An important task in image processing is the process of filling in missing parts of damaged images based on the information obtained from the surrounding areas. It is called inpainting. The goals of inpainting are numerous such as removing scratches in old photographic image, removing text and logos, restoration of damaged paintings. In this paper we present a nonlinear diffusion model for image inpainting based on a nonlinear partial differential equation as first proposed by Perona and Malik in [8]. In our previous work [3] the existence, uniqueness and regularity of the solution for the proposed mathematical model are established in an Hilbert space. The discretization of the partial differential equation of the proposed model is performed using finite elements method and finite differences method. For finite differences method our model is very simple to implement, fast, and produces nearly identical results to more complex, and usually slower, known methods. However for finite elements method we observe that it requires large computational cost, especially for high-resolution images. To avoid this slowing problem, domain decomposition algorithm has been proposed, aiming to split one large problem into many smaller problems. To illustrate the effective performance of our method, we present some experimental results on several images.

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Abstract:The generation process of medical images is inevitably accompanied by a certain noise which degrades the quality of the image and assigns the final clinical diagnosis. Therefore, the denoising step plays an important role in the treatment of medical images in order to prepare the steps of diagnosis and therapy. In this paper, we propose a nonlinear diffusion model for denoising of large size images. The numerical approach to this problem is based on an algorithm combining the methods of finite element and of domain decomposition. Numerical simulations show that the proposed algorithm is a useful alternative for the treatment of degraded images large size.

A new approach to inpainting problems that combines domain decomposition methods (DDM) with deep neural networks (DNN) to solve partial differential equations (PDE) is presented. First, this article examines different existing and emerging approaches to inpainting while emphasizing their advantages and disadvantages in a unified framework. After that, we introduce an algorithm that highlights the combination of DDM and DNN techniques for solving PDEs of a proposed mathematical inpainting model. For this model, the modified approach that has been adopted uses the DNN method which is based on convolutional neural networks (CNN) to reduce the computational cost in our