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An improved Newton projection method for nonnegative deblurring of Poisson-corrupted images with Tikhonov regularization. (English) Zbl 1241.65059
Summary: A quasi-Newton projection method for image deblurring is presented. The image restoration problem is mathematically formulated as a nonnegatively constrained minimization problem where the objective function is the sum of the Kullback-Leibler divergence, used to express fidelity to the data in the presence of Poisson noise, and of a Tikhonov regularization term. The Hessian of the objective function is approximated so that the Newton system can be efficiently solved by using Fast Fourier Transforms. The numerical results show the potential of the proposed method both in terms of relative error reduction and computational efficiency.

65K05Mathematical programming (numerical methods)
90C30Nonlinear programming
90C53Methods of quasi-Newton type
94A08Image processing (compression, reconstruction, etc.)
Full Text: DOI
[1] Anconelli, B., Bertero, M., Boccacci, P., Carbillet, M., Lanteri, H.: Iterative methods for the reconstruction of astronomical images with high dynamic range. J. Comput. Appl. Math. 198, 321--331 (2007) · Zbl 1145.65089 · doi:10.1016/j.cam.2005.06.049
[2] Bardsley, J., N’djekornom, L.: Tikhonov regularized poisson likelihood estimation: theoretical justification and a computational method. Inverse Probl. Sci. Eng. 16(2), 199--215 (2008) · Zbl 1258.35206 · doi:10.1080/17415970701404235
[3] Bardsley, J., N’djekornom, L.: An analysis of regularization by diffusion for ill-posed poisson likelihood estimation. Inverse Probl. Sci. Eng. 17(4), 537--550 (2009) · Zbl 1167.65074 · doi:10.1080/17415970802231594
[4] Bardsley, J.M., Vogel, C.R.: A nonnegatively constrained convex programming method for image reconstruction. SIAM J. Sci. Comput. 25, 1326--1343 (2003) · Zbl 1061.65047 · doi:10.1137/S1064827502410451
[5] Bertsekas, D.: Constrained Optimization and Lagrange Multiplier Methods. Academic, New York (1982) · Zbl 0572.90067
[6] Bertsekas, D.: Projected Newton methods for optimization problem with simple constraints. SIAM J. Control Optim. 20(2), 221--245 (1982) · Zbl 0507.49018 · doi:10.1137/0320018
[7] Bertsekas, D.: Nonlinear Programming, 2nd ed. Athena Scientific, Belmont, Massachusetts (1999) · Zbl 1015.90077
[8] Conchello, J.A., McNally, J.G.: Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy. Proc. SPIE 2655, 199--208 (1996)
[9] Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control. Optim. 22, 936--964 (1984) · Zbl 0555.90086 · doi:10.1137/0322061
[10] Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imag. 9, 84--93 (1990) · doi:10.1109/42.52985
[11] Hanke, M., Nagy, J.G., Vogel, C.: Quasi Newton approach to nonnegative image restorations. Linear Algebra Appl. 316, 222--333 (2000) · Zbl 0960.65071 · doi:10.1016/S0024-3795(00)00116-6
[12] Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring images: matrices, spectra and filtering. SIAM, Philadelphia (2006) · Zbl 1112.68127
[13] Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999) · Zbl 0934.90082
[14] Landi, G., Loli Piccolomini, E.: A projected Newton-CG method for nonnegative astronomical image deblurring. Numer. Algorithms 48, 279--300 (2008) · Zbl 1151.65053 · doi:10.1007/s11075-008-9198-3
[15] Lant’eri, H., Roche, M., Aime, C.: Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl. 18, 1397--1419 (2002) · Zbl 1023.62099 · doi:10.1088/0266-5611/18/5/313
[16] Bertero, M, Boccacci, P., Desiderà, G., Vicidomini, G.: Image deblurring with Poisson data: from cells to galaxies. Inverse Probl. 25, 123006 (2009) · Zbl 1186.85001 · doi:10.1088/0266-5611/25/12/123006
[17] Lanteri, H., Bertero, M., Zanni, L.: Iterative image reconstruction: a point of view. In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT) (CRM Series, vol. 7), pp. 37--63 (2008)
[18] Markham, J., Conchello, J.A.: Fast maximum-likelihood image-restoration algorithms for threedimensional fluorescence microscopy. J. Opt. Soc. Am. A 18, 1062--1071 (2001). · doi:10.1364/JOSAA.18.001062
[19] Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999) · Zbl 0930.65067
[20] Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imag. 1, 113--122 (1982) · doi:10.1109/TMI.1982.4307558
[21] Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002) · Zbl 1008.65103
[22] Vogel, C.R., Oman, M.E.: Fast, robust total variation--based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813--824 (1998) · Zbl 0993.94519 · doi:10.1109/83.679423
[23] Wang, Z., Bovik, A.C.: Mean squared error: love it or leave it? IEEE Signal Process. Mag. 98, 98--117 (2009) · doi:10.1109/MSP.2008.930649
[24] Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600--612 (2004) · Zbl 05453404 · doi:10.1109/TIP.2003.819861