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Introduction (.ppt, 7.58 MB)
Welcome to non-rigid world - Rock, Paper and Scissors - Invariance - Similarity and correspondence - Analysis and synthesis - Landscape - Course in a nutshell
- A taste of geometry (.ppt, 14.6 MB)
Distances - Metric - Metric balls - Topology - Connectedness - Compactness - Convergence - Continuity - Homeomorphisms - Lipschitz continuity - Isometries - Isometry groups - Symmetry - Dilation - Distortion - Almost isometries - Length spaces - Completeness - Restricted and induced metric - Convexity - Manifolds - Charts and atlases - Smooth manifolds - Manifolds with boundary - Embedded surfaces - Tangent plane - Normal - Orientability - First fundamental form - Intrinsic geometry - Area - Curvature - Principal curvatures - Directional derivative - Shape operator - Second fundamental form - Mean and Gaussian curvatures - Extrinsic geometry - Riemannian geometry - Nash's embedding theorem - Bending and rigidity - Intrinsic definition of Gaussian curvature - Theorema egregium - Gauss-Bonnet formula - Intrinsic invariants
- Discrete geometry (.ppt, 9.39 MB)
Sampling - Farthest point sampling - Voronoi tessellation - Sufficient sampling density conditions - Optimal sampling - Centroidal Voronoi tessellation - Lloyd-Max algorithm - Neighborhood - Connectivity - Delaunay tessellation - Triangular meshes - Barycentric coordinates - Manifold meshes - Geometry images - Geometric validity - Skeleton - Local feature size - Approximation quality - Schwarz lantern
- Shortest paths and fast marching methods (.ppt, 10.5 MB)
Shapes as graphs - Shortest paths in graphs - Bellman's principle of optimality - Dijkstra's algorithm - Metrication errors - Distance map on surfaces - Intrinsic and extrinsic gradients - Eikonal equation - Viscosity solution - Fast marching methods - Update step - Causality condition - Monotonicity condition - Fast marching on obtuse meshes - Eikonal equation on parametric surfaces - Fast marching on parametric surfaces - Raster scan fast marching - Parallel marching - Minimal geodesic
- Optimization (.ppt, 3.43 MB)
Optimization problems - Local and global minimum - Convex functions - One-dimensional optimality conditions - Gradient - Gradient of a matrix function - Hessian - Optimality conditions - Optimization algorithms - Stopping criteria - Line search - Armijo rule - Steepest descent - Condition number - Q-norm - Preconditioning - Newton method - Non-convex optimization - Iterative majorization - Constrained optimization - Lagrange multipliers - KKT conditions - Penalty methods
- In the rigid kingdom (.ppt, 9.36 MB)
Cinderela's tale - Exstrinsic shape similarity - Principal components - Geometric moments - Signal decomposition intuition - Moments distance - Other moments - Iterative closest point algorithms - Shape-to-shape distance - Point-to-point distance - Point-to-plane distance - Second-order point-to-shape distance - Closest points - Approximate nearest neighbors - Convergence - Enter numerical optimization - Local quadratic approximant - Iterative closest point algorithm revisited
- Multidimensional scaling (.ppt, 17.1 MB)
Cinderella 2.0 - If it doesn't fit, you must acquit! - Metric model - Intrinsic vs extrinsic similarity - Canonical forms - Mapmaker's problem - Linial's example - Minimum distortion embedding - Multidimensional scaling - Stress - Matrix expression of L2-stress - LS-MDS - Gradient of L2-stress - Complexity - Majorizing inequality - Iterative majorization - SMACOF algorithm - Multiresolution MDS - Correction - Modified stress - Two-grid MDS - Multigrid MDS - A trouble with topology - Topological noise - Joint intrinsic/extrinsic similarity - How to fit a glove? - Computation of joint similarity
- Spectral methods (.ppt, 3.6 MB)
Gram matrices - Classic MDS - Local methods - Laplacian matrix - Minimum eigenvalue problems - Laplacian eigenmaps - Laplace-Beltrami operator - Chladni plates - Laplace-Beltrami spectrum - Shape DNA - To hear the shape of the drum - GPS embedding - Discrete Laplace-Beltrami operator - No free lunch
- Non-Euclidean embedding (.ppt, 6.3 MB)
Why non-Euclidean? - Non-Euclidean MDS - Spherical geometry - Spherical embedding - Generalized multidimensional scaling - Local representation - Geodesic distance approximation - Un ballo a quattro passi - Minimization algorithm - Quadratic stress - How to move to adjacent triangles? - Point on edge - Point on vertex - MDS vs GMDS - Multiresolution
- Isometry-invariant similarity (.ppt, 4.7 MB)
Equivalence - Similarity - Isometry-invariant distance - Discrete isometry-invariant distance - Canonical forms distance - Gromov-Hausdorff distance - Discrete Gromov-Hausdorff distance - Connection to ICP distance - Connection to canonical form distance - Self-similarity (symmetry) - Symmetry: extrinsic vs. intrinsic
- Partial similarity (.ppt, 12.7 MB)
Greek mythology - Partial similarity - Human vision example - Visual agnosia - Recognition by parts - Significance - Multicriterion optimization - Pareto optimality - Set-valued partial similarity - Order relations - Scalar-valued partial similarity - Characteristic functions - Fuzzy sets - Partial similarity computation - Intrinsic partial similarity - Extrinsic partial similarity - Not only size matters - Boundary regularization - Mumford-Shah functional - Regularized partial similarity
- Non-rigid correspondence and calculus of shapes (.ppt, 13.9 MB)
Natural correspondence - Correspondence between curves - Invariant parametrization - Canonical forms - Image processing insight - Intrinsic regularization - Cauchy-Green tensor - Physical insight - Harmonic maps - Minimum-distortion correspondence - Texture transfer - Calculus of shapes - Temporal super-resolution - Motion-compensated interpolation - Texture substitution - Metamorphing - Face caricaturization - Killing field - As isometric as possible morph
- Three-dimensional face recognition (.ppt, 9.64 MB)
Biometrics in the age of Patriarchs - First face recognition - Face recognition today - Recognition accuracy - What is a face? - The curse of expressions - Isometric model of expressions - How to canonize a person? - Telling identical twins apart - Comparing photometric properties - Spherical embedding
- Conclusion (.ppt, 757 KB)
Non-rigid world - Three pillars - Messages to take home
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