Abstract
Convolutional sparse coding improves on the standard sparse approximation by incorporating a global shift-invariant model. The most efficient convolutional sparse coding methods are based on the alternating direction method of multipliers and the convolution theorem. The only major difference between these methods is how they approach a convolutional least-squares fitting subproblem. This letter presents a solution to this subproblem, which improves the efficiency of the state-of-the-art algorithms. We also use the same approach for developing an efficient convolutional dictionary learning method. Furthermore, we propose a novel algorithm for convolutional sparse coding with a constraint on the approximation error.
Highlights
S PARSE representations are widely used in various applications of signal and image processing [1]–[6]
The contributions of this letter are summarized as follows: (i) we present an efficient approach for solving the convolutional least-squares fitting which leads to a constant improvement on the complexity of the existing Convolutional sparse coding (CSC) algorithms; (ii) we
All methods are based on the same alternating approach explained in Section II-D and use alternating direction method of multipliers (ADMM) in both phases (CSC and dictionary update)
Summary
S PARSE representations are widely used in various applications of signal and image processing [1]–[6]. A common formulation of the sparse coding problem is given as minimize x 1 s.t. where D is the dictionary, x ∈ Rm is the sparse representation vector, s ∈ Rn is the signal, and represents the upper bound on the approximation error. The applications of sparse representations and dictionary learning usually involve either or both extraction and estimation of local features This is handled by a prior decomposition of the original signal into vectorized overlapping blocks (e.g., patches in image processing). A common approach for convolutional dictionary learning (CDL) entails optimizing the filters and the sparse coefficient maps using a batch of P training signals [23]–[26]. MATLAB implementations of the proposed algorithms are available at GitHub repository [33]
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