×

Total variation-penalized Poisson likelihood estimation for ill-posed problems. (English) Zbl 1171.94001

Summary: The noise contained in data measured by imaging instruments is often primarily of Poisson type. This motivates, in many cases, the use of the Poisson negative-log likelihood function in place of the ubiquitous least squares data fidelity when solving image deblurring problems. We assume that the underlying blurring operator is compact, so that, as in the least squares case, the resulting minimization problem is ill-posed and must be regularized. In this paper, we focus on total variation regularization and show that the problem of computing the minimizer of the resulting total variation-penalized Poisson likelihood functional is well-posed. We then prove that, as the errors in the data and in the blurring operator tend to zero, the resulting minimizers converge to the minimizer of the exact likelihood function. Finally, the practical effectiveness of the approach is demonstrated on synthetically generated data, and a nonnegatively constrained, projected quasi-Newton method is introduced.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

Software:

KELLEY
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems 10, 1217–1229 (1994) · Zbl 0809.35151
[2] Asaki, T., Chartrand, R., Le, T.: A Variational Approach to Reconstructing Images Corrupted by Poisson Noise. UCLA CAM Report 05-49 (November 2005)
[3] Bardsley, J.M.: A limited memory, quasi-newton preconditioner for nonnegatively constrained image reconstruction. J. Opt. Soc. Amer. A 21(5), 724–731 (2004)
[4] Bardsley, J.M., Laobeul, N.: Tikhonov regularized Poisson likelihood estimation: theoretical justification and a computational method. Inverse Probl. Sci. Eng. 16(2), 199–215 (January 2008) · Zbl 1258.35206
[5] Bardsley, J.M., Vogel, C.R.: A nonnnegatively constrained convex programming method for image reconstruction. SIAM J. Sci. Comput. 25(4), 1326–1343 (2004) · Zbl 1061.65047
[6] Berterro, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. Institute of Physics Publishing, Bristol (1998) · Zbl 0914.65060
[7] Bertsekas, D.P.: Projected newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20, 221–246 (1982) · Zbl 0507.49018
[8] Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997) · Zbl 0874.68299
[9] Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, Heidelberg (1990) · Zbl 0706.46003
[10] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC, Boca Raton (1992) · Zbl 0804.28001
[11] Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhauser Verlag, Basel (1984) · Zbl 0545.49018
[12] Green, M.: Statistics of images, the TV algorithm of Rudin-Osher-Fatemi for image denoising, and an improved denoising algorithm. CAM Report 02-55, UCLA (October 2002)
[13] Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999) · Zbl 0934.90082
[14] Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corruted by poisson noise. J. Math. Imaging Vision 27(3), 257–263 (2007)
[15] Moré, J.J., Toraldo, G.: On the solution of large quadratic programming problems with bound constraints. SIAM J. Optim. 1, 93–113 (1991) · Zbl 0752.90053
[16] Nocedal, J., Wright, S.: Numerical Optimization. Springer, Heidelberg (1999) · Zbl 0930.65067
[17] Nowak, R., Kolaczyk, E.: A statistical multiscale framework for poisson inverse problems. IEEE Trans. Inform. Theory 46(5), 1811–1825 (2000) · Zbl 0999.94004
[18] Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992) · Zbl 0780.49028
[19] Sardy, S., Antoniadis, A., Tseng, P.: Automatic smoothing with wavelets for a wide class of distributions. J. Comput. Graph. Statist. 13(2), 1–23 (2004)
[20] Sardy, S., Tseng, P.: On the statistical analysis of smoothing by maximizing dirty Markov random field posterior distributions. J. Amer. Statist. Assoc. 99(465), 191–204(14) (2004) · Zbl 1089.62518
[21] Snyder, D.L., Hammoud, A.M., White, R.L.: Image recovery from data acquired with a charge-coupled-device camera. J. Opt. Soc. Amer. A 10, 1014–1023 (1993)
[22] Snyder, D.L., Helstrom, C.W., Lanterman, A.D., Faisal, M., White, R.L.: Compensation for readout noise in CCD images. J. Opt. Soc. Amer. A 12, 272–283 (1995)
[23] Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic, Boston (1990) · Zbl 0831.65059
[24] Yu, D.F., Fessler, J.A.: Edge-preserving tomographic reconstruction with nonlocal regularization. IEEE Trans. Med. Imag. 21(2), 159–173 (2002)
[25] Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002) · Zbl 1008.65103
[26] Vogel, C.R., Oman, M.E.: A fast, robust algorithm for total variation based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824 (1998) · Zbl 0993.94519
[27] Zeidler, E.: Applied Functional Analysis: Main Principles and their Applications. Springer, New York (1995) · Zbl 0834.46003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.