  Preconditioned Proximal Point (PPP) algorithms provide a unified framework
for splitting methods in image restoration. Recent advancements with RED
(Regularization by Denoising) and PnP (Plug-and-Play) priors have achieved
state-of-the-art performance in this domain, emphasizing the need for a
meaningful particular solution. However, degenerate PPP algorithms typically
exhibit weak convergence in infinite-dimensional Hilbert space, leading to
uncertain solutions. To address this issue, we propose the Halpern-type
Preconditioned Proximal Point (HPPP) algorithm, which leverages the strong
convergence properties of Halpern iteration to achieve a particular solution.
Based on the implicit regularization defined by gradient RED, we further
introduce the Gradient REgularization by Denoising via HPPP called GraRED-HP3
algorithm. The HPPP algorithm is shown to have the regularity converging to a
particular solution by a toy example. Additionally, experiments in image
deblurring and inpainting validate the effectiveness of GraRED-HP3, showing it
surpasses classical methods such as Chambolle-Pock (CP), PPP, RED, and RED-PRO.
