Abstract
Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed.
Highlights
In the field of computed tomography (CT) [1,2,3,4], image reconstruction is generally considered an inverse problem, namely, obtaining pixel values of an image from measured projections and a known projection operator
In X-ray CT, an image is seriously degraded by, for example, metal and ring artifacts [5,6,7] when a certain part of the projections received by the detector is inaccurate or missing
Interpolation projection is a well-known method for reducing such artifacts [5, 7]; that is, inaccurate projections are interpolated from their neighboring data and replaced by synthesized values
Summary
In the field of computed tomography (CT) [1,2,3,4], image reconstruction is generally considered an inverse problem, namely, obtaining pixel values of an image from measured projections and a known projection operator. To reconstruct an image with pixel values and to estimate certain parts of projections instead of inaccurate part q, unknown variables x ∈ RJ+ and y ∈ RM + are treated, respectively. It solves an initialvalue problem of differential equations consisting of state variables including parts of projections as well as image pixel values. To validate the theoretical results given by the proposed method in comparison with results obtained by linear interpolation and a reduced system, reduction of metal artifacts is numerically simulated by using a large-sized model
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