IFT LAPLACIAN APPLICATIONS. Mikhail Bessmeltsev

Transcription

1 IFT LAPLACIAN APPLICATIONS Mikhail Bessmeltsev

2 Rough Intuition You can learn a lot about a shape by hitting it (lightly) with a hammer!

3 Spectral Geometry What can you learn about its shape from vibration frequencies and oscillation patterns?

4 THE COTANGENT LAPLACIAN

5 Laplacian is sparse! Induced by the connectivity of the triangle mesh.

6 How can we use L? (useful properties of the Laplacian) In Computer Graphics and Geometry Modeling/Processing In Machine Learning

7 How can we use L? (useful properties of the Laplacian) In Computer Graphics and Geometry Modeling/Processing In Machine Learning

8 One Object, Many Interpretations Deviation from neighbors

9 One Object, Many Interpretations Decreasing E Images made by E. Vouga Dirichlet energy: Measures smoothness

10 One Object, Many Interpretations Vibration modes

11 Key Observation (in discrete case)

12 After (More) Trigonometry mage/formula in Functional Characterization of Instrinsic and Extrinsic Geometry, TOG 2017 (Corman et al.) Laplacian only depends on edge lengths

13 Isometry Bending without stretching.

14 Lots of Interpretations Global isometry Local isometry

15 Intrinsic Techniques Isometry invariant

16 Isometry Invariance: Hope

17 Isometry Invariance: Reality Few shapes can deform isometrically

18 Isometry Invariance: Reality isometries? Few shapes can deform isometrically

19 Useful Fact

20 Beware Image from: Raviv et al. Volumetric Heat Kernel Signatures. 3DOR Not the same.

21 Why Study the Laplacian? Encodes intrinsic geometry Edge lengths on triangle mesh, Riemannian metric on manifold Multi-scale Filter based on frequency Geometry through linear algebra Linear/eigenvalue problems, sparse positive definite matrices Connection to physics Heat equation, wave equation, vibration,

22 How can we use L? (useful properties of the Laplacian) In Computer Graphics and Geometry Modeling/Processing In Machine Learning

23 Example Task: Shape Descriptors Pointwise quantity

24 Descriptor Tasks Characterize local geometry Feature/anomaly detection Describe point s role on surface Symmetry detection, correspondence

25 Descriptors We ve Seen Before Gaussian and mean curvature

26 Desirable Properties Distinguishing Provides useful information about a point Stable Numerically and geometrically Intrinsic No dependence on embedding

27 Intrinsic Descriptors Invariant under Rigid motion Bending without stretching

28 Intrinsic Descriptor Theorema Egregium ( Totally Awesome Theorem ): Gaussian curvature is intrinsic.

29 End of the Story? Second derivative quantity

30 Desirable Properties Incorporates neighborhood information in an intrinsic fashion Stable under small deformation

31 Shape Context

32 Shape Context + Translational invariance + Scale invariance - Rotational invariance

33 Shape Context Idea! Compute angles relative to the tangent + Translational invariance + Scale invariance + Rotational invariance

34 Connection to Physics Heat equation fall/lectureslides/11_shape_matching.pdf

35 Intrinsic Observation Heat diffusion patterns are not affected if you bend a surface.

36 Global Point Signature Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Rustamov, SGP 2007

37 Global Point Signature If surface does not self-intersect, neither does the GPS embedding. Proof: Laplacian eigenfunctions span LL 22 (ΣΣ); if GPS(p)=GPS(q), then all functions on ΣΣ would be equal at p and q.

38 Global Point Signature GPS is isometry-invariant. Proof: Comes from the Laplacian.

39 Drawbacks of GPS Assumes unique λ s Potential for eigenfunction switching Nonlocal feature

40 PDE Applications of the Laplacian Heat equation fall/lectureslides/11_shape_matching.pdf

41 PDE Applications of the Laplacian Image courtesy G. Peyré Wave equation

42 PDE Applications of the Laplacian Image courtesy G. Peyré Wave equation

43 Solutions in the LB Basis Heat equation

44 Heat Kernel Signature (HKS) Continuous function of tt [00, ) How much heat diffuses from x to itself in time t?

45 Heat Kernel Signature (HKS) A concise and provably informative multi-scale signature based on heat diffusion Sun, Ovsjanikov, and Guibas; SGP 2009

46 Heat Kernel Signature (HKS) Good properties: Isometry-invariant Multiscale Not subject to switching Easy to compute Related to curvature at small scales

