Abstract
Interpreting the celebrated Rudin-Osher-Fatemi (ROF) model in a Bayesian framework has led to interesting new variants for Total Variation image denoising in the last decade. The Posterior Mean variant avoids the so-called staircasing artifact of the ROF model but is computationally very expensive. Another recent variant, called TV-ICE (for Iterated Conditional Expectation), delivers very similar images but uses a much faster fixed-point algorithm. In the present work, we consider the TV-ICE approach in the case of a Poisson noise model. We derive an explicit form of the recursion operator, and show linear convergence of the algorithm, as well as the absence of staircasing effect. We also provide a numerical algorithm that carefully handles precision and numerical overflow issues, and show experiments that illustrate the interest of this Poisson TV-ICE variant.
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References
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1), 259–268 (1992)
Caselles, V., Chambolle, A., Novaga, M.: Total variation in imaging. In: Handbook of Mathematical Methods in Imaging, pp. 1016–1057. Springer, New York (2011)
Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. Theoretical foundations and numerical methods for sparse recovery 9, 263–340 (2010)
Darbon, J., Sigelle, M.: Image restoration with discrete constrained total variation part I: Fast and exact optimization. J. Math. Imag. Vis. 26(3), 261–276 (2007)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vis. 40(1), 120–145 (2011)
Buades, A., Coll, B., Morel, J.-M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4(2), 490–530 (2005)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Processing 16(8), 2080–2095 (2007)
Louchet, C., Moisan, L.: Total variation denoising using posterior expectation. In: Proc, European Signal Processing Conf (2008)
Louchet, C., Moisan, L.: Posterior Expectation of the Total Variation model: Properties and Experiments. SIAM J. Imaging Sci. 6(4), 2640–2684 (2013)
Louchet, C., Moisan, L.: Total variation denoising using iterated conditional expectation. In: Proc, European Signal Processing Conf (2014)
Setzer, S., Steidl, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Comm. Image Representation 21(3), 193–199 (2010)
Deledalle, C., Tupin, F., Denis, L.: Poisson NL means: Unsupervised non local means for poisson noise. In: Proc. Int. Conf. Imag. Processing, pp. 801–804 (2010)
Schmidt, K.D.: On the covariance of monotone functions of a random variable. Unpublished note, University of Dresden (2003)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)
NIST Digital Library of Mathematical Functions (2014). http://dlmf.nist.gov/ (release 1.0.9 of August 29, 2014)
Cuyt, A., Petersen, V.B., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of continued fractions for special functions. Springer, New York (2008)
Jones, W.B., Thron, W.J.: Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, vol. 11. Addison-Wesley Publishing Co., Reading, MA (1980)
Numerical recipes: The art of scientific computing, 2nd edn. Cambridge University (2007)
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Dover Publications no. 55 (1972)
Csiszar, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19, 2032–2066 (1991)
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Abergel, R., Louchet, C., Moisan, L., Zeng, T. (2015). Total Variation Restoration of Images Corrupted by Poisson Noise with Iterated Conditional Expectations. In: Aujol, JF., Nikolova, M., Papadakis, N. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2015. Lecture Notes in Computer Science(), vol 9087. Springer, Cham. https://doi.org/10.1007/978-3-319-18461-6_15
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DOI: https://doi.org/10.1007/978-3-319-18461-6_15
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