Statistical Signal Processing and Estimation for Spatial andMultidimensional Data: Multiresolution Methods, Efficient Algorithms and Geometric Reconstruction

威尔斯基的这项研究计划有三个主要组成部分，其总体目标是开发统计信号处理和空间和多维数据估计的新算法。第一个是发展一维和多维多尺度信号分析和处理的统计模型和方法。正在解决的问题包括发展与新兴的多分辨率信号分解和小波变换理论相对应的统计方法，研究迭代、多重网格信号处理算法，以及将这些方法应用于从逆重建问题到信号或图像分割问题的各种主题。第二是开发高效的空间数据处理算法。本课题的重点是开发非因果模型的结构，例如那些由偏差方程或马尔可夫随机场描述的模型，以开发极其高效和高度并行的算法。正在研究的具体问题包括用于多维信号处理的径向向内和向外递归的使用、基于多维信号处理的空间划分的并行处理结构、以及用于跟踪时空随机场中的运动和其他时间变化的高效算法的开发。第三是发展统计方法，用于在给定各种量的不确定测量的情况下估计或重建多维数据中的几何特征，例如2-D或3-D中凸起对象的支撑，或3-D对象的2-D轮廓。

Willsky There are three major components of this research program which has as its overall objective the development of new classes of algorithms for statistical signal processing and estimation of spatial and multidimensional data. The first is the development of statistical models and methods for multiscale signal analysis and processing in one and several dimensions. Problems that are being addressed include the development of a statistical counterpart to the emerging theory of multiresolution signal decompositions and wavelet transforms, the investigation of iterative, multigrid signal processing algorithms, and applications of these methods to topics ranging from inverse reconstruction problems to problems of signal or image segmentation. The second is the development of efficient algorithms for processing spatial data. This topic focuses on the exploitation of the structure of noncausal models, such as those described by partial difference equations or Markov random fields, in order to develop extremely efficient and highly parallel algorithms. Specific problems being investigated include the employment of radially inward and outward recursions for multidimensional signal processing, parallel processing structures based on spatial partitioning of multidimensional signal processing, and the development of efficient algorithms for tracking motion and other temporal changes in space-time random fields. The third is the development of statistical methods for estimating or reconstructing geometric features in multidimensional data given uncertain measurements of various quantities, such as the support of a convex object in 2-D or 3-D, or the 2-D silhouette of 3-D objects.