Convergence of block cyclic projection and Cimmino algorithms for compressed sensing based tomography

Area of Science:

• Applied Mathematics
• Image Reconstruction
• Optimization Theory

Background:

• Convex feasibility and optimization problems are fundamental in applied mathematics.
• Existing projection methods have limitations in convergence speed and accuracy, especially under perturbations.
• Compressed sensing and total variation minimization offer powerful tools for improving image reconstruction in areas like computerized tomography (CT).

Purpose of the Study:

• To propose novel projection-based algorithms for convex feasibility and optimization problems within a compressed sensing framework.
• To theoretically derive the convergence properties of these new algorithms.
• To demonstrate the practical applicability and convergence behavior of the proposed methods through an illustrative example.

Main Methods:

• Development of a varying block cyclic projection method.
• Introduction of a block Cimmino's algorithm.
• Application of the convergence theorem for amalgamated projection methods to analyze the new algorithms.
• Utilizing compressed sensing principles and total variation minimization.

Main Results:

• The proposed varying block cyclic projection method and block Cimmino's algorithm exhibit convergence.
• Theoretical convergence is derived based on the amalgamated projection method's convergence theorem.
• The new algorithms are shown to be effective within the compressed sensing framework.

Conclusions:

• The novel block projection algorithms offer improved convergence and accuracy for convex feasibility and optimization problems.
• These methods provide a valuable extension to existing projection techniques, particularly for compressed sensing applications.
• The theoretical framework and practical examples support the efficacy of the proposed varying block cyclic projection and block Cimmino's algorithms.