Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression

Jumakulyyev, Ikram; Schultz, Thomas

doi:10.1007/978-3-030-56215-1_5

Cited by 4 publications

(2 citation statements)

References 29 publications

(56 reference statements)

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“…Let us also note that FED and FSI are mainly applied to problems in image processing or computer vision, see, e.g., [4,32,33]. However, the schemes can be applied to many parabolic problems, including time-dependent boundary condition, also in an engineering context as demonstrated in this work.…”

Section: Fast Semi-iterative Diffusionmentioning

confidence: 91%

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Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Bähr

Breuß

2022

Mathematics

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Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.

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Section: Fast Semi-iterative Diffusionmentioning

confidence: 91%

“…Time integration methods are typically used to design a numerical scheme. Here, we recalled the first-order EE scheme (31) and the second-order CN method (33). These baseline schemes are mainly used in our work for comparison purposes.…”

Section: Summary On Discretisationmentioning

confidence: 99%

Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Bähr

Breuß

2022

Mathematics

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Lossless PDE-based Compression of 3D Medical Images

Jumakulyyev

Schultz

2021

Lecture Notes in Computer Science

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Deep spatial and tonal data optimisation for homogeneous diffusion inpainting

Peter

Schrader

Alt

et al. 2023

Pattern Anal Applic

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Diffusion-based inpainting can reconstruct missing image areas with high quality from sparse data, provided that their location and their values are well optimised. This is particularly useful for applications such as image compression, where the original image is known. Selecting the known data constitutes a challenging optimisation problem, that has so far been only investigated with model-based approaches. So far, these methods require a choice between either high quality or high speed since qualitatively convincing algorithms rely on many time-consuming inpaintings. We propose the first neural network architecture that allows fast optimisation of pixel positions and pixel values for homogeneous diffusion inpainting. During training, we combine two optimisation networks with a neural network-based surrogate solver for diffusion inpainting. This novel concept allows us to perform backpropagation based on inpainting results that approximate the solution of the inpainting equation. Without the need for a single inpainting during test time, our deep optimisation accelerates data selection by more than four orders of magnitude compared to common model-based approaches. This provides real-time performance with high quality results.

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Fourth-Order Anisotropic Diffusion for Inpainting and Image Compression

Cited by 4 publications

References 29 publications

Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Lossless PDE-based Compression of 3D Medical Images

Deep spatial and tonal data optimisation for homogeneous diffusion inpainting

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