Deflated preconditioned Conjugate Gradient methods for noise filtering of low-field MR images

Xiujie Shan*, Martin B. van Gijzen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)
30 Downloads (Pure)


We study efficient implicit methods to denoise low-field MR images using a nonlinear diffusion operator as a regularizer. This problem can be formulated as solving a nonlinear reaction–diffusion equation. After discretization, a lagged-diffusion approach is used which requires a linear system solve in every nonlinear iteration. The choice of diffusion model determines the denoising properties, but it also influences the conditioning of the linear systems. As a solution method, we use Conjugate Gradient (CG) in combination with a suitable preconditioner and deflation technique. We consider four different preconditioners in combination with subdomain deflation. We evaluate the methods for four commonly used denoising operators: standard Laplace operator, two Perona–Malik type operators, and the Total Variation (TV) operator. We show that a Discrete Cosine Transform (DCT) preconditioner works best for problems with a slowly varying diffusion coefficient, while Jacobi preconditioning with subdomain deflation works best for a strongly varying diffusion, as happens for the TV operator. This research is part of a larger effort that aims to provide low-cost MR imaging capabilities for low-resource settings. We have evaluated the algorithms on low-field MRI images using inexpensive commodity hardware. With a suitable preconditioner for the chosen diffusion model, we are able to limit the time to denoise three-dimensional images of more than 2 million pixels to less than 15 s, which is fast enough to be used in practice.

Original languageEnglish
Article number113730
Number of pages17
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 2022

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.


  • DPCG
  • Image denoising
  • Low-field MRI
  • PDE


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