
Louis L. ScharfColorado State University | CSU · Department of Mathematics
Louis L. Scharf
Doctor of Philosophy
About
327
Publications
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Introduction
My currentresearch interests are statistical signal processing and machine learning for space-time matrix analysis in multi-sensor arrays that record a time-series at each sensor. Typical methodologies include group-invariant subspace methods for detection, estimation, error analysis, and frequency-wavenumber spectrum analysis. Current applications include passive and active radar and sonar; SAR and pulsed-Doppler radar; hyperspectral imaging, and group-invariant image classification.
Publications
Publications (327)
Recent work by Ram\'irez et al. [2] has introduced Multi-Channel Factor Analysis (MFA) as an extension of factor analysis to multi-channel data that allows for latent factors common to all channels as well as factors specific to each channel. This paper validates the MFA covariance model and analyzes the statistical properties of the MFA estimators...
This paper addresses the passive detection of a common rank-one subspace signal received in two multi-sensor arrays. We consider the case of a one-antenna transmitter sending a common Gaussian signal, independent Gaussian noises with arbitrary spatial covariance, and known channel subspaces. The detector derived in this paper is a generalized likel...
Recent work by Ramírez et al. [2] has introduced Multi-Channel Factor Analysis (MFA) as an extension of factor analysis to multi-channel data that allows for latent factors common to all channels as well as factors specific to each channel. This paper validates the MFA covariance model and analyzes the statistical properties of the MFA estimators....
In this paper the standard formulas for instantaneous and average power in sinusoidal systems are generalized to non-sinusoidal systems by appealing to the Hilbert Transform and Bedrosian's Theorem for a general class of non-sinusoidal currents and voltages. It is shown that a complex representation of real instantaneous power is the sum of complex...
This chapter begins with the Hilbert space geometry of quadratic performance bounds and then specializes these results to the Euclidean geometry of the Cramér-Rao bound for parameters that are carried in the mean value or the covariance matrix of a MVN model. Coherence arises naturally. A concluding section on information geometry ties the Cramér-R...
In this chapter, we illustrate the use of coherence and its generalizations to other application domains, namely, compressed sensing, multiset CCA, kernel methods, and time-frequency modeling. The concept of coherence in compressed sensing and matrix completion is made clear by the restricted isometry property and the concept of coherence index, wh...
All distances between subspaces are functions of the principal angles between them and thus can ultimately be interpreted as measures of coherence between pairs of subspaces. In this chapter, we first review the geometry of the Grassmann and Stiefel manifolds, in which q-dimensional subspaces and q-dimensional frames live, respectively. Then, we as...
In this chapter, we establish many basic results concerning inference and hypothesis testing in the proper, complex, multivariate normal distribution. We consider in particular second-order measurement models in which the unknown covariance matrix belongs to a cone. This is often the case in signal processing and machine learning. Two important res...
This chapter considers the detection of a common subspace signal in two multi-sensor channels. This problem is usually referred to as passive detection. We study second-order detectors where the unknown transmitted signal is modeled as a zero-mean Gaussian and averaged out or marginalized and first-order detectors where the unknown transmitted sign...
This chapter is devoted to the detection of signals that are constrained to lie in a subspace. The subspace may be known, or known only by its dimension. The probability distribution for the measurements may carry the signal in a parameterization of the mean or in a parameterization of the covariance matrix. Likelihood ratio detectors are derived,...
This chapter extends the problem of null hypothesis testing for linear independence between random variables to the problem of testing for linear independence between times series. When the time series are approximated with finite-dimensional random vectors, then this is a problem of null hypothesis testing for block-structured covariance matrices....
This chapter begins with a review of least squares and Procrustes problems and continues with a discussion of least squares in the linear separable model, model order determination, and total least squares. A section on oblique projections addresses the problem of resolving a few modes in the presence of many. Sections on multidimensional scaling a...
This chapter opens with the estimate and plug (EP) adaptations of the detectors in Chap. 5. These solutions adapt matched subspace detectors to unknown noise covariance matrices by constructing covariance estimates from a secondary channel of signal-free measurements. Then the Kelly and Should we say ACE (adaptive coherence estimator) detectors, an...
