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A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science
Abstract
We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed by Zhu and Chan in [An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, CAM Report 08-34, UCLA, Los Angeles, CA, 2008] to a broader class of convex optimization problems. In addition, we survey several closely related methods and explain the connections to PDHG. We point out convergence results for a modified version of PDHG that has a similarly good empirical convergence rate for total variation (TV) minimization problems. We also prove a convergence result for PDHG applied to TV denoising with some restrictions on the PDHG step size parameters. We show how to interpret this special case as a projected averaged gradient method applied to the dual functional. We discuss the range of parameters for which these methods can be shown to converge. We also present some numerical comparisons of these algorithms applied to TV denoising, TV deblurring, and constrained $l_1$ minimization problems.
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Citations
Cites
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
Beck, A., & Teboulle, M. (2009) · SIAM Journal on Imaging Sciences · doi:10.1137/080716542 · Cited by 11,952
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
Wang, Y., Yang, J., Yin, W., & Zhang, Y. (2008) · SIAM Journal on Imaging Sciences · doi:10.1137/080724265 · Cited by 1,975
Cited by
Parikh, N., & Boyd, S. (2014) · Foundations and Trends® in Optimization · doi:10.1561/2400000003 · Cited by 4,102
Condat, L. (2013) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-012-0245-9 · Cited by 912
Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding
O’Donoghue, B., Chu, E., Parikh, N., & Boyd, S. (2016) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-016-0892-3 · Cited by 664
Image super-resolution: the techniques, applications, and future
Yue, L., Shen, H., Li, J., Yuan, Q., Zhang, H., & Zhang, L. (2016) · Signal Processing · doi:10.1016/j.sigpro.2016.05.002 · Cited by 554
Komodakis, N., & Pesquet, J.-C. (2015) · IEEE Signal Processing Magazine · doi:10.1109/msp.2014.2377273 · Cited by 419
Metadata
- Authors
- Esser, E., Zhang, X., & Chan, T. F.
- Venue
- SIAM Journal on Imaging Sciences
- Published
- 2010-01
- Volume
- 3
- Issue
- 4
- Pages
- 1015-1046
- ISSN
- 1936-4954 (ISSN-L)
- OpenAlex
- W1981112674
- Cited by
- 695
- References
- 2
- FWCI
- 28.17
- Type
- journal article
- Language
- en
- Open access
- closed
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