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We introduce a fast iterative shrinkage algorithm for patch-smoothness regularization of inverse problems in medical imaging. This approach is enabled by the reformulation of current non-local regularization schemes as an alternating algorithm to minimize a global criterion. The proposed algorithm alternates between evaluating the denoised inter-patch differences by shrinkage and computing an image that is consistent with the denoised inter-patch differences and measured data. We derive analytical shrinkage rules for several penalties that are relevant in non-local regularization. The redundancy in patch comparisons used to evaluate the shrinkage steps are exploited using convolution operations. The resulting algorithm is observed to be considerably faster than current alternating non-local algorithms. The proposed scheme is applicable to a large class of inverse problems including deblurring, denoising, and Fourier inversion. The comparisons of the proposed scheme with state-of-the-art regularization schemes in the context of recovering images from undersampled Fourier measurements demonstrate a considerable reduction in alias artifacts and preservation of edges. [ABSTRACT FROM PUBLISHER]

We propose a novel blind compressive sensing (BCS) frame work to recover dynamic magnetic resonance images from undersampled measurements. This scheme models the dynamic signal as a sparse linear combination of temporal basis functions, chosen from a large dictionary. In contrast to classical compressed sensing, the BCS scheme simultaneously estimates the dictionary and the sparse coefficients from the undersampled measurements. Apart from the sparsity of the coefficients, the key difference of the BCS scheme with current low rank methods is the nonorthogonal nature of the dictionary basis functions. Since the number of degrees-of-freedom of the BCS model is smaller than that of the low-rank methods, it provides improved reconstructions at high acceleration rates. We formulate the reconstruction as a constrained optimization problem; the objective function is the linear combination of a data consistency term and sparsity promoting \ell1 prior of the coefficients. The Frobenius norm dictionary constraint is used to avoid scale ambiguity. We introduce a simple and efficient majorize–minimize algorithm, which decouples the original criterion into three simpler subproblems. An alternating minimization strategy is used, where we cycle through the minimization of three simpler problems. This algorithm is seen to be considerably faster than approaches that alternates between sparse coding and dictionary estimation, as well as the extension of K-SVD dictionary learning scheme. The use of the \ell1 penalty and Frobenius norm dictionary constraint enables the attenuation of insignificant basis functions compared to the \ell0 norm and column norm constraint assumed in most dictionary learning algorithms; this is especially important since the number of basis functions that can be reliably estimated is restricted by the available measurements. We also observe that the proposed scheme is more robust to local minima compared to K-SVD method, which relies on greedy sparse coding. Our phase transition experiments demonstrate that the BCS scheme provides much better recovery rates than classical Fourier-based CS schemes, while being only marginally worse than the dictionary aware setting. Since the overhead in additionally estimating the dictionary is low, this method can be very useful in dynamic magnetic resonance imaging applications, where the signal is not sparse in known dictionaries. We demonstrate the utility of the BCS scheme in accelerating contrast enhanced dynamic data. We observe superior reconstruction performance with the BCS scheme in comparison to existing low rank and compressed sensing schemes. [ABSTRACT FROM AUTHOR]