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

曲面形状分析、合成和模拟的新计算方法

• 批准号: 0713012
• 负责人: Washington Mio
• 金额: $ 65.6万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713012&HistoricalAwards=false
• 关键词: Novel; Computational; Methods; Analysis; Synthesis

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.

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得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曲线形状计算研究取得了长足的进展。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，这将为表面形状的定量研究提供自然环境。一套完整的计算工具将被设计和实现来量化形状的相似性和差异，从样本中开发统计模型，从学习模型中合成形状，并分析和模拟形状动力学。技术将被开发，以转换一个噪声点云表示的曲面属零到一个最小失真的参数化在标准球。对齐算法将被设计为最好地匹配表面的几何特征，并提取最优参数来建模一系列形状。从加权Sobolev空间继承的黎曼度量将捕捉形状之间任意顺序的几何相似性和差异。该项目将侧重于一阶度量，因为它们在几何精度和计算鲁棒性之间提供了很好的平衡。由于曲面几何的典型复杂性，许多算法将采用从粗到精的方法来处理点云和三角形网格。在频率或时空域的球形定位也将用于统计建模和实现计算效率。提出对三维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所驱动的。形状是与几何数据中产生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，支持者将使该项目下开发的形状建模和分析工具可供更广泛的研究团体使用，并将积极寻求与这些领域的研究人员合作。