Gaussian and Mean Curvature Planar points: Zero Gaussian curvature and zero mean curvature Tangent plane intersects surface at infinity points Gauss C

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1 Outline Shape Analysis Basics COS 526, Fall 21 Curvature PCA Distance Features Some slides from Rusinkiewicz Curvature Curvature Curvature κof a curve is reciprocal of radius of circle that best approximates it Defined at a point p in a direction w Line has κ= = Principal Curvatures Principal Curvatures The curvature at a point varies between some minimum and maximum these are the principal curvatures κ 1 and κ 2 They occur in the principal directions d 1 and d 2 which are perpendicular to each other Minimum Curvature κ 1 Maximum Curvature Maximum Curvature κ 2 1

2 Gaussian and Mean Curvature Planar points: Zero Gaussian curvature and zero mean curvature Tangent plane intersects surface at infinity points Gauss Curvature K=κ1 κ2 Mean Curvature H=½ H=½ (κ1 + κ2) Parabolic points: Zero Gaussian curvature, nonnon-zero mean curvature Tangent plane intersects surface along 1 curves Elliptical points: Positive Gaussian curvature Convex/concave depending on sign of mean curvature Tangent plane intersects surface at 1 point Hyperbolic points: Negative Gaussian curvature Tangent plane intersects surface along 2 curves Mesh Saliency: Motivated by models of perceptual salience Difference between mean curvature blurred with σ and blurred with 2σ Lee5 2

3 λ Outline Principal Component Analysis (PCA) Curvature PCA Distance Features Tensor voting Extract points {q i } in neighborhood Compute covariance matrix M Analyze eigenvalues and eigenvectors of M (via SVD) Eigenvectors are Principal Axes x x x y x z q n i 1 y x y y y z M = n = i 1 z x z y z z t M = USU λa S = Covariance Matrix Ax λb U = Bx λ c Cx A y By C y Az Bz C z Eigenvalues& Eigenvectors Principal Component Analysis (PCA) What Does PCA Tell Us? Tensor voting Extract points {q i } in neighborhood Compute covariance matrix M Analyze eigenvalues and eigenvectors of M (via SVD) x x x y x z q n i 1 y x y y y z M = n = i 1 z x z y z z t M = USU λa S = Covariance Matrix Ax λb U = Bx λ c Cx A y By C y Az Bz C z Eigenvalues& Eigenvectors Eigenvectors are Principal Axes of Inertia Eigenvalues are variances of the point distribution in those directions What Does PCA Tell Us? Helps differentiate nearly plane-like, like, from stick-like, from sphere-like, etc. What Does PCA Tell Us? Helps differentiate nearly plane-like, like, from stick-like, from sphere-like, etc. 2 / ( / ( λ 1+ λ 2 + λ 3) 3

4 What Does PCA Tell Us? Outline Helps us construct a local coordinate frame for every point Map ê 1 to X axis Map ê 2 to Y axis Map ê 3 to Z axis Curvature PCA Distance Features Distances on Meshes Distances on Meshes How close are two points p and q? Desirable properties: Surazhsky5 Geodesic Distance Geodesic Distance Length of shortest path between p and q on surface Can be computed exactly in O(n 2 logn) [Mitchell87] Often approximated with Dijkstra salgorithm algorithm on vertex graph Surazhsky5 Properties: 4

5 Diffusion Distance Diffusion Distance Amount of heat that diffuses from p to q in time t Related to probability of random walk starting at p arriving at q after time t Affected by all paths from p to q Sun9 Lafon/ Sun9 Diffusion Distance Diffusion Distance Can be computed by eigenanalysis of Laplacian Related to Euclidean distance in Spectral embedding Can be approximated for long times by first few eigenfunctions Properties: Diffusion Distance Biharmonic Distance Properties: Related to solution to biharmonic equations Can be computed by eigenanalysis of Laplacian 5

6 Biharmonic Distance Distance Comparison Properties: Biharmonic Geodesic Diffusion Biharmonic Geodesic Diffusion Distance Comparison Distance Comparison Biharmonic Geodesic Diffusion Biharmonic Geodesic Diffusion Outline Features Curvature PCA Distance Features Definition (Merriam-Webster) a prominent part or characteristic 6

7 Point Features Applications: Maintaining shape features as process mesh Matching shape features as align meshes Reasoning about part decomposition Selecting viewpoints Visualization etc. Algorithmic methods Iteratively choose furthest point Others? Analytic methods Extremaof DoG of curvature (saliency) Extrema of Gauss curvature Extrema of HKS Extrema of AGD etc. Analytic methods Many methods consider scale-space space persistence Zhang8 Zou8 Analytic methods Many methods consider scale-space space persistence Provides robustness to noise Still difficult to detect semantic points Zou9 Zou9 7

8 Your assignment 8