Convergence and comparison results for double splittings of Hermitian positive definite matrices

Unsupervised classification plays an important role in understanding polarimetric synthetic aperture radar (PolSAR) images. One of the typical representations of PolSAR data is in the form of Hermitian positive definite (HPD) covariance matrices. Most algorithms for unsupervised classification using this representation either use statistical distribution models or adopt polarimetric target decompositions. In this paper, we propose an unsupervised classification method by introducing a sparsity-based similarity measure on HPD matrices. Specifically, we first use a novel Riemannian sparse coding scheme for representing each HPD covariance matrix as sparse linear combinations of other HPD matrices, where the sparse reconstruction loss is defined by the Riemannian geodesic distance between HPD matrices. The coefficient vectors generated by this step reflect the neighborhood structure of HPD matrices embedded in the Euclidean space and hence can be used to define a similarity measure. We apply the scheme for PolSAR data, in which we first oversegment the images into superpixels, followed by representing each superpixel by an HPD matrix. These HPD matrices are then sparse coded, and the resulting sparse coefficient vectors are then clustered by spectral clustering using the neighborhood matrix generated by our similarity measure. The experimental results on different fully PolSAR images demonstrate the superior performance of the proposed classification approach against the state-of-the-art approaches.

Hermitian positive definite (HPD) covariance matrices form one of the most widely-used data representations in PolSAR applications. However, most of these applications either use statistical distribution models on the PolSAR covariance matrices or polarimetric target decomposition. In this paper, we study HPD matrices for PolSAR image classification in the context of sparse coding. More specifically, the PolSAR HPD matrices are first represented as sparse linear combinations of elements from a dictionary, where each element itself is an HPD matrix and the representation loss is measured by the affine-invariant Riemannian metric. We then introduce a sparsity induced similarity measure between two HPD matrices. Finally, we propose a supervised classification scheme using support vector machines on the Riemannian sparse codes and an unsupervised classification scheme encompassing a sparsity induced similarity measure followed by spectral clustering. The proposed methods are validated on the NASA/JPL AIRSAR fully PolSAR data. The experimental results demonstrate the effectiveness of our methods.

Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory. MSC:15A45, 15A57.

This paper presents a pre-processing procedure associated with a Riemannian geometry method to carry out a target detection in a radar system. Firstly, the sample data in each range cell in one coherent processing interval is modeled as a Hermitian positive-definite (HPD) matrix. All these matrices constitute a complex Riemannian manifold. Each point on this manifold is an HPD matrix. Then, the new radar observation data for each range cell is an HPD matrix, and a pre-processing procedure, called the weighted averaging filter, is imposed on these HPD matrices. The covariance matrix in each range cell is replaced by a weighted average of covariance matrices of its surrounding cells. The decision rule is made by comparing the distance between the Riemannian mean estimated by the covariance matrices of neighboring locations and the covariance matrix of cell under test with a threshold. The superiority of the proposed algorithm is verified by numerical experiments.

Geometric target detection based on total Bregman divergence

In this paper, we propose a K‐nearest neighbor manifold filter on the Riemannian manifold and apply it to signal detection within clutter. In particular, the correlation and power of sample data in each cell are modeled as an Hermitian positive definite (HPD) matrix. A K‐nearest neighbor filter that performs the weight average of the set of K‐nearest neighbor HPD matrices of each HPD matrix is proposed to reduce the clutter power. Then, the clutter covariance matrix is estimated as the Riemannian mean of a set of secondary HPD matrices. Signal detection is considered as distinguishing the matrices of clutter and target signal on the Riemannian manifold. Moreover, to speed up the convergence of matrix equation of Riemannian mean, we exploit a strategy to choose the initial input matrix and step size of this equation. Numerical results show that the proposed detector achieves a detection performance improvement over the conventional detector as well as its state‐of‐the‐art counterpart in nonhomogeneous clutter.

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm compared with traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.

Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners. For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different monotone matrices.

