# American Institute of Mathematical Sciences

November  2012, 6(4): 565-598. doi: 10.3934/ipi.2012.6.565

## Some proximal methods for Poisson intensity CBCT and PET

 1 Aix-Marseille Univ, LATP, UMR 7353, F-13453 Marseille, France, France 2 Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France 3 Aix-Marseille Univ, CPPM, UMR 7346, F-13288 Marseille, France

Received  November 2011 Revised  April 2012 Published  November 2012

Cone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information on a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed. It turns out that in this low-dose context, i.e. for low intensity signals acquired by photon counting devices, noise should not be approximated anymore by a Gaussian distribution, but is following a Poisson distribution. We investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with well-established methods.
Citation: Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems and Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
##### References:
 [1] S. Ahn and J. Fessler, Globally convergent image reconstruc- tion for emission tomography using relaxed ordered subsets algorithms, IEEE Trans. Med. Imag., 22 (2003), 613-626. [2] S. Alenius and U. Ruotsalainen, Bayesian image reconstruction for emission tomography based on median root prior, Europ. J. of Nucl. Med. and Molec. Im., 24 (1998), 258-265. [3] F. J. Andscombe, The transformation of Poisson, binomial and non negative-binomial data, Biometrika, 35 (1948), 246-254. [4] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE TIP, 18 (2009), 2419-2434. [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. on Imag. Sci., 2 (2009), 183-202. [6] M. Bertero, P. Boccacci, G. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20 pp. [7] S. Bonettini and V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 095001, 26 pp. [8] L. M. Briceño-Arias and P. L. Combettes, Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery, Numerical Mathematics: Theory, Methods, and Applications, 2 (2009), 485-508. [9] F. C. Brunner, J.-C. Clemens, C. Hemmer and C. Morel, Imaging performance of the hybrid pixel detectors xpad3-s, Physics in Medicine and Biology, 54 (2009), 1773-1789. [10] A. Chambolle, An algorithm for total variation minimization and applications, JMIV, 20 (2004), 89-97. [11] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, JMIV, 40 (2011), 120-145. [12] G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101. [13] P. L. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multi. Model. and Simu., 4 (2005), 1168-1200. [14] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Com. P. & A. Math., 57 (2004), 1413-1457. [15] A. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. of the Roy. Stat. Soc. Ser. B, 39 (1977), 1-38. [16] Y. K. Dewaraja, K. F. Koral and J. A. Fessler, Regularized reconstruction in quantitative spect using CT side information from hybrid imaging, Phys. Med. Biol., 55 (2010), 2523-2539. [17] F.-X. Dupé, J. Fadili and J.-L. Starck, A proximal iteration for deconvolving Poisson noisy images using sparse representations, IEEE TIP, 18 (2009), 310-321. [18] H. Erdoğan and J. Fessler, Monotonic algorithms for transmission tomography, IEEE TMI, 18 (1999), 801-814. [19] L. Feldkamp, L. Davis and J. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A., 1 (1984), 612-619. [20] M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145. [21] M. Fisz, The limiting distribution of a function of two independant random variables and its statistical application, Colloquium Mathematicum, 3 (1955), 138-146. [22] Kenneth M. Hanson and George W. Wecksung, Local basis-function approach to computed tomography, Appl. Opt., 24 (1985), 4028-4039. [23] Z. Harmany, R. Marcia and R. Willett, This is SPIRAL-TAP: Sparse Poisson intensity Reconstruction ALgorithms- Theory and Practice, IEEE Trans. Image Process., 21 (2010), 1084-1096. [24] S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Orlando, FL, 1984. [25] Julia Herzen, Tilman Donath, Franz Pfeiffer, Oliver Bunk, Celestiste, Felix Beckmann, Andreas Schreyer and Christian David, Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source, Opt. Express, 17 (2009), 10010-10018. [26] H. M. Hudson and R. S. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Trans. Med. Imag., 13 (1994), 601-609. [27] P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981. [28] J.-J.Moreau, Proximité et dualité dans un espace hilbertien, Bulletin Soc. Math. France, 93 (1965), 273-299. [29] R. Khoury, A. Bonissent, J.-C. Clémens, C. Meessen, E. Vigeolas, M. Billault and C. Morel, A geometrical calibration method for the PIXSCAN micro-CT scanner, Journal of Instrumentation, 4 (2009), P07016. [30] K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomo., 8 (1984), 306-316. [31] C. Lartizien, N. Costes, A. Reilhac, M. Janier and D. Sappey-Marinier, The clearPET project: Development of a 2nd generation high-performance small animal PET scanner, in "Second AMI Meeting," Madrid, Spain, Sept., 2003. [32] J.-B. Mosset, O. Devroede, M. Krieguer, M. Rey, J.-M. Vieira, J. H. Jung, C. Kuntner, M. Streun, K. Ziemons, E. Auffray, P. Sempere-Roldan, P. Lecoq, P. Bruyndonckx, J.-F. Loude, S. Tavernier and C. Morel, Development of an optimized LSO/LuYAP phoswich detector head for the Lausanne ClearPET demonstrator, Nuclear Science, IEEE Transactions on, 53 (2006), 25-29. [33] Yu. Nesterov, "Introductory Lectures on Convex Optimization: A Basic Course," Optimization, 87, Kluwer Ac. Pub., Boston, MA, 2004. [34] Yu. Nesterov, Smooth minimization of non-smooth functions, Math. Progr., 103 (2005), 127-152. [35] Y. Nesterov, Gradient methods for minimizing composite objective function, Ecore discussion paper, 2007. [36] S. Nicol, S. Karkar, C. Hemmer, A. Dawiec, D. Benoit, P. Breugnon, B. Dinkespiler, F. Riviere, J.-P. Logier, M. Niclas, J. Royon, C. Meessen, F. Cassol, J.-C. Clemens, A. Bonissent, F. Debarbieux, E. Vigeolas, P. Delpierre and C. Morel, Design and construction of the ClearPET/XPAD small animal PET/CT scanner, Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE, Oct. 24 2009-Nov. 1 2009, 3311-3314. [37] P. Pangaud, S. Basolo, N. Boudet, J.-F. Berar, B. Chantepie, P. Delpierre, B. Dinkespiler, S. Hustache, M. Menouni and C. Morel, XPAD3: A new photon counting chip for X-ray CT-scanner, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 571 (2007), 321-324; Proceedings of the 1st International Conference on Molecular Imaging Technology - EuroMedIm, 2006. [38] A. R. De Pierro, A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography, IEEE TMI, 14 (1995), 132-137. [39] N. Pustelnik, C. Chaux and J.-C. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE TIP, 20 (2011), 2450-2462. [40] N. Pustelnik, C. Chaux, J.-C. Pesquet and C. Comtat, Parallel algorithm and hybrid regularization for dynamic PET reconstruction, in "IEEE Med. Im. Conf.," Knoxville, TN, 2010. [41] M. Rey, S. Jan, J.-M. Vieira, J.-B. Mosset, M. Krieguer, C. Comtat and C. Morel, Count rate performance study of the Lausanne ClearPET scanner demonstrator, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 571 (2007), 207-210; Proceedings of the 1st International Conference on Molecular Imaging Technology - EuroMedIm, 2006. [42] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [43] L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction in positron emission tomography, IEEE TMI, 1 (1982), 113-122. [44] E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807. [45] Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE TIP, 13 (2004), 600-612. [46] P. Weiss, G. Aubert and L. Blanc-Féraud, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. on Sci. Comp., 31 (2009), 2047-2080.

