Digital shapes can be defined as objects having a visual appearance that exists in some Euclidean space (of dimension two, three, or higher). Digital shapes have contains geometric information (namely the spatial extent of the object) and can be described by structures, which decompose them into meaningful parts, namely, points and critical lines of a scalar field. We can distinguish between two types of representations for digital shapes, namely geometric and structural. Examples of geometric representations are given by models of surfaces, solid models, 2D images, 3D and 4D models (animations, descriptions of timevarying scalar fields), representations of shapes defined by combining data sampled at medium and high dimensions. A structural representation (also called morphological) is defined by making explicit the association between shapes components. Examples are given by skeletons, critical networks for terrains, segmentations and contours of 3D shapes, stratifications.
In the research field of images processing and analysis, many researches have focused on the modeling, analysis, understanding, and retrieval of 2D shapes. There have been also several proposals for geometric representations of 2.5D shapes (terrains), and 3D shapes, describing contours and boundaries of solid objects. A huge amount of data, describing 2D and 3D shapes, has led to a growing interest in tools for analysis and understanding of shapes. These tools are important in order to define representations of objects, which are needed as a basis for the contentbased retrieval in many applications that deal with digital shapes, like mechanical engineering and medicine, as well as entertainment, gaming and simulation.
In any case, digital data, which are spatially correlated, are produced and created with increasing frequency, and they are changing science and applications nowadays. In this context, the main challenge consists of defining software tools, able to handling data complexity and information abundance. Such data often consist of huge collections of disorganized samples, and there is the need to distiguish and retrieve relevant information and discard not meaningful details. As consequence, the size of these data is the most challenging problem. Thus, it is needed to define and create computational tools that process large data sets efficiently and to generate synthetic structural descriptors for spatial objects.
Overall objective of this research consists of developing geometric and structural representations of digital shapes in three, four, and higher dimensions, by exploiting their discretizations as simplicial and cell complexes, as well as algorithms and tools for analyzing shapes of dimension three and higher. Specifically, this project aimed to work on three interrelated issues:
 Design and development of efficient representations of simplicial and cell complexes  Size of discrete representations of digital shapes is an interesting problem, while representing shapes in medium and high dimensions (like triangulated hypersurfaces, extracted from static and dynamic 3D scalar fields) as well as simplicial and cell shapes, obtained by connecting a subset of Euclidean points in three or higher dimensions. Moreover, there are few libraries, freely available, which are able to represent and manipulate shapes in medium and high dimensions.
 Development of new structural representations and algorithms for analyzing shapes in arbitrary dimension  Combinatorial tools are becoming mandatory in order to efficiently represent and manipulate huge amount of data in high dimensions. These tools may avoid expensive numeric simulations and may be easily generalized to high dimensions. Overall objective of this project is to define tools, provided by computational topology, and, in particular, to define and construct shapes decompositions, based on the discrete Morse theory.
 Development of hierarchical representations for analyzing the structure of shapes in arbitrary dimensions  Objective consists of designing and developing multiresolution models, which are based on structural representations of shapes in arbitrary dimension, from which it is possible to extract and retrieve structural descriptions at different level of astraction with the aim of making structural representations more effective and computationally more efficient.
