Browsing by Author "Wang, Yilun"
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Item A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration(2008-07) Yang, Junfeng; Yin, Wotao; Zhang, Yin; Wang, YilunWe 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.Item A New Alternating Minimization Algorithm for Total Variation Image Reconstruction(2007-06) Wang, Yilun; Yang, Junfeng; Yin, Wotao; Zhang, YinWe 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.Item Enhanced compressed sensing using iterative support detection(2009) Wang, Yilun; Yin, Wotao; Zhang, YinI present a new compressive reconstruction algorithm, which aims to simultaneously achieve low measurement requirement and fast reconstruction. This algorithm alternates between detecting partial support information of the true signal and solving a resulting truncated ℓ 1 minimization problem. I generalize Null Space Property to Truncated Null Space Property and exploit it for theoretical analysis of this truncated ℓ 1 minimization algorithm with Iterative Support Detection (abbreviated as ISD). Numerical results indicate the advantages of ISD over many other state of the art algorithms such as the basis pursuit (BP) model, the iterative reweighted ℓ 1 minimization algorithm (IRL1) and the iterative reweighted least squares algorithm (IRLS). Meanwhile, its limitation is demonstrated by both theoretical and experimental results.Item Fast algorithms for total variation minimization with applications to image deconvolution and compressed sensing(2007) Wang, Yilun; Yin, Wotao; Zhang, YinThis thesis presents two fast algorithms for total variation based image restoration. As an important branch of imaging sciences, image restoration aims to recover the original images from the degraded observations or measurements and usually some kind of regularization techniques are required for numerical stability and reducing aliasing artifacts of the recovered images. As a non-smoothing regularization technique, total variation minimization has been widely used in many image restoration fields because it allows for discontinuities in the images but at the same time disfavors oscillations. Moreover, total variation minimization can exploit the spareness of the gradient of the recovered images in many fields such as medical imaging to reduce the aliasing artifacts of the recovered images. In spite of the outstanding performance of total variation minimization in modeling many natural images, it is still a great challenge to develop efficient algorithms for it because of the high nonlinearity of total variation. Many progresses have been made to develop efficient algorithms for total variation minimization but the existing algorithms are still far from satisfying in many fields. In this thesis, we present two fast algorithms for total variation minimization in two important applications: image deconvolution and image reconstruction in compressed sensing. In image deconvolution, we present a simple algorithmic framework for recovering images from blurry and noisy observations when a blurring point-spread function is given. This framework introduces an augmented variable and construct a new functional based on the original variable and the augmented variable. We alternatively minimize the new functional based on these two variables alternatively. Basically, our algorithm is an iterative procedure of alternately solving a pair of easy subproblems associated with an increasing sequence of penalty parameter values. The main computation at each iteration is three Fast Fourier Transforms (FFTs). For image reconstruction for compressed sensing, our new algorithm is a fixed point iteration algorithm that is based on two powerful algorithmic ideas: operator-splitting and graph-based algorithms. The operator-splitting technique allows us to solve one easy subproblem and a relatively hard subproblem in each fixed point iteration associated with an increasing sequence of penalty parameter values. The newly emerging graph-based algorithms efficiently generate solutions of the hard subproblem. For both algorithms, global convergence is well established and their practical performances are also demonstrated by several typical numerical experiments.Item Sparse Signal Reconstruction via Iterative Support Detection(2009-09) Wang, Yilun; Yin, WotaoWe present a novel sparse signal reconstruction method "ISD", aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical L1 minimization approach. ISD addresses failed reconstructions of L1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new reconstruction by solving the minimization problem min{sum_{i not in I} |x_i| : Ax = b}, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit (OMP) method, as well as its variants, because (i) the index set I in ISD is not necessarily nested or increasing and (ii) the minimization problem above updates all the components of x at the same time. We generalize the Null Space Property to Truncated Null Space Property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called threshold-ISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that threshold-ISD has significant advantages over the classical L1 minimization approach, as well as two state-of-the-art algorithms: the iterative reweighted L1 minimization algorithm (IRL1) and the iterative reweighted least-squares algorithm (IRLS). MATLAB code is available for download from: http://www.caam.rice.edu/~optimization/L1/ISD/.