Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent 1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches.
This paper proposes a deep neural network structure that exploits edge information in addressing representative lowlevel vision tasks such as layer separation and image filtering. Unlike most other deep learning strategies applied in this context, our approach tackles these challenging problems by estimating edges and reconstructing images using only cascaded convolutional layers arranged such that no handcrafted or application-specific image-processing components are required. We apply the resulting transferrable pipeline to two different problem domains that are both sensitive to edges, namely, single image reflection removal and image smoothing. For the former, using a mild reflection smoothness assumption and a novel synthetic data generation method that acts as a type of weak supervision, our network is able to solve much more difficult reflection cases that cannot be handled by previous methods. For the latter, we also exceed the state-of-the-art quantitative and qualitative results by wide margins. In all cases, the proposed framework is simple, fast, and easy to transfer across disparate domains.
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