The ordered subsets mirror descent optimization method with applications to tomography (Q2784404)

There is a tendency in medical diagnostics to include the metabolic changes in the cells of organs and tissues into the anatomical imaging modality [\textit{C. Boesch}, Molecular aspects of magnetic resonance imaging and spectroscopy. Molecular Aspects of Medicine 20, 185-318 (1999); \textit{W.J. Schempp}, Magnetic resonance imaging: Mathematical foundations and applications. (1998; Zbl 0930.92015)]. The paper under review deals with the modality of positron emission tomography (PET) as a clinical diagnostic imaging technique for measuring the metabolic activity of cells in the human body. In contrast to clinical magnetic resonance spectroscopy (MRS), PET imaging depends upon the radioactive tracing of positron emitters and therefore allows to formulate the image reconstruction problem as a parameter estimation problem rather than an inverse problem. The prototype of an approach to an inverse, and as such ill--posed imaging problem, is the filtered back-projection method of \(X\)-ray computerized tomography (CT) whereas MRS reconstruction is based on fast Fourier inversion.NEWLINENEWLINENEWLINEThe goal of the maximum likelihood estimation, as applied to emission tomography, is to find the expected number of annihilations by maximizing the probability of the set of coincidence events registered by the two--fold covering of the circular detector array.NEWLINENEWLINENEWLINEThe mathematical model of the detection process is based on the quantum optical assumption that photon counts represent a Poisson process [\textit{J. Bertoin}, Lévy processes. (1996; Zbl 0938.60005); \textit{K.-i. Sato}, Lévy processes and infinitely divisible distributions. (1999; Zbl 0973.60001)]. The optimal solution to the PET incomplete data problem is the maximum likelihood estimate of the discretized density of the tracer. This approach improves the original filtered back--projection PET image reconstruction method. The most promising method to overcome the tremendous size of the PET image reconstruction data is a gradient descent type method.NEWLINENEWLINENEWLINEThe authors implemented a specific PET algorithm, an accelerated version of the mirror descent method called ordered (OS) subsets technique [\textit{H.M. Hudson} and \textit{R.S. Larkin}, IEEE Trans. Med. Imag. 13, 601-609 (1994); \textit{C. Kamphuis} and \textit{F.J. Beekman}, ibid. 17, 1101-1105 (1998); \textit{S.H. Manglos} et al., Phys. Med. Biol. 40, 1225-1241 (1995)]. They demonstrate that the OS mirror descent approach is suited for solving the PET reconstruction problem of minimizing a convex function of several million variables on the standard simplex. A series of pictures of phantoms and Alzheimer patients illustrate the statistical iterative reconstruction approach which compares favorably with the currently commercially used methods.