Multidimensional Size Functions for Shape Comparison

@article{Biasotti2008MultidimensionalSF,
  title={Multidimensional Size Functions for Shape Comparison},
  author={Silvia Biasotti and Andrea Cerri and Patrizio Frosini and Daniela Giorgi and Claudia Landi},
  journal={Journal of Mathematical Imaging and Vision},
  year={2008},
  volume={32},
  pages={161-179}
}
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 ℝ. The potentialities of the k-dimensional setting, that is using functions with values in ℝk, were not explored until now for lack of an efficient computational… 
ADVANCES IN MULTIDIMENSIONAL SIZE THEORY
TLDR
Some recent results about size functions in this multidimensional setting are surveyed, with particular reference to the localization of their discontinuities.
ADVANCES IN MULTIDIMENSIONAL SIZE THEORY
TLDR
This work surveys some recent results about size functions in this multidimensional setting, with particular reference to the localization of their discontinuities.
k-dimensional Size Functions for Shape Description and Comparison
TLDR
This paper advises the use of k-dimensional size functions for comparison and retrieval in the context of multidimensional shapes, taking into account different properties expressed by a multivalued real function defined on the shape.
Describing shapes by geometrical-topological properties of real functions
TLDR
This survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner.
Robustness and Modularity of 2-Dimensional Size Functions - An Experimental Study
TLDR
The aim of the present paper is to validate, through some experiments on 3D-models, a computational framework recently introduced to deal with 2-dimensional Size Theory, and show that the cited framework is modular and robust with respect to noise, non-rigid and non-metric-preserving shape transformations.
A new approximation Algorithm for the Matching Distance in Multidimensional Persistence
TLDR
This paper proposes a new computational framework to deal with the multidimensional matching distance, by proving some new theoretical results and using them to formulate an algorithm for computing such a distance up to an arbitrary threshold error.
Invariance properties of the multidimensional matching distance in Persistent Topology and Homology
TLDR
This paper shows that the multidimensional matching distance is actually invariant with respect to such a choice, and formally depends on a subset of $\R^n\times-valued filtering functions inducing a parameterization of these half-planes.
Suspension models for testing shape similarity methods
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 46 REFERENCES
On the use of size functions for shape analysis
TLDR
It is shown that the representation of shape in terms of size functions can be tailored to suit the invariance of the problem at hand and is stable against small qualitative and quantitative changes of the viewed shape.
Computing Size Functions from Edge Maps
TLDR
It is shown that size functions can actually be defined without making assumptions on the topological structure of the viewed shape, so that they can be profitably used even in the presence of fragmented edge maps.
Describing shapes by geometrical-topological properties of real functions
TLDR
This survey is to provide a clear vision of what has been developed so far, focusing on methods that make use of theoretical frameworks that are developed for classes of real functions rather than for a single function, even if they are applied in a restricted manner.
Using matching distance in size theory: A survey
TLDR
The link between reduced size functions and the dissimilarity measure δ is established by a theorem, stating that the matching distance provides an easily computable lower bound for δ.
Metric-topological approach to shape representation and recognition
Measuring shapes by size functions
  • P. Frosini
  • Mathematics, Computer Science
    Other Conferences
  • 1992
TLDR
The concept of deformation distance between manifolds is presented, a distance which measures the `difference in shape' of two manifolds and the link between deformation distances and size functions is pointed out.
Invariant Size Functions
  • A. Verri, C. Uras
  • Mathematics
    Applications of Invariance in Computer Vision
  • 1993
TLDR
It is concluded that size functions can be useful for viewpoint invariant recognition of natural shapes by means of size functions invariant for Euclidean, affine, or projective transformations.
Shape Recognition Via an a Contrario Model for Size Functions
TLDR
This article proposes a statistical method, namely an a contrario method, to merge features derived from several families of size functions, which leads to a global shape recognition method dedicated to perceptual similarities.
The Use of Size Functions for Comparison of Shapes Through Differential Invariants
TLDR
This study focuses on a projective differential invariant which allows to decide if one shape can be considered as the deformation of another one by a rotation of the camera.
Morse homology descriptor for shape characterization
  • M. Allili, D. Corriveau, D. Ziou
  • Computer Science
    Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004.
  • 2004
TLDR
A new topological method for shape description is proposed that is suitable for any multi-dimensional data set that can be modelled as a manifold and Classical Morse theory is used to establish a link between the topology of a pair of lower level sets of f and its critical points lying between the two levels.
...
1
2
3
4
5
...