Maximal Invariant Statistic Research Articles

Invariance Theory for Adaptive Radar Detection in Heterogeneous Environment

Adaptive detection of a signal competing with Gaussian interference with spatially varying power is addressed in this letter. Invariance principle is used to focus the attention on a class of decision rules exhibiting some specific symmetries, which represent a sufficient condition to ensure the constant false alarm rate property. The class of invariant decision rules is characterized via the design of a maximal invariant statistic together with its equivalent in the parameter space. Finally, it is proved that the adaptive normalized matched filter using a suitable recursive estimate of the interference covariance structure coincides with a maximal invariant component.

Polarimetric Covariance Eigenvalues Classification in SAR Images

This letter proposes a novel technique for automatic classification of the dominant scattering mechanisms associated with the pixels of polarimetric SAR images. Focusing on the heterogeneous scenario wherein the polarimetric image pixels share the same covariance but different power levels, the original data are replaced by a maximal invariant statistic in order to remove the dependence on the scaling factors. Then, the classification problem is formulated as a multiple hypothesis test which is addressed by applying the model order selection rules. The performance analysis is conducted on both simulated and measured data and points out the effectiveness of the proposed approach.

On Multiple Covariance Equality Testing with Application to SAR Change Detection

This paper deals with the problem of testing the equality of $M$ covariance matrices. We first identify a suitable group of transformations leaving the problem invariant and obtain the corresponding maximal invariant statistic. Then, the Generalized Likelihood Ratio Test (GLRT) is recalled and explicit expressions for Rao, Wald, Gradient, and Durbin tests are provided. Also, equivalences among them and with other well-known tests proposed in open literature (mostly for the real-valued case) are analyzed and compared. Finally, the application of the proposed framework to the relevant signal processing application of multipass Coherent Change Detection (CCD) in polarimetric synthetic aperture radar is demonstrated both on simulated and on live data.

On Time-Reversal Imaging by Statistical Testing

This letter is focused on the design and analysis of computational wideband time-reversal imaging algorithms, designed to be adaptive with respect to the noise levels pertaining to the frequencies being employed for scene probing. These algorithms are based on the concept of cell-by-cell processing and are obtained as theoretically-founded decision statistics for testing the hypothesis of single-scatterer presence (absence) at a specific location. These statistics are also validated in comparison with the maximal invariant statistic for the proposed problem.

On the Statistical Invariance for Adaptive Radar Detection in Partially Homogeneous Disturbance Plus Structured Interference

This paper deals with the problem of adaptive vector subspace signal detection in partially homogeneous Gaussian disturbance and structured (unknown) deterministic interference within the framework of invariance theory. It is first proved that the Maximal Invariant Statistic (MIS) for the problem at hand is scalar-valued and coincides with the well-known adaptive normalized matched filter evaluated after data projection in the complementary subspace of the interfering signal. Second, the statistical characterization of the MIS under both hypotheses is derived. Then, it is shown the statistical equivalence of (two-step) generalized-likelihood ratio test, Rao and Wald tests, as well as the more recently considered Durbin and Gradient test, to the above statistic. Finally, simulation results are provided to confirm our findings and analyze the performance trend of the MIS with the relevant parameters.

On the Maximal Invariant Statistic for Adaptive Radar Detection in Partially Homogeneous Disturbance With Persymmetric Covariance

This letter deals with the problem of adaptive signal detection in partially-homogeneous and persymmetric Gaussian disturbance within the framework of invariance theory. First, a suitable group of transformations leaving the problem invariant is introduced and the Maximal Invariant Statistic (MIS) is derived. Then, it is shown that the (Two-step) Generalized-Likelihood Ratio test, Rao and Wald tests can be all expressed in terms of the MIS, thus proving that they all ensure a Constant False-Alarm Rate (CFAR).

Invariance Theory for Adaptive Detection in Interference With Group Symmetric Covariance Matrix

This paper describes the adaptive detection of point-like targets in a Gaussian environment assuming a group symmetric structure for the interference covariance matrix. This special configuration enforces block-sparsity and permits splitting of the original observation space into lower-dimensional subspaces, each characterized by its own nuisance parameters. Hence, the Principle of Invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered. All the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper. Finally, some common design criteria are applied to come up with adaptive architectures that share invariance, and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes.

Forcing Scale Invariance in Multipolarization SAR Change Detection

This paper considers the problem of coherent (in the sense that both amplitudes and relative phases of the polarimetric returns are used to construct the decision statistic) multipolarization synthetic aperture radar change detection starting from the availability of image pairs exhibiting possible power mismatches/miscalibrations. The principle of invariance is used to characterize the class of scale-invariant decision rules which are insensitive to power mismatches and ensure the constant false alarm rate property. A maximal invariant statistic is derived together with the induced maximal invariant in the parameter space which significantly compresses the data/parameter domain. A generalized likelihood ratio test is synthesized both for the cases of two- and three-polarimetric channels. Interestingly, for the two-channel case, it is based on the comparison of the condition number of a data-dependent matrix with a suitable threshold. Some additional invariant decision rules are also proposed. The performance of the considered scale-invariant structures is compared to those from two noninvariant counterparts using both simulated and real radar data. The results highlight the robustness of the proposed method and the performance tradeoff involved.