This chapter proceeds from a discussion of scalar-valued coherence to multiple coherence to matrix-valued coherence. Connections are established with principal angles and with canonical correlations. The study of factorizations of two-channel covariance matrices leads to filtering formulas for MMSE filters and their error covariances. When covarian...
This paper is devoted to the performance analysis of the detectors proposed in the companion paper where a comprehensive design framework is presented for the adaptive detection of subspace signals. The framework addresses four variations on subspace detection: the subspace may be known or known only by its dimension; consecutive visits to the subs...
This paper addresses the problem of detecting multidimensional subspace signals in noise of unknown covariance. It is assumed that a primary channel of measurements, possibly consisting of signal plus noise, is augmented with a secondary channel of measurements containing only noise. The noises in these two channels share a common covariance matrix...
This paper is devoted to the performance analysis of the detectors proposed in the companion paper [1] where a comprehensive design framework is presented for the adaptive detection of subspace signals. The framework addresses four variations on subspace detection: the subspace may be known or known only by its dimension; consecutive visits to the...
Consider the set of possible observations turned out by geometric and radiometric transformations of an object. This set is generally a manifold in the ambient space of observations. It has been shown
[1]
that in those cases where the geometric deformations are affine and the radiometric deformations are monotonic, the radiometry invariant univer...
This paper addresses the problem of detecting multidimensional subspace signals, which model range-spread targets, in noise of unknown covariance. It is assumed that a primary channel of measurements, possibly consisting of signal plus noise, is augmented with a secondary channel of measurements containing only noise. The noises in these two channe...
Occupancy grids encode for hot spots on a map that is represented by a two dimensional grid of disjoint cells. The problem is to recursively update the probability that each cell in the grid is occupied, based on a sequence of sensor measurements from a moving platform. In this paper, we provide a new Bayesian framework for generating these probabi...
The problem is to detect a multi-dimensional source transmitting an unknown sequence of complex-valued symbols to a multi-sensor array. In some cases the channel subspace is known, and in others only its dimension is known. Should the unknown transmissions be treated as unknowns in a first-order statistical model, or should they be assigned a prior...
This work presents a generalization of classical factor analysis (FA). Each of
${M}$
channels carries measurements that share factors with all other channels, but also contains factors that are unique to the channel. Furthermore, each channel carries an additive noise whose covariance is diagonal, as is usual in factor analysis, but is otherwise...
Occupancy grids encode for hot spots on a map that is represented by a two dimensional grid of disjoint cells. The problem is to recursively update the probability that each cell in the grid is occupied, based on a sequence of sensor measurements. In this paper, we provide a new Bayesian framework for generating these probabilities that does not as...
In this paper, we address the problem of subspace averaging, with special emphasis placed on the question of estimating the dimension of the average. The results suggest that the enumeration of sources in a multi-sensor array, which is a problem of estimating the dimension of the array manifold, and as a consequence the number of radiating sources,...
Radar tracking of aircraft targets at low elevation angles can be complicated by clutter from wind turbines, which we define to be clutter. In this paper we study detection in a staring pulse-Doppler radar. By exploiting the second-order correlation structure of this wind-turbine clutter we derive a target detector that uses a sequence of adaptive...
In this work, we consider a two-channel multiple-input multiple-output (MIMO) passive detection problem, in which there is a surveillance array and a reference array. The reference array is known to carry a linear combination of broadband noise and a subspace signal of known dimension, but unknown basis. The question is whether the surveillance cha...
Sensor fusion maybe used to improve detection performance in applications. The idea is to make decisions locally, and then transmit them to a global fusion centre where the global decision is made. For global decision making, Bayes or Neyman-Pearson reasoning determines the optimal use of the local decision variables. However, the determination of...
In this work, we consider a two-channel multiple-input multiple-output (MIMO) passive detection problem, in which there is a surveillance array and a reference array. The reference array is known to carry a linear combination of broadband noise and a subspace signal of known dimension but unknown basis. The question is whether the surveillance chan...
