American Institute of Mathematical Sciences

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

Some proximal methods for Poisson intensity CBCT and PET

Sandrine Anthoine 1, , Jean-François Aujol 2, , Yannick Boursier 3, and  Clothilde Mélot 1,

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.
Keywords: total variation, wavelet $l_1$-regularization., CT, proximal methods, PET, Poisson noise, Tomography.
Mathematics Subject Classification: Primary: 68U10, 94A08; Secondary: 47N1.
Citation: Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & 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. Google Scholar

[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. Google Scholar

[3]

F. J. Andscombe, The transformation of Poisson, binomial and non negative-binomial data, Biometrika, 35 (1948), 246-254. Google Scholar

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A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE TIP, 18 (2009), 2419-2434.  Google Scholar

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A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. on Imag. Sci., 2 (2009), 183-202.  Google Scholar

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M. Bertero, P. Boccacci, G. Talenti, R. Zanella and L. Zanni, A discrepancy principle for Poisson data, Inverse Problems, 26 (2010), 105004, 20 pp.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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A. Chambolle, An algorithm for total variation minimization and applications, JMIV, 20 (2004), 89-97.  Google Scholar

[11]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, JMIV, 40 (2011), 120-145.  Google Scholar

[12]

G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101.  Google Scholar

[13]

P. L. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multi. Model. and Simu., 4 (2005), 1168-1200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

H. Erdoğan and J. Fessler, Monotonic algorithms for transmission tomography, IEEE TMI, 18 (1999), 801-814. Google Scholar

[19]

L. Feldkamp, L. Davis and J. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A., 1 (1984), 612-619. Google Scholar

[20]

M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145.  Google Scholar

[21]

M. Fisz, The limiting distribution of a function of two independant random variables and its statistical application, Colloquium Mathematicum, 3 (1955), 138-146.  Google Scholar

[22]

Kenneth M. Hanson and George W. Wecksung, Local basis-function approach to computed tomography, Appl. Opt., 24 (1985), 4028-4039. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[28]

J.-J.Moreau, Proximité et dualité dans un espace hilbertien, Bulletin Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[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. Google Scholar

[30]

K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomo., 8 (1984), 306-316. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

Yu. Nesterov, "Introductory Lectures on Convex Optimization: A Basic Course," Optimization, 87, Kluwer Ac. Pub., Boston, MA, 2004.  Google Scholar

[34]

Yu. Nesterov, Smooth minimization of non-smooth functions, Math. Progr., 103 (2005), 127-152.  Google Scholar

[35]

Y. Nesterov, Gradient methods for minimizing composite objective function, Ecore discussion paper, 2007. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

A. R. De Pierro, A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography, IEEE TMI, 14 (1995), 132-137. Google Scholar

[39]

N. Pustelnik, C. Chaux and J.-C. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE TIP, 20 (2011), 2450-2462.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[43]

L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction in positron emission tomography, IEEE TMI, 1 (1982), 113-122. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

F. J. Andscombe, The transformation of Poisson, binomial and non negative-binomial data, Biometrika, 35 (1948), 246-254. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

A. Chambolle, An algorithm for total variation minimization and applications, JMIV, 20 (2004), 89-97.  Google Scholar

[11]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, JMIV, 40 (2011), 120-145.  Google Scholar

[12]

G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101.  Google Scholar

[13]

P. L. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multi. Model. and Simu., 4 (2005), 1168-1200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[18]

H. Erdoğan and J. Fessler, Monotonic algorithms for transmission tomography, IEEE TMI, 18 (1999), 801-814. Google Scholar

[19]

L. Feldkamp, L. Davis and J. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A., 1 (1984), 612-619. Google Scholar

[20]

M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Transactions on Image Processing, 19 (2010), 3133-3145.  Google Scholar

[21]

M. Fisz, The limiting distribution of a function of two independant random variables and its statistical application, Colloquium Mathematicum, 3 (1955), 138-146.  Google Scholar

[22]

Kenneth M. Hanson and George W. Wecksung, Local basis-function approach to computed tomography, Appl. Opt., 24 (1985), 4028-4039. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

P. J. Huber, "Robust Statistics," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[28]

J.-J.Moreau, Proximité et dualité dans un espace hilbertien, Bulletin Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[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. Google Scholar

[30]

K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomo., 8 (1984), 306-316. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

Yu. Nesterov, "Introductory Lectures on Convex Optimization: A Basic Course," Optimization, 87, Kluwer Ac. Pub., Boston, MA, 2004.  Google Scholar

[34]

Yu. Nesterov, Smooth minimization of non-smooth functions, Math. Progr., 103 (2005), 127-152.  Google Scholar

[35]

Y. Nesterov, Gradient methods for minimizing composite objective function, Ecore discussion paper, 2007. Google Scholar

[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. Google Scholar

[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. Google Scholar

[38]

A. R. De Pierro, A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography, IEEE TMI, 14 (1995), 132-137. Google Scholar

[39]

N. Pustelnik, C. Chaux and J.-C. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE TIP, 20 (2011), 2450-2462.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[43]

L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction in positron emission tomography, IEEE TMI, 1 (1982), 113-122. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

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