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 \$\R^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 thecomparison of multidimensional size functions can be reduced tothe \$1\$-dimensional case by a suitable change of variables.Indeed, a foliation in half-planes can be given, suchthat the restriction of a multidimensional size function to eachof these half-planes turns out to be a classical size function intwo scalar variables. This leads to the definition of a newdistance between multidimensional size functions, and to the proofof their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.

Multidimensional size functions for shape comparison / S., Biasotti; A., Cerri; P., Frosini; D., Giorgi; Landi, Claudia. - In: JOURNAL OF MATHEMATICAL IMAGING AND VISION. - ISSN 0924-9907. - STAMPA. - 32:(2008), pp. 161-179. [10.1007/s10851-008-0096-z]

### Multidimensional size functions for shape comparison

#### Abstract

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 \$\R^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 thecomparison of multidimensional size functions can be reduced tothe \$1\$-dimensional case by a suitable change of variables.Indeed, a foliation in half-planes can be given, suchthat the restriction of a multidimensional size function to eachof these half-planes turns out to be a classical size function intwo scalar variables. This leads to the definition of a newdistance between multidimensional size functions, and to the proofof their stability with respect to that distance. Experiments are carried out to show the feasibility of the method.
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Multidimensional size functions for shape comparison / S., Biasotti; A., Cerri; P., Frosini; D., Giorgi; Landi, Claudia. - In: JOURNAL OF MATHEMATICAL IMAGING AND VISION. - ISSN 0924-9907. - STAMPA. - 32:(2008), pp. 161-179. [10.1007/s10851-008-0096-z]
S., Biasotti; A., Cerri; P., Frosini; D., Giorgi; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11380/589947`
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