One of the challenges in automatic detection and classification of underwater targets in sonar imagery is variation of the target returns and features with respect to target aspect. This paper adopts a framework for target classification that offers local invariance properties with respect to target aspect. Sonar image snippets of a target type at...
In this work we consider a two-channel passive detection problem, in which there is a surveillance array where the presence/absence of a target signal is to be detected, and a reference array that provides a noise-contaminated version of the target signal. We assume that the transmitted signal is an unknown rank-one signal, and that the noises are...
The problem of estimating a low-dimensional subspace from a collection of experimentally measured subspaces arises in many applications of statistical signal processing. In this paper we address this problem, and give a solution for the average subspace that minimizes an extrinsic mean-squared error, defined by the squared Frobenius norm between pr...
This letter considers the problem of threshold selection for a correlation test among multiple (>2) random vectors. The generalized likelihood ratio test (GLRT) for this problem uses a generalized Hadamard ratio to test for block diagonality in a composite covariance matrix. As the number of realizations used to estimate the composite covariance ma...
In this paper, we investigate threshold effects associated with the swapping of signal and noise subspaces in estimating signal parameters from compressed noisy data. The term threshold effect refers to a sharp departure of mean-squared error from the Cramar-Rao bound when the signal-to-noise ratio falls below a threshold SNR. In many cases, the th...
The principle of Robust Principal Component Analysis (RPCA) is to additively resolve a matrix into a low-rank and a sparse component. The question that arises in the application of this principle to experimental data is, “when is this resolution an identification of the actual low-rank and sparse components of the data?” We report several experimen...
In this paper, we study threshold effects associated with swapping of signal and noise subspaces in estimating signal parameters from co-prime arrays. A subspace swap occurs when the measured data is better approximated by a subset of components of an orthogonal subspace than by the components of the noise-free signal subspace, and is known to be t...
In this work, we study the impact of compressive sampling with random matrices on Fisher information and the Cramer-Rao bound (CRB) for nonlinear parameter estimation in a complex multivariate normal measurement model. We consider the class of random compression matrices whose distribution is invariant to right-unitary transformations. For this cla...
This paper derives an asymptotic generalized likelihood ratio test (GLRT) and an asymptotic locally most powerful invariant test (LMPIT) for two hypothesis testing problems: 1) Is a vector-valued random process cyclostationary (CS) or is it wide-sense stationary (WSS)? 2) Is a vector-valued random process CS or is it nonstationary? Our approach use...
Suppose that there is a ground set which consists of a large number of
vectors in a Hilbert space. Consider the problem of selecting a subset of the
ground set such that the projection of a vector of interest onto the subspace
spanned by the vectors in the chosen subset reaches the maximum norm. This
problem is generally NP-hard, and alternative ap...
We consider the problem of estimating p damped complex exponentials from spatial samples of their weighted sum, taken by a sparse sensor array. Our focus is on a particular sparse array geometry, where the array can be thought of as a subsampled version of a dense (with half-wavelength spacings) uniform line array, plus an extra sensor that is posi...
We derive an estimator of the cycle period of a univariate cyclostationary process based on an information-theoretic criterion. Transforming the univariate cyclostationary process into a vector-valued wide-sense stationary process allows us to obtain the structure of the covariance matrix, which is block-Toeplitz, and its block size depends on the...
Let a measurement consist of a linear combination of damped complex
exponential modes, plus noise. The problem is to estimate the parameters of
these modes, as in line spectrum estimation, vibration analysis, speech
processing, system identification, and direction of arrival estimation. Our
results differ from standard results of modal analysis to...
In this paper, we analyze the impact of compressed sensing with complex
random matrices on Fisher information and the Cram\'{e}r-Rao Bound (CRB) for
estimating unknown parameters in the mean value function of a complex
multivariate normal distribution. We consider the class of random compression
matrices whose distribution is right-orthogonally inv...
The advent of compressed sensing theory has revolutionized our view of imaging, as it demonstrates that subsampling has manageable consequences for image inversion, provided that the image is sparse in an apriori known dictionary. For imaging problems in spectrum analysis (estimating complex exponential modes), and passive and active radar/sonar (e...
