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An efficient computational method for total variationpenalized Poisson likelihood estimation
1.  Department of Mathematical Sciences, University of Montana Missoula, Montana 59812, United States 
In the Bayesian pointofview taken in this paper, a negativelog prior (or regularization) function is added to the negativelog likelihood function, and the resulting function is minimized. We focus on the case where the negativelog prior is the wellknown total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negativelog likelihood is used, the total variation term yields a minimization problem that is computationally challenging. The primary result of this work is the efficient computational method that is presented for the solution of such problems, together with its convergence analysis. With the computational method in hand, we then perform experiments that indicate that the Poisson negativelog likelihood yields a more computationally efficient method than does the use of the least squares function. We also present results that indicate that this may even be the case when the data noise is i.i.d. Gaussian, suggesting that regardless of noise statistics, using the Poisson negativelog likelihood can yield a more computationally tractable problem when total variation regularization is used.
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J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction, SIAM Journal on Scientific Computing, 25 (2004), 13261343 (electronic). doi: 10.1137/S1064827502410451. 
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Johnathan M. Bardsley and Aaron Luttman, Total variationpenalized Poisson likelihood estimation for illposed problems, accepted, Advances in Computational Mathematics, Special Issue on Mathematical Imaging, University of Montana Technical Report #8, 2006. 
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P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems, Mathematical Programming, 39 (1987), 93116. doi: 10.1007/BF02592073. 
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D. Calvetti, G. Landi, L. Reichel and S. Sgallari, Nonnegativity and iterative methods for illposed problems, Inverse Problems, 20 (2004), 17471758. doi: 10.1088/02665611/20/6/003. 
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Daniela Calvetti and Erkki Somersalo, A Gaussian hypermodel for recovering blocky objects, Inverse Problems, 23 (2007), 733754. doi: 10.1088/02665611/23/2/016. 
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J. W. Goodman, "Introduction to Fourier Optics," 2nd Edition, McGrawHill, 1996. 
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M. Green, Statistics of images, the TV algorithm of RudinOsherFatemi for image denoising, and an improved denoising algorithm, CAM Report 0255, UCLA, October 2002. 
[11] 
Jinggang Huang and David Mumford, Statistics of natural images and models, in "Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition," 1999, 541547. 
[12] 
Jari Kaipio and Erkki Somersalo, "Satistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, SpringerVerlag, New York, 2005. 
[13] 
C. T. Kelley, "Iterative Methods for Optimization," Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. 
[14] 
J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM Journal on Optimization, 1 (1991), 93113. doi: 10.1137/0801008. 
[15] 
J. Nagy and Z. Strakoš, Enforcing nonnegativity in image reconstruction algorithms, Mathematical Modeling, Estimation, and Imaging, David C. Wilson, et.al., Eds., 4121 (2000), 182190. 
[16] 
J. Nocedal and S. Wright, "Numerical Optimization," Series in Operations Research. SpringerVerlag, New York, 1999. doi: 10.1007/b98874. 
[17] 
R. T. Rockafellar, "Convex Analysis," Princeton University Press, 1970. 
[18] 
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259268. doi: 10.1016/01672789(92)90242F. 
[19] 
D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a chargecoupleddevice camera, Journal of the Optical Society of America A, 10 (1993), 10141023. doi: 10.1364/JOSAA.10.001014. 
[20] 
C. R. Vogel, Computational methods for inverse problems, With a foreword by H. T. Banks. Frontiers in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. 
[21] 
C. R. Vogel and M. E. Oman, Fast, robust total variationbased reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813824. doi: 10.1109/83.679423. 
show all references
References:
[1] 
D. P. Bertsekas, On the GoldsteinLevitinPoljak gradient projection method, IEEE Transactions on Automatic Control, 21 (1976), 174184. doi: 10.1109/TAC.1976.1101194. 
[2] 
Johnathan M. Bardsley and James G. Nagy, Covariancepreconditioned iterative methods for nonnegatively constrained astronomical imaging, SIAM Journal on Matrix Analysis and Applications, 27 (2006), 11841197. doi: 10.1137/040615043. 
[3] 
J. M. Bardsley and C. R. Vogel, A nonnnegatively constrained convex programming method for image reconstruction, SIAM Journal on Scientific Computing, 25 (2004), 13261343 (electronic). doi: 10.1137/S1064827502410451. 
[4] 
Johnathan M. Bardsley and Aaron Luttman, Total variationpenalized Poisson likelihood estimation for illposed problems, accepted, Advances in Computational Mathematics, Special Issue on Mathematical Imaging, University of Montana Technical Report #8, 2006. 
[5] 
P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems, Mathematical Programming, 39 (1987), 93116. doi: 10.1007/BF02592073. 
[6] 
D. Calvetti, G. Landi, L. Reichel and S. Sgallari, Nonnegativity and iterative methods for illposed problems, Inverse Problems, 20 (2004), 17471758. doi: 10.1088/02665611/20/6/003. 
[7] 
Daniela Calvetti and Erkki Somersalo, A Gaussian hypermodel for recovering blocky objects, Inverse Problems, 23 (2007), 733754. doi: 10.1088/02665611/23/2/016. 
[8] 
Torbjørn Eltoft and Taesu Kim, On the multivariate Laplace distribution, IEEE Signal Processing Letters, 13 (2006), 300303. doi: 10.1109/LSP.2006.870353. 
[9] 
J. W. Goodman, "Introduction to Fourier Optics," 2nd Edition, McGrawHill, 1996. 
[10] 
M. Green, Statistics of images, the TV algorithm of RudinOsherFatemi for image denoising, and an improved denoising algorithm, CAM Report 0255, UCLA, October 2002. 
[11] 
Jinggang Huang and David Mumford, Statistics of natural images and models, in "Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition," 1999, 541547. 
[12] 
Jari Kaipio and Erkki Somersalo, "Satistical and Computational Inverse Problems," Applied Mathematical Sciences, 160, SpringerVerlag, New York, 2005. 
[13] 
C. T. Kelley, "Iterative Methods for Optimization," Frontiers in Applied Mathematics, 18, SIAM, Philadelphia, 1999. 
[14] 
J. J. Moré and G. Toraldo, On the solution of large quadratic programming problems with bound constraints, SIAM Journal on Optimization, 1 (1991), 93113. doi: 10.1137/0801008. 
[15] 
J. Nagy and Z. Strakoš, Enforcing nonnegativity in image reconstruction algorithms, Mathematical Modeling, Estimation, and Imaging, David C. Wilson, et.al., Eds., 4121 (2000), 182190. 
[16] 
J. Nocedal and S. Wright, "Numerical Optimization," Series in Operations Research. SpringerVerlag, New York, 1999. doi: 10.1007/b98874. 
[17] 
R. T. Rockafellar, "Convex Analysis," Princeton University Press, 1970. 
[18] 
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259268. doi: 10.1016/01672789(92)90242F. 
[19] 
D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a chargecoupleddevice camera, Journal of the Optical Society of America A, 10 (1993), 10141023. doi: 10.1364/JOSAA.10.001014. 
[20] 
C. R. Vogel, Computational methods for inverse problems, With a foreword by H. T. Banks. Frontiers in Applied Mathematics, 23, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. 
[21] 
C. R. Vogel and M. E. Oman, Fast, robust total variationbased reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813824. doi: 10.1109/83.679423. 
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