Geometry Processing for IsoSurfaces

无论是人脸的几何形状、脑瘤的边界还是多变量金融数据的决策界面，等值面都是理解和分析所有类型数据的重要对象。行进立方体算法是从科学数据集中提取等值面的一种非常有效和广泛使用的技术。本研究项目的目的是利用行进立方体算法产生的等值面的一些特殊性质，以便开发出特别有效的几何处理工具来分析、操纵和处理这些曲面。该研究项目的智力价值包括开发了新的方法，用于通过新的有效的数字几何处理工具从数据中提取含义和理解，用于由行进立方体算法产生的等值面的特殊情况。由于等值面在许多科学领域中的广泛应用，本研究的结果将对数据分析产生广泛的影响。最近发现，由行进立方体(MC)算法产生的等值面具有一些特殊的性质，这使得开发特别高效的几何处理工具成为可能。这些性质包括(1)正交多边形网络的一些特殊方面，它允许将曲线技巧提升到曲面上；(2)由原始MC方法的修改提供的局部函数性质，它允许将传统的逼近技术应用于这些特殊的等值面。这项研究将集中在曲面几何处理的三个方面，即(1)纹理映射、网格化、压缩和许多其他有用的曲面操作所必需的参数化；(2)估计曲率和其他相关曲面属性的方法，这些方法将作为许多有用的高级曲面处理技术的基本工具；(3)平滑和光顺曲面的算法。

Whether it is the geometry of a human face, the boundary of a brain tumor or the decision interface of multivariate financial data, isosurfaces are important objects for understanding and analyzing all types of data. The marching cubes algorithm is a very effective and widely used technique for extracting isosurfaces from scientific data sets. The purpose of this research project is to exploit some special properties of the isosurfaces produced by the marching cubes algorithm in order to develop particularly efficient geometry processing tools for the analysis, manipulation and processing of these surfaces. The intellectual merit of the research project consists of the development of new methods for extracting meaning and understanding from data by new and efficient digital geometry processing tools for the special case of isosurfaces produced by the marching cubes algorithm. Since isosurfaces are widely used in many areas of science, the results of this research will have broad impact on data analysis in general.It has recently been discovered that the isosurfaces produced by the marching cubes (MC) algorithm have some special properties that allow for the development of particularly efficient geometry processing tools. These properties include (1) some special aspects of a network of orthogonal polygons which allow curve techniques to be lifted to surfaces and (2) a local function property provided by a modification of the original MC method which allows conventional approximation techniques to be applied to these special isosurfaces. This research will concentrate on three aspects of geometry processing for surfaces, namely (1) parameterizations which are necessary for texture mapping, remeshing, compression and many other useful surface operations; (2) methods of estimating curvature and other related surface properties which will serve as the basic tools for many useful higher level surface processing techniques and (3) algorithms for smoothing and fairing surfaces.