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颖的计算模型和策略来分析欧几里得 3 空间中的球面形状。 近年来，基于无限维曲线空间几何的方法论对曲线形状的计算研究取得了实质性进展。然而，将这些方法扩展到表面的尝试遇到了很大的障碍。在该项目中，提出了一种有效的计算解决方案，涵盖了问题的所有基本方面。将构建配备测地度量的形状空间，这将为表面形状的定量研究提供自然环境。将设计和实现一整套计算工具来量化形状相似性和发散性，从样本中开发统计模型，从学习模型中合成形状，以及分析和模拟形状动力学。将开发技术将零属表面的噪声点云表示转换为标准球体上的最小失真参数化。对齐算法将被设计为最好地匹配表面的几何特征，并提取用于建模一系列形状的最佳参数化。从加权索博列夫空间继承的黎曼度量将捕获形状之间的几何相似性和差异，以任何所需的顺序。该项目将重点关注一阶指标，因为它们在几何精度和计算鲁棒性之间提供了良好的平衡。由于表面几何形状的典型复杂性，许多算法将采用从粗到细的方法来处理点云和三角网格。频率或时空域中球形形状的定位也将用于统计建模并实现计算效率。对 3D 物体的形状和形式的研究是由计算机视觉、医学成像和计算生物学等领域出现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，例如从各种图像模式中识别对象或目标、大脑解剖和功能建模、生物生长和运动的模拟以及与疾病和衰老相关的解剖变化。因此，支持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，并将积极寻求与这些领域的研究人员合作。

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.