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Line 95: By imposing the sparsity prior in the inherent structure of <math display="inline">\mathbf{x}</math>, strong conditions for a unique representation and feasible methods for estimating it are granted. Similarly, such a constraint can be applied to its representation itself, generating a cascade of sparse representations: Each code is defined by a few atoms of a given set of convolutional dictionaries. Based on these criteria, yet another extension denominated ~~mlti~~multi-layer convolutional sparse coding (ML-CSC) is proposed. A set of analytical dictionaries <math display="inline">\{\mathbf{D}\}_{k=1}^{K}</math> can be efficiently designed, where sparse representations at each layer <math display="inline">\{\mathbf{\Gamma}\}_{k=1}^{K}</math> are guaranteed by imposing the sparsity prior over the dictionaries themselves.<ref name="papyan_2017_convolutional" /> In other words, by considering dictionaries to be stride convolutional matrices i.e. atoms of the local dictionaries shift <math display="inline">m</math> elements instead of a single one, where <math display="inline">m</math> corresponds to the number of channels in the previous layer, it is guaranteed that the <math display="inline">\\|\mathbf{\Gamma}\\|_{0,\infty}</math> norm of the representations along layers is bounded. For example, given the dictionaries <math display="inline">\mathbf{D}_{1} \in \mathbb{R}^{N\times Nm_{1}}, \mathbf{D}_{2} \in \mathbb{R}^{Nm_{1}\times Nm_{2}}</math>, the signal is modeled as <math display="inline">\mathbf{D}_{1}\mathbf{\Gamma}_{1}= \mathbf{D}_{1}(\mathbf{D}_{2}\mathbf{\Gamma}_{2})</math>, where <math display="inline">\mathbf{\Gamma}_{1}</math> is its sparse code, and <math display="inline">\mathbf{\Gamma}_{2}</math> is the sparse code of <math display="inline">\mathbf{\Gamma}_{1}</math>. Then, the estimation of each representation is formulated as an optimization problem for both noise-free and noise-corrupted scenarios, respectively. Assuming <math display="inline">\mathbf{\Gamma}_{0}=\mathbf{x}</math>: <math display="block">\begin{aligned}

Convolutional sparse coding: Difference between revisions