Coincidence of Maximal Invariants for Two Adaptive Radar Detection Problems

This letter deals with adaptive radar detection of targets embedded in Gaussian clutter plus range-distributed subspace-structured jamming by exploiting the invariance theory. The class of invariant detectors which ensure the constant false alarm rate property has been characterized by designing the maximal invariant statistic for the studied detection problem. The achievement of this letter is the coincidence of the obtained maximal invariant with that derived assuming range-concentrated jamming.

Design and Analysis of Invariant Receivers for Gaussian Targets

The purpose of this paper is to elaborate on the invariant framework proposed in “An invariant approach to adaptive radar detection under covariance persymmetry” (A. De Maio and D. Orlando, IEEE Trans. Signal Processing, vol. 63, no. 5, pp. 1297–1309, May 2015) so as to account for target fluctuations at the design stage. To this end, we assume that the radar cross section of the target is ruled by an exponential distribution (Swerling 1 target model) and derive the complete statistical characterization of a maximal invariant statistic. Hence, we exploit the obtained probability density functions to synthesize both the optimum and the locally optimum (in the low Signal-to-Interference-plus-Noise Ratio regime) invariant receivers. Finally, we discuss some invariant sub-optimum decision rules based on theoretically solid design criteria and analyze their performances in comparison with the benchmark invariant test.

Adaptive Radar Detection of a Subspace Signal Embedded in Subspace Structured Plus Gaussian Interference Via Invariance

This paper deals with adaptive radar detection of a subspace signal competing with two sources of interference. The former is Gaussian with unknown covariance matrix and accounts for the joint presence of clutter plus thermal noise. The latter is structured as a subspace signal and models coherent pulsed jammers impinging on the radar antenna. The problem is solved via the Principle of Invariance which is based on the identification of a suitable group of transformations leaving the considered hypothesis testing problem invariant. A maximal invariant statistic, which completely characterizes the class of invariant decision rules and significantly compresses the original data domain, as well as its statistical characterization are determined. Thus, the existence of the optimum invariant detector is addressed together with the design of practically implementable invariant decision rules. At the analysis stage, the performance of some receivers belonging to the new invariant class is established through the use of analytic expressions.

A Unifying Framework for Adaptive Radar Detection in Homogeneous Plus Structured Interference— Part I: On the Maximal Invariant Statistic

This paper deals with the problem of adaptive multidimensional/multichannel signal detection in homogeneous Gaussian disturbance with unknown covariance matrix and structured deterministic interference. The aforementioned problem corresponds to a generalization of the well-known Generalized Multivariate Analysis of Variance (GMANOVA). In this first part of the work, we formulate the considered problem in canonical form and, after identifying a desirable group of transformations for the considered hypothesis testing, we derive a Maximal Invariant Statistic (MIS) for the problem at hand. Furthermore, we provide the MIS distribution in the form of a stochastic representation. Finally, strong connections to the MIS obtained in the open literature in simpler scenarios are underlined.

A Unifying Framework for Adaptive Radar Detection in Homogeneous Plus Structured Interference— Part II: Detectors Design

This paper deals with the problem of adaptive multidimensional/multichannel signal detection in homogeneous Gaussian disturbance with unknown covariance matrix and structured (unknown) deterministic interference. The aforementioned problem extends the well-known Generalized Multivariate Analysis of Variance (GMANOVA) tackled in the open literature. In a companion paper, we have obtained the Maximal Invariant Statistic (MIS) for the problem under consideration, as an enabling tool for the design of suitable detectors which possess the Constant False-Alarm Rate (CFAR) property. Herein, we focus on the development of several theoretically-founded detectors for the problem under consideration. First, all the considered detectors are shown to be function of the MIS, thus proving their CFARness property. Secondly, coincidence or statistical equivalence among some of them in such a general signal model is proved. Thirdly, strong connections to well-known simpler scenarios found in adaptive detection literature are established. Finally, simulation results are provided for a comparison of the proposed receivers.

Invariant Rules for Multipolarization SAR Change Detection

This paper deals with coherent (in the sense that both amplitudes and relative phases of the polarimetric returns are used to construct the decision statistic) multipolarization synthetic aperture radar (SAR) change detection assuming the availability of reference and test images collected from $N$ multiple polarimetric channels. At the design stage, the change detection problem is formulated as a binary hypothesis testing problem, and the principle of invariance is used to come up with decision rules sharing the constant false alarm rate property. The maximal invariant statistic and the maximal invariant in the parameter space are obtained. Hence, the optimum invariant test is devised proving that a uniformly most powerful invariant detector does not exist. Based on this, the class of suboptimum invariant receivers, which also includes the generalized likelihood ratio test, is considered. At the analysis stage, the performance of some tests, belonging to the aforementioned class, is assessed and compared with the optimum clairvoyant invariant detector. Finally, detection maps on real high-resolution SAR data are computed showing the effectiveness of the considered invariant decision structures.