show all references

##### References:
 [1] S. Ahn and J. Fessler, Globally convergent image reconstruc- tion for emission tomography using relaxed ordered subsets algorithms, IEEE Trans. Med. Imag., 22 (2003), 613-626. [2] S. Alenius and U. Ruotsalainen, Bayesian image reconstruction for emission tomography based on median root prior, Europ. J. of Nucl. Med. and Molec. Im., 24 (1998), 258-265. [3] F. J. Andscombe, The transformation of Poisson, binomial and non negative-binomial data, Biometrika, 35 (1948), 246-254. [4] A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE TIP, 18 (2009), 2419-2434. [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. on Imag. Sci., 2 (2009), 183-202. [6] M. Bertero, P. Boccacci, G. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20 pp. [7] S. Bonettini and V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 095001, 26 pp. [8] L. M. Briceño-Arias and P. L. Combettes, Convex variational formulation with smooth coupling for multicomponent signal decomposition and recovery, Numerical Mathematics: Theory, Methods, and Applications, 2 (2009), 485-508. [9] F. C. Brunner, J.-C. Clemens, C. Hemmer and C. Morel, Imaging performance of the hybrid pixel detectors xpad3-s, Physics in Medicine and Biology, 54 (2009), 1773-1789. [10] A. Chambolle, An algorithm for total variation minimization and applications, JMIV, 20 (2004), 89-97. [11] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, JMIV, 40 (2011), 120-145. [12] G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101. [13] P. L. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multi. Model. and Simu., 4 (2005), 1168-1200. [14] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Com. P. & A. Math., 57 (2004), 1413-1457. [15] A. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. of the Roy. Stat. Soc. Ser. B, 39 (1977), 1-38. [16] Y. K. Dewaraja, K. F. Koral and J. A. Fessler, Regularized reconstruction in quantitative spect using CT side information from hybrid imaging, Phys. Med. Biol., 55 (2010), 2523-2539. [17] F.-X. Dupé, J. Fadili and J.-L. Starck, A proximal iteration for deconvolving Poisson noisy images using sparse representations, IEEE TIP, 18 (2009), 310-321. [18] H. Erdoğan and J. Fessler, Monotonic algorithms for transmission tomography, IEEE TMI, 18 (1999), 801-814. [19] L. Feldkamp, L. Davis and J. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A., 1 (1984), 612-619. [20] M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145. [21] M. Fisz, The limiting distribution of a function of two independant random variables and its statistical application, Colloquium Mathematicum, 3 (1955), 138-146. [22] Kenneth M. Hanson and George W. Wecksung, Local basis-function approach to computed tomography, Appl. Opt., 24 (1985), 4028-4039. [23] Z. Harmany, R. Marcia and R. Willett, This is SPIRAL-TAP: Sparse Poisson intensity Reconstruction ALgorithms- Theory and Practice, IEEE Trans. Image Process., 21 (2010), 1084-1096. [24] S. Helgason, "Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions," Pure and Applied Mathematics, 113, Academic Press, Orlando, FL, 1984. [25] Julia Herzen, Tilman Donath, Franz Pfeiffer, Oliver Bunk, Celestiste, Felix Beckmann, Andreas Schreyer and Christian David, Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source, Opt. Express, 17 (2009), 10010-10018. [26] H. M. Hudson and R. S. Larkin, Accelerated image reconstruction using ordered subsets of projection data, IEEE Trans. Med. Imag., 13 (1994), 601-609. [27] P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981. [28] J.-J.Moreau, Proximité et dualité dans un espace hilbertien, Bulletin Soc. Math. France, 93 (1965), 273-299. [29] R. Khoury, A. Bonissent, J.-C. Clémens, C. Meessen, E. Vigeolas, M. Billault and C. Morel, A geometrical calibration method for the PIXSCAN micro-CT scanner, Journal of Instrumentation, 4 (2009), P07016. [30] K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomo., 8 (1984), 306-316. [31] C. Lartizien, N. Costes, A. Reilhac, M. Janier and D. Sappey-Marinier, The clearPET project: Development of a 2nd generation high-performance small animal PET scanner, in "Second AMI Meeting," Madrid, Spain, Sept., 2003. [32] J.-B. Mosset, O. Devroede, M. Krieguer, M. Rey, J.-M. Vieira, J. H. Jung, C. Kuntner, M. Streun, K. Ziemons, E. Auffray, P. Sempere-Roldan, P. Lecoq, P. Bruyndonckx, J.-F. Loude, S. Tavernier and C. Morel, Development of an optimized LSO/LuYAP phoswich detector head for the Lausanne ClearPET demonstrator, Nuclear Science, IEEE Transactions on, 53 (2006), 25-29. [33] Yu. Nesterov, "Introductory Lectures on Convex Optimization: A Basic Course," Optimization, 87, Kluwer Ac. Pub., Boston, MA, 2004. [34] Yu. Nesterov, Smooth minimization of non-smooth functions, Math. Progr., 103 (2005), 127-152. [35] Y. Nesterov, Gradient methods for minimizing composite objective function, Ecore discussion paper, 2007. [36] S. Nicol, S. Karkar, C. Hemmer, A. Dawiec, D. Benoit, P. Breugnon, B. Dinkespiler, F. Riviere, J.-P. Logier, M. Niclas, J. Royon, C. Meessen, F. Cassol, J.-C. Clemens, A. Bonissent, F. Debarbieux, E. Vigeolas, P. Delpierre and C. Morel, Design and construction of the ClearPET/XPAD small animal PET/CT scanner, Nuclear Science Symposium Conference Record (NSS/MIC), 2009 IEEE, Oct. 24 2009-Nov. 1 2009, 3311-3314. [37] P. Pangaud, S. Basolo, N. Boudet, J.-F. Berar, B. Chantepie, P. Delpierre, B. Dinkespiler, S. Hustache, M. Menouni and C. Morel, XPAD3: A new photon counting chip for X-ray CT-scanner, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 571 (2007), 321-324; Proceedings of the 1st International Conference on Molecular Imaging Technology - EuroMedIm, 2006. [38] A. R. De Pierro, A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography, IEEE TMI, 14 (1995), 132-137. [39] N. Pustelnik, C. Chaux and J.-C. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE TIP, 20 (2011), 2450-2462. [40] N. Pustelnik, C. Chaux, J.-C. Pesquet and C. Comtat, Parallel algorithm and hybrid regularization for dynamic PET reconstruction, in "IEEE Med. Im. Conf.," Knoxville, TN, 2010. [41] M. Rey, S. Jan, J.-M. Vieira, J.-B. Mosset, M. Krieguer, C. Comtat and C. Morel, Count rate performance study of the Lausanne ClearPET scanner demonstrator, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 571 (2007), 207-210; Proceedings of the 1st International Conference on Molecular Imaging Technology - EuroMedIm, 2006. [42] T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. [43] L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction in positron emission tomography, IEEE TMI, 1 (1982), 113-122. [44] E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807. [45] Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE TIP, 13 (2004), 600-612. [46] P. Weiss, G. Aubert and L. Blanc-Féraud, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. on Sci. Comp., 31 (2009), 2047-2080.
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