Computing Size Functions from Edge Maps
International Journal of Computer Vision
Proceedings of the Second Joint European - US Workshop on Applications of Invariance in Computer Vision
Topological persistence and simplification
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Topologies for the digital spaces Z2 and Z3
Computer Vision and Image Understanding
The Use of Size Functions for Comparison of Shapes Through Differential Invariants
Journal of Mathematical Imaging and Vision
Morse Homology Descriptor for Shape Characterization
ICPR '04 Proceedings of the Pattern Recognition, 17th International Conference on (ICPR'04) Volume 4 - Volume 04
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
Families of cut-graphs for bordered meshes with arbitrary genus
Graphical Models
Stability of Persistence Diagrams
Discrete & Computational Geometry
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Shape recognition via an a contrario model for size functions
ICIAR'06 Proceedings of the Third international conference on Image Analysis and Recognition - Volume Part II
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Recognition of Occluded Shapes Using Size Functions
ICIAP '09 Proceedings of the 15th International Conference on Image Analysis and Processing
Size functions for the morphological analysis of melanocytic lesions
Journal of Biomedical Imaging - Special issue on mathematical methods for images and surfaces
The Global-Local transformation for noise resistant shape representation
Computer Vision and Image Understanding
A global method for reducing multidimensional size graphs
GbRPR'11 Proceedings of the 8th international conference on Graph-based representations in pattern recognition
Robustness and modularity of 2-dimensional size functions: an experimental study
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
Persistent betti numbers for a noise tolerant shape-based approach to image retrieval
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
A new algorithm for computing the 2-dimensional matching distance between size functions
Pattern Recognition Letters
The hitchhiker's guide to the galaxy of mathematical tools for shape analysis
ACM SIGGRAPH 2012 Courses
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
Pattern Recognition Letters
A study of monodromy in the computation of multidimensional persistence
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
Comparison of persistent homologies for vector functions: From continuous to discrete and back
Computers & Mathematics with Applications
From a Non-Local Ambrosio-Tortorelli Phase Field to a Randomized Part Hierarchy Tree
Journal of Mathematical Imaging and Vision
Suspension models for testing shape similarity methods
Computer Vision and Image Understanding
Classification of hepatic lesions using the matching metric
Computer Vision and Image Understanding
PHOG: photometric and geometric functions for textured shape retrieval
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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Size Theory has proven to be a useful framework for shape analysis in the context of pattern recognition. Its main tool is a shape descriptor called size function. Size Theory has been mostly developed in the 1-dimensional setting, meaning that shapes are studied with respect to functions, defined on the studied objects, with values in 驴. The potentialities of the k-dimensional setting, that is using functions with values in 驴 k , were not explored until now for lack of an efficient computational approach. In this paper we provide the theoretical results leading to a concise and complete shape descriptor also in the multidimensional case. This is possible because we prove that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.