Over the last decade, 3D shape analysis has become a topic of increasing interest in the computer vision community. Nevertheless, when attempting to apply current image analysis methods to 3D shapes (feature-based description, registration, recognition, indexing, etc.) one has to face fundamental differences between images and geometric objects. Shape analysis poses new challenges that are non-existent in image analysis. The purpose of the tutorial is to overview the foundations of shape analysis and to formulate state-of-the-art theoretical and computational methods for shape description based on their intrinsic geometric properties. The emerging field of diffusion geometry provides a generic framework for many methods in the analysis of geometric shapes and objects. The tutorial will present in a new light the problems of shape analysis based on diffusion geometric constructions such as manifold embeddings using the Laplace-Beltrami and heat operator, heat kernel local descriptors, diffusion and compute-time metrics.
- Local and global diffusion geometric structures in shape analysis
Dimensions of media - Shapes vs images - Shape analysis and synthesis - Invariant similarity and correspondence - Metric model - Isometry - Laplace-Beltrami operator - To see the sound - Shape DNA - Diffusion kernel - Spectral properties - Heat kernel - Diffusion distance - Scale invariance - Commute-time distance - Spectral shape distance - Heat kernel signatures - ShapeGoogle - Discrete Laplacian
- Discrete diffusion in graphs
Graph Laplacian - Discrete heat operator - Spectral properties - PCA - Dimensionality reduction - Shape analysis applications
- M. M. Bronstein, A. M. Bronstein, "Shape recognition with spectral distances", IEEE Trans. Pattern Analysis and Machine Intelligence (PAMI), 2010.
- A. M. Bronstein, M. M. Bronstein, R. Kimmel, Numerical geometry of non-rigid shapes, Springer, 2008
- Annotated bibliography