Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging

SIAM Journal on Imaging Sciences

Volumn 8, Issue 1, 2015, Pages 426-457

  Hesse, Robert ^a   Russell Luke, D ^a   Sabach, Shoham ^a   Tam, Matthew K ^b  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

^b UNIVERSITY OF NEWCASTLE   (Australia)

Author keywords

Alternating minimization; Deconvolution; Kurdyka ojasiewicz; Nonconvex nonsmooth minimization; Ptychography

Indexed keywords

DECONVOLUTION; OPTIMIZATION;

ALTERNATING MINIMIZATION; ALTERNATING MINIMIZATION ALGORITHMS; IMPLICIT-EXPLICIT METHODS; MATHEMATICAL FRAMEWORKS; NONCONVEX; NONCONVEX OPTIMIZATION PROBLEM; NUMERICAL COMPARISON; PTYCHOGRAPHY;

ALGORITHMS;

EID: 84926328511     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/14098168X     Document Type: Article

References (37)

1. F Abboud, E Chouzenoux, J.-C Pesquet, J.-H Chenot, and L Laborelli, A hybrid alternating proximal method for blind video restoration, in 2014 Proceedings of the 22nd European Signal Processing Conference (EUSIPCO), Lisbon, Portugal, IEEE, Piscataway, NJ, 2014, pp. 1811-1815.
2. H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), pp. 5-16.
3. H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality, Math. Oper. Res., 35 (2010), pp. 438-457.
4. H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods, Math. Program., 137 (2013), pp. 91-129.
5. H. H. Bauschke, P. L. Combettes, and D. R. Luke, Finding best approximation pairs relative to two closed convex sets in Hilbert spaces, J. Approx. Theory, 127 (2004), pp. 178-192.
6. A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183-202.
7. T. Blumensath and M. Davies, Iterative hard thresholding for compressed sensing, Appl. Comput. Harmon. Anal., 27 (2009), pp. 265-274.
8. T. Blumensath and M. Davies, Normalised iterative hard thresholding; guaranteed stability and performance, IEEE J. Sel. Top. Signal Process., 4 (2010), pp. 298-309.
9. J. Bolte, A. Daniilidis, and A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), pp. 1205-1223.
10. J. Bolte, A. Daniilidis, A. Lewis, and M. Shiota, Clarke subgradients of stratifiable functions, SIAM J. Optim., 18 (2007), pp. 556-572.
11. J. Bolte, S. Sabach, and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146 (2014), pp. 459-494.
12. E Chouzenoux, J.-C Pesquet, and A Repetti, A Block Coordinate Variable Metric Forward- Backward Algorithm, http://www.optimization-online.org/DBHTML/2013/12/4178.html (2013).
13. P. L. Combettes and V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168-1200.
14. J. Douglas and H. H. Rachford, On the numerical solution of heat conduction problems in two or three space variables. Trans. Amer. Math. Soc., 82 (1956), pp. 421-439.
15. J. R. Fienup, Phase retrieval algorithms: A comparison, Appl. Opt., 21 (1982), pp. 2758-2769.
16. P. Frankel, G. Garrigos, and J. Peypouquet, Splitting methods with variable metric for Kurdyka-Łojasiewicz functions and general convergence rates, J. Optim. Theory Appl., to appear.
17. R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of phase from image and diffraction plane pictures, Optik, 35 (1972), pp. 237-246.
18. R. A. Gonsalves, Phase retrieval and diversity in adaptive optics, Opt. Eng., 21 (1982), pp. 829-832.
19. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, Efficient subpixel image registration algorithms, Opt. Lett., 33 (2008), pp. 156-158.
20. J. Hagemann, A.-L. Robisch, D. R. Luke, C. Homann, T. Hohage, P. Cloetens, H. Suhonen, and T. Salditt, Reconstruction of wave front and object for inline holography from a set of detection planes, Opt. Express, 22 (2014), pp. 11552-11569.
21. R. Hesse, D. R. Luke, and P. Neumann, Alternating projections and Douglas-Rachford for sparse affine feasibility, IEEE Trans. Signal Process., 62 (2014), pp. 4868-4881.
22. R. Hegerl and W. Hoppe, Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen prim¨arstrahlwellenfeld, Ber. Bunsenges. Phys. Chem, 74 (1970), pp. 1148-1154.
23. P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), pp. 964-979.
24. D. R. Luke, Finding best approximation pairs relative to a convex and a prox-regular set in a Hilbert space, SIAM J. Optim., 19 (2008), pp. 714-739.
25. D. R. Luke, J. V. Burke, and R. G. Lyon, Optical wavefront reconstruction: Theory and numerical methods, SIAM Rev., 44 (2002), pp. 169-224.
26. D. R. Luke, Relaxed averaged alternating reflections for diffraction imaging, Inverse Problems, 21 (2005), pp. 37-50.
27. A. M. Maiden and J. M. Rodenburg, An improved ptychographical phase retrieval algorithm for diffractive imaging, Ultramicroscopy, 109 (2009), pp. 1256-1262.
28. P. Mattsson, Projection Methods in Ptychographic Imaging, Masters Thesis, Georg-August-Universität, Göttingen, 2013.
29. L. Meyer, Blocking Strategies for Large-Scale Split Feasibility, Masters Thesis, Georg-August-Universität, Göttingenn, 2014.
30. J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, Efficient algorithms for ptychographic phase retrieval, in Inverse Problems and Applications, Contemp. Math. 615, (2014), pp. 261-280.
31. R. T. Rockafellar and R. J. Wets, Variational Analysis, Grundlehren Math. Wiss., Springer-Verlag, Berlin, 1998.
32. J. M. Rodenburg, Ptychography and related diffractive imaging methods, Adv. Imaging Electron Phys., 150 (2008), pp. 87-184.
33. J. M. Rodenburg and R. H. T. Bates, The theory of super-resolution electron microscopy via Wignerdistribution deconvolution, R. Soc. London Philos. Trans. Ser. A Math. Phys. Eng. Sci., 339 (1992), pp. 521-553.
34. J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, Hard-X-ray lensless imaging of extended objects, Phys. Rev. Lett., 98 (2007), 034801.
35. Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, Phase retrieval with application to optical imaging, IEEE Signal Process. Magazine, to appear.
36. P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, Probe retrieval in ptychographic coherent diffractive imaging, Ultramicroscopy, 109 (2009), pp. 338-343.
37. R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, Hard X-ray imaging of bacterial cells: Nano-diffraction and ptychographic reconstruction, Opt. Express, 20 (2012), pp. 19232-19254.