The main goal of this project is to develop novel computational models and strategies to analyze the shapes of spherical surfaces in Euclidean 3-space. In recent years, there has been a substantial progress in the computational study of shapes of curves with methodology based on the geometry of infinite-dimensional spaces of curves. However, attempts to extend these approaches to surfaces have encountered tall obstacles. In this project, an effective computational solution is proposed that encompasses all fundamental aspects of the problem. Shape spaces will be constructed equipped with geodesic metrics, which will provide a natural environment for the quantitative study of shapes of surfaces. A full set of computational tools will be designed and implemented to quantify shape similarity and divergence, to develop statistical models from samples, to synthesize shapes from learned models, and to analyze and simulate shape dynamics. Techniques will be developed to convert a noisy point-cloud representation of a surface of genus zero to a minimum-distortion parametrization over the standard sphere. Alignment algorithms will be designed to best match the geometric features of surfaces and to extract optimal parametrizations for modeling a family of shapes. Riemannian metrics inherited from weighted Sobolev spaces will capture geometric similarities and discrepancies between shapes to any desired order. The project will focus on first-order metrics, as they offer a good balance between geometric accuracy and robustness for computations. Due to the typical complexity of the geometry of surfaces, many algorithms will employ a coarse-to-fine approach both for the processing of point clouds and triangular meshes. Localization of spherical shapes in the frequency or spatio-temporal domains will also be employed for statistical modeling and to achieve computational efficiency.
The proposed research on shapes and forms of 3D objects is motivated by a series of problems arising in areas such as computer vision, medical imaging, and computational biology. Shape is a key attribute associated with patterns arising in geometric data and its effective computational representation and analysis will have an impact on application domains such as the recognition of objects or targets from various modalities of images, modeling brain anatomy and functions, the simulation of biological growth and motion, and anatomical changes associated with diseases and aging. As such, the proponents will make the tools of shape modeling and analysis developed under this project available to the broader research community and will also actively pursue collaborations with researchers in these areas.
This project dealt primarily with quantification of morphology and development of computational methods for the analysis and interpretation of variation of shape and form of objects such as surfaces and solids in 3-dimensional space. Characterization of morphological variation is a problem of basic relevance in many areas of science, including developmental and evolutionary biology, paleontology and neuroscience, as well as in problems arising in medical imaging and computer vision. Since a variety of imaging modalities are employed in the acquisition of morphological data, to ensure broad applicability of the methods, only weak assumptions were made on the way shape data were structured. One of the main contributions to the broad area of research related to this project was the development of computationally feasible and principled approaches to shape modeling. An important component of the project was the construction of shape spaces, which yield an environment for statistical shape analysis once these spaces are equipped with metrics that allow quantification of morphological similarity and divergence. With these basic elements in place, the project investigated techniques to model spatial and temporal shape variation, which were used both for analysis and inference. The methodology was applied to neuroimaging data acquired for the study of normal and pathological variation in brain anatomy and to the development of models of shape variation that can potentially help to identify mechanisms that control phenotypic plasticity and provide a better understanding of inheritance and evolution of phenotypic traits. Education and training of graduate and undergraduate students were integral parts of the project through student participation in multiple research activities throughout the duration of the project.
