# American Institute of Mathematical Sciences

2014, 8(3): 733-760. doi: 10.3934/ipi.2014.8.733

## Bayesian image restoration for mosaic active imaging

 1 LTCI, CNRS UMR5141, Institut Mines-Télécom, Télécom ParisTech, Paris, France 2 IMT, CNRS UMR5219, Université de Toulouse, Toulouse, France 3 ONERA - The French Aerospace Lab, F-31055 Toulouse, France, France

Received  December 2012 Revised  February 2014 Published  August 2014

In this paper, we focus on the restoration of images acquired with a new active imaging concept. This new instrument generates a mosaic of active imaging acquisitions. We first describe a simplified forward model of this so-called mosaic active imaging''. We also assume a prior on the distribution of images, using the (TV), and deduce a restoration algorithm. This algorithm is a two-stage iterative process which alternates between: i) the estimation of the restored image; ii) the estimation of the acquisition parameters. We then provide the details useful to the implementation of these two steps. In particular, we show that the image estimation can be performed with graph cuts. This allows a fast resolution of this image estimation step. Finally, we detail numerical experiments showing that acquisitions made with a mosaic active imaging device can be restored even under severe noise levels, with few acquisitions.
Citation: Nicolas Lermé, François Malgouyres, Dominique Hamoir, Emmanuelle Thouin. Bayesian image restoration for mosaic active imaging. Inverse Problems & Imaging, 2014, 8 (3) : 733-760. doi: 10.3934/ipi.2014.8.733
##### References:
 [1] J. Bect, L. Blanc-Féraud, G. Aubert and A. Chambolle, A L1-unified variational framework for image restoration,, in Computer Vision - ECCV 2004, (2004), 1. [2] D. Bertsekas, Nonlinear Programming,, 2nd edition, (2003). [3] Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,, Pattern Analysis And Machine Intelligence, 26 (2004), 1124. doi: 10.1109/TPAMI.2004.60. [4] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. [5] A. Chambolle, Total variation minimization and a class of binary MRF models,, in Energy Minimization Methods in Computer Vision and Pattern Recognition, (3757), 136. doi: 10.1007/11585978_10. [6] A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows,, International Journal of Computer Vision, 84 (2009), 288. doi: 10.1007/s11263-009-0238-9. [7] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. [8] T. Chan and J. Shen, Variational image inpainting,, Communications in Pure and Applied Math., 58 (2005), 579. doi: 10.1002/cpa.20075. [9] F. Chassat, Optical Propagation through Atmospheric Turbulence: Moral Study and Application of Anisoplanatism in Adaptive Optics,, PhD thesis, (1992). [10] R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement,, Applied and Computational Harmonic Analysis, 9 (2000), 1. doi: 10.1006/acha.2000.0299. [11] J. Darbon and M. Sigelle, Image restoration with discrete constrained total variation. I. Fast and exact optimization,, Journal of Mathematical Imaging and Vision, 26 (2006), 261. doi: 10.1007/s10851-006-8803-0. [12] G. Demoment, Image reconstruction and restoration: Overview