Two dimensional noncausal AR-ARCH model: Stationary conditions, parameter estimation and its application to anomaly detection

• We introduced a new statistical model (i.e. AR-ARCH) for modeling background of sonar images.
• We provided the stationarity condition of the model and proposed a method for parameter estimation of the model.
• We showed that the parameter estimation method is consistent.
• We used the model in order to detect anomalies in sonar images.
• Simulation results show the performance of the proposed method.

Image anomaly detection is the process of extracting a small number of clustered pixels which are different from the background. The type of image, its characteristics and the type of anomalies depend on the application at hand. In this paper, we introduce a new statistical model called noncausal autoregressive–autoregressive conditional heteroscedasticity (AR-ARCH) model for background in sonar images. Based on this background model, we propose a novel anomaly detection technique in sonar images. This new statistical model (i.e. noncausal ARCH) is an extension of the conventional ARCH model. We provide sufficient stationarity conditions and develop a computationally efficient method for estimating the model parameters which reduces to solving two sets of linear equations. We show that this estimator is asymptotically consistent. Using matched subspace detector (MSD) along with noncausal AR-ARCH modeling of the background in the wavelet domain, we propose an anomaly detection algorithm for sonar images, which is computationally efficient and less dependent on the image orientation. Simulation results demonstrate the performance of the proposed parameter estimation and the anomaly detection algorithm.