# American Institute of Mathematical Sciences

November  2015, 9(4): 1139-1169. doi: 10.3934/ipi.2015.9.1139

## Bilevel optimization for calibrating point spread functions in blind deconvolution

 1 Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany, Germany

Received  October 2014 Revised  March 2015 Published  October 2015

Blind deconvolution problems arise in many imaging modalities, where both the underlying point spread function, which parameterizes the convolution operator, and the source image need to be identified. In this work, a novel bilevel optimization approach to blind deconvolution is proposed. The lower-level problem refers to the minimization of a total-variation model, as is typically done in non-blind image deconvolution. The upper-level objective takes into account additional statistical information depending on the particular imaging modality. Bilevel problems of such type are investigated systematically. Analytical properties of the lower-level solution mapping are established based on Robinson's strong regularity condition. Furthermore, several stationarity conditions are derived from the variational geometry induced by the lower-level problem. Numerically, a projected-gradient-type method is employed to obtain a Clarke-type stationary point and its convergence properties are analyzed. We also implement an efficient version of the proposed algorithm and test it through the experiments on point spread function calibration and multiframe blind deconvolution.
Citation: Michael Hintermüller, Tao Wu. Bilevel optimization for calibrating point spread functions in blind deconvolution. Inverse Problems & Imaging, 2015, 9 (4) : 1139-1169. doi: 10.3934/ipi.2015.9.1139
##### References:
 [1] M. S. C. Almeida and L. B. Almeida, Blind and semi-blind deblurring of natural images,, IEEE Trans. Image Process., 19 (2010), 36. doi: 10.1109/TIP.2009.2031231. Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing,, Springer, (2002). Google Scholar [3] J. Bardsley, S. Jefferies, J. Nagy and R. Plemmons, A computational method for the restoration of images with an unknown, spatially-varying blur,, Opt. Express, 14 (2006), 1767. doi: 10.1364/OE.14.001767. Google Scholar [4] M. Burger and O. Scherzer, Regularization methods for blind deconvolution and blind source separation problems,, Math. Control Signals Systems, 14 (2001), 358. doi: 10.1007/s498-001-8041-y. Google Scholar [5] J.-F. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring using multiple images,, J. Comput. Phys., 228 (2009), 5057. doi: 10.1016/j.jcp.2009.04.022. Google Scholar [6] P. Campisi and K. Egiazarian, eds., Blind image deconvolution: Theory and applications,, CRC press, (2007). doi: 10.1201/9781420007299. Google Scholar [7] A. S. Carasso, Direct blind deconvolution,, SIAM J. Appl. Math., 61 (2001), 1980. doi: 10.1137/S0036139999362592. Google Scholar [8] _________, The APEX method in image sharpening and the use of low exponent Lévy stable laws,, SIAM J. Appl. Math., 63 (2002), 593. doi: 10.1137/S0036139901389318. Google Scholar [9] _________, APEX blind deconvolution of color Hubble space telescope imagery and other astronomical data,, Optical Engineering, 45 (2006). Google Scholar [10] _________, False characteristic functions and other pathologies in variational blind deconvolution: A method of recovery,, SIAM J. Appl. Math., 70 (2009), 1097. doi: 10.1137/080737769. Google Scholar [11] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, J. Math. Imaging Vis., 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [12] T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods,, SIAM, (2005). doi: 10.1137/1.9780898717877. Google Scholar [13] T. F. Chan and C.-K. Wong, Total variation blind deconvolution,, IEEE Trans. Image Process., 7 (1998), 370. doi: 10.1109/83.661187. Google Scholar [14] R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing,, in Proceedings of the IEEE International Conference on Acoustics, (2008), 3869. Google Scholar [15] S. Cho, Y. Matsushita and S. Lee, Removing non-uniform motion blur from images,, in IEEE 11th International Conference on Computer Vision, (2007), 1. doi: 10.1109/ICCV.2007.4408904. Google Scholar [16] J. C. De los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization,, Inverse Problems and Imaging, 7 (2013), 1183. doi: 10.3934/ipi.2013.7.1183. Google Scholar [17] A. L. Dontchev and R. T. Rockafellar, Robinson's implicit function theorem and its extensions,, Math. Program., 117 (2009), 129. doi: 10.1007/s10107-007-0161-1. Google Scholar [18] D. A. Fish, A. M. Brinicombe and E. R. Pike, Blind deconvolution by means of the Richardson-Lucy algorithm,, J. Opt. Soc. Am. A, 12 (1995), 58. doi: 10.1364/JOSAA.12.000058. Google Scholar [19] R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints,, SIAM J. Optim., 17 (2006), 259. doi: 10.1137/S1052623402407382. Google Scholar [20] R. W. Freund and N. M. Nachtigal, QMR: A quasi-minimal residual method for non-Hermitian linear systems,, Numer. Math., 60 (1991), 315. doi: 10.1007/BF01385726. Google Scholar [21] M. Fukushima, Z.