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Preconditioned block Kaczmarz methods for linear equations with an application to computed tomography.

Duo Liu1, Wenli Wang1, Gangrong Qu1

  • 1School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, China.

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This study introduces convergence conditions for preconditioned Kaczmarz methods in computed tomography (CT) image reconstruction. A novel block strategy and preconditioners accelerate convergence, outperforming traditional methods.

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Area of Science:

  • Medical imaging
  • Computational mathematics
  • Image reconstruction algorithms

Background:

  • Preconditioned Kaczmarz methods are crucial for image reconstruction.
  • Establishing convergence conditions and efficient block strategies is essential for practical application.
  • Computed tomography (CT) image reconstruction benefits from accelerated methods.

Purpose of the Study:

  • To establish convergence conditions for preconditioned Kaczmarz methods.
  • To design an efficient block strategy and preconditioners for CT image reconstruction.
  • To accelerate the convergence of Kaczmarz methods in CT.

Main Methods:

  • Establishing convergence conditions for preconditioned block Kaczmarz methods.
  • Proving the dependence of convergence on the initial guess.
  • Proposing a new method with a novel block strategy and specific preconditioners for CT.

Main Results:

  • Demonstrated effective acceleration of CT image reconstruction using the proposed block strategy and preconditioners.
  • Maintained satisfactory image quality in numerical experiments with phantoms and real CT data.
  • Showcased superior performance compared to Landweber and non-preconditioned block Kaczmarz iterations.

Conclusions:

  • The proposed method, integrating a designed block strategy and specific preconditioners, significantly accelerates CT image reconstruction.
  • The new approach offers improved performance over traditional iterative methods.
  • This work contributes to more efficient and effective CT image reconstruction techniques.