This paper considers the problem of testing for the independence among multiple random vectors with each random vector representing a time series captured at one sensor. Implementing the Generalized Likelihood Ratio Test involves testing the null hypothesis that the composite covariance matrix of the channels is block-diagonal through the use of a...
We derive the generalized likelihood ratio test (GLRT) for detecting cyclostationarity in scalar-valued time series. The main idea behind our approach is Gladyshev's relationship, which states that when the scalar-valued cyclostationary signal is blocked at the known cycle period it produces a vector-valued wide-sense stationary process. This resul...
In this paper, we examine adaptive compression policies, when the sequence of vector-valued measurements to be compressed is noisy and the compressed variables are themselves noisy. The optimization criterion is information gain. In the case of sequential scalar compressions, the unit-norm compression vectors that greedily maximize per-stage inform...
This paper is motivated by sensing and wireless communication, where data compression or dimension reduction may be used to reduce the required communication bandwidth. High-dimensional measurements are converted into low-dimensional representations through linear compression. Our aim is to compress a noisy sensor measurement, allowing for the fact...
This paper addresses the problem of testing for the independence among multiple ( $geq 2$) random vectors. The generalized likelihood ratio test tests the null hypothesis that the composite covariance matrix of the channels is block-diagonal, using a generalized Hadamard ratio. Using the theory of Gram determinants, we show that this Hadamard ratio...
Spectrum sensing is an important building block to realize the cognitive radio concept. In order to combat fading in the wireless environment, cooperation among the sensing users is usually employed. In this paper, we develop a closed-form optimal local decision threshold for cooperative spectrum sensing in cognitive radio systems via large deviati...
In this paper, we investigate threshold effects associated with swapping of signal and noise subspaces in estimating signal parameters from compressed noisy data. The term threshold effect refers to a catastrophic increase in mean-squared error when the signal-to-noise ratio falls below a threshold SNR. In many cases, the threshold effect is caused...
This paper is motivated by sensing and wireless communication, where data compression or dimension reduction may be used to reduce the required communication bandwidth. High-dimensional measurements are converted into low-dimensional representations through linear compression. Our aim is to compress a noisy sensor measurement, allowing for the fact...
In this paper, we analyze the impact of compressed sensing with random matrices on Fisher information and the CRB for estimating unknown parameters in the mean value function of a multivariate normal distribution. We consider the class of random compression matrices that satisfy a version of the Johnson-Lindenstrauss lemma, and we derive analytical...
This paper considers the problem of testing for the independence among multiple (≥ 2) random vectors with each random vector representing a time series captured at one sensor. Implementing the Generalized Likelihood Ratio Test involves testing the null hypothesis that the composite covariance matrix of the channels is block-diagonal through the use...
This paper is motivated by the problem of integrating multiple sources of measurements. We consider two multiple-input-multiple-output (MIMO) channels, a primary channel and a secondary channel, with dependent input signals. The primary channel carries the signal of interest, and the secondary channel carries a signal that shares a joint distributi...
This paper is motivated by the problem of integrating multiple sources of
measurements. We consider two multiple-input-multiple-output (MIMO) channels, a
primary channel and a secondary channel, with dependent input signals. The
primary channel carries the signal of interest, and the secondary channel
carries a signal that shares a joint distributi...
A theory and algorithm for detecting and classifying weak, distributed patterns in network data is presented. The patterns we consider are anomalous temporal correlations between signals recorded at sensor nodes in a network. We use robust matrix completion and second order analysis to detect distributed patterns that are not discernible at the lev...
Cooperation among secondary users can greatly improve the spectrum sensing performance in cognitive radio. In our previous work [1], we proposed the diversity as a measure of the cooperative gain and developed cooperative spectrum sensing schemes based on some pre-defined local threshold selections. In this paper, on the other hand, we start the co...
We examine greedy adaptive measurement policies in the context of a linear Gaussian measurement model with an information-based optimization criterion. We provide sufficient conditions under which the greedy policy is optimal in the sense of maximizing the net information gain. We also discuss an example where the greedy policy is not optimal.
