Estimation and Signal Processing for Spatial Data: Efficient Algorithms, Inverse Problems, and Computational Vision and Geometry

本文的研究重点是空间数据的估计和信号处理。我们考虑了三个主要方面。第一个是关于空间数据处理的有效算法。要解决的问题包括：(I)使用新的递归概念以及空间分解来开发递归和高度并行的算法，以估计由非因果模型描述的二维过程以及用于其他二维数字滤波应用；以及(Ii)利用某些类型的随机场(例如各向同性场)中存在的对称性来开发高效的递归和谱估计程序。第二部分是关于几个维度的逆问题和信号重建问题算法的发展。要研究的问题包括：(1)发展有效的广义层析成像方法以精确和近似地解决2-D和3-D逆散射问题；(2)发展基于系统识别的反问题方法，特别强调在多个空间尺度上工作的算法。第三个领域涉及计算视觉和几何。描述了一些研究问题，包括(I)基于估计的算法的发展，用于计算视觉问题，如运动和深度估计；以及(Ii)计算几何问题的系统和估计理论公式的研究，例如给出不同量的不确定测量的对象的重建，如凸对象的支承、其轮廓、其内部等。这项研究侧重于空间数据的估计和信号处理。讨论的三个主要领域是(I)空间数据处理的有效算法(Ii)二维和三维逆问题和信号重建问题的算法的发展以及(Iii)与计算视觉和几何有关的问题。

This research focuses on estimation and signal processing for spatial data. Three major areas are considered. The first deals with efficient algorithms for the processing of spatial data. Problems to be addressed include (i) the employment of novel notions of recursion as well as spatial decompositions to develop recursive and highly parallel algorithms for the estimation of 2-D processes described by noncausal models and for other 2-D digital filtering applications; and (ii) the exploitation of the symmetries present in some classes of random fields (e.g. isotropic fields) to develop efficient procedures for recursive and spectral estimation. The second deals with the development of algorithms for inverse and signal reconstruction problems in several dimensions. Among the problems to be investigated are (i) the development of efficient and generalized-tomographic methods for exact and approximate solution of 2-D and 3-D inverse scattering problems; and (ii) the development of system- identification-based approaches to inverse problems with particular emphasis on algorithms that work at multiple spatial scales. The third area deals with computational vision and geometry. A number of research problems are described including (i) the development of estimation-based algorithms for problems of computational vision such as motion and depth estimation; and (ii) the investigation of system and estimation-theoretic formulations of problems in computational geometry such as the reconstruction of objects given uncertain measurements of various quantities such as the support of a convex object, its silhouette, its interior, etc. This research focuses on estimation and signal processing for spatial data. The three major areas considered are (i) efficient algorithms for the processing of spatial data (ii) the development of algorithms for inverse and signal reconstruction problems in two and three dimensions and (iii) questions related to computational vision and geometry.