The novel two-dimensional adaptive filter algorithms with the performance analysis

• The establishment of the novel 2D adaptive filters such as 2D-SPU-NLMS, 2D-SPU-APA, and 2D-SRAPA which lead to the reduction of the computational complexity in 2D applications.
• The mean-square performance analysis of 2D adaptive filter algorithms and studying the transient and steady-state performance of them.
• Demonstration of the presented algorithms in 2D system identification and 2D-ANC applications.

Two-dimensional (2D) adaptive filtering is a technique that can be applied to many images, and signal processing applications. This paper extends the one-dimensional adaptive filter algorithms to 2D structures and the novel 2D adaptive filters are established. Based on this extension, the 2D selective partial update NLMS (2D-SPU-NLMS), the 2D selective partial update APA (2D-SPU-APA) and the 2D selective regressor APA (2D-SR-APA) are presented. In 2D-SPU adaptive algorithms, the filter coefficients are partially updated, and in 2D-SR-APA, the recent regressors of input signal are optimally selected in each time iteration. These algorithms reduce the computational complexity in 2D adaptive filter applications. In the following, a unified approach for the establishment and mean-square performance analysis of the family of 2D adaptive filter algorithms is presented. This analysis is based on energy conservation arguments and does not need to assume a Gaussian or white distribution for the regressors. We demonstrate the good performance of the proposed algorithms through several simulation results in 2D system identification and 2D adaptive noise cancellation (2D-ANC) for image restoration. The results are compared with the classical 2D adaptive filters such as 2D-LMS, 2D-NLMS, and 2D-APA. Also we show that the derived theoretical expressions are useful in predicting the steady-state and transient performance of 2D adaptive filter algorithms.