Noisy Image Reconstruction Via Fast Linearized Lagrangian Dual Alternating Direction Method of Multipliers

In this paper, an efficient noisy image reconstruction algorithm based on compressed sensing in the wavelet domain is proposed. The new algorithm is composed of three steps. Firstly, the noisy image is represented with its coefficients using the discrete wavelet transform. Secondly, the measurement is obtained by using a random Gaussian matrix. Finally, a fast linearized Lagrangian dual alternating direction method of multipliers is proposed to reconstruct the sparse coefficients, which will be converted by the inverse discrete wavelet transform to the reconstructed image. Our experimental results show that the proposed reconstruction algorithm yields a slightly higher peak signal to noise ratio reconstructed image as well as a much faster convergence rate as compared to some existing reconstruction algorithms.

[1]  Gitta Kutyniok,et al.  1 . 2 Sparsity : A Reasonable Assumption ? , 2012 .

[2]  Philippe L. Toint,et al.  Approximating Hessians in unconstrained optimization arising from discretized problems , 2011, Comput. Optim. Appl..

[3]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[4]  Hou Biao,et al.  Development and Prospect of Compressive Sensing , 2011 .

[5]  Yaakov Tsaig,et al.  Breakdown of equivalence between the minimal l1-norm solution and the sparsest solution , 2006, Signal Process..

[6]  J. Romberg,et al.  Imaging via Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[7]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[8]  Shiqian Ma,et al.  Fast alternating linearization methods for minimizing the sum of two convex functions , 2009, Math. Program..

[9]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[10]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

[11]  José M. Bioucas-Dias,et al.  Fast Image Recovery Using Variable Splitting and Constrained Optimization , 2009, IEEE Transactions on Image Processing.

[12]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[13]  Hong Zhu,et al.  Primal and dual alternating direction algorithms for ℓ1-ℓ1-norm minimization problems in compressive sensing , 2012, Computational Optimization and Applications.

[14]  Yin Zhang,et al.  Fixed-Point Continuation for l1-Minimization: Methodology and Convergence , 2008, SIAM J. Optim..

[15]  Junfeng Yang,et al.  Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization , 2012, Math. Comput..

[16]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[17]  Junfeng Yang,et al.  A Fast Alternating Direction Method for TVL1-L2 Signal Reconstruction From Partial Fourier Data , 2010, IEEE Journal of Selected Topics in Signal Processing.

[18]  Michael P. Friedlander,et al.  Probing the Pareto Frontier for Basis Pursuit Solutions , 2008, SIAM J. Sci. Comput..

[19]  José M. Bioucas-Dias,et al.  An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems , 2009, IEEE Transactions on Image Processing.

[20]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[21]  Lawrence Carin,et al.  Bayesian Compressive Sensing , 2008, IEEE Transactions on Signal Processing.

[22]  D. L. Donoho,et al.  Compressed sensing , 2006, IEEE Trans. Inf. Theory.

[23]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[24]  Necdet Serhat Aybat,et al.  A First-Order Augmented Lagrangian Method for Compressed Sensing , 2010, SIAM J. Optim..

[25]  Michael Elad,et al.  On the stability of the basis pursuit in the presence of noise , 2006, Signal Process..

[26]  B. Mercier,et al.  A dual algorithm for the solution of nonlinear variational problems via finite element approximation , 1976 .

[27]  Kim-Chuan Toh,et al.  A coordinate gradient descent method for ℓ1-regularized convex minimization , 2011, Comput. Optim. Appl..

[28]  Yaakov Tsaig,et al.  Extensions of compressed sensing , 2006, Signal Process..

[29]  Emmanuel J. Candès,et al.  NESTA: A Fast and Accurate First-Order Method for Sparse Recovery , 2009, SIAM J. Imaging Sci..

[30]  Yun-Hai Xiao,et al.  An Inexact Alternating Directions Algorithm for Constrained Total Variation Regularized Compressive Sensing Problems , 2011, Journal of Mathematical Imaging and Vision.

[31]  Masashi Sugiyama,et al.  Dual-Augmented Lagrangian Method for Efficient Sparse Reconstruction , 2009, IEEE Signal Processing Letters.

[32]  Junfeng Yang,et al.  Alternating Direction Algorithms for 1-Problems in Compressive Sensing , 2009, SIAM J. Sci. Comput..