Fusion is widely used to improve the overall detection performance in applications such as radar, wireless sensor networks, wireless communications, spectrum sensing and so on. While the optimum fusion strategy for any preset local decision performance can be easily obtained by the Neyman-Pearson lemma, the selection of the local detection strategy...
A family of multi-sensor detection problems is proposed in which the number of hypotheses to be resolved can be traded off against the probability of error, the signal-to-noise ratio (SNR), and the number of sensors. Using large deviations and an approximation of the large deviation rate function, it is shown that the number of hypotheses resolvabl...
Spectrum sensing has become one of the main components of a cognitive transmitter. Conventional detectors suffer from noise power uncertainties and multiantenna detectors have been proposed to overcome this difficulty, and to improve the detection performance. However, most of the proposed multiantenna detectors are based on non-optimal techniques,...
We present a theory and algorithm for detecting and classifying weak,
distributed patterns in network data that provide actionable information
with quantiable measures of uncertainty. Our work demonstrates the
eectiveness of space-time inference on graphs, robust matrix completion,
and second order analysis for the detection of distributed patterns...
In this paper we describe an approach for the detection and classication
of weak, distributed patterns in sensor networks. Of course, before one
can begin development of a pattern detection algorithm, one must rst
dene the term "pattern", which by nature is a broad and inclusive term.
One of the key aspects of our work is a denition of pattern that...
In this paper we study the existence of locally most powerful invariant tests
(LMPIT) for the problem of testing the covariance structure of a set of
Gaussian random vectors. The LMPIT is the optimal test for the case of close
hypotheses, among those satisfying the invariances of the problem, and in
practical scenarios can provide better performanc...
The purpose of this article is to examine the greedy adaptive measurement
policy in the context of a linear Guassian measurement model with an
optimization criterion based on information gain. In the special case of
sequential scalar measurements, we provide sufficient conditions under which
the greedy policy actually is optimal in the sense of max...
The phasor measurement units (PMU) are expected to enhance state estimation in the power grid by providing accurate and timely measurements. However, due to communication errors and equipment failures, some detrimental data can occur among the measurements. The largest residual removal (LRR) algorithm is commonly used for phasor state estimation wi...
Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that complex random signals are proper or circular. A proper complex random variable is uncorrelated with its complex conjugate, and a circular complex random variable has a probabil...
In [1]-[4], we considered the question of basis mismatch in compressive sensing. Our motivation was to study the effect of mismatch between the mathematical basis (or frame) in which a signal was assumed to be sparse and the physical basis in which the signal was actually sparse. We were motivated by the problem of inverting a complex space-time ra...
A simple fusion rule with multiple thresholds is presented for a distributed system to resolve multiple hypotheses. In contrast to the common assumption that the data is conditionally independent and identically distributed, only conditional independence under each hypothesis is assumed here. This allows the modeling of situations in which differen...
In this work, we derive a maximum likelihood formula for beamsteering in a multi-sensor array. The novelty of the work is that the impinging signal and noises are wide sense stationary (WSS) time series with unknown power spectral densities, unlike in previous work that typically considers white signals. Our approach naturally provides a way of fus...
Compressed sensing theory suggests that successful inversion of an image of the physical world from its modal parameters can be achieved at measurement dimensions far lower than the image dimension, provided that the image is sparse in an a priori known basis. The assumed basis for sparsity typically corresponds to a gridding of the parameter space...
For an improper complex signal x, its complementary covariance ExxT is not
zero and thus it carries useful statistical information about x. Widely linear
processing exploits Hermitian and complementary covariance to improve
performance. In this paper we extend the existing theory of widely linear
complex Kalman filters (WLCKF) and unscented WLCKFs...
This work addresses the problem of deciding whether a set of realizations of a vector-valued time series with unknown temporal correlation are spatially correlated or not. Specifically, the spatial correlation is induced by a colored source over a frequency-flat single-input multiple-output (SIMO) channel distorted by independent and identically di...
This paper considers linear precoding for time-varying multiple input multiple-output (MIMO) channels. We show that linear minimum mean-squared error (LMMSE) equalization based on the conjugate gradient (CG) method can result in significantly reduced complexity compared with conventional approaches. This reduction is achieved by incorporating a con...