We provide an analysis of the algorithms necessary for the optimal use of multidimensional signal reconstruction from multichannel acquisition. First, we provide computable conditions to test the matrix invertibility and propose algorithms to find a particular inverse. Second, we determine the existence of perfect reconstruction systems for given FIR analysis filters with some sampling matrices and some FIR synthesis polyphase matrices. Then, we present the development of an efficient algorithm designed to find a sampling matrix with maximum sampling rate and FIR synthesis polyphase matrix for given FIR analysis filters so that the system provides a perfect reconstruction. Once a particular synthesis matrix is found, we can characterize all synthesis matrices and find an optimal one according to a design criterion.

We study the invertibility of M-variate polynomial (respectively : Laurent polynomial) matrices of size N by P. Such matrices represent multidimensional systems in various settings including filter banks, multiple-input multiple-output systems, and multirate systems. The main result of this paper is to prove that when N - P ges M, then H(z) is generically invertible; whereas when N - P Lt M, then H(z) is generically noninvertible. As a result, we can have an alternative approach in design of the multidimensional systems.

We consider the problem of simultaneously segmenting data samples drawn from multiple linear subspaces and estimating model parameters for those subspaces. This "subspace segmentation" problem naturally arises in many computer vision applications such as motion and video segmentation, and in the recognition of human faces, textures, and range data. Generalized Principal Component Analysis (GPCA) has provided an effective way to resolve the strong coupling between data segmentation and model estimation inherent in subspace segmentation. Essentially, GPCA works by first finding a global algebraic representation of the unsegmented data set, and then decomposing the model into irreducible components, each corresponding to exactly one subspace. We provide a summary of important algebraic properties and statistical facts that are crucial for making GPCA both efficient and robust, even when the given data are corrupted with noise or contaminated by outliers. We demonstrate the effectiveness of GPCA using a large testbed of synthetic and real experiments.

This paper develops a new mathematical framework for studying the subspace-segmentation problem. We examine some important algebraic properties of subspace arrangements that are closely related to the subspace-segmentation problem. More specifically, we introduce an important class of invariants given by the Hilbert functions. We show that there exist rich relations between subspace arrangements and their corresponding Hilbert functions. We propose a new subspace-segmentation algorithm, and showcase two applications to demonstrate how the new theoretical revelation may solve subspace segmentation and model selection problems under less restrictive conditions with improved results.