TY - JOUR

T1 - A Direct Algorithm for Optimization Problems with the Huber Penalty

AU - Xu, Jingyan

AU - Noo, Frédéric

AU - Tsui, Benjamin M.W.

N1 - Funding Information:
Manuscript received June 2, 2017; revised September 15, 2017 and September 29, 2017; accepted September 30, 2017. Date of publication October 5, 2017; date of current version December 29, 2017. This work was supported by the National Cancer Institute of the National Institutes of Health under Award R21CA211035. The work of F. Noo was also supported by Siemens Medical Solutions, USA. (Corresponding author: Jingyan Xu.) J. Xu and B. M. W. Tsui are with the Division of Medical Imaging Physics, Department of Radiology, Johns Hopkins University, Baltimore, MD 21287-0859 USA (e-mail: jxu18@jhmi.edu).
Funding Information:
This work was supported by the National Cancer Institute of the National Institutes of Health under Award R21CA211035. The work of F. Noo was also supported by Siemens Medical Solutions, USA. We thank Michael Mozdy at University of Utah for proofreading the manuscript. The concepts presented in this paper are based on research and are not commercially available. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
Publisher Copyright:
© 2017 IEEE.

PY - 2018/1

Y1 - 2018/1

N2 - We present a direct (noniterative) algorithm for 1-D quadratic data fitting with neighboring intensity differences penalized by the Huber function. Applications of such an algorithm include 1-D processing of medical signals, such as smoothing of tissue time concentration curves in kinetic data analysis or sinogram preprocessing, and using it as a subproblem solver for 2-D or 3-D image restoration and reconstruction. dynamic programming was used to develop the direct algorithm. The problem was reformulated as a sequence of univariate optimization problems, for k = 1, ⋯ ,N, where N is the number of data points. The solution to the univariate problem at index k is parameterized by the solution at k + 1, except at k = N. Solving the univariate optimization problem at k = N yields the solution to each problem in the sequence using back-tracking. Computational issues and memory cost are discussed in detail. Two numerical studies, tissue concentration curve smoothing and sinogram preprocessing for image reconstruction, are used to validate the direct algorithm and illustrate its practical applications. In the example of 1-D curve smoothing, the efficiency of the direct algorithm is compared with four iterative methods: the iterative coordinate descent, Nesterov's accelerated gradient descent algorithm, FISTA, and an off-the-shelf second order method. The first two methods were applied to the primal problem, the others to the dual problem. The comparisons show that the direct algorithm outperforms all other methods by a significant factor, which rapidly grows with the curvature of the Huber function. The second example, sinogram preprocessing, showed that robustness and speed of the direct algorithm are maintained over a wide range of signal variations, and that noise and streaking artifacts could be reduced with almost no increase in computation time. We also outline how the proposed 1-D solver can be used for imaging applications.

AB - We present a direct (noniterative) algorithm for 1-D quadratic data fitting with neighboring intensity differences penalized by the Huber function. Applications of such an algorithm include 1-D processing of medical signals, such as smoothing of tissue time concentration curves in kinetic data analysis or sinogram preprocessing, and using it as a subproblem solver for 2-D or 3-D image restoration and reconstruction. dynamic programming was used to develop the direct algorithm. The problem was reformulated as a sequence of univariate optimization problems, for k = 1, ⋯ ,N, where N is the number of data points. The solution to the univariate problem at index k is parameterized by the solution at k + 1, except at k = N. Solving the univariate optimization problem at k = N yields the solution to each problem in the sequence using back-tracking. Computational issues and memory cost are discussed in detail. Two numerical studies, tissue concentration curve smoothing and sinogram preprocessing for image reconstruction, are used to validate the direct algorithm and illustrate its practical applications. In the example of 1-D curve smoothing, the efficiency of the direct algorithm is compared with four iterative methods: the iterative coordinate descent, Nesterov's accelerated gradient descent algorithm, FISTA, and an off-the-shelf second order method. The first two methods were applied to the primal problem, the others to the dual problem. The comparisons show that the direct algorithm outperforms all other methods by a significant factor, which rapidly grows with the curvature of the Huber function. The second example, sinogram preprocessing, showed that robustness and speed of the direct algorithm are maintained over a wide range of signal variations, and that noise and streaking artifacts could be reduced with almost no increase in computation time. We also outline how the proposed 1-D solver can be used for imaging applications.

KW - Denoising

KW - Huber penalty

KW - dynamic programming

KW - robust estimation

KW - smoothing

KW - total variation

UR - http://www.scopus.com/inward/record.url?scp=85031768683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031768683&partnerID=8YFLogxK

U2 - 10.1109/TMI.2017.2760104

DO - 10.1109/TMI.2017.2760104

M3 - Article

C2 - 28981412

AN - SCOPUS:85031768683

VL - 37

SP - 162

EP - 172

JO - IEEE Transactions on Medical Imaging

JF - IEEE Transactions on Medical Imaging

SN - 0278-0062

IS - 1

M1 - 8058471

ER -