Learning overcomplete dictionaries for sparse signal representation has become a hot topic fascinated by many researchers in the recent years, while most of the existing approaches have a serious problem that they always lead to local minima. In this article, we present a novel augmented Lagrangian multi-scale dictionary learning algorithm (ALM-DL), which is achieved by first recasting the constrained dictionary learning problem into an AL scheme, and then updating the dictionary after each inner iteration of the scheme during which majorizationminimization technique is employed for solving the inner subproblem. Refining the dictionary from low scale to high makes the proposed method less dependent on the initial dictionary hence avoiding local optima. Numerical tests for synthetic data and denoising applications on real images demonstrate the superior performance of the proposed approach.
Image reconstruction from sparse fan-beam projection data would result in image error. In this paper, a hybrid imaging algorithm from sparse fan-beam projections is proposed. The first, high exact sparse spectrum data is extracted from image reconstruction from sparse fan-beam projections by filtered back-projection (FBP), and then image is reconstructed from the data using the iterative next neighbor regridding (INNG) algorithm combined with total variation (TV) gradient descent method. The INNG step can restrict image distortions around the model and the TV gradient descent step can remove small oscillations in the model while preserving edges. The combined method is compared with the original INNG algorithm and TV gradient descent method. Computer simulation results demonstrate that the hybrid algorithm is effective for sparse fan-beam projection reconstruction.
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