Short Communication to SMI 2011: Affine-invariant geodesic geometry of deformable 3D shapes

Authors:
Dan Raviv;Alexander M. Bronstein;Michael M. Bronstein;Ron Kimmel;Nir Sochen
Affiliations:
Department of Computer Science, Technion, Israel;Department of Electrical Engineering, Tel Aviv University, Israel;Institute of Computational Science, Faculty of Informatics, Universití della Svizzera Italiana, Lugano, Switzerland;Department of Computer Science, Technion, Israel;Department of Applied Mathematics, Tel Aviv University, Israel
Venue:
Computers and Graphics
Year:
2011

Citing 13
Cited 3

Affine Invariant Flows in the Beltrami Framework

Journal of Mathematical Imaging and Vision
Fast exact and approximate geodesics on meshes

ACM SIGGRAPH 2005 Papers
A Theoretical and Computational Framework for Isometry Invariant Recognition of Point Cloud Data

Foundations of Computational Mathematics
A Comparison of Affine Region Detectors

International Journal of Computer Vision
Short note: O(N) implementation of the fast marching algorithm

Journal of Computational Physics
Efficient Computation of Isometry-Invariant Distances Between Surfaces

SIAM Journal on Scientific Computing
Parallel algorithms for approximation of distance maps on parametric surfaces

ACM Transactions on Graphics (TOG)
Symmetry of Shapes Via Self-similarity

ISVC '08 Proceedings of the 4th International Symposium on Advances in Visual Computing, Part II
Partial intrinsic reflectional symmetry of 3D shapes

ACM SIGGRAPH Asia 2009 papers
Global intrinsic symmetries of shapes

SGP '08 Proceedings of the Symposium on Geometry Processing
Gromov-Hausdorff stable signatures for shapes using persistence

SGP '09 Proceedings of the Symposium on Geometry Processing
On bending invariant signatures for surfaces

IEEE Transactions on Pattern Analysis and Machine Intelligence
A general framework for low level vision

IEEE Transactions on Image Processing

Affine-invariant curvature estimators for implicit surfaces

Computer Aided Geometric Design
Affine-invariant photometric heat kernel signatures

EG 3DOR'12 Proceedings of the 5th Eurographics conference on 3D Object Retrieval
Equi-affine invariant geometries of articulated objects

Proceedings of the 15th international conference on Theoretical Foundations of Computer Vision: outdoor and large-scale real-world scene analysis

Quantified Score

Hi-index	0.00

Visualization

Abstract

Natural objects can be subject to various transformations yet still preserve properties that we refer to as invariants. Here, we use definitions of affine-invariant arclength for surfaces in R^3 in order to extend the set of existing non-rigid shape analysis tools. We show that by re-defining the surface metric as its equi-affine version, the surface with its modified metric tensor can be treated as a canonical Euclidean object on which most classical Euclidean processing and analysis tools can be applied. The new definition of a metric is used to extend the fast marching method technique for computing geodesic distances on surfaces, where now, the distances are defined with respect to an affine-invariant arclength. Applications of the proposed framework demonstrate its invariance, efficiency, and accuracy in shape analysis.