CGV: Small: Modeling and Analysis of Multidimensional Shapes

媒体世界可以分为一维媒体，如文本和声音，以及多维媒体。在后者中，数字形状的特点是视觉外观和几何性质。形状的例子有图片、图像、固体物体的3D模型、视频、动画等。该项目的总体目标是定义和开发用于建模和分析随时间演变的形状的创新工具，例如3D对象的动画，或来自科学模拟的时变现象的描述。这项研究的应用领域包括医学、生物学、地球科学、物理和化学，在图像处理和视觉中的2D形状的建模和分析，在计算机图形学和地理数据处理中的2.5D和3D形状，以及在科学可视化中的3.5D形状(表示体数据集的图形)已经做了大量的研究。这方面的研究大多集中在形状的几何表示上。最近，基于拓扑工具的结构表示被开发出来。该项目正在开发用于建模和分析4D形状和4.5D形状的几何和结构表示和算法，4D形状描述形状随时间的演变，而4.5D形状表示4D标量场的图形。后者的例子是时变的动态三维标量场，前者的例子是随时间变化的3D标量场的动画序列或等值面。具体地说，该项目正在开发有效的4D标量场和4D形状的多分辨率几何表示，基于Morse理论和广义曲率概念的此类形状的分割新方法，以及基于分割的4D和4.5D形状的多尺度结构建模。

Title: Modeling and Analysis of Multidimensional ShapesAbstractThe media world can be classified into one-dimensional media, like text and sound, and multidimensional media. Among these latter, digital shapes are characterized by a visual appearance and by a geometric nature. Examples of shapes are pictures, images, 3D models of solid objects, videos, animations, etc. The general goal of the project is defining and developing innovative tools for modeling and analyzing shapes that evolve over time, such as animations of 3D objects, or descriptions of time-varying phenomena arising from scientific simulations. Application areas benefitting from this research include medicine, biology, earth sciences, physics, and chemistry.A large amount of research has been done on modeling and analyzing 2D shapes in image processing and vision, 2.5D and 3D shapes in computer graphics and geographic data processing, and 3.5D shapes (representing the graph of volume data sets) in scientific visualization. Most of this research has been focused on a geometric representation of a shape. More recently structural representations based on topological tools have been developed. This project is developing geometric and structural representations and algorithms for modeling and analyzing 4D shapes, which describe the evolution of a shape over time, and 4.5D shapes, which represent the graph of 4D scalar fields. Examples of the latter are time-varying dynamic three-dimensional scalar fields, examples of the former are animation sequences or isosurfaces of 3D scalar fields varying over time. Specifically, the project is developing effective multi-resolution geometric representations for 4D scalar fields and 4D shapes, new approaches to the segmentation of such shapes based on Morse theory and a generalized notion of curvature, and multi-scale structural modeling of 4D and 4.5D shapes based on segmentation.