American Institute of Mathematical Sciences

May  2008, 2(2): 167-185. doi: 10.3934/ipi.2008.2.167

An efficient computational method for total variation-penalized Poisson likelihood estimation

 1 Department of Mathematical Sciences, University of Montana Missoula, Montana 59812, United States

Received  February 2007 Revised  January 2008 Published  April 2008

Approximating non-Gaussian noise processes with Gaussian models is standard in data analysis. This is due in large part to the fact that Gaussian models yield parameter estimation problems of least squares form, which have been extensively studied both from the theoretical and computational points of view. In image processing applications, for example, data is often collected by a CCD camera, in which case the noise is a Guassian/Poisson mixture with the Poisson noise dominating for a sufficiently strong signal. Even so, the standard approach in such cases is to use a Gaussian approximation that leads to a negative-log likelihood function of weighted least squares type.
In the Bayesian point-of-view taken in this paper, a negative-log prior (or regularization) function is added to the negative-log likelihood function, and the resulting function is minimized. We focus on the case where the negative-log prior is the well-known total variation function and give a statistical interpretation. Regardless of whether the least squares or Poisson negative-log 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 negative-log 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 negative-log likelihood can yield a more computationally tractable problem when total variation regularization is used.
Citation: Johnathan M. Bardsley. An efficient computational method for total variation-penalized Poisson likelihood estimation. Inverse Problems & Imaging, 2008, 2 (2) : 167-185. doi: 10.3934/ipi.2008.2.167
References:
 [1] D. P. Bertsekas, On the Goldstein-Levitin-Poljak gradient projection method,, IEEE Transactions on Automatic Control, 21 (1976), 174. doi: 10.1109/TAC.1976.1101194. [2] Johnathan M. Bardsley and James G. Nagy, Covariance-preconditioned iterative methods for nonnegatively constrained astronomical imaging,, SIAM Journal on Matrix Analysis and Applications, 27 (2006), 1184. 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), 1326. doi: 10.1137/S1064827502410451. [4] Johnathan M. Bardsley and Aaron Luttman, Total variation-penalized Poisson likelihood estimation for ill-posed problems,, accepted, (2006). [5] P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems,, Mathematical Programming, 39 (1987), 93. doi: 10.1007/BF02592073. [6] D. Calvetti, G. Landi, L. Reichel and S. Sgallari, Non-negativity and iterative methods for ill-posed problems,, Inverse Problems, 20 (2004), 1747. doi: 10.1088/0266-5611/20/6/003. [7] Daniela Calvetti and Erkki Somersalo, A Gaussian hypermodel for recovering blocky objects,, Inverse Problems, 23 (2007), 733. doi: 10.1088/0266-5611/23/2/016. [8] Torbjørn Eltoft and Taesu Kim, On the multivariate Laplace distribution,, IEEE Signal Processing Letters, 13 (2006), 300. doi: 10.1109/LSP.2006.870353. [9] J. W. Goodman, "Introduction to Fourier Optics,", 2nd Edition, (1996). [10] M. Green, Statistics of images, the TV algorithm of Rudin-Osher-Fatemi for image denoising, and an improved denoising algorithm,, CAM Report 02-55, (2002), 02. [11] Jinggang Huang and David Mumford, Statistics of natural images and models,, in, (1999), 541. [12] Jari Kaipio and Erkki Somersalo, "Satistical and Computational Inverse Problems,", Applied Mathematical Sciences, (2005). [13] C. T. Kelley, "Iterative Methods for Optimization,", Frontiers in Applied Mathematics, (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), 93. doi: 10.1137/0801008. [15] J. Nagy and Z. Strakoš, Enforcing nonnegativity in image reconstruction algorithms,, Mathematical Modeling, 4121 (2000), 182. [16] J. Nocedal and S. Wright, "Numerical Optimization,", Series in Operations Research. Springer-Verlag, (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), 259. doi: 10.1016/0167-2789(92)90242-F. [19] D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera,, Journal of the Optical Society of America A, 10 (1993), 1014. 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, (2002). [21] C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images,, IEEE Transactions on Image Processing, 7 (1998), 813. doi: 10.1109/83.679423.

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References:
 [1] D. P. Bertsekas, On the Goldstein-Levitin-Poljak gradient projection method,, IEEE Transactions on Automatic Control, 21 (1976), 174. doi: 10.1109/TAC.1976.1101194. [2] Johnathan M. Bardsley and James G. Nagy, Covariance-preconditioned iterative methods for nonnegatively constrained astronomical imaging,, SIAM Journal on Matrix Analysis and Applications, 27 (2006), 1184. 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), 1326. doi: 10.1137/S1064827502410451. [4] Johnathan M. Bardsley and Aaron Luttman, Total variation-penalized Poisson likelihood estimation for ill-posed problems,, accepted, (2006). [5] P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems,, Mathematical Programming, 39 (1987), 93. doi: 10.1007/BF02592073. [6] D. Calvetti, G. Landi, L. Reichel and S. Sgallari, Non-negativity and iterative methods for ill-posed problems,, Inverse Problems, 20 (2004), 1747. doi: 10.1088/0266-5611/20/6/003. [7] Daniela Calvetti and Erkki Somersalo, A Gaussian hypermodel for recovering blocky objects,, Inverse Problems, 23 (2007), 733. doi: 10.1088/0266-5611/23/2/016. [8] Torbjørn Eltoft and Taesu Kim, On the multivariate Laplace distribution,, IEEE Signal Processing Letters, 13 (2006), 300. doi: 10.1109/LSP.2006.870353. [9] J. W. Goodman, "Introduction to Fourier Optics,", 2nd Edition, (1996). [10] M. Green, Statistics of images, the TV algorithm of Rudin-Osher-Fatemi for image denoising, and an improved denoising algorithm,, CAM Report 02-55, (2002), 02. [11] Jinggang Huang and David Mumford, Statistics of natural images and models,, in, (1999), 541. [12] Jari Kaipio and Erkki Somersalo, "Satistical and Computational Inverse Problems,", Applied Mathematical Sciences, (2005). [13] C. T. Kelley, "Iterative Methods for Optimization,", Frontiers in Applied Mathematics, (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), 93. doi: 10.1137/0801008. [15] J. Nagy and Z. Strakoš, Enforcing nonnegativity in image reconstruction algorithms,, Mathematical Modeling, 4121 (2000), 182. [16] J. Nocedal and S. Wright, "Numerical Optimization,", Series in Operations Research. Springer-Verlag, (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), 259. doi: 10.1016/0167-2789(92)90242-F. [19] D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera,, Journal of the Optical Society of America A, 10 (1993), 1014. 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, (2002). [21] C. R. Vogel and M. E. Oman, Fast, robust total variation-based reconstruction of noisy, blurred images,, IEEE Transactions on Image Processing, 7 (1998), 813. doi: 10.1109/83.679423.
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