Maximal Invariant and Weakly Equivariant Estimators

Equivariant functions can be useful for constructing of maximal invariant statistic. In this article, we discuss construction of maximal invariants based on a given weakly equivariant function under some additional conditions. The theory easily extends to the case of two or more weakly equivariant functions. Also, we derive a maximal invariant statistic when the group contains a sharply transitive and a characteristic subgroup. Finally, we consider the independence of invariant and weakly equivariant functions under some special conditions.

An Invariant Approach to Adaptive Radar Detection Under Covariance Persymmetry

In this paper, we propose a systematic and unifying framework to deal with adaptive radar detection in the presence of Gaussian interference sharing a persymmetric covariance structure. First, we determine the group of transformations which leaves the considered hypothesis testing problem unaltered; then, after reduction by sufficiency, we determine a maximal invariant statistic which is a four dimensional vector and significantly compresses the original observation space. Its first two components are the one-step and the two-step Generalized Likelihood Ratio Test decision statistics, respectively, whereas its last two entries represent an ancillary statistic. We provide also the exact statistical characterization of the maximal invariant which is exploited to synthesize both the optimum and the locally optimum (in the low Signal-to-Interference-plus-Noise Ratio regime) invariant receivers. Finally, some sub-optimum decision rules based on theoretically solid design criteria are discussed and their performances are analyzed in comparison with the benchmark invariant test.

Invariance and Optimality of CFAR Detectors in Binary Composite Hypothesis Tests

We investigate the relationship between constant false alarm rate (CFAR) and invariant tests. We introduce the minimal invariant group (MIG). We show that for a family of distributions, the unknown parameters are eliminated from the distribution of the maximal invariant statistic under the MIG while the maximum information of the observed signal is preserved. We prove that any invariant test with respect to MIG is CFAR and conversely, for any CFAR test an invariant statistic exists with respect to an MIG under some mild conditions. Moreover for a given CFAR test, we propose a systematic method for deriving an enhanced test, i.e., a function of observations exists such that the likelihood ratio (LR) of the maximal invariant of its MIG gives an enhanced test. Furthermore, we introduce the uniformly most powerful-CFAR (UMP-CFAR) test as the optimal CFAR bound among all CFAR tests. We then prove that the UMP-CFAR test for the minimally invariant hypothesis testing problem is given by the LR of the maximal invariant under MIG. For some problems, this test (MP-CFAR) depends on the unknown parameters of the alternative hypothesis, however, provides an upper-performance bound for all suboptimal CFAR tests. We also propose three suboptimal novel CFAR tests among which one is asymptotically optimal.

Maximal Invariants and Performance of Some Invariant Hypothesis Tests for an Adaptive Detection Problem

Maximal invariants for adaptive detection of a signal in unknown interference from multiple observations is derived. Given coherent samples from P sets of observations, it is shown that a maximal invariant statistic for the detection problem is a 2P × 1-dimensional vector comprising the eigenvalues of two Hermitian positive definite matrices obtained from the data set. Two invariant detectors, well known for P=1, are generalized for the case of multiple observations and closed form expressions for the probability of detection and probability of false alarm are derived along with the distributions of the signal-to-interference-plus-noise loss factors. Several novel invariant detectors are constructed from the maximal invariants and the receiver operating characteristics of the detectors compared.

Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors

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 performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires one to find the maximal invariant statistic for the detection problem and then derive its distribution under both hypotheses, which in general is a rather involved procedure. As an alternative, Wijsman's theorem provides the ratio of the maximal invariant densities without even finding an explicit expression for the maximal invariant. We first consider the problem of testing whether a set of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional Gaussian random vectors are uncorrelated or not, and show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. Second, we study the case in which the vectors under the null hypothesis are uncorrelated and identically distributed, that is, the sphericity test for Gaussian vectors, for which we show that the LMPIT is given by the Frobenius norm of a normalized version of the sample covariance matrix. Finally, some numerical examples illustrate the performance of the proposed tests, which provide better results than their GLRT counterparts.

Constructing Locally Best Invariant Tests of the Linear Regression Model Using the Density Function of a Maximal Invariant

In the context of the linear regression model in which som e regression coefficients are of interest and others are purely nuisance parameters, we derive the density function of a maximal invariant statistic after eliminating the nuisance parameters by the principle of invariance argu ment. This allows the construction of a range of optimal test statistics including the locally best invariant (LBI) test wh ich is equivalent to the well-known one-sided t-test. The resultant LBI test is also found to be uniformly most powerfu l invariant (UMPI).