Structured Total Least Squares Problem Research Articles

The structured total least squares algorithm research for passive location based on angle information

Based on the constrained total least squares (CTLS) passive location algorithm with bearing-only measurements, in this paper, the same passive location problem is transformed into the structured total least squares (STLS) problem. The solution of the STLS problem for passive location can be obtained using the inverse iteration method. It also expatiates that both the STLS algorithm and the CTLS algorithm have the same location mean squares error under certain condition. Finally, the article presents a kind of location and tracking algorithm for moving target by combining STLS location algorithm with Kalman filter (KF). The efficiency and superiority of the proposed algorithms can be confirmed by computer simulation results.

A Global Solution for the Structured Total Least Squares Problem with Block Circulant Matrices

We study the structured total least squares (STLS) problem of system of linear equations Ax = b, where A has a block circulant structure with N blocks. We show that by applying the discrete Fourier transform (DFT), the STLS problem decomposes into Nunstructured total least squares (TLS) problems. The N solutions of these problems are then assembled to generate the optimal global solution of the STLS problem. Similar results are obtained for elementary block circulant matrices. Here the optimal solution is obtained by assembling two solutions: one of an unstructured TLS problem and the second of a multidimensional TLS problem.

Implementation of the regularized structured total least squares algorithms for blind image deblurring

The structured total least squares (STLS) problem has been introduced to handle problems involving structured matrices corrupted by noise. Often the problem is ill-posed. Recently, regularization has been proposed in the STLS framework to solve ill-posed blind deconvolution problems encountered in image deblurring when both the image and the blurring function have uncertainty. The kernel of the regularized STLS (RSTLS) problem is a least squares problem involving Block–Toeplitz–Toeplitz–Block matrices. In this paper an algorithm is described to solve this problem, based on a particular implementation of the generalized Schur Algorithm (GSA). It is shown that this new implementation improves the computational efficiency of the straightforward implementation of GSA from O( N 2.5) to O( N 2), where N is the number of pixels in the image.

A regularized structured total least squares algorithm for high-resolution image reconstruction

High-resolution image reconstruction is an important problem in image processing. In general, the blurring matrices are ill-conditioned, and it is necessary to compute a regularized solution. Moreover, error exists not only in the blurred image but also the blurring matrix, thus the total least squares method tends to give better results than the ordinary least squares method. Since the blurring matrices are also structured, it is more appropriate to apply Structured Total Least Squares (STLS). Ng et al. [Int. J. Imaging Systems Technol. 12 (2002) 35] recently proposed a Regularized Constrained Total Least Squares (RCTLS) algorithm for this problem. RCTLS is essentially a different name for Regularized Structured Total Least Squares (RSTLS). However, Ng et al. solved a problem that approximates the RCTLS problem. The algorithm proposed in this paper solves the exact regularization of the STLS problem, and it is a faster algorithm. Also proposed is a preconditioner for the linear systems encountered in our RSTLS algorithm.

Efficient implementation of a structured total least squares based speech compression method

We present a fast implementation of a recently proposed speech compression scheme, based on an all-pole model of the vocal tract. Each frame of the speech signal is analyzed by storing the parameters of the complex damped exponentials deduced from the all-pole model and its initial conditions. In mathematical terms, the analysis stage corresponds to solving a structured total least squares (STLS) problem. It is shown that by exploiting the displacement rank structure of the involved matrices the STLS problem can be solved in a very fast way. Synthesis is computationally very cheap since it consists of adding the complex damped exponentials based on the transmitted parameters.The compression scheme is applied on a speech signal. The speed improvement of the fast vocoder analysis scheme is demonstrated. Furthermore, the quality of the compression scheme is compared with that of a standard coding algorithm, by using the segmental signal-to-noise ratio.

The structured total least‐squares approach for non‐linearly structured matrices

AbstractIn this paper, an extension of the structured total least‐squares (STLS) approach for non‐linearly structured matrices is presented in the so‐called ‘Riemannian singular value decomposition’ (RiSVD) framework. It is shown that this type of STLS problem can be solved by solving a set of Riemannian SVD equations. For small perturbations the problem can be reformulated into finding the smallest singular value and the corresponding right singular vector of this Riemannian SVD. A heuristic algorithm is proposed. Some examples of Vandermonde‐type matrices are used to demonstrate the improved accuracy of the obtained parameter estimator when compared to other methods such as least squares (LS) or total least squares (TLS). Copyright © 2002 John Wiley & Sons, Ltd.

IQML-like algorithms for solving structured total least squares problems: a unified view

The structured total least squares (STLS) problem is an extension of the total least squares (TLS) problem for solving an overdetermined system of equations Ax≈ b. In many cases the extended data matrix [A b] has a special structure (Hankel, Toeplitz,…). In those cases the use of STLS is often required if a maximum likelihood (ML) estimate of x is desired. The main objective of this manuscript is to clarify the difference between several IQML-like algorithms—for solving STLS problems—that have been proposed by different authors and within different frameworks. Some of these algorithms yield suboptimal solutions while others yield optimal solutions. An important result is that the classicial IQML algorithm yields suboptimal solutions to the STLS problem. We illustrate this on a specific STLS problem, namely the estimation of the parameters of superimposed exponentially damped cosines in noise. We also indicate when this suboptimality starts playing an important role.

Analysis of the structured total least squares problem for Hankel/Toeplitz matrices

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) approach when structured matrices are involved and a similarly structured rank deficient approximation of that matrix is desired. In many of those cases the STLS approach yields a Maximum Likelihood (ML) estimate as opposed to, e.g., TLS. In this paper we analyze the STLS problem for Hankel matrices (the theory can be extended in a straightforward way to Toeplitz matrices, block Hankel and block Toeplitz matrices). Using a particular parametrisation of rank-deficient Hankel matrices, we show that this STLS problem suffers from multiple local minima, the properties of which depend on the parameters of the new parametrisation. The latter observation makes initial estimates an important issue in STLS problems and a new initialization method is proposed. The new initialization method is applied to a speech compression example and the results confirm the improved performance compared to other previously proposed initialization methods.

Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares Problem

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators.