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A novel reconstruction method for compressive spectral imaging is designed by assuming that the spectral image of interest is sufficiently smooth on a collection of graphs. Since the graphs are not known in advance, we propose to infer them from a panchromatic image using a state-of-the-art graph learning method. Our approach leads to solutions with closed-form that can be found efficiently by solving multiple sparse systems of linear equations in parallel. Extensive simulations and an experimental demonstration show the merits of our method in comparison with traditional methods based on sparsity and total variation and more recent methods based on low-rank minimization and deep-based plug-and-play priors. Our approach may be instrumental in designing efficient methods based on deep neural networks and covariance estimation.more » « less
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Let $$\Gamma$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$G=\mathrm{Cay}(\Gamma,T)$$ be a Cayley graph of $$\Gamma$$. The graph $$G$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for \textcolor[rgb]{0,0,1}{the second (largest) eigenvalue} of the adjacency matrix of the graph $$G$$ in terms of the second eigenvalues of certain subgraphs of $$G$$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$.more » « less
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In this paper, we calculate the optimal sampling sets for bandlimited signals on cographs. We take into account the tree structure of the cograph to derive closed form results for the uniqueness sets of signals with a given bandwidth. These results do not require expensive spectral decompositions and represent a promising tool for the analysis of signals on graphs that can be approximated by cographs.more » « less
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This paper discusses the generalization of the concept of blue noise sampling from traditional halftoning to signal processing on graphs. Making use of the spatial properties of blue noise, we generate sampling patterns that provide reconstruction errors that are similar to the ones obtained with state of the art approaches. This sampling scheme presents an alternative to those techniques that require spectral decompositions.more » « less
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Let $$G$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$\Gamma=\mathrm{Cay}(G,T)$$ be a Cayley graph of $$G$$. The graph $$\Gamma$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $$\Gamma$$ in terms of the second eigenvalues of certain subgraphs of $$\Gamma$$. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$, and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$.more » « less
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