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
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show all references

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IEEE Trans. Med. Imag., 22 (2003), 613-626. Google Scholar

[2]

Europ. J. of Nucl. Med. and Molec. Im., 24 (1998), 258-265. Google Scholar

[3]

Biometrika, 35 (1948), 246-254. Google Scholar

[4]

IEEE TIP, 18 (2009), 2419-2434.  Google Scholar

[5]

SIAM J. on Imag. Sci., 2 (2009), 183-202.  Google Scholar

[6]

Inverse Problems, 26 (2010), 105004, 20 pp.  Google Scholar

[7]

Inverse Problems, 27 (2011), 095001, 26 pp.  Google Scholar

[8]

Numerical Mathematics: Theory, Methods, and Applications, 2 (2009), 485-508.  Google Scholar

[9]

Physics in Medicine and Biology, 54 (2009), 1773-1789. Google Scholar

[10]

JMIV, 20 (2004), 89-97.  Google Scholar

[11]

JMIV, 40 (2011), 120-145.  Google Scholar

[12]

Mathematical Programming, 64 (1994), 81-101.  Google Scholar

[13]

Multi. Model. and Simu., 4 (2005), 1168-1200.  Google Scholar

[14]

Com. P. & A. Math., 57 (2004), 1413-1457.  Google Scholar

[15]

J. of the Roy. Stat. Soc. Ser. B, 39 (1977), 1-38.  Google Scholar

[16]

Phys. Med. Biol., 55 (2010), 2523-2539. Google Scholar

[17]

IEEE TIP, 18 (2009), 310-321.  Google Scholar

[18]

IEEE TMI, 18 (1999), 801-814. Google Scholar

[19]

J. Opt. Soc. Am. A., 1 (1984), 612-619. Google Scholar

[20]

IEEE Transactions on Image Processing, 19 (2010), 3133-3145.  Google Scholar

[21]

Colloquium Mathematicum, 3 (1955), 138-146.  Google Scholar

[22]

Appl. Opt., 24 (1985), 4028-4039. Google Scholar

[23]

IEEE Trans. Image Process., 21 (2010), 1084-1096.  Google Scholar

[24]

Pure and Applied Mathematics, 113, Academic Press, Orlando, FL, 1984.  Google Scholar

[25]

Opt. Express, 17 (2009), 10010-10018. Google Scholar

[26]

IEEE Trans. Med. Imag., 13 (1994), 601-609. Google Scholar

[27]

Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1981.  Google Scholar

[28]

Bulletin Soc. Math. France, 93 (1965), 273-299.  Google Scholar

[29]

Journal of Instrumentation, 4 (2009), P07016. Google Scholar

[30]

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[31]

in "Second AMI Meeting," Madrid, Spain, Sept., 2003. Google Scholar

[32]

Nuclear Science, IEEE Transactions on, 53 (2006), 25-29. Google Scholar

[33]

Optimization, 87, Kluwer Ac. Pub., Boston, MA, 2004.  Google Scholar

[34]

Math. Progr., 103 (2005), 127-152.  Google Scholar

[35]

Ecore discussion paper, 2007. Google Scholar

[36]

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SIAM J. on Sci. Comp., 31 (2009), 2047-2080.  Google Scholar

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