-Q. Luo and J.-S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Comput. Optim. Appl., 10 (1998), 5. doi: 10.1023/A:1018359900133. Google Scholar [22] E. M. Gafni and D. P. Bertsekas, Convergence of a Gradient Projection Method,, Laboratory for Information and Decision Systems Report LIDS-P-1201, (1982). Google Scholar [23] L. He, A. Marquina and S. J. Osher, Blind deconvolution using TV regularization and Bregman iteration,, International Journal of Imaging Systems and Technology, 15 (2005), 74. doi: 10.1002/ima.20040. Google Scholar [24] M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm,, SIAM J. Optim., 20 (2009), 868. doi: 10.1137/080720681. Google Scholar [25] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem,, SIAM J. Appl. Math., 64 (2004), 1311. doi: 10.1137/S0036139903422784. Google Scholar [26] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration,, SIAM J. Sci. Comput., 28 (2006), 1. doi: 10.1137/040613263. Google Scholar [27] M. Hintermüller and T. Surowiec, A bundle-free implicit programming approach for a class of MPECs in function space,, preprint, (2014). Google Scholar [28] M. Hintermüller and T. Wu, Nonconvex $TV^q$-models in image restoration: Analysis and a trust-region regularization-based superlinearly convergent solver,, SIAM J. Imaging Sci., 6 (2013), 1385. doi: 10.1137/110854746. Google Scholar [29] _________, A superlinearly convergent $R$-regularized Newton scheme for variational models with concave sparsity-promoting priors,, Comput. Optim. Appl., 57 (2014), 1. doi: 10.1007/s10589-013-9583-2. Google Scholar [30] K. Ito and K. Kunisch, An active set strategy based on the augmented Lagrangian formulation for image restoration,, Mathematical Modelling and Numerical Analysis, 33 (1999), 1. doi: 10.1051/m2an:1999102. Google Scholar [31] L. Justen, Blind Deconvolution: Theory, Regularization and Applications,, Ph.D. thesis, (2006). Google Scholar [32] L. Justen and R. Ramlau, A non-iterative regularization approach to blind deconvolution,, Inverse Problems, 22 (2006), 771. doi: 10.1088/0266-5611/22/3/003. Google Scholar [33] D. Kundur and D. Hatzinakos, Blind image deconvolution,, IEEE Signal Process. Mag., 13 (1996), 43. doi: 10.1109/79.489268. Google Scholar [34] ________, Blind image deconvolution revisited,, IEEE Signal Process. Mag., 13 (1996), 61. Google Scholar [35] K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models,, SIAM J. Imaging Sci., 6 (2013), 938. doi: 10.1137/120882706. Google Scholar [36] A. Levin, Blind motion deblurring using image statistics,, Advances in Neural Information Processing Systems, 19 (2006), 841. Google Scholar [37] A. B. Levy, Solution sensitivity from general principles,, SIAM J. Control Optim., 40 (2001), 1. doi: 10.1137/S036301299935211X. Google Scholar [38] Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658. Google Scholar [39] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications,, Springer, (2006). Google Scholar [40] J. Nocedal and S. Wright, Numerical optimization,, 2nd ed., (2006). Google Scholar [41] J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,, Kluwer Academic Publishers, (1998). doi: 10.1007/978-1-4757-2825-5. Google Scholar [42] J. V. Outrata, A generalized mathematical program with equilibrium constraints,, SIAM J. Control Optim., 38 (2000), 1623. doi: 10.1137/S0363012999352911. Google Scholar [43] S. M. Robinson, Strongly regular generalized equations,, Math. Oper. Res., 5 (1980), 43. doi: 10.1287/moor.5.1.43. Google Scholar [44] ________, Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity,, Math. Programming Stud., 30 (1987), 45. Google Scholar [45] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis,, Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar [46] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [47] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity,, Math. Oper. Res., 25 (2000), 1. doi: 10.1287/moor.25.1.1.15213. Google Scholar [48] S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM J. Optim., 11 (2001), 918. doi: 10.1137/S1052623499361233. Google Scholar [49] Q. Shan, J. Jia and A. Agarwala, High-quality motion deblurring from a single image,, ACM T. Graphic, 27 (2008). doi: 10.1145/1399504.1360672. Google Scholar [50] A. Shapiro, Sensitivity analysis of parameterized variational inequalities,, Math. Oper. Res., 30 (2005), 109. doi: 10.1287/moor.1040.0115. Google Scholar [51] J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints,, J. Math. Anal. Appl., 307 (2005), 350. doi: 10.1016/j.jmaa.2004.10.032. Google Scholar [52] J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems,, SIAM J. Optim., 7 (1997), 481. doi: 10.1137/S1052623493257344. Google Scholar [53] Y.-L. You and M. Kaveh, A regularization approach to joint blur identification and image restoration,, IEEE Trans. Image Process., 5 (1996), 416. Google Scholar

