Introduction and Motivations

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.

“…l’universo …é scritto in lingua matematica, e i caratteri sono

…figure geometriche …;

senza questi é un aggirarsi per un oscuro laberinto.”

[The universe …is written in a mathematical language, and its characters are

…geometrical figures…;

without those it is a mere wandering in vain around a dark maze.]

Galileo Galilei, Saggiatore (VI, 232), 1623