Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computing Size Functions from Edge Maps
International Journal of Computer Vision
ACM Computing Surveys (CSUR)
Content-Based Image Retrieval at the End of the Early Years
IEEE Transactions on Pattern Analysis and Machine Intelligence
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Rotation invariant spherical harmonic representation of 3D shape descriptors
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
The Use of Size Functions for Comparison of Shapes Through Differential Invariants
Journal of Mathematical Imaging and Vision
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Retrieval of trademark images by means of size functions
Graphical Models - Special issue on the vision, video and graphics conference 2005
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
k-dimensional Size Functions for Shape Description and Comparison
ICIAP '07 Proceedings of the 14th International Conference on Image Analysis and Processing
Size functions for comparing 3D models
Pattern Recognition
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
A survey of content based 3D shape retrieval methods
Multimedia Tools and Applications
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Computing Multidimensional Persistence
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
A concise and provably informative multi-scale signature based on heat diffusion
SGP '09 Proceedings of the Symposium on Geometry Processing
Shape google: Geometric words and expressions for invariant shape retrieval
ACM Transactions on Graphics (TOG)
The hitchhiker's guide to the galaxy of mathematical tools for shape analysis
ACM SIGGRAPH 2012 Courses
Multi-scale approximation of the matching distance for shape retrieval
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
Pattern Recognition Letters
Comparison of persistent homologies for vector functions: From continuous to discrete and back
Computers & Mathematics with Applications
Suspension models for testing shape similarity methods
Computer Vision and Image Understanding
Comparing shapes through multi-scale approximations of the matching distance
Computer Vision and Image Understanding
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Size Theory has proven to be a useful geometrical/topological approach to shape comparison. Originally introduced by considering 1-dimensional properties of shapes, described by means of real-valued functions, it has recently been generalized to taking into account multi-dimensional properties coded by functions valued in R^k. This has led to the introduction of a shape descriptor called k-dimensional size function, and the k-dimensional matching distance to compare size functions. This paper presents new theoretical results about the 2-dimensional matching distance, leading to the formulation of an algorithm for its approximation up to an arbitrary error threshold. Experiments on 3D object comparison are shown to discuss the efficacy and effectiveness of the algorithm.