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Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions

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Abstract

Image inpainting involves filling in damaged or missing regions of an image by utilizing information from the surrounding areas. In this paper, we investigate a fully nonlinear partial differential equation inspired by the modified Cahn–Hilliard equation. Instead of using standard potentials that depend solely on pixel intensities, we consider morphological image enhancement filters that are based on a variant of the shock filter:

$$\begin{aligned} \partial _t u&= \Delta \left( -\nu \arctan (\Delta u)|\nabla u| - \mu \Delta u \right) + \lambda (u_0 - u). \end{aligned}$$

This is referred to as the Shock Filter Cahn–Hilliard Equation. The equation is nonlinear with respect to the highest-order derivative, which poses significant mathematical challenges. To address these, we make use of a specific approximation argument, establishing the existence of a family of approximate solutions through the Leray–Schauder fixed point theorem and the Aubin–Lions lemma. In the limit, we obtain a solution strategy wherein we can prove the existence and uniqueness of solutions. Proving the latter involves the Kruzhkov entropy type-admissibility conditions. Additionally, we use a numerical method based on the convexity splitting idea to approximate solutions of the nonlinear partial differential equation and achieve fast inpainting results. To demonstrate the effectiveness of our approach, we apply our method to standard binary images and compare it with variations of the Cahn–Hilliard equation commonly used in the field.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the quality of this work. This work was supported in part by the Croatian Science Foundation under project number HRZZ-MOBODL-2023-08-7617 and IP-2022-10-7261 Analysis of Partial Differential Equations and Shape Optimization (ADESO), the Austrian Science Foundation (FWF) Stand Alone Project number P-35508-N, and the Croatian-Austrian bilateral projects From PDEs to Deep Learning: Advancing Medical Image Processing and Mathematical Aspects of Granular Hydro-dynamics - Modelling, Analysis, and Numerics.

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Authors and Affiliations

  1. Department of Mathematics, University of Vienna, Oskar Morgenstern Platz-1, Vienna, 1090, Austria

    Darko Mitrovic & Andrej Novak

  2. Department of Physics, Faculty of Science, University of Zagreb, Bijenicka cesta 32, Zagreb, 10000, Croatia

    Andrej Novak

  3. Department of Mathematics and Natural Sciences, University of Montenegro, Cetinjski put bb, Podgorica, 81000, Montenegro

    Darko Mitrovic

  4. Dubrava University Hospital, Avenija Gojka Šuška 6, Zagreb, 10000, Croatia

    Andrej Novak

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Mitrovic, D., Novak, A. Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions. Arch Rational Mech Anal 248, 105 (2024). https://doi.org/10.1007/s00205-024-02057-w

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