47 Heat Kernel Signature (HKS) Bad properties: Issues remain with repeated eigenvalues Theoretical guarantees require (near-)isometry

48 Wave Kernel Signature (WKS) Average probability over time that particle is at x. Initial energy distribution The Wave Kernel Signature: A Quantum Mechanical Approach to Shape Analysis Aubry, Schlickewei, and Cremers; ICCV Workshops 2012

49 Wave Kernel Signature (WKS) HKS WKS vision.in.tum.de/_media/spezial/bib/aubry-et-al-4dmod11.pdf

50 Wave Kernel Signature (WKS) Good properties: [Similar to HKS] Localized in frequency Stable under some non-isometric deformation Some multi-scale properties

51 Wave Kernel Signature (WKS) Bad properties: [Similar to HKS] Can filter out large-scale features

52 Many Others Lots of spectral descriptors in terms of Laplacian eigenstructure.

53 Combination with Machine Learning Learn f rather than defining it Learning Spectral Descriptors for Deformable Shape Correspondence Litman and Bronstein; PAMI 2014

54 Application: Feature Extraction Maxima of k t (x,x) over x for large t. A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion Sun, Ovsjanikov, and Guibas; SGP 2009 Feature points

55 Preview: Correspondence 10.pdf

56 Descriptor Matching Simply match closest points in descriptor space.

57 Descriptor Matching Problem Symmetry

58 Heat Kernel Map How much heat diffuses from p to x in time t? One Point Isometric Matching with the Heat Kernel Ovsjanikov et al. 2010

59 Heat Kernel Map Theorem: Only have to match one point! One Point Isometric Matching with the Heat Kernel Ovsjanikov et al. 2010

60 Self-Map: Symmetry Intrinsic symmetries become extrinsic in GPS space! Global Intrinsic Symmetries of Shapes Ovsjanikov, Sun, and Guibas 2008 Discrete intrinsic symmetries

61 All Over the Place Laplacians appear everywhere in shape analysis and geometry processing.

62 Biharmonic Distances Biharmonic distance Lipman, Rustamov & Funkhouser, 2010

63 Geodesic Distances Geodesics in heat Crane, Weischedel, and Wardetzky; TOG 2013

64 Finding geodesics Crane, Weischedel, and Wardetzky. Geodesics in Heat. TOG, 2013.

65 Mean Curvature Flow

66 Mean Curvature Flow Implicit fairing of irregular meshes using diffusion and curvature flow Desbrun et al., 1999

67 Mean Curvature Flow Changes at each time step Implicit fairing of irregular meshes using diffusion and curvature flow Desbrun et al., 1999

68 Recall:

69 Another fairing Screened Poisson Equation

70 Useful Technique Choice: Evaluate at time T nconditionally stable, but not necessarily accurate for large T! Implicit time stepping

71 Parameterization: Harmonic Map Multiresolution analysis of arbitrary meshes Eck et al., 1995 (and many others!)

72 Others Shape retrieval from Laplacian eigenvalues Shape DNA [Reuter et al., 2006] Quadrangulation Nodal domains [Dong et al., 2006] Surface deformation As-rigid-as-possible [Sorkine & Alexa, 2007]

73 How can we use L? (useful properties of the Laplacian) In Computer Graphics and Geometry Modeling/Processing In Machine Learning

74 Semi-Supervised Learning Semi-supervised learning using Gaussian fields and harmonic functions Zhu, Ghahramani, & Lafferty 2003

75 Semi-Supervised Technique irichlet energy Linear system of equations (Poisson)

76 Related Method Step 1: Build k-nn graph Step 2: Compute p smallest Laplacian eigenvectors Step 3: Solve semi-supervised problem in subspace Using Manifold Structure for Partially Labelled Classification Belkin and Niyogi; NIPS 2002

77 Manifold Regularization Loss function Regularizer Dirichlet energy Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples Belkin, Niyogi, and Sindhwani; JMLR 2006

78 Examples of Manifold Regularization Laplacian-regularized least squares (LapRLS) Laplacian support vector machine (LapSVM) On Manifold Regularization Belkin, Niyogi, Sindhwani; AISTATS 2005

79 Diffusion Maps Embedding from first k eigenvalues/vectors: Roughly: ΨΨ tt xx ΨΨ tt yy is probability that x, y diffuse to the same point in time t. Diffusion Maps Coifman and Lafon; Applied and Computational Harmonic Analysis, 2006 Robust to sampling and noise Image from