The media world can be classified into one-dimensional media, like text and sound, and multidimensional media. Among these latter, digital shapes are characterized by a visual appearance and by a geometric nature. Examples of shapes are pictures, images, 3D models of solid objects, videos, animations, etc. The general goal of the project is defining and developing innovative tools for modeling and analyzing shapes that evolve over time, such as animations of 3D objects, or descriptions of time-varying phenomena arising from scientific simulations. Application areas benefitting from this research include medicine, biology, earth sciences, physics, and chemistry.
A large amount of research has been done on modeling and analyzing 2D shapes in image processing and vision, 2.5D and 3D shapes in computer graphics and geographic data processing, and 3.5D shapes (representing the graph of volume data sets) in scientific visualization. Most of this research has been focused on a geometric representation of a shape. More recently structural representations based on topological tools have been developed. This project is developing geometric and structural representations and algorithms for modeling and analyzing 4D shapes, which describe the evolution of a shape over time, and 4.5D shapes, which represent the graph of 4D scalar fields. Examples of the latter are time-varying dynamic three-dimensional scalar fields, examples of the former are animation sequences or isosurfaces of 3D scalar fields varying over time. Specifically, the project is developing effective multi-resolution geometric representations for 4D scalar fields and 4D shapes, new approaches to the segmentation of such shapes based on Morse theory and a generalized notion of curvature, and multi-scale structural modeling of 4D and 4.5D shapes based on segmentation.