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Notes on the primal-dual algorithm for convex optimization applied to X-ray tomographic image reconstruction

Emil Y. Sidky, Xiaochuan Pan

Abstract

The purpose of these notes is to provide background on understanding the primal-dual algorithm of Chambolle and Pock [1] for imaging scientists. The presentation focuses on providing intuition and an algorithmic system that is amenable to pre-conditioning. The document aims to be self-contained, providing background on the essential facts of non-smooth convex analysis.[2]

Notes on the primal-dual algorithm for convex optimization applied to X-ray tomographic image reconstruction

Abstract

The purpose of these notes is to provide background on understanding the primal-dual algorithm of Chambolle and Pock [1] for imaging scientists. The presentation focuses on providing intuition and an algorithmic system that is amenable to pre-conditioning. The document aims to be self-contained, providing background on the essential facts of non-smooth convex analysis.[2]
Paper Structure (35 sections, 210 equations, 24 figures, 4 algorithms)

This paper contains 35 sections, 210 equations, 24 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Methods
  3. CPPD background
  4. Heuristics for the CPPD algorithm and preconditioning
  5. Forward Euler iteration
  6. Backward Euler iteration
  7. Approximate backward Euler iteration
  8. Generalization to matrix-mapping steps
  9. Preconditioning based on inverse of $(A^\top A)$
  10. Heurstic derivation of CPPD
  11. Results
  12. Optimization problems and solvers
  13. simulation parameters
  14. Least-squares convergence studies for full-sampling
  15. Total-variation constrained least-squares convergence studies for full-sampling
  16. ...and 20 more sections

Figures (24)

  • Figure 1: Computerized breast phantom and its gradient magnitude image (GMI). In the GMI it is apparent that the phantom has a high degree of gradient sparsity even though the tissue borders are highly irregular. The gradient sparsity is useful for testing ideal recovery from under-sampled data by use of sparsity regularization with TV. The attenuation values in this phantom or taken to be 0.194 cm$^{-1}$ and 0.233 cm$^{-1}$ for fat and fibro-glandular tissue, respectively. The phantom and its GMI are shown in a gray scale window of [0.174, 0.253] cm$^{-1}$ and [0.0, 0.05] cm$^{-1}$, respectively.
  • Figure 2: Plots of $\|r_\sigma\|_2$ and $\|r_\tau\|_2$, see Eqs. (\ref{['cc1']}) and (\ref{['cc2']}) in \ref{['app:convergence']}, as a function of iteration number and for different values of the step-size ratio parameter $\rho$.
  • Figure 3: Plots of the LSQ objective function gradient magnitude for CPPD-LSQ, GD-LSQ, and CGLS as a function of iteration number.
  • Figure 4: Plots of the LSQ objective function for CPPD-LSQ, GD-LSQ, and CGLS as a function of iteration number.
  • Figure 5: Plots of the image root-mean-square-error (RMSE) discrepancy for CPPD-LSQ, GD-LSQ, and CGLS as a function of iteration number.
  • ...and 19 more figures