We first make a brief review of GMRES convergence results. Then we derive new bounds for the GMRES residual norm by making use of a unitary matrix U and a Hermitian positive definite matrix P, which are GMRES-equivalent to the coefficient matrix A with respect to the initial residual r/sub 0/. The existence of such U and P was proved by Leonid (2000). As a GMRES residual norm bound for linear systems with Hermitian positive definite coefficient matrices is known and a GMRES residual norm bound for linear systems with unitary coefficient matrices can be readily derived from Liesen's (2000) work, our new bounds follow from the fact that two GMRES-equivalent matrices make the same residual.

This article deals with the problem of detecting maritime targets embedded in nonhomogeneous sea clutter, where the limited number of secondary data is available due to the heterogeneity of sea clutter. A class of linear discriminant analysis (LDA)-based matrix information geometry (MIG) detectors is proposed in the supervised scenario. As customary, Hermitian positive-definite (HPD) matrices are used to model the observational sample data, and the clutter covariance matrix of the received dataset is estimated as the geometric mean of the secondary HPD matrices. Given a set of training HPD matrices with class labels, which are elements of a higher dimensional HPD matrix manifold, the LDA manifold projection learns a mapping from the higher dimensional HPD matrix manifold to a lower dimensional one subject to maximum discrimination. In this study, the LDA manifold projection, with the cost function maximizing between-class distance while minimizing within-class distance, is formulated as an optimization problem in the Stiefel manifold. Four robust LDA-MIG detectors corresponding to different geometric measures are proposed. Numerical results based on both simulated radar clutter with interferences and real IPIX radar data show the advantage of the proposed LDA-MIG detectors against their counterparts without using LDA and the state-of-the-art maritime target detection methods in nonhomogeneous sea clutter.

A new fixed point theorem in [formula omitted]-metric space and its application to solve a class of nonlinear matrix equations

Matrix information geometry (MIG) detector, which converts sample data to Hermitian positive definite (HPD) matrices located on HPD matrix manifold, provides an innovative scheme for target detection. In this paper, a subband MIG detector is proposed to detect target submerged into heterogeneous clutter background with short pulses. More precisely, subband filtering is firstly performed to suppress strong clutter by designing a discrete Fourier transform modulated filter bank. Then, a set of HPD matrices are modeled by the filtered data and a HPD matrix manifold is formed. In each subband, the detection from geometric consideration on the manifold is derived. Thus, a subband MIG detector is formulated and information divergence is utilized to measure the dissimilarity of the observed data and the clutter. Finally, numerical experiments based on simulated data and real sea clutter data show that the proposed method can achieve better detection performance compared with the traditional methods.

Convergence and comparison theorems for double splittings of matrices

This study explores the application of geometric measures‐based means and medians on the Riemannian manifold of Hermitian positive‐definite (HPD) matrix to target detection problems in radar systems. Firstly, the slow‐time dimension of radar received clutter data in each cell is modelled and mapped to HPD matrix space, which can be described as a complex Riemannian manifold. Each point of this manifold is an HPD matrix. Then, several geometric measures are presented for measuring closeness between two HPD matrices. According to these measures, the means and medians of a finite collection of HPD matrices are deduced, and various matrix constant false alarm rate (CFAR) detectors are designed. The principle of target detection is that if a location has enough dissimilarity from the geometric mean or median estimated by its neighbouring locations, targets are supposed to appear at this location. Moreover, different distance measures adopted in detector can result in different performance of detection. These differences are owing to different measure structure, reflected by the anisotropy of a location on the Riemannian manifold. Numerical experiments are given to demonstrate the relationship between anisotropy of the geometric measures and the detection performance of their corresponding matrix CFAR detectors.

A class of the iteration method from the double splitting of coefficient matrix for solving the linear system is further investigated. By structuring a new matrix, the iteration matrix of the corresponding double splitting iteration method is presented. On the basis of convergence and comparison theorems for single splittings, we present some new convergence and comparison theorems on spectral radius for splittings of matrices.