Fast multidimensional dictionary learning algorithms and their application in 3D inverse synthetic aperture radar image restoration and noise reduction

Abstract

By generalising dictionary learning (DL) algorithms to multidimensional (MD) mode and using them in applications where signals are inherently multidimensional, such as in three-dimensional (3D) inverse synthetic aperture radar (ISAR) imaging, it is possible to achieve much higher speed and less computational complexity. In this study, the formulation of the multidimensional dictionary learning (MDDL) problem is expressed and two algorithms are proposed to solve it. The first one is based on the method of optimum directions (MOD) algorithm for 1D dictionary learning (1DDL), which uses alternating minimisation and gradient projection approach. As the MDDL problem is non-convex, the second algorithm approximates the non-convex objective with a new jointly convex function and efficiently solves it. As an application, we use the proposed methods to restore and denoise the ISAR image. Numerical experiments highlight that the proposed algorithms, in addition to reducing the computational complexity and the amount of required memory, also entail less training data for learning the dictionary, and enjoy higher convergence speed in comparison to their one-dimensional (1D) counterparts. Specifically, convergence speed of MD algorithms, depending on the size of the training data, is up to at least 10.7 times faster than the equivalent 1DDL algorithm. According to the simulation results, the SNR value achieved by the proposed algorithms is higher than the case where we use the 3D-IFFT for image reconstruction and the case of fixed dictionaries, by approximately 12 and 4 dB, respectively.