noise and distortions, requiring sophisticated pro-cessing methods to improve the model quality. In spite of recentadvances made in mesh optimization, traditional mesh processingalgorithms approach their limits, since triangle primitives implic-itly store information about local surface topology including ver-tex valence or adjacency. This leads to a substantial additionaloverhead in computation time and memory costs. With increasingmodel size we thus experience a shift from triangle mesh represen-tations towards purely point-based surface descriptions. Forinstance, recent work concentrated on point-based rendering pipe-lines [18, 19], where point samples without connectivity are pro-posed as rendering primitives. Surprisingly, however, little workhas been done so far on direct processing or manipulation of point-sampled geometry. In this paper we present a new framework forspectral analysis and processing of point-sampled objects. Themethod operates directly on irregular point sets with normals anddoes not require any a priori connectivity information. Our frame-work extends so-called windowed Fourier transforms - a conceptbeing well known from signal processing - to geometry.The Fourier transform is a powerful and widely used tool fordata analysis and manipulation. In particular, image processingtechniques successfully exploit frequency representations toimplement a variety of advanced spectral processing algorithmscomprising noise removal, enhancement, feature detection andextraction, up/down-sampling, etc. [7]. Extending this approach togeneral geometric models is difficult due to a number of intrinsiclimitations of the conventional Fourier transform: First, it requiresa global parameterization on which the basis functions are defined.Second, most FT algorithms require a regular sampling pattern[17]. These prerequisites are usually not satisfied by common dis-crete geometry, rendering the standard Fourier transforminoperable. A further limitation of traditional Fourier representa-tions is the lack of spatial localization making it impractical forlocal data analysis. We will show how these limitations can beovercome and present a generalization of the windowed FT to gen-eral 2-manifolds. The basic idea behind our framework is to pre-process the raw irregular point cloud into a model representationthat describes the object surface with a set of regularly resampledheight fields. These surface patches form “windows” in which wecompute a discrete Fourier transform to obtain a set of local fre-quency spectra. Although being confined to individual surfacepatches, our windowed FT provides a powerful and versatilemechanism for both local and global processing. The concept offrequency on point-sampled geometry gives us access to the vastspace of sophisticated spectral methods resulting from tens ofyears of research in signal processing. In this paper we will focus on two classes of such methods:Spectral filtering and resampling. We will point out how sophisti-cated filtering operations can be implemented elegantly by analyz-ing and modifying the coefficients of the frequency spectrum.Possible applications include noise removal, analysis of the sur-face microstructure and enhancement. Further we present a fastalgorithm for adaptively resampling point-based geometry, usingthe spectral representation to determine optimal sampling rates.This method is particularly useful for reducing the complexity ofoverly dense point-sampled models. By using FFT and other signalprocessing algorithms our framework is efficient in computationand memory costs, amenable to hardware acceleration and allowsus to process hundreds of millions of points on contemporary PCs.Figure 1: Spectral processing pipeline. Processing stages are depicted as rectangles, rounded boxes represent input/output data of each stage. Gray back-ground color indicates the preprocessing phase.Regular Grid FourierSpectrum ModifiedSpectrumFiltered PatchCreate Patch LayoutSDADFTSpectralAnalysisInverseDFT Re-samplingProcessed PatchesInput DataRaw Patchesinal [email protected] [email protected] ZürichPermission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copiesare not made or distributed for profit or commercial advantage and thatcopies bear this notice and the full citation on the first page. To copyotherwise, to republish, to post on servers or to redistribute to lists,requires prior specific permission and/or a fee.ACM SIGGRAPH 2001, 12-17 August 2001, Los Angeles, CA, USA© 2001 ACM 1-58113-374-X/01/08...$5.003791.1 Previous WorkExtending the concept of frequency onto geometry has gainedincreasing attention over the last years. Conceptually, this general-ization can be accomplished by the eigenfunctions of the Lapla-cian. Taubin [20] pioneered spectral methods for irregular meshesusing a discrete Laplacian to implement iterative Gaussiansmoothing for triangle meshes. This method has later beenimproved by Desbrun et al. [4] who tackled the


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Princeton COS 598B - Spectral Processing of Point-Sampled Geometry

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