News & Announcements

Dr. Xiuwen Liu recieves NSF grantSeptember 17, 2007

Dr. Liu recieves 4-year NSF grant with total of $655981 and first increment of 157,697.

ABSTRACT

Project: Novel Computational Methods for the Analysis, Synthesis and
Simulation of Shapes of Surfaces

Proposal Number: DMS-0713012

PI: Washington Mio
Co-PI: Xiuwen Liu

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.