Hindawi Publishing Corporation Journal of Electrical and Computer Engineering Volume 2012, Article ID 628479, 7 pages doi:10.1155/2012/628479 Research Article Target Detection Using Nonsingular Approximations for a Singular Covariance Matrix Nir Gorelik, 1 Dan Blumberg, 2 Stanley R. Rotman, 1 and Dirk Borghys 3 1 Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 2 Department of Geography and Environmental Development, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 3 Signal and Image Centre, Royal Military Academy, 1000 Brussels, Belgium Correspondence should be addressed to Nir Gorelik, nir.gorelik@gmail.com Received 1 April 2012; Accepted 7 June 2012 Academic Editor: Xiaofei Hu Copyright © 2012 Nir Gorelik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Accurate covariance matrix estimation for high-dimensional data can be a difficult problem. A good approximation of the covari- ance matrix needs in most cases a prohibitively large number of pixels, that is, pixels from a stationary section of the image whose number is greater than several times the number of bands. Estimating the covariance matrix with a number of pixels that is on the order of the number of bands or less will cause not only a bad estimation of the covariance matrix but also a singular covariance matrix which cannot be inverted. In this paper we will investigate two methods to give a sufficient approximation for the covariance matrix while only using a small number of neighboring pixels. The first is the quasilocal covariance matrix (QLRX) that uses the variance of the global covariance instead of the variances that are too small and cause a singular covariance. The second method is sparse matrix transform (SMT) that performs a set of K-givens rotations to estimate the covariance matrix. We will compare results from target acquisition that are based on both of these methods. An improvement for the SMT algorithm is suggested. 1. Introduction The most widely used algorithms for target detection are traditionally based on the covariance matrix [1]. This matrix estimates the direction and magnitude of the noise in an image. In the equation for a matched filter presented in [1] we have R = t T Φ −1 G (x − m), (1) x is the examined pixel, m is the estimate of that pixel based on the surroundings, Φ G is the global covariance matrix, and t is the target signature. In words, we can say that our matched filter for target detection will detect the target in a particular pixel x if x is different than its surroundings (x − m), unlike the noise (controlled by Φ −1 G ) and in the direction of the target. If the target signature is unknown, then the RX algorithm uses the target residual (x − m) as its own match, that is, R = (x − m) T Φ −1 G (x − m), (2) Φ G is traditionally calculated as follows: Φ G = 1 N N  i=1 (x i − m)(x i − m) T . (3) Although the equation is theoretically justified if the back- ground is stationary, it is often used in cases where this is not true. In target detection, the image is not normally statistically stationary; it will however have quasistationary “patches” which connect to each other at the edges. When one estimates the mean and covariance matrix of the background of a particular pixel, the local neighboring pixels will have provided a better estimate than the pixels of the entire image. In [2], we show that much better results can be obtained if one uses a “quasilocal covariance matrix” (QLRX). In general terms, it uses the eigenvectors of the overall global matrix, but the eigenvalues are taken locally. This tends to lower the matched filter scores at edges in the data (when the image is going from one stationary distribution to another), but allows for accurate detection in less noisy areas.