show all references

##### References:
 [1] M. S. C. Almeida and L. B. Almeida, Blind and semi-blind deblurring of natural images,, IEEE Trans. Image Process., 19 (2010), 36. doi: 10.1109/TIP.2009.2031231. Google Scholar [2] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing,, Springer, (2002). Google Scholar [3] J. Bardsley, S. Jefferies, J. Nagy and R. Plemmons, A computational method for the restoration of images with an unknown, spatially-varying blur,, Opt. Express, 14 (2006), 1767. doi: 10.1364/OE.14.001767. Google Scholar [4] M. Burger and O. Scherzer, Regularization methods for blind deconvolution and blind source separation problems,, Math. Control Signals Systems, 14 (2001), 358. doi: 10.1007/s498-001-8041-y. Google Scholar [5] J.-F. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring using multiple images,, J. Comput. Phys., 228 (2009), 5057. doi: 10.1016/j.jcp.2009.04.022. Google Scholar [6] P. Campisi and K. Egiazarian, eds., Blind image deconvolution: Theory and applications,, CRC press, (2007). doi: 10.1201/9781420007299. Google Scholar [7] A. S. Carasso, Direct blind deconvolution,, SIAM J. Appl. Math., 61 (2001), 1980. doi: 10.1137/S0036139999362592. Google Scholar [8] _________, The APEX method in image sharpening and the use of low exponent Lévy stable laws,, SIAM J. Appl. Math., 63 (2002), 593. doi: 10.1137/S0036139901389318. Google Scholar [9] _________, APEX blind deconvolution of color Hubble space telescope imagery and other astronomical data,, Optical Engineering, 45 (2006). Google Scholar [10] _________, False characteristic functions and other pathologies in variational blind deconvolution: A method of recovery,, SIAM J. Appl. Math., 70 (2009), 1097. doi: 10.1137/080737769. Google Scholar [11] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, J. Math. Imaging Vis., 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [12] T. F. Chan and J. Shen, Image Processing and Analysis: Variational, PDE, Wavelet, and Stochastic Methods,, SIAM, (2005). doi: 10.1137/1.9780898717877. Google Scholar [13] T. F. Chan and C.-K. Wong, Total variation blind deconvolution,, IEEE Trans. Image Process., 7 (1998), 370. doi: 10.1109/83.661187. Google Scholar [14] R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing,, in Proceedings of the IEEE International Conference on Acoustics, (2008), 3869. Google Scholar [15] S. Cho, Y. Matsushita and S. Lee, Removing non-uniform motion blur from images,, in IEEE 11th International Conference on Computer Vision, (2007), 1. doi: 10.1109/ICCV.2007.4408904. Google Scholar [16] J. C. De los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization,, Inverse Problems and Imaging, 7 (2013), 1183. doi: 10.3934/ipi.2013.7.1183. Google Scholar [17] A. L. Dontchev and R. T. Rockafellar, Robinson's implicit function theorem and its extensions,, Math. Program., 117 (2009), 129. doi: 10.1007/s10107-007-0161-1. Google Scholar [18] D. A. Fish, A. M. Brinicombe and E. R. Pike, Blind deconvolution by means of the Richardson-Lucy algorithm,, J. Opt. Soc. Am. A, 12 (1995), 58. doi: 10.1364/JOSAA.12.000058. Google Scholar [19] R. Fletcher, S. Leyffer, D. Ralph and S. Scholtes, Local convergence of SQP methods for mathematical programs with equilibrium constraints,, SIAM J. Optim., 17 (2006), 259. doi: 10.1137/S1052623402407382. Google Scholar [20] R. W. Freund and N. M. Nachtigal, QMR: A quasi-minimal residual method for non-Hermitian linear systems,, Numer. Math., 60 (1991), 315. doi: 10.1007/BF01385726. Google Scholar [21] M. Fukushima, Z.-Q. Luo and J.-S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Comput. Optim. Appl., 10 (1998), 5. doi: 10.1023/A:1018359900133. Google Scholar [22] E. M. Gafni and D. P. Bertsekas, Convergence of a Gradient Projection Method,, Laboratory for Information and Decision Systems Report LIDS-P-1201, (1982). Google Scholar [23] L. He, A. Marquina and S. J. Osher, Blind deconvolution using TV regularization and Bregman iteration,, International Journal of Imaging Systems and Technology, 15 (2005), 74. doi: 10.1002/ima.20040. Google Scholar [24] M. Hintermüller and I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm,, SIAM J. Optim., 20 (2009), 868. doi: 10.1137/080720681. Google Scholar [25] M. Hintermüller and K. Kunisch, Total bounded variation regularization as a bilaterally constrained optimization problem,, SIAM J. Appl. Math., 64 (2004), 1311. doi: 10.1137/S0036139903422784. Google Scholar [26] M. Hintermüller and G. Stadler, An infeasible primal-dual algorithm for total bounded variation-based inf-convolution-type image restoration,, SIAM J. Sci. Comput., 28 (2006), 1. doi: 10.1137/040613263. Google Scholar [27] M. Hintermüller and T. Surowiec, A bundle-free implicit programming approach for a class of MPECs in function space,, preprint, (2014). Google Scholar [28] M. Hintermüller and T. Wu, Nonconvex $TV^q$-models in image restoration: Analysis and a trust-region regularization-based superlinearly convergent solver,, SIAM J. Imaging Sci., 6 (2013), 1385. doi: 10.1137/110854746. Google Scholar [29] _________, A superlinearly convergent $R$-regularized Newton scheme for variational models with concave sparsity-promoting priors,, Comput. Optim. Appl., 57 (2014), 1. doi: 10.1007/s10589-013-9583-2. Google Scholar [30] K. Ito and K. Kunisch, An active set strategy based on the augmented Lagrangian formulation for image restoration,, Mathematical Modelling and Numerical Analysis, 33 (1999), 1. doi: 10.1051/m2an:1999102. Google Scholar [31] L. Justen, Blind Deconvolution: Theory, Regularization and Applications,, Ph.D. thesis, (2006). Google Scholar [32] L. Justen and R. Ramlau, A non-iterative regularization approach to blind deconvolution,, Inverse Problems, 22 (2006), 771. doi: 10.1088/0266-5611/22/3/003. Google Scholar [33] D. Kundur and D. Hatzinakos, Blind image deconvolution,, IEEE Signal Process. Mag., 13 (1996), 43. doi: 10.1109/79.489268. Google Scholar [34] ________, Blind image deconvolution revisited,, IEEE Signal Process. Mag., 13 (1996), 61. Google Scholar [35] K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models,, SIAM J. Imaging Sci., 6 (2013), 938. doi: 10.1137/120882706. Google Scholar [36] A. Levin, Blind motion deblurring using image statistics,, Advances in Neural Information Processing Systems, 19 (2006), 841. Google Scholar [37] A. B. Levy, Solution sensitivity from general principles,, SIAM J. Control Optim., 40 (2001), 1. doi: 10.1137/S036301299935211X. Google Scholar [38] Z.-Q. Luo, J.-S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658. Google Scholar [39] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications,, Springer, (2006). Google Scholar [40] J. Nocedal and S. Wright, Numerical optimization,, 2nd ed., (2006). Google Scholar [41] J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints,, Kluwer Academic Publishers, (1998). doi: 10.1007/978-1-4757-2825-5. Google Scholar [42] J. V. Outrata, A generalized mathematical program with equilibrium constraints,, SIAM J. Control Optim., 38 (2000), 1623. doi: 10.1137/S0363012999352911. Google Scholar [43] S. M. Robinson, Strongly regular generalized equations,, Math. Oper. Res., 5 (1980), 43. doi: 10.1287/moor.5.1.43. Google Scholar [44] ________, Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity,, Math. Programming Stud., 30 (1987), 45. Google Scholar [45] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis,, Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar [46] L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [47] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity,, Math. Oper. Res., 25 (2000), 1. doi: 10.1287/moor.25.1.1.15213. Google Scholar [48] S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM J. Optim., 11 (2001), 918. doi: 10.1137/S1052623499361233. Google Scholar [49] Q. Shan, J. Jia and A. Agarwala, High-quality motion deblurring from a single image,, ACM T. Graphic, 27 (2008). doi: 10.1145/1399504.1360672. Google Scholar [50] A. Shapiro, Sensitivity analysis of parameterized variational inequalities,, Math. Oper. Res., 30 (2005), 109. doi: 10.1287/moor.1040.0115. Google Scholar [51] J. J. Ye, Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints,, J. Math. Anal. Appl., 307 (2005), 350. doi: 10.1016/j.jmaa.2004.10.032. Google Scholar [52] J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems,, SIAM J. Optim., 7 (1997), 481. doi: 10.1137/S1052623493257344. Google Scholar [53] Y.-L. You and M. Kaveh, A regularization approach to joint blur identification and image restoration,, IEEE Trans. Image Process., 5 (1996), 416. Google Scholar
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