ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H -1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation.
Original languageEnglish (US)
Pages (from-to)931-957
Number of pages27
JournalDiscrete and Continuous Dynamical Systems
Issue number3
StatePublished - Aug 20 2013
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Carola-Bibiane Schonlieb acknowledges the financial support provided by the Cambridge Centre for Analysis (CCA), the Royal Society International Exchanges Award IE110314 for the project High-order Compressed Sensing for Medical Imaging, the EPSRC first grant Nr. EP/J009539/1 Sparse & Higher-order Image Restoration, and the EPSRC / Isaac Newton Trust Small Grant on Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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