Matched subspace detectors

Abstract

and the noise is MVN with mean ScjJ and covariance R = a 2 R o:where J-L = 0 under H o and J-L > 0 under H 1 • This is the standard detection problem wherein the polarity of the signal x is assumed known. Near the end of Section V we replacewhere polarity is unknown.We shall assume that the signal x obeys the linear subspaceThe detection problems to be studied in this paper may be described as follows. We are given N samples from a real, scalar time series {y(n), n = 0,1, 0 0 " N -I} which are assembled into the N-dimensional measurement vectorBased on these data, we must decide between two possible hypotheses regarding how the data was generated. The null hypothesis H o says that the data consist of noise tI only. The alternative hypothesis H 1 says that the data consist of a sum of signal J-LX and noise tI; that is, (2.2) (2.1)to include subspace interferences. These problems involve unknown parameters in the mean and covariance of a multivariate normal (MVN) distribution. For each problem in the class, we establish invariances for the GLR and find that they are identical to the natural invariances for the problem. We show that a monotone function of the GLRT equals one of the uniformly most powerful invariant (UMP-invariant) tests derived in [1]. This means that the GLRT is itself UMPinvariant. In addition to tying up the theories of invariance and the GLRT, our results generalize and extend previous work on these problems published in [1]-[6]. We begin our development by establishing the invariances of the GLRT in the MVN problem. We then specialize our results for structured means in order to derive UMP-invariant GLRT detectors for matched subspace filtering in subspace interference. The GLRT produces an UMP-invariant detector, which is CFAR if the noise variance is unknown. As we shall find, the optimum detector may be interpreted as a null steering or interference rejecting processor followed by a matched subspace detector.

Detection Power, Estimation Efficiency, and Predictability in Event-Related fMRI

“…For most fMRI experiments, S should at the very least contain a constant term and a linear trend term, e.g., the zeroth-and first-order Legendre polynomials. Following Scharf and Friedlander (1994), we refer to the subspaces spanned by the columns of X and S as the signal subspace ͗X͘ and the interference subspace ͗S͘, respectively. These subspaces lie within the N-dimensional space spanned by the data.…”

Anomaly subspace detection based on a multi-scale Markov random field model

“…Scharf and Friedlander [21] formulated a MSD for the general problem of detecting subspace signals in subspace interference and additive white Gaussian noise. Here, the anomaly detection is based on a statistical model which better describes the background clutter.…”

“…The MSD, proposed by Scharf and Friedlander [21], is applied to the estimated innovations process. By analyzing the performance of this method and comparing it to Method III we examine the contribution of the proposed multi-scale model to the detection performance.…”

“…In many detection problems, the information about the targets is a subspace in which the targets lie. In these applications, the matched signal detector is replaced by a matched subspace detector (MSD), a generalization of the matched filter, which was formulated by Scharf and Friedlander [21]. The MSD is used for detecting subspace signals in subspace interference and additive noise, using the principle of the generalized likelihood ratio test (GLRT) .…”

“…The MSD incorporates the available a priori information about the targets into the detection process and thus improves the detection performance. The MSD was originally developed for signal detection in subspace interference and white Gaussian noise [21]. Here, we formulate an MSD for signal detection in subspace interference and noise which follows the multiscale GMRF model.…”