This focused study proposes to analyze the structure induced on spaces of shapes by the action of groups of diffeomorphisms equipped with a right invariant metric. The project contains four components related to these spaces: their geometric analysis, the development of appropriate statistical methods, the required numerical analysis, and the application of the results to medical imaging and computational anatomy. The general framework is addressed by a new approach which in some sense formalizes the mechanics of shapes. Lie groups with right invariant metrics indeed are structures on which classical laws of mechanics can be shown to hold, and in particular the conservation of momentum along paths of minimal energy. It turns out that this momentum is a key to the representation and characterization of deformations in this context. It is albeit difficult to handle, because it is usually singular, as a measure, or a distribution on a singular support. This and the numerical difficulty it creates is probably one of the main challenges that we address in our study. Other important aspects are the study of the geometry such an approach induces on shape spaces, including a study of their curvatures, and the existence and stability of normal coordinates. This will be related to open issues in shape statistics, and applied in particular to biomedical imaging problems.
This approach is therefore designed to provide new tools for describing and analyzing shapes. Although shapes are prevalent in the outside world and in science, this is a difficult problem. For the human mind, there is an intuitive notion of what shapes are, why they differ or look alike, or when they present abnormalities with respect to ordinary observations. Sculpture is the art of rendering existing shapes, or creating new ones, and the fact that artists are still able to provide unambiguous instances of subjects through distorted or schematic representations is a strong indication of the robustness of the human shape recognition engine. However, an analytical description of a shape is much less obvious, and humans are much less efficient for this task, as if the understanding and recognition of forms work without an accurate extraction of their constituting components. We can recognize a squash from an eggplant or a pepper via a simple outline, and even provide a series of discriminative features which distinguish them, but it is much harder to phrase a verbal description of any of them, accurate enough, say for a painter to reproduce it. It is therefore not surprising that, for mathematics, shape description remains mostly a challenge. There are however very important applications which depend on progresses made in this field, one of them being the computerized analysis of biomedical shapes (computational anatomy), which analyzes the impact of diseases on shapes of organs, obtained from modern techniques of non-invasive 3D imagery. The last fifty years of research in computer vision has shown a amazingly large variety of points of view and techniques designed for this purpose: 2D or 3D sets they delineate (via either volume or boundary), moment-based features, medial axes or surfaces, null sets of polynomials, configurations of points of interest (landmarks), to name but a few. Yet, it does not seem that any of these methods has emerged as ideal, neither conceptually nor computationally, for describing shapes. An important aspect of our study will be to describe shapes with an indirect approach, from the way they can be deformed. This is not a new idea, and can be traced back to the seminal works of D'Arcy Thompson at the beginning of the 20th century, but its mathematical formalization and the design of practical algorithms is a comprehensive task, still offering many open problems, that the present group will try to address and convey to the scientific community.
