Learning Stable Nonlinear Cross-Diffusion Models for Image Restoration

Barbeiro, Sílvia; Lobo, Diogo

doi:10.1007/s10851-019-00931-x

Cited by 6 publications

(6 citation statements)

References 16 publications

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“…Theorem 1 states the stability conditions for this finite difference scheme, which is an extension of a previous result presented in [BL20]. Define λ max as the supreme of both λ 1 and λ 2 .…”

Section: Explicit and Semi-implicit Implementations And Stability Res...mentioning

confidence: 55%

“…Their use is widespread in areas like population dynamics (see [ABRB11] and references therein), but has recently attracted some interest in the image processing community [ABCD17a, ABCD17b, ABCD17c, BL20] as a natural extension to the complex diffusion methods proposed by Gilboa et al [GSZ04], where the image is represented by a complex function and the filtering process is governed by a nonlinear PDE of diffusion type with a complex-valued diffusion coefficient. This equation can be written as a cross-diffusion system for the 1) real and 2) imaginary parts of the image, and enhanced imaging possibilities emerge if we drop the complex point of view and work with 1) the image to be processed and 2) information about that image (e.g., edge locations) [ABCD17c,BL20].…”

Section: Introductionmentioning

confidence: 99%

“…It is known that implicit schemes have better stability properties when compared to their explicit counterparts [BSJ14], allowing larger time steps and, in the particular case of cross-diffusion, a wider range of diffusion coefficients [ABCD17c,BL20]. However, the computational complexity of traditional implicit finite difference schemes turns their use impracticable for tasks that aim for on-the-fly results, such as image processing in medical contexts.…”

Section: Introductionmentioning

confidence: 99%

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Fast stable finite difference schemes for nonlinear cross-diffusion

Lobo¹

2021

Preprint

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The dynamics of cross-diffusion models leads to a high computational complexity for implicit difference schemes, turning them unsuitable for tasks that require results in real-time. We propose the use of two operator splitting schemes for nonlinear cross-diffusion processes in order to lower the computational load, and establish their stability properties using discrete L 2 energy methods. Furthermore, by attaining a stable factorization of the system matrix as a forward-backward pass, corresponding to the Thomas algorithm for self-diffusion processes, we show that the use of implicit cross-diffusion can be competitive in terms of execution time, widening the range of viable cross-diffusion coefficients for on-the-fly applications.

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Section: Explicit and Semi-implicit Implementations And Stability Res...mentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast stable finite difference schemes for nonlinear cross-diffusion

Lobo¹

2021

Preprint

Self Cite

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“…Chen and Pock [18] train flux functions and derivative filters of a diffusion-inspired model to obtain exceptional denoising results. Other authors only train nonlinearities of models [19,20,21] or learn PDEs directly with symbolic approaches [22,23,24]. Instead of focusing on denoising performance, we want arXiv:2010.10888v1 [eess.IV] 21 Oct 2020 to obtain insights into the scale behaviour of our model.…”

Section: Related Workmentioning

confidence: 99%

Learning Integrodifferential Models for Image Denoising

Alt

Weickert

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We introduce an integrodifferential extension of the edge-enhancing anisotropic diffusion model for image denoising. By accumulating weighted structural information on multiple scales, our model is the first to create anisotropy through multiscale integration. It follows the philosophy of combining the advantages of model-based and data-driven approaches within compact, insightful, and mathematically well-founded models with improved performance. We explore trained results of scale-adaptive weighting and contrast parameters to obtain an explicit modelling by smooth functions. This leads to a transparent model with only three parameters, without significantly decreasing its denoising performance. Experiments demonstrate that it outperforms its diffusion-based predecessors. We show that both multiscale information and anisotropy are crucial for its success.

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“…Barbeiro and Lobo studied the optimized partial differential equation (PDE) model for image filtering. The gray image is represented by the vector field of two real-valued functions, and the image restoration problem is modeled through the evolution process, so that the restored image meets the cross initial boundary value problem at any time [6]. The purpose of Rajalakshmi and Prince's research is to develop a mathematical model of the retinal layer with a complex neural structure that can detect incoming signals and convert the signals into an equivalent peak sequence.…”

Section: Introductionmentioning

confidence: 99%

Design of Knitted Garment Design Model Based on Mathematical Image Theory

2022

Journal of Sensors

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Iconology is an approach used by Western art theorists and art history researchers to explain plastic art and even relative to other various plastic art activities. This article is aimed at researching and discussing the design of knitted garment design model based on mathematical image theory. This article first discusses the application of digital image processing in the field of knitted clothing; digital image processing technology has a wide range of applications in the field of knitted clothing, including four aspects of fiber, yarn, fabric, and clothing, and proposes digital image filtering technology and digital image fuzzy filtering algorithm and then researches and analyzes the drape performance of knitted fabrics. Fabric drape is an important performance that reflects the beauty of the appearance of the fabric. The design of knitted garments is discussed based on mathematical imagery, and finally, the pressure performance of knitted garments is tested and analyzed. The experimental results in this paper show that the data correlation of the clothing pressure measured in different situations of the upper body is 0.87328, 0.779832, and 0.780213, respectively, under the plain weave, rib, and mesh fabric clothing fabrics. It shows that under the pressure of this degree of knitted clothing, it is in line with the needs of human healthy growth, because a certain degree of clothing pressure can maintain human physical and mental health and improve the quality of life.

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Learning Stable Nonlinear Cross-Diffusion Models for Image Restoration

Cited by 6 publications

References 16 publications

Fast stable finite difference schemes for nonlinear cross-diffusion

Fast stable finite difference schemes for nonlinear cross-diffusion

Learning Integrodifferential Models for Image Denoising

Design of Knitted Garment Design Model Based on Mathematical Image Theory

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