Detection of multiple correlated time series and its application in synthetic aperture sonar imagery

Abstract

Detecting the presence of a common but unknown signal among two or more data channels is a problem that finds its uses in many applications, including collaborative sensor networks, geological monitoring of seismic activity, radar, and sonar. Some detection systems in such situations use decision fusion to combine individual detection decisions into one global decision. However, this detection paradigm can be sub-optimal as local decisions are based on the perspective of a single sensory system. Thus, methods that capture the coherent or mutual information among multiple data sets are needed. This work considers the problem of testing for the independence among multiple (≥ 2) random vectors. The solution is attained by considering a Generalized Likelihood Ratio Test (GLRT) that tests the null hypothesis that the composite covariance matrix of the channels, a matrix containing all inter and intra-channel second-order information, is block-diagonal. The test statistic becomes a generalized Hadamard ratio given by the ratio of the determinant of the estimate of this composite covariance matrix over the product of the determinant of its diagonal blocks. One important question in the practical application of any likelihood ratio test is the values of the test statistic needed to achieve sufficient evidence in support of the decision to reject the null hypothesis. To gain some understanding of the false alarm probability or size of the test for the generalized Hadamard ratio, we employ the theory of Gram determinants to show that the likelihood ratio can be written as a product of ratios of the squared residual from two linear prediction problems. This expression for the likelihood ratio leads quite simply to the fact that the generalized Hadamard ratio is stochastically equivalent to a product of independently distributed beta random variables under the null hypothesis. Asymptotically, the scaled logarithm of the generalized Hadamard ratio converges in distribution to a chi-squared random variable as the number of samples used to estimate the composite covariance matrix grows large. The degrees of freedom for this chi-squared distribution are closely related to the dimensions of the parameter spaces considered in the development of the GLRT. Studies of this asymptotic distribution seem to indicate, however, that the rate of convergence is particularly slow for all but the simplest of problems and may therefore lack practicality. For this reason, we consider the use of saddlepoint approximations as a practical alternative for this problem. This leads to methods that can be used to determine the threshold needed to approximately achieve a desired false alarm probability. We next turn our attention to an alternative implementation of the generalized Hadamard ratio for 2-dimensional wide-sense stationary random processes. Although the true GLRT for this problem would impose a Toeplitz structure (more specifically, a Toeplitz-block-Toeplitz structure) on the estimate of the composite covariance matrix, an intractable problem with no closed-form solution, the asymptotic theory of large Toeplitz matrices shows that the generalized Hadamard ratio converges to a broadband coherence statistic as the size of the composite covariance matrix grows large. Although an asymptotic result, simulations of several applications show that even finite dimensional implementations of the broadband coherence statistic can provide a significant improvement in detection performance. This improvement in performance is most likely attributed to the fact that, by constraining the model to incorporate stationarity, we have alleviated some of the difficulties associated with estimating highly parameterized models. Although more generally applicable, the unconstrained covariance estimates used in the generalized Hadamard ratio require the estimation of a much larger number of parameters. These methods are then applied to the detection of underwater targets in pairs of high frequency and broadband sonar images coregistered over the seafloor. This is a difficult problem due to various factors such as variations in the operating and environmental conditions, presence of spatially varying clutter, and variations in target shapes, compositions, and orientation. A comprehensive study of these methods is conducted using three sonar imagery datasets. The first two datasets are actual images of objects lying on the seafloor and are collected at different geographical locations with the environments from each presenting unique challenges. These two datasets will be used to demonstrate the usefulness of results pertaining to the null distribution of the generalized Hadamard ratio and to study the effects different clutter environments can have on its applicability. They are also used to compare the performance of the broadband coherence detector to several alternative detection techniques. The third dataset used in these studies contains actual images of the seafloor with synthetically generated targets of different geometrical shapes inserted into the images. The primary purpose of this dataset is to study the proposed detection technique's robustness to deviations from coregistration which may occur in practice due to the disparities in high frequency and broadband sonar. Using the results of this section, we will show that the fundamental principle of detecting underwater targets using coherence-based approaches is itself a very useful solution for this problem and that the broadband coherence statistic is adequately adept at achieving this.