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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi> <mml:mfenced><mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>ω</mml:mi></mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>o</mml:mi> <mml:mfenced><mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow> <mml:mrow><mml:mi>ω</mml:mi></mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> in terms of the function value and the distance between consecutive iterates, respectively, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo></mml:math> 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.