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

该项目的主要目标是开发新的计算模型和策略来分析欧几里德三维空间中球面的形状。近年来，基于无限维曲线空间几何的方法在曲线形状的计算研究方面取得了长足的进展。然而，将这些方法扩展到表面的尝试遇到了巨大的障碍。在这个项目中，提出了一个有效的计算解决方案，涵盖了问题的所有基本方面。形状空间的构建将配备测地线度量，这将为曲面形状的定量研究提供一个自然环境。将设计和实施一整套计算工具，以量化形状的相似性和差异性，根据样本建立统计模型，根据学习的模型合成形状，以及分析和模拟形状动力学。将开发技术来将零亏格曲面的噪声点云表示转换为标准球面上的最小失真参数化。对齐算法将被设计为与曲面的几何特征最佳匹配，并提取用于造型一族形状的最佳参数。从加权Sobolev空间继承的黎曼度量将捕捉形状之间的几何相似和差异，以达到任何所需的顺序。该项目将专注于一阶度量，因为它们在几何精度和计算稳健性之间提供了良好的平衡。由于曲面几何的典型复杂性，许多算法都会采用从粗到精的方法来处理点云和三角网格。球体形状在频域或时空域的局部化也将被用于统计建模和提高计算效率。三维物体形状和形状的研究是由计算机视觉、医学成像和计算生物学等领域产生的一系列问题推动的。形状是与几何数据中出现的模式相关的关键属性，其有效的计算表示和分析将对应用领域产生影响，如从各种形式的图像中识别物体或目标，建模大脑解剖和功能，模拟生物生长和运动，以及与疾病和衰老相关的解剖变化。因此，倡导者将向更广泛的研究界提供在该项目下开发的形状建模和分析工具，并将积极寻求与研究人员在这些领域的合作。

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.