Abstract — In some types of magnetic resonance (MR) imaging, particularly functional brain scans, the conventional Fourier model for the measurements is inaccurate. Magnetic field inhomogeneities, caused by imperfect main fields and by magnetic susceptibility variations, induce distortions in images that are reconstructed by conventional Fourier methods. These artifacts hamper the use of functional MR imaging (fMRI) in brain regions near air/tissue interfaces. Recently, iterative methods that combine the conjugate gradient (CG) algorithm with nonuniform FFT (NUFFT) operations have been shown to provide considerably improved image quality relative to the conjugatephase method. However, for non-Cartesian k-space trajectories, each CG-NUFFT iteration requires numerous k-space interpolations, operations that are computationally expensive and poorly suited to fast hardware implementations. This paper proposes a faster iterative approach to field-corrected MR image reconstruction based on the CG algorithm and certain Toeplitz matrices. This CG-Toeplitz approach requires k-space interpolations only for the initial iteration; thereafter only FFTs are required. Simulation results show that the proposed CG-Toeplitz approach produces equivalent image quality as the CG-NUFFT method with significantly reduced computation time. Index Terms — fMRI imaging, spiral trajectory, magnetic susceptibility, non-Cartesian sampling I.

Abstract—Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an 1-regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the “optimal” E-spline parameters (poles and zeros) and we investigate their

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MPI für biologische Kybernetik, AGBS;

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We develop the theory behind the Expectation Maximization algorithm and an exact inversion formula for the attenuated Radon transform, two reconstruction methods used in SPECT. We also implement both methods and present a few

These annotated slides were prepared by Jeff Fessler for attendees of the ISBI tutorial on statistical image reconstruction methods. The purpose of the annotation is to provide supplemental details, and particularly to provide extensive literature references for further study. For a fascinating history of tomography, see [1]. For broad coverage of image science, see [2]. For further references on image reconstruction, see review papers and chapters, e.g., [3–9].