The problem of hypothesis testing in the Neyman–Pearson formulation is considered from a geometric viewpoint. In particular, a concise geometric interpretation of deterministic and random signal detection in the philosophy of information geometry is presented. In such a framework, both hypotheses and detectors can be treated as geometrical objects on the statistical manifold of a parameterized family of probability distributions. Both the detector and detection performance are geometrically elucidated in terms of the Kullback–Leibler divergence. Compared to the likelihood ratio test, the geometric interpretation provides a consistent but more comprehensive means to understand and deal with signal detection problems in a rather convenient manner. Example of the geometry based detector in radar constant false alarm rate (CFAR) detection is presented, which shows its advantage over the classical processing method.

Radar coincidence imaging (RCI) is a high-resolution staring imaging technique without the limitation of relative motion between target and radar. The sparsity-driven approaches are commonly used in RCI, while the prior knowledge of imaging models needs to be known accurately. However, as one of the major model errors, the gain-phase error exists generally, and may cause inaccuracies of the model and defocus the image. In the present report, the sparse auto-calibration method is proposed to compensate the gain-phase error in RCI. The method can determine the gain-phase error as part of the imaging process. It uses an iterative algorithm, which cycles through steps of target reconstruction and gain-phase error estimation, where orthogonal matching pursuit (OMP) and Newton’s method are used, respectively. Simulation results show that the proposed method can improve the imaging quality significantly and estimate the gain-phase error accurately.

This paper proposes a class of covariance estimators based on information divergences in heterogeneous environments. In particular, the problem of covariance estimation is reformulated on the Riemannian manifold of Hermitian positive-definite (HPD) matrices. The means associated with information divergences are derived and used as the estimators. Without resorting to the complete knowledge of the probability distribution of the sample data, the geometry of the Riemannian manifold of HPD matrices is considered in mean estimators. Moreover, the robustness of mean estimators is analyzed using the influence function. Simulation results indicate the robustness and superiority of an adaptive normalized matched filter with our proposed estimators compared with the existing alternatives.

In this paper, Bayesian nonlinear filtering is considered from the viewpoint of information geometry and a novel filtering method is proposed based on information geometric optimization. Under the Bayesian filtering framework, we derive a relationship between the nonlinear characteristics of filtering and the metric tensor of the corresponding statistical manifold. Bayesian joint distributions are used to construct the statistical manifold. In this case, nonlinear filtering can be converted to an optimization problem on the statistical manifold and the adaptive natural gradient descent method is used to seek the optimal estimate. The proposed method provides a general filtering formulation and the Kalman filter, the Extended Kalman filter (EKF) and the Iterated Extended Kalman filter (IEKF) can be seen as special cases of this formulation. The performance of the proposed method is evaluated on a passive target tracking problem and the results demonstrate the superiority of the proposed method compared to various Kalman filter methods.

This paper presents a covariance matrix estimation method based on information geometry in a heterogeneous clutter. In particular, the problem of covariance estimation is reformulated as the computation of geometric median for covariance matrices estimated by the secondary data set. A new class of total Bregman divergence is presented on the Riemanian manifold of Hermitian positive-definite (HPD) matrix, which is the foundation of information geometry. On the basis of this divergence, total Bregman divergence medians are derived instead of the sample covariance matrix (SCM) of the secondary data. Unlike the SCM, resorting to the knowledge of statistical characteristics of the sample data, the geometric structure of matrix space is considered in our proposed estimators, and then the performance can be improved in a heterogeneous clutter. At the analysis stage, numerical results are given to validate the detection performance of an adaptive normalized matched filter with our estimator compared with existing alternatives.

The measurement update stage in the nonlinear filtering is considered in the viewpoint of information geometry, and the filtered state is considered as an optimization estimation in parameter space has been corresponded with the iteration in the statistical manifold, then a recursive method is proposed in this paper. This method is derived based on the natural gradient descent on the statistical manifold, which constructed by the posterior probability density function (PDF) of state conditional on the measurement. The derivation procedure is processing in the geometric viewpoint, and gives a geometric interpretation for the iteration update. Besides, the proposed method can be seen as an extended for the Kalman filter and its variants. For the one step in our proposed method, it is identical to the Extended Kalman filter (EKF) in the nonlinear case, while traditional Kalman filter in the linear case. Benefited from the natural gradient descent used in the update stage, our proposed method performs better than the existing methods, and the results have showed in the numerical experiments.