This paper describes an optimization framework for reconstructing nonnegative image intensities from linear projections contaminated with Poisson noise. Such Poisson inverse problems arise in a variety of applications, ranging from medical imaging to astronomy. A total variation regularization term is used to counter the ill-posedness of the inverse problem and results in reconstructions that are piecewise smooth. The proposed algorithm sequentially approximates the objective function with a regularized quadratic surrogate which can easily be minimized. Unlike alternative methods, this approach ensures that the natural nonnegativity constraints are satisfied without placing prohibitive restrictions on the nature of the linear projections to ensure computational tractability. The resulting algorithm is computationally efficient and outperforms similar methods using wavelet-sparsity or partition-based regularization.
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"Additionally imposing prior information on the solution, e.g. that the solutions has a small total variation, leads to an extension of the EM algorithm, e.g. the EM-TV algorithm  . See also     for related approaches and   for extensions of EM-TV to Bregmanized total variation regularization. "
[Show abstract][Hide abstract]ABSTRACT: The aim of this paper is to test and analyze a novel technique for image
reconstruction in positron emission tomography, which is based on (total
variation) regularization on both the image space and the projection space. We
formulate our variational problem considering both total variation penalty
terms on the image and on an idealized sinogram to be reconstructed from a
given Poisson distributed noisy sinogram. We prove existence, uniqueness and
stability results for the proposed model and provide some analytical insight
into the structures favoured by joint regularization.
For the numerical solution of the corresponding discretized problem we employ
the split Bregman algorithm and extensively test the approach in comparison to
standard total variation regularization on the image. The numerical results
show that an additional penalty on the sinogram performs better on
reconstructing images with thin structures.
[Show abstract][Hide abstract]ABSTRACT: We propose a flexible and computationally efficient method to solve the non-homogeneous Poisson (NHP) model for grayscale and color images within the TV framework. The NHP model is relevant to image restoration in several applications, such as PET, CT, MRI, etc. The proposed algorithm uses a novel method to spatially adapt the regularization parameter; it also uses a quadratic approximation of the negative log-likelihood function to pose the original problem as a non-negative quadratic programming problem. The reconstruction quality of the proposed algorithm outperforms state of the art algorithms for grayscale image restoration corrupted with Poisson noise. Moreover, it places no prohibitive restriction on the forward operator, and to best of our knowledge, the proposed algorithm is the only one that explicitly includes the NHP model for color images and that spatially adapts its regularization parameter within the TV framework.
[Show abstract][Hide abstract]ABSTRACT: Computerized tomography (CT) plays an important role in medical imaging, especially for diagnosis and therapy. However, higher radiation dose from CT will result in increasing of radiation exposure in the population. There-fore, the reduction of radiation from CT is an essential issue. Expectation maximization (EM) is an iterative method used for CT image reconstruction that maximizes the likelihood function under Poisson noise assump-tion. Total variation regularization is a technique used frequently in image restoration to preserve edges, given the assumption that most images are piecewise constant. Here, we propose a method combining expectation maximization and total variation regularization, called EM+TV. This method can reconstruct a better image using fewer views in the computed tomography setting, thus reducing the overall dose of radiation. The numeri-cal results in two and three dimensions show the efficiency of the proposed EM+TV method by comparison with those obtained by filtered back projection (FBP) or by EM only.
Full-text · Article · Mar 2011 · Proceedings of SPIE - The International Society for Optical Engineering