# 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…

## 80 Citations

ADVANCES IN MULTIDIMENSIONAL SIZE THEORY

- Computer Science, Mathematics
- 2011

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

- Computer Science, Mathematics
- 2011

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

- Mathematics14th International Conference on Image Analysis and Processing (ICIAP 2007)
- 2007

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

- MathematicsCSUR
- 2008

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.

A new algorithm for computing the 2-dimensional matching distance between size functions

- Computer SciencePattern Recognit. Lett.
- 2011

Robustness and Modularity of 2-Dimensional Size Functions - An Experimental Study

- Mathematics, Materials ScienceCAIP
- 2011

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

- Computer Science
- 2011

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

- MathematicsArXiv
- 2010

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.

The Global-Local transformation for noise resistant shape representation

- Computer ScienceComput. Vis. Image Underst.
- 2011

Suspension models for testing shape similarity methods

- Computer ScienceComput. Vis. Image Underst.
- 2014

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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.

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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.

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