ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE
The scope of this paper is to offer an overview of the main results obtained by the authors in recent literature in connection with the introduction of a Delta formalism, á la Dirac-Schwartz, for random generalized functions (distributions) associated with random closed sets, having an integer Hausd...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Slovenian Society for Stereology and Quantitative Image Analysis
2011-05-01
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| Series: | Image Analysis and Stereology |
| Subjects: | |
| Online Access: | http://www.ias-iss.org/ojs/IAS/article/view/805 |
| _version_ | 1818930041193496576 |
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| author | Vincenzo Capasso Elena Villa |
| author_facet | Vincenzo Capasso Elena Villa |
| author_sort | Vincenzo Capasso |
| collection | DOAJ |
| description | The scope of this paper is to offer an overview of the main results obtained by the authors in recent literature in connection with the introduction of a Delta formalism, á la Dirac-Schwartz, for random generalized functions (distributions) associated with random closed sets, having an integer Hausdorff dimension n lower than the full dimension d of the environment space Rd. A concept of absolute continuity of random closed sets arises in a natural way in terms of the absolute continuity of suitable mean content measures, with respect to the usual Lebesgue measure on Rd. Correspondingly mean geometric densities are introduced with respect to the usual Lebesgue measure; approximating sequences are provided, that are of interest for the estimation of mean geometric densities of lower dimensional random sets such as fbre processes, surface processes, etc. Many models in material science and in biomedicine include time evolution of random closed sets, describing birthand-growth processes; the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the relevant kinetic parameters. |
| first_indexed | 2024-12-20T03:54:23Z |
| format | Article |
| id | doaj.art-6186272ff66e4af7b4e920b6eabef9bb |
| institution | Directory Open Access Journal |
| issn | 1580-3139 1854-5165 |
| language | English |
| last_indexed | 2024-12-20T03:54:23Z |
| publishDate | 2011-05-01 |
| publisher | Slovenian Society for Stereology and Quantitative Image Analysis |
| record_format | Article |
| series | Image Analysis and Stereology |
| spelling | doaj.art-6186272ff66e4af7b4e920b6eabef9bb2022-12-21T19:54:22ZengSlovenian Society for Stereology and Quantitative Image AnalysisImage Analysis and Stereology1580-31391854-51652011-05-01261233610.5566/ias.v26.p23-36777ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINEVincenzo CapassoElena VillaThe scope of this paper is to offer an overview of the main results obtained by the authors in recent literature in connection with the introduction of a Delta formalism, á la Dirac-Schwartz, for random generalized functions (distributions) associated with random closed sets, having an integer Hausdorff dimension n lower than the full dimension d of the environment space Rd. A concept of absolute continuity of random closed sets arises in a natural way in terms of the absolute continuity of suitable mean content measures, with respect to the usual Lebesgue measure on Rd. Correspondingly mean geometric densities are introduced with respect to the usual Lebesgue measure; approximating sequences are provided, that are of interest for the estimation of mean geometric densities of lower dimensional random sets such as fbre processes, surface processes, etc. Many models in material science and in biomedicine include time evolution of random closed sets, describing birthand-growth processes; the Delta formalism provides a natural framework for deriving evolution equations for mean densities at all (integer) Hausdorff dimensions, in terms of the relevant kinetic parameters.http://www.ias-iss.org/ojs/IAS/article/view/805approximation of geometric densitiesbirth-and-growth processesgeometric densitiesgeometric measure theoryrandom distributionsrandom measuresstochastic geometry |
| spellingShingle | Vincenzo Capasso Elena Villa ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE Image Analysis and Stereology approximation of geometric densities birth-and-growth processes geometric densities geometric measure theory random distributions random measures stochastic geometry |
| title | ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE |
| title_full | ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE |
| title_fullStr | ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE |
| title_full_unstemmed | ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE |
| title_short | ON MEAN DENSITIES OF INHOMOGENEOUS GEOMETRIC PROCESSES ARISING IN MATERIAL SCIENCE AND MEDICINE |
| title_sort | on mean densities of inhomogeneous geometric processes arising in material science and medicine |
| topic | approximation of geometric densities birth-and-growth processes geometric densities geometric measure theory random distributions random measures stochastic geometry |
| url | http://www.ias-iss.org/ojs/IAS/article/view/805 |
| work_keys_str_mv | AT vincenzocapasso onmeandensitiesofinhomogeneousgeometricprocessesarisinginmaterialscienceandmedicine AT elenavilla onmeandensitiesofinhomogeneousgeometricprocessesarisinginmaterialscienceandmedicine |