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In this paper, we propose a model for parallel magnetic resonance imaging (pMRI) reconstruction, regularized by a carefully designed tight framelet system, that can lead to reconstructed images with much less artifacts in comparison to those from existing models. Our model is motivated from the observations that each receiver coil in a pMRI system is more sensitive to the specific object nearest to the coil, and all coil images are correlated. To exploit these observations, we first stack all coil images together as a 3-dimensional (3D) data matrix, and then design a 3D directional Haar tight framelet (3DHTF) to represent it. After analyzing sparse information of the coil images provided by the high-pass filters of the 3DHTF, we separate the high-pass filters into effective ones and ineffective ones, and we then devise a 3D directional Haar semi-tight framelet (3DHSTF) from the 3DHTF by replacing its ineffective filters with only one filter. This 3DHSTF is tailor-made for coil images, meanwhile, giving a significant saving in computation comparing to the 3DHTF. With the 3DHSTF, we propose an l1-3DHSTF model for pMRI reconstruction. Numerical experiments for MRI phantom and in-vivo data sets are provided to demonstrate the superiority of our l1-3DHSTF model in terms of the efficiency of reducing aliasing artifacts in the reconstructed images. 

In this work, we consider a distributed online convex optimization problem, with time-varying (potentially adversarial) constraints. A set of nodes, jointly aim to minimize a global objective function, which is the sum of local convex functions. The objective and constraint functions are revealed locally to the nodes, at each time, after taking an action. Naturally, the constraints cannot be instantaneously satisfied. Therefore, we reformulate the problem to satisfy these constraints in the long term. To this end, we propose a distributed primal-dual mirror descent-based algorithm, in which the primal and dual updates are carried out locally at all the nodes. This is followed by sharing and mixing of the primal variables by the local nodes via communication with the immediate neighbors. To quantify the performance of the proposed algorithm, we utilize the challenging, but more realistic metrics of dynamic regret and fit. Dynamic regret measures the cumulative loss incurred by the algorithm compared to the best dynamic strategy, while fit measures the long term cumulative constraint violations. Without assuming the restrictive Slater’s conditions, we show that the proposed algorithm achieves sublinear regret and fit under mild, commonly used assumptions. 

This work considers a super-resolution framework forovercomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the ℓ1 norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of ℓ1 norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors. 

This work considers a super-resolution framework for overcomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the ℓ1 norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of ℓ1 norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors. 

Sparsity promoting functions (SPFs) are commonly used in optimization problems to find solutions which are sparse in some basis. For example, the [Formula: see text]-regularized wavelet model and the Rudin–Osher–Fatemi total variation (ROF-TV) model are some of the most well-known models for signal and image denoising, respectively. However, recent work demonstrates that convexity is not always desirable in SPFs. In this paper, we replace convex SPFs with their induced nonconvex SPFs and develop algorithms for the resulting model by exploring the intrinsic structures of the nonconvex SPFs. These functions are defined as the difference of the convex SPF and its Moreau envelope. We also present simulations illustrating the performance of a special SPF and the developed algorithms in image denoising. 

This paper studies a new convex variational model for denoising and deblurring images with multiplicative noise. Considering the statistical property of the multiplicative noise following Nakagami distribution, the denoising model consists of a data fidelity term, a quadratic penalty term, and a total variation regularization term. Here, the quadratic penalty term is mainly designed to guarantee the model to be strictly convex under a mild condition. Furthermore, the model is extended for the simultaneous denoising and deblurring case by introducing a blurring operator. We also study some mathematical properties of the proposed model. In addition, the model is solved by applying the primal-dual algorithm. The experimental results show that the proposed method is promising in restoring (blurred) images with multiplicative noise.