Inverse Problems and Imaging (IPI)

Some proximal methods for Poisson intensity CBCT and PET

Pages: 565 - 598, Volume 6, Issue 4, November 2012      doi:10.3934/ipi.2012.6.565

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Sandrine Anthoine - Aix-Marseille Univ, LATP, UMR 7353, F-13453 Marseille, France (email)
Jean-François Aujol - Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France (email)
Yannick Boursier - Aix-Marseille Univ, CPPM, UMR 7346, F-13288 Marseille, France (email)
Clothilde Mélot - Aix-Marseille Univ, LATP, UMR 7353, F-13453 Marseille, France (email)

Abstract: 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:  Tomography, CT, PET, proximal methods, Poisson noise, total variation, wavelet $l_1$-regularization.
Mathematics Subject Classification:  Primary: 68U10, 94A08; Secondary: 47N10.

Received: November 2011;      Revised: April 2012;      Available Online: November 2012.