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An accelerated preconditioned proximal gradient algorithm with a generalized Nesterov momentum for PET image reconstruction

Yizun Lin, Yongxin He, C. Ross Schmidtlein, Deren Han

TL;DR

Numerical results presented in this work indicate that as ω∈(0,1] increase, APPGA converges at a progressively faster rate, and exhibits superior performance compared to the PPGA and the preconditioned Krasnoselskii–Mann algorithm.

Abstract

This paper presents an Accelerated Preconditioned Proximal Gradient Algorithm (APPGA) for effectively solving a class of Positron Emission Tomography (PET) image reconstruction models with differentiable regularizers. We establish the convergence of APPGA with the Generalized Nesterov (GN) momentum scheme, demonstrating its ability to converge to a minimizer of the objective function with rates of $o(1/k^{2ω})$ and $o(1/k^ω)$ in terms of the function value and the distance between consecutive iterates, respectively, where $ω\in(0,1]$ is the power parameter of the GN momentum. To achieve an efficient algorithm with high-order convergence rate for the higher-order isotropic total variation (ITV) regularized PET image reconstruction model, we replace the ITV term by its smoothed version and subsequently apply APPGA to solve the smoothed model. Numerical results presented in this work indicate that as $ω\in(0,1]$ increase, APPGA converges at a progressively faster rate. Furthermore, APPGA exhibits superior performance compared to the preconditioned proximal gradient algorithm and the preconditioned Krasnoselskii-Mann algorithm. The extension of the GN momentum technique for solving a more complex optimization model with multiple nondifferentiable terms is also discussed.

An accelerated preconditioned proximal gradient algorithm with a generalized Nesterov momentum for PET image reconstruction

TL;DR

Numerical results presented in this work indicate that as ω∈(0,1] increase, APPGA converges at a progressively faster rate, and exhibits superior performance compared to the PPGA and the preconditioned Krasnoselskii–Mann algorithm.

Abstract

This paper presents an Accelerated Preconditioned Proximal Gradient Algorithm (APPGA) for effectively solving a class of Positron Emission Tomography (PET) image reconstruction models with differentiable regularizers. We establish the convergence of APPGA with the Generalized Nesterov (GN) momentum scheme, demonstrating its ability to converge to a minimizer of the objective function with rates of and in terms of the function value and the distance between consecutive iterates, respectively, where is the power parameter of the GN momentum. To achieve an efficient algorithm with high-order convergence rate for the higher-order isotropic total variation (ITV) regularized PET image reconstruction model, we replace the ITV term by its smoothed version and subsequently apply APPGA to solve the smoothed model. Numerical results presented in this work indicate that as increase, APPGA converges at a progressively faster rate. Furthermore, APPGA exhibits superior performance compared to the preconditioned proximal gradient algorithm and the preconditioned Krasnoselskii-Mann algorithm. The extension of the GN momentum technique for solving a more complex optimization model with multiple nondifferentiable terms is also discussed.
Paper Structure (10 sections, 10 theorems, 82 equations, 9 figures, 1 table)

This paper contains 10 sections, 10 theorems, 82 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. PET image reconstruction
  3. SHOITV regularized PET image reconstruction model
  4. Accelerated preconditioned proximal gradient algorithm
  5. Convergence analysis of APPGA
  6. Numerical results
  7. Simulation setup for APPGA
  8. Simulation results for APPGA
  9. Extension of the generalized Nesterov momentum
  10. Concluding remarks

Key Result

Proposition 1

Let operator $\mathcal{T}$ be defined by defmT, where ${\bm P}\in{\mathbb D}_{++}^{d}$. Then ${\bm f}^*$ is a solution of model model:SHOTV2 if and only if it is a fixed point of $\mathcal{T}$.

Figures (9)

  • Figure 1: (a) The original uniform phantom: uniform background with six uniform hot spheres of distinct radii; (b) the image reconstructed by using ATV regularization; (c) the image reconstructed by using ITV regularizatio.
  • Figure 2: (a) The original brain phantom: high quality clinical PET brain image; (b) the unregularized reconstructed image; (c) the image reconstructed by using first-order TV regularization; (d) the image reconstructed by using higher-order TV regularization.
  • Figure 3: (a) Brain phantom: high quality clinical PET brain image; (b) uniform phantom: uniform background with six uniform hot spheres of distinct radii.
  • Figure 4: NOFV (left) and PSNR (right) versus iteration number by PKMA, PPGA and APPGA ($\omega=1/4,1/2,3/4,1$).
  • Figure 5: Reconstructed brain images of PKMA, PPGA and APPGA($\omega$ =1): top to bottom rows are reconstructed by 25, 50 and 100 iterations, respectively.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 3
  • proof
  • Lemma 4: Descent lemma
  • Lemma 5
  • proof
  • proof : Proof of Proposition \ref{['prop1forconverg']}
  • Proposition 6
  • ...and 7 more