New Dictionary Learning Methods for Two-Dimensional Signals

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

“…Problem (12) is jointly non‐convex over $${\left\{{\mathbf{D}}_{d}\right\}}_{d=1}^{D}$$ and $${\left\{{\mathcal{X}}_{i}\right\}}_{i=1}^{L}$$, so we can use alternating minimisation to solve it [24]. This approach alternates between SC for MD training signals where dictionaries $${\left\{{\mathbf{D}}_{d}\right\}}_{d=1}^{D}$$ are kept fixed, and then updating dictionaries $${\left\{{\mathbf{D}}_{d}\right\}}_{d=1}^{D}$$ where $${\left\{{\mathcal{X}}_{i}\right\}}_{i=1}^{L}$$ are fixed.…”

“…This method is based on the MOD algorithm for 1DDL [15]. In [24], MOD is generalised for two‐dimensional signals. In this section, we generalise the MOD algorithm for MD signals, so we call the proposed algorithm fast multidimensional DL based on MOD (FMDL‐MOD).…”

“…The proposed optimization approach is a gradient descent based on a first order Taylor series approximation. This idea is given in [24] for the 2D case, and in this paper we generalise it to the MD case. To obtain this approximation, consider the function F (.)…”

“…To solve the problem (23), alternating minimisation [24] is used. Specifically, the minimisation is iteratively performed over $${\left\{{\mathbf{D}}_{d}\right\}}_{d=1}^{D}$$ and $${\left\{{\mathcal{X}}_{i}\right\}}_{i=1}^{L}$$, where $${\left\{{\mathbf{D}}_{d}^{a}\right\}}_{d=1}^{D}$$ and $${\left\{{\mathcal{X}}_{i}^{a}\right\}}_{i=1}^{L}$$ represent previous values of these parameters and are considered as constant parameters in optimization.…”

“…Unlike the SC algorithms whose generalisation to MD mode has been investigated in a lot of research, a little development has been done on the MDDL algorithms. For example, in [21][22][23][24], 1DDL algorithms are generalised to the two-dimensional (2D) mode, and in [25] the 2D DL (2DDL) algorithm proposed in [21] is generalised to the MD mode. Our goal in this paper is to fill this gap by providing some fast MDDL algorithms.…”