Image reconstruction with imperfect forward models and applications in deblurring. (English) Zbl 1401.94018
Summary: We present and analyze an approach to image reconstruction problems with imperfect forward models based on partially ordered spaces – Banach lattices. In this approach, errors in the data and in the forward models are described using order intervals. The method can be characterized as the lattice analogue of the residual method, where the feasible set is defined by linear inequality constraints. The study of this feasible set is the main contribution of this paper. Convexity of this feasible set is examined in several settings, and modifications for introducing additional information about the forward operator are considered. Numerical examples demonstrate the performance of the method in deblurring with errors in the blurring kernel.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 49N45 | Inverse problems in optimal control |
| 49N30 | Problems with incomplete information (optimization) |
Keywords:
inverse problems; imperfect forward models; residual method; deblurring; blind deblurring; deconvolution; blind deconvolution; uncertainty quantificationSoftware:
CVXReferences:
| [1] | Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Grad. Stud. Math. 50, AMS, Providence, RI, 2002. · Zbl 1022.47001 |
| [2] | M. Ackermann, M. Ajello, A. Allafort, K. Asano, W. B. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri et al., Determination of the point-spread function for the Fermi large area telescope from on-orbit data and limits on pair halos of active galactic nuclei, Astrophys. J., 765 (2013), 54, . |
| [3] | M. Bertero, P. Boccacci, G. Desiderá, and G. Vicidomini, Image deblurring with Poisson data: From cells to galaxies, Inverse Problems, 25 (2009), 123006, · Zbl 1186.85001 |
| [4] | D. Bertsimas, D. B. Brown, and C. Caramanis, Theory and applications of robust optimization, SIAM Rev., 53 (2011), pp. 464–501, . · Zbl 1233.90259 |
| [5] | K. Bredies, K. Kunisch, and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), pp. 492–526, . · Zbl 1195.49025 |
| [6] | K. Bredies, K. Kunisch, and T. Valkonen, Properties of \(L^1\)-\({TGV}^2\): The one-dimensional case, J. Math. Anal. Appl., 398 (2013), pp. 438–454, . · Zbl 1253.49024 |
| [7] | M. Burger, K. Papafitsoros, E. Papoutsellis, and C.-B. Schönlieb, Infimal convolution regularisation functionals of \(BV\) and \(L^p\) spaces, J. Math. Imaging Vision, 55 (2016), pp. 343–369, . · Zbl 1342.49014 |
| [8] | T. F. Chan and S. Esedoḡlu, Aspects of total variation regularized \(L^1\) function approximation, SIAM J. Appl. Math., 65 (2005), pp. 1817–1837, . · Zbl 1096.94004 |
| [9] | T. F. Chan and C.-K. Wong, Total variation blind deconvolution, IEEE Trans. Image Process., 7 (1998), pp. 370–375, . |
| [10] | S. Esedoḡlu and S. J. Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, Comm. Pure Appl. Math., 57 (2004), pp. 1609–1626, . · Zbl 1083.49029 |
| [11] | M. Grant and S. Boyd, Graph implementations for nonsmooth convex programs, in Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds., Lect. Notes Control Inf. Sci. 371, Springer-Verlag Limited, 2008, pp. 95–110. . · Zbl 1205.90223 |
| [12] | M. Grant and S. Boyd, CVX: MATLAB Software for Disciplined Convex Programming, Version 2.1, , 2014. |
| [13] | M. Grasmair, M. Haltmeier, and O. Scherzer, The residual method for regularizing ill-posed problems, Appl. Math. Comput., 218 (2011), pp. 2693–2710, . · Zbl 1247.65076 |
| [14] | V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications, De Gruyter, Berlin, Boston, 2013. · Zbl 0489.65035 |
| [15] | Y. Korolev, Making use of a partial order in solving inverse problems: II, Inverse Problems, 30 (2014), 085003. · Zbl 1298.47021 |
| [16] | Y. Korolev and A. Yagola, Making use of a partial order in solving inverse problems, Inverse Problems, 29 (2013), 095012. · Zbl 1285.65031 |
| [17] | D. Perrone and P. Favaro, A clearer picture of total variation blind deconvolution, IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), pp. 1041–1055, . |
| [18] | R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401 |
| [19] | L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D., 60 (1992), pp. 259–268, . · Zbl 0780.49028 |
| [20] | P. Sarder and A. Nehorai, Deconvolution methods for 3-D fluorescence microscopy images, IEEE Signal Process. Mag., 23 (2006), pp. 32–45, . |
| [21] | H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974. · Zbl 0296.47023 |
| [22] | P. J. Shaw and D. J. Rawlins, The point-spread function of a confocal microscope: Its measurement and use in deconvolution of 3-D data, J. Microscopy, 163 (1991), pp. 151–165, . |
| [23] | J. L. Starck, E. Pantin, and F. Murtagh, Deconvolution in astronomy: A review, Publ. Astronom. Soc. Pacific, 114 (2002), p. 1051, . |
| [24] | A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. · Zbl 0831.65059 |
| [25] | B. Zhang, J. Zerubia, and J.-C. Olivo-Marin, Gaussian approximations of fluorescence microscope point-spread function models, Appl. Opt., 46 (2007), pp. 1819–1829, . |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
