Browsing by Author "Yang, Junfeng"

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    A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration
    (2008-07) Yang, Junfeng; Yin, Wotao; Zhang, Yin; Wang, Yilun
    We generalize the alternating minimization algorithm recently proposed in [32] to effciently solve a general, edge-preserving, variational model for recovering multichannel images degraded by within- and cross-channel blurs, as well as additive Gaussian noise. This general model allows the use of localized weights and higher-order derivatives in regularization, and includes a multichannel extension of total variation (MTV) regularization as a special case. In the MTV case, we show that the model can be derived from an extended half-quadratic transform of Geman and Yang [14]. For color images with three channels and when applied to the MTV model (either locally weighted or not), the per-iteration computational complexity of this algorithm is dominated by nine fast Fourier transforms. We establish strong convergence results for the algorithm including finite convergence for some variables and fastﾠq-linear convergence for the others. Numerical results on various types of blurs are presented to demonstrate the performance of our algorithm compared to that of the MATLAB deblurring functions. We also present experimental results on regularization models using weighted MTV and higher-order derivatives to demonstrate improvements in image quality provided by these models over the plain MTV model.
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    A Fast TVL1-L2 Minimization Algorithm for Signal Reconstruction from Partial Fourier Data
    (2008-10) Yang, Junfeng; Zhang, Yin; Yin, Wotao
    Recent compressive sensing results show that it is possible to accurately reconstruct certain compressible signals from relatively few linear measurements via solving nonsmooth convex optimization problems. In this paper, we propose a simple and fast algorithm for signal reconstruction from partial Fourier data. The algorithm minimizes the sum of three terms corresponding to total variation, $\ell_1$-norm regularization and least squares data fitting. It uses an alternating minimization scheme in which the main computation involves shrinkage and fast Fourier transforms (FFTs), or alternatively discrete cosine transforms (DCTs) when available data are in the DCT domain. We analyze the convergence properties of this algorithm, and compare its numerical performance with two recently proposed algorithms. Our numerical simulations on recovering magnetic resonance images (MRI) indicate that the proposed algorithm is highly efficient, stable and robust.
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    A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
    (2007-06) Wang, Yilun; Yang, Junfeng; Yin, Wotao; Zhang, Yin
    We propose, analyze and test an alternating minimization algorithm for recovering images from blurry and noisy observa- tions with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also isotropic forms of total variation discretizations. The per-iteration computational complexity of the algorithm is three Fast Fourier Transforms (FFTs). We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or q-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the Lagged Diffusivity algorithm for total-variation-based deblurring. Some extensions of our algorithm are also discussed.
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    Alternating Direction Algorithms for L1-Problems in Compressive Sensing
    (2009-11) Yang, Junfeng; Zhang, Yin
    In this paper, we propose and study the use of alternating direction algorithms for several L1-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis-pursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the L1-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an L1-problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several state-of-the-art algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the l1-norm fidelity should be the fidelity of choice in compressive sensing.
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    Practical Compressive Sensing with Toeplitz and Circulant Matrices
    (2010-01) Yin, Wotao; Morgan, Simon; Yang, Junfeng; Zhang, Yin
    Compressive sensing encodes a signal into a relatively small number of incoherent linear measurements. In theory, the optimal incoherence is achieved by completely random measurement matrices. However, such matrices are difficult and/or costly to implement in hardware realizations. After summarizing recent results of how random Toeplitz and circulant matrices can be easily (or even naturally) realized in various applications, we introduce fast algorithms for reconstructing signals from incomplete Toeplitz and circulant measurements. We present computational results showing that Toeplitz and circulant matrices are not only as effective as random matrices for signal encoding, but also permit much faster signal decoding.