Accelerating ordered-subsets X-ray CT image reconstruction using the linearized augmented Lagrangian framework

The augmented Lagrangian (AL) optimization method has drawn more attention recently in imaging applications due to its decomposable structure for composite cost functions and empirical fast convergence rate under weak conditions. However, for problems, e.g., X-ray computed tomography (CT) image reconstruction, where the inner least-squares problem is challenging, the AL method can be slow due to its iterative inner updates. In this paper, using a linearized AL framework, we propose an ordered-subsets (OS) accelerable linearized AL method, OS-LALM, for solving penalized weighted least-squares (PWLS) X-ray CT image reconstruction problems. To further accelerate the proposed algorithm, we also propose a deterministic downward continuation approach for fast convergence without additional parameter tuning. Experimental results show that the proposed algo- rithm significantly accelerates the “convergence” of X-ray CT image reconstruction with negligible overhead and exhibits excellent gradient error tolerance when using many subsets for OS acceleration.

[1]  Zhixun Su,et al.  Linearized Alternating Direction Method with Adaptive Penalty for Low-Rank Representation , 2011, NIPS.

[2]  Stanley Osher,et al.  A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration , 2010, J. Sci. Comput..

[3]  Jeffrey A. Fessler,et al.  Ordered subsets acceleration using relaxed momentum for X-ray CT image reconstruction , 2013, 2013 IEEE Nuclear Science Symposium and Medical Imaging Conference (2013 NSS/MIC).

[4]  J. Fessler,et al.  Combining Augmented Lagrangian Method with Ordered Subsets for X-Ray CT Reconstruction , 2013 .

[5]  Jeffrey A. Fessler,et al.  Ordered subsets with momentum for accelerated X-ray CT image reconstruction , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[6]  Xavier Bresson,et al.  Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction , 2010, SIAM J. Imaging Sci..

[7]  Jeffrey A. Fessler,et al.  Reduced memory augmented Lagrangian algorithm for 3D iterative x-ray CT image reconstruction , 2012, Medical Imaging.

[8]  Jeffrey A. Fessler,et al.  Accelerating X-ray CT ordered subsets image reconstruction with Nesterov ’ s first-order methods , 2013 .

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

[10]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

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

[12]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[13]  Jeffrey A. Fessler,et al.  A Splitting-Based Iterative Algorithm for Accelerated Statistical X-Ray CT Reconstruction , 2012, IEEE Transactions on Medical Imaging.

[14]  Hakan Erdogan,et al.  Ordered subsets algorithms for transmission tomography. , 1999, Physics in medicine and biology.

[15]  Tony F. Chan,et al.  A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science , 2010, SIAM J. Imaging Sci..

[16]  Jeffrey A. Fessler,et al.  Fast X-Ray CT Image Reconstruction Using a Linearized Augmented Lagrangian Method With Ordered Subsets , 2014, IEEE Transactions on Medical Imaging.

[17]  Xiangfeng Wang,et al.  The Linearized Alternating Direction Method of Multipliers for Dantzig Selector , 2012, SIAM J. Sci. Comput..

[18]  Yunhai Xiao,et al.  Splitting and linearizing augmented Lagrangian algorithm for subspace recovery from corrupted observations , 2013, Adv. Comput. Math..

[19]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[20]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

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

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

[23]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

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