Topology of Nonisolated Singularities and Scale-based Geometry

DMS-0103862 James N. Damon教授的研究将应用奇点理论中的无穷小和分层方法来研究一类高度非孤立的奇异空间的拓扑和变形性质。 这些空间是某些自然普遍变体的非线性部分。 例如，映射在各种等价下的几何和变形性质可以被这样的泛变量所捕获。 切向量场的结构将用于推导某些基本拓扑不变量的代数公式。 这些将给予在某些自然代数和几何多重性，测量奇异行为的映射相对于相关的几何结构，如foliations第二，他将完善这些想法的问题，在计算机成像的发展几何结构相关的对象和功能的图像方面，这种高度奇异的空间。 图像中的离散性、噪声和失真的存在需要一种适用于不可微函数、测度甚至分布的“基于尺度的几何”。 这样的几何将适用于给定类型中的“几乎所有”对象，并且将在尺度空间中产生稳定的几何结构。 这将允许图像的几何分析，使用函数和措施区分图像中的各种特征。Damon教授的研究的第一部分将确定特定类型的非线性方程组的解集的定性属性。 这些可以从某些通用的方程组中得到。 他建议使用某些无穷小对称的方程推导属性的一套解决方案的代数不变量，反映了这两个属性的普遍方程和如何具体的方程涉及到普遍的。 其次，研究将应用于计算机成像问题。 对于图像中的对象和特征，人们可以将捕捉它们的属性的几何结构相关联以用于各种成像目的。 这样的结构使用如上的方程组来定义。 图像中离散性、噪声和失真的存在干扰了几何特征的识别。 该研究将通过“基于尺度”的版本改进上述方法，该版本引入了克服这些困难的鲁棒几何结构，然后可以用于各种计算机成像问题。

DMS-0103862James N. DamonProfessor Damon's research will apply infinitesimal and stratification methods from singularity theory to investigatethe topology and deformation properties of a general class of highly nonisolated singular spaces. These spaces arise as nonlinear sections of certain natural universal varieties. For instance, geometric and deformation properties of mappings under various equivalences can be captured by such universal varieties. The structure of the tangent vector fields will be used to deduce algebraic formulae for certain fundamental topological invariants. These will be given in terms of certain natural algebraic and geometric multiplicities, measuring the singular behavior of mappings relative to associated geometric structures such as foliations Second, he will refine these ideas for questions in computer imaging by developing geometric structures associated to objects and features in images in terms of such highly singular spaces. The presence of discreteness, noise, and distortions in images require a "scale-based geometry" which is applicable to nondifferentiable functions, measures, and even distributions. Such a geometry will apply to "almost all" objects in a given type, and will yield stable geometric structures in scale space. This will allow the geometric analysis of images using functions and measures discriminating various features in images.The first part of Professor Damon's research will determine for specific types of systems of nonlinear equations, the qualitative properties of the set of solutions. These can be obtained from certain universal systems of equations. He proposes to use certain infinitesimal symmetries of the equations to deduce properties of the set of solutions in terms of algebraic invariants which reflect both properties of the universal equations and how the specific equations relate to the universal ones. Second, the research will be applied to problems in computer imaging. To objects and features in images, one may associate geometric structures capturing their properties for various imaging purposes. Such structures are defined using systems of equations as above. The presence of discreteness, noise and distortions in images interferes with identifying geometric features. The research will refine the methods described above via "scale-based" versions which introduce robust geometric structures overcoming these difficulties, which can then beused for a variety of computer imaging problems.