Algorithmic Riemannian Geometry for a Statistical Analysis of Images

用于图像统计分析的算法黎曼几何摘要该项目涉及图像的新算法表示和用于图像内容自动分析的几何信号处理技术的研究。研究人员利用微分几何和统计学衍生的方法和工具，开发了一种基于图像对象形状和纹理的外观分析的新框架。由于在感兴趣的图像中经常遇到的形状和纹理的巨大可变性，统计公式是必不可少的。微分几何方法在图像处理中的应用仍处于起步阶段，但前景广阔，因为有确凿的证据表明，这种方法特别适合于研究多维、非线性特征，如形状和纹理。近年来，研究人员开发了一个统计形状分析程序；形状被视为形状空间的元素，其几何形状被用于形状分析。研究人员以类似的方式处理纹理，通过创建纹理的黎曼流形，并将两种表示集成到一个单一的形状-纹理模型中，用于图像内容的算法分析。将图像分解为光谱分量，将局部光谱直方图作为具有非参数费雪信息诱导的几何结构的无限维统计流形的元素。微分几何结构被用来开发算法：(i)统计推断和形状纹理特征的学习；（ii）利用形状-纹理先验对物体进行贝叶斯检测和识别；（三）有效处理的降维技术。

ALGORITHMIC RIEMANNIAN GEOMETRY FOR A STATISTICALANALYSIS OF IMAGESAbstractThis project is concerned with the investigation of novel algorithmic representations of images and geometrical signal processing techniques for the automated analysis of image content. The investigators develop a new framework for an appearance-based analysis of imaged objects in terms of their shapes and textures using methods and tools derived from differential geometry and statistics. A statistical formulation is of the essence due to the large variability of shapes and textures frequently encountered in imagery of interest. The use of differential geometric methods in image processing is still incipient, but very promising, as solid evidence exists that such methodology is particularly well suited for the study of multidimensional, nonlinear features such as shapes and textures.In recent years, the investigators have developed a statistical shape analysis program; shapes are viewed as elements of a shape space whose geometry is exploited for shape analysis. The investigators treat textures in a similar manner by creating a Riemannian manifold of textures and integrate both representations into a single shape-texture model for the algorithmic analysis of image content. Images are decomposed into their spectral components and local spectral histograms are treated as elements of an infinite-dimensional statistical manifold equipped with a geometric structure induced by non-parametric Fisher information. Differential geometric constructs are utilized to develop algorithms for: (i) statistical inferences and learning of shape-texture features; (ii) Bayesian detection and recognition of objects using shape-texture priors; (iii) dimensionality reduction techniques for efficient processing.