Tomographic Inverse Problem with Estimating Missing Projections

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  • Masashi Kimura
    Graduate School of Health Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan
  • Yusaku Yamaguchi
    National Hospital Organization, Shikoku Medical Center for Children and Adults, 2-1-1 Senyu, Zentsuji 765-8507, Japan
  • Omar M. Abou Al-Ola
    Faculty of Science, Tanta University, El-Giesh St., Tanta, Gharbia 31527, Egypt
  • Tetsuya Yoshinaga
    Institute of Biomedical Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan

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Abstract

<jats:p>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.</jats:p>

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