of common estimation structures and problems,, IEEE, 37 (1989), 2024. doi: 10.1109/29.45551. [13] M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing,, Springer, (2010). doi: 10.1007/978-1-4419-7011-4. [14] R. Fante, Electromagnetic beam propagation in turbulent media,, Proceedings of the IEEE, 63 (1975), 1669. doi: 10.1109/PROC.1975.10035. [15] D. Fried, Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,, Journal of the Optical Society of America, 56 (1966), 1372. doi: 10.1364/JOSA.56.001372. [16] D. Greig, B. Porteous and A. Seheult, Exact maximum a posteriori estimation for binary images,, Journal of the Royal Statistical Society, 51 (1989), 271. [17] D. Hamoir, Procédé et système d'imagerie active à champ large: Method and system for active imaging with a large field,, Patent WO 2010119225, (2010). [18] L. Hespel, M.-T. Velluet, A. Bonnefois, N. Rivière, M. Fraces, D. Hamoir, B. Tanguy, B. Duchenne and J. Isbert, Comparison of a physics-based BIL simulator with experiments,, in International Symposium on Photoelectronic Detection and Imaging, (2009). [19] J. Kiefer, Sequential minimax search for a maximum,, Proceedings of the American Mathematical Society, 4 (1953), 502. doi: 10.1090/S0002-9939-1953-0055639-3. [20] A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, Doklady Akademii Nauk SSSR, 66 (1949). [21] V. Kolmogorov and R. Zabih, What energy functions can be minimized via graph cuts?,, Pattern Analysis And Machine Intelligence, 26 (2004), 147. doi: 10.1109/TPAMI.2004.1262177. [22] F. Luisier, T. Blu and M. Unser, Image denoising in mixed Poisson-Gaussian noise,, IEEE Transactions on Image Processing, 20 (2011), 696. doi: 10.1109/TIP.2010.2073477. [23] F. Malgouyres and F. Guichard, Edge direction preserving image zooming: A mathematical and numerical analysis,, SIAM Journal on Numerical Analysis, 39 (2001), 1. doi: 10.1137/S0036142999362286. [24] Y. Mao and J. Gilles, Non rigid geometric distortions correction - Application to atmospheric turbulence stabilization,, Inverse Problems and Imaging, 6 (2012), 531. doi: 10.3934/ipi.2012.6.531. [25] Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations,, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, (2001). [26] M. Mäkitalo and A. Foi, Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise,, IEEE Transactions on Image Processing, 22 (2013), 91. doi: 10.1109/TIP.2012.2202675. [27] M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM Journal of Applied Mathematics, 61 (2000), 633. doi: 10.1137/S0036139997327794. [28] J. Picard and H. Ratliff, Minimum cuts and related problems,, Networks, 5 (1975), 357. doi: 10.1002/net.3230050405. [29] N. Rivière, L. Hespel, M.-T. Velluet, Y.-M. Frédéric, P. Barillot and F. Hélias, Modeling of an active burst illumination imaging system: Comparison between experimental and modelled 3d scene,, in Society of Photo-Optical Instrumentation Engineers, (7382), 783509. doi: 10.1117/12.864694. [30] F. Roddier, The effects of atmospheric turbulence in optical astronomy,, Progress in Optics, 19 (1981), 281. doi: 10.1016/S0079-6638(08)70204-X. [31] 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.

