Mesh Smoothing - pmp-book.org

1 2 EMESHINGFORIMPROVINGMESHQUALITY3 IMPLIFICATIONFORCOMPLEXITYREDUCTION3 URFACESMOOTHINGFORNOISEREMOVAL!NALYSISOF SURFACEQUALITY&REEFORMANDMULTIRESOLUTION MODELING2 EMOVALOFTOPOLOGICALANDGEOMETRICALERRORS) NPUT$ATA#!$2 ANGE 3 CAN4 OMOGRAPHY0 ARAMETERIZATIONMesh SmoothingMark PaulyMark PaulyMotivation Filter out high frequency components for noise removalDesbrun, Meyer, Schroeder, Barr: Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 99 Mark Pauly3 Motivation Advanced Filtering / Signal ProcessingGuskow, Sweldens, Schroeder: Multiresolution Signal Processing for Meshes, SIGGRAPH 99 Pauly, Kobbelt, Gross: Point-Based Multi-Scale Surface Representation, ACM TOG 2006 Mark Pauly4 Motivation Fair Surface DesignMark PaulyMotivation Hole-filling with energy-minimizing patches5 Mark Pauly6 Motivation Mesh deformationBotsch, Kobbelt.

2 An intuitive framework for real-time freeform modeling, SIGGRAPH 04 Kobbelt, Campagna, Vorsatz, Seidel: Interactive Multi-Resolution Modeling on Arbitrary Meshes, SIGGRAPH 98 Mark PaulyOutline Motivation Smoothing as Diffusion Smoothing as Energy Minimization Alternative Approaches7 Mark PaulyOutline Motivation Smoothing as Diffusion Spectral Analysis Laplacian Smoothing Curvature Flow Smoothing as Energy Minimization Alternative Approaches8 Mark Pauly9 Filter Design Assume high frequency components = noise Low-pass filterspatial domainfrequency domainlow passMark Pauly10 Filter Design Assume high frequency components = noise Low-pass filterreconstruction = filtered signalspatial domainMark PaulyFilter Design Assume high frequency components = noise Low-pass filter damps high frequencies (ideal.)

3 Cut off) , by convolution with Gaussian (spatial domain) =!multiply with Gaussian (frequency domain) Fourier Transform11 Mark Pauly12 Spectral Analysis and Filter Design Univariate: Fourier Analysis Example: Low-pass filter Damp (ideally cut off high frequencies) Multiply F with Gaussian (= convolve f with Gaussian) Are there "geometric frequencies"?spatial domainfrequency domainF( )=12 f(t)e i tdtMark Pauly13 Spectral Analysis and Filter Design Univariate: Fourier Analysis Generalization are eigenfunctions of the Laplacian use them as basis functions for geometry ei t= 2 t2ei t= 2ei tei tF( )=12 f(t)e i tdtMark PaulySpectral Analysis Eigenvalues of Laplacian frequenciesB.

4 Vallet, B. Levy. Spectral geometry processing with manifold harmonics. Technical report, INRIA-ALICE, PaulySpectral Analysis Low-pass filter reconstruction from eigenvectors associated with low frequenciesB. Vallet, B. Levy. Spectral geometry processing with manifold harmonics. Technical report, INRIA-ALICE, PaulySpectral Analysis Eigenvalues of Laplace matrix frequencies Low-pass filter reconstruction from eigenvectors associated with low frequencies Decomposition in frequency bands is used for mesh deformation often too expensive for direct use in practice!difficult to compute eigenvalues efficiently For Smoothing apply PaulyOutline Motivation Smoothing as Diffusion Spectral Analysis Laplacian Smoothing Curvature Flow Smoothing as Energy Minimization Alternative Approaches17 Mark Pauly18 Diffusion Diffusion equation tx= xdiffusion constantLaplace operatorMark Pauly19 Diffusion Diffusion equation tx= xdiffusion constantLaplace operatorMark Pauly20 Laplacian Smoothing Discretization of diffusion equation Leads to simple update rule iterate until convergenceexplicit Euler integration tpi= pip i=pi+ dt piMark Pauly21A Simple ExampleMark Pauly22A Simple ExampleMark Pauly23A Simple ExampleMark Pauly24A Simple ExampleMark

5 Pauly25A Simple ExampleMark Pauly26A Simple ExampleFlow of vertex positionsMark Pauly27 Laplacian Smoothing0 Iterations5 Iterations20 IterationsMark PaulyOutline Motivation Smoothing as Diffusion Spectral Analysis Laplacian Smoothing Curvature Flow Smoothing as Energy Minimization Alternative Approaches28 Mark Pauly29 Curvature Flow Curvature is independent of parameterization Flow equation We havemean curvaturesurface normalLaplace-Beltrami operator tp= 2 Hn Sp= 2 HnMark Pauly30 Curvature Flow Mean curvature flow use discrete Laplace-Beltrami operator (cot weights) Compare to uniform discretization of LaplacianUmbrellaLaplace-Beltramitangent ial driftvertices move onlyalong normal tp= SpMark Pauly31 ComparisonOriginal Umbrella Laplace-BeltramiMark PaulyOutline Motivation Smoothing as Diffusion Smoothing as Energy Minimization membrane energy thin-plate energy Alternative Approaches32 Mark Pauly33 Energy Minimization Penalize "un-aesthetic behavior" Measure fairness principle of the simplest shape physical interpretation Minimize energy functional examples.

6 Membrane / thin plate energyMark PaulyNon-Linear Energies Membrane energy (surface area) Thin-plate surface (curvature) Too simplify energies Sds minwith S=c S 21+ 22ds minwith S=c,n( S)=d34 Mark PaulyMembrane Surfaces Surface parameterization Membrane energy (surface area)p: IR2 IR3 pu 2+ pv 2dudv min35 Mark PaulyMembrane Surfaces Surface parameterization Membrane energy (surface area) Variational calculusp: IR2 IR3 pu 2+ pv 2dudv min p=036 Mark PaulyThin-Plate Surfaces Surface parameterization Thin-plate energy (curvature) Variational calculusp: IR2 IR3 puu 2+2 puv 2+ pvv 2dudv min 2p=037 Mark PaulyEnergy FunctionalsMembrane Sp=0 Thin Plate 2Sp=0 3Sp=038 Mark PaulyAnalysis Minimizer surfaces satisfy Euler-Lagrange PDE They are stationary surfaces of Laplacian flows Explicit flow integration corresponds to iterative solution of linear system kSp=0 p t= kSp39 Mark PaulyOutline Motivation Smoothing as Diffusion Smoothing as Energy Minimization Alternative Approaches40 Mark Pauly41 Alternative Approaches Anisotropic Diffusion Data-dependent Non-linear Normal filtering Smooth normal field and reconstruct (mesh editing)

7 Non-linear PDEs Avoid parameter dependence for fair surface design Bilateral Filteringdiffusion tensor tx=divD xMark Pauly42 Example of Bilateral FilteringJones, Durand, Desbrun: Non-iterative feature preserving mesh Smoothing , SIGGRAPH 2003 Mark Pauly43 Literature Ta u b i n : A signal processing approach to fair surface design, SIGGRAPH 1996 Desbrun, Meyer, Schroeder, Barr: Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, SIGGRAPH 99 Botsch, Kobbelt: An Intuitive Framework for Real-Time Freeform Modeling, SIGGRAPH 2004 Fleishman, Drori, Cohen-Or: Bilateral mesh denoising, SIGGRAPH 2003 Jones, Durand, Desbrun: Non-iterative feature preserving mesh Smoothing , SIGGRAPH 2003