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Introduction and Motivations

  • Vincenzo Capasso
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Many real phenomena may be modelled as random sets in \(\mathbb {R}^d\), and in several situations as evolving random sets. Application areas include tumor driven angiogenesis, crystallization processes, patterns in Biology, etc.

All quoted processes may be described by time dependent random sets of different Hausdorff dimensions (for instance, crystallization processes are modelled in general by full dimensional growing sets, and lower dimensional interfaces, while angiogenesis by systems of random curves). In many cases all these kinds of phenomena may be modelled as space-time structured stochastic processes whose geometric structure is of great relevance.

A rigorous definition of the relevant geometric quantities in a stochastic setting of the above systems (fibres for angiogenesis, dislocations for crystalline materials, etc.) is very important for statistical applications and in mean field approximations.

On the other hand, the diagnosis of a pathology may significantly depend upon the shapes present in images of cells, organs, biological systems, etc., so that Statistical Shape Analysis is the required mathematical approach.

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Copyright information

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  1. 1.ADAMSS (Centre for Advanced Applied Mathematical and Statistical Sciences)Universitá degli Studi di Milano La StataleMilanoItaly

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