show all references

##### References:
 [1] J. Bect, L. Blanc-Féraud, G. Aubert and A. Chambolle, A L1-unified variational framework for image restoration,, in Computer Vision - ECCV 2004, (2004), 1. [2] D. Bertsekas, Nonlinear Programming,, 2nd edition, (2003). [3] Y. Boykov and V. Kolmogorov, An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,, Pattern Analysis And Machine Intelligence, 26 (2004), 1124. doi: 10.1109/TPAMI.2004.60. [4] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. [5] A. Chambolle, Total variation minimization and a class of binary MRF models,, in Energy Minimization Methods in Computer Vision and Pattern Recognition, (3757), 136. doi: 10.1007/11585978_10. [6] A. Chambolle and J. Darbon, On total variation minimization and surface evolution using parametric maximum flows,, International Journal of Computer Vision, 84 (2009), 288. doi: 10.1007/s11263-009-0238-9. [7] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. [8] T. Chan and J. Shen, Variational image inpainting,, Communications in Pure and Applied Math., 58 (2005), 579. doi: 10.1002/cpa.20075. [9] F. Chassat, Optical Propagation through Atmospheric Turbulence: Moral Study and Application of Anisoplanatism in Adaptive Optics,, PhD thesis, (1992). [10] R. R. Coifman and A. Sowa, Combining the calculus of variations and wavelets for image enhancement,, Applied and Computational Harmonic Analysis, 9 (2000), 1. doi: 10.1006/acha.2000.0299. [11] J. Darbon and M. Sigelle, Image restoration with discrete constrained total variation. I. Fast and exact optimization,, Journal of Mathematical Imaging and Vision, 26 (2006), 261. doi: 10.1007/s10851-006-8803-0. [12] G. Demoment, Image reconstruction and restoration: Overview of common estimation structures and problems,, IEEE, 37 (1989), 2024. doi: 10.1109/29.45551. [13] M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing,, Springer, (2010). doi: 10.1007/978-1-4419-7011-4. [14] R. Fante, Electromagnetic beam propagation in turbulent media,, Proceedings of the IEEE, 63 (1975), 1669. doi: 10.1109/PROC.1975.10035. [15] D. Fried, Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,, Journal of the Optical Society of America, 56 (1966), 1372. doi: 10.1364/JOSA.56.001372. [16] D. Greig, B. Porteous and A. Seheult, Exact maximum a posteriori estimation for binary images,, Journal of the Royal Statistical Society, 51 (1989), 271. [17] D. Hamoir, Procédé et système d'imagerie active à champ large: Method and system for active imaging with a large field,, Patent WO 2010119225, (2010). [18] L. Hespel, M.-T. Velluet, A. Bonnefois, N. Rivière, M. Fraces, D. Hamoir, B. Tanguy, B. Duchenne and J. Isbert, Comparison of a physics-based BIL simulator with experiments,, in International Symposium on Photoelectronic Detection and Imaging, (2009). [19] J. Kiefer, Sequential minimax search for a maximum,, Proceedings of the American Mathematical Society, 4 (1953), 502. doi: 10.1090/S0002-9939-1953-0055639-3. [20] A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, Doklady Akademii Nauk SSSR, 66 (1949). [21] V. Kolmogorov and R. Zabih, What energy functions can be minimized via graph cuts?,, Pattern Analysis And Machine Intelligence, 26 (2004), 147. doi: 10.1109/TPAMI.2004.1262177. [22] F. Luisier, T. Blu and M. Unser, Image denoising in mixed Poisson-Gaussian noise,, IEEE Transactions on Image Processing, 20 (2011), 696. doi: 10.1109/TIP.2010.2073477. [23] F. Malgouyres and F. Guichard, Edge direction preserving image zooming: A mathematical and numerical analysis,, SIAM Journal on Numerical Analysis, 39 (2001), 1. doi: 10.1137/S0036142999362286. [24] Y. Mao and J. Gilles, Non rigid geometric distortions correction - Application to atmospheric turbulence stabilization,, Inverse Problems and Imaging, 6 (2012), 531. doi: 10.3934/ipi.2012.6.531. [25] Y. Meyer, Oscillating Patterns in Image Processing and in Some Nonlinear Evolution Equations,, The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, (2001). [26] M. Mäkitalo and A. Foi, Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise,, IEEE Transactions on Image Processing, 22 (2013), 91. doi: 10.1109/TIP.2012.2202675. [27] M. Nikolova, Local strong homogeneity of a regularized estimator,, SIAM Journal of Applied Mathematics, 61 (2000), 633. doi: 10.1137/S0036139997327794. [28] J. Picard and H. Ratliff, Minimum cuts and related problems,, Networks, 5 (1975), 357. doi: 10.1002/net.3230050405. [29] N. Rivière, L. Hespel, M.-T. Velluet, Y.-M. Frédéric, P. Barillot and F. Hélias, Modeling of an active burst illumination imaging system: Comparison between experimental and modelled 3d scene,, in Society of Photo-Optical Instrumentation Engineers, (7382), 783509. doi: 10.1117/12.864694. [30] F. Roddier, The effects of atmospheric turbulence in optical astronomy,, Progress in Optics, 19 (1981), 281. doi: 10.1016/S0079-6638(08)70204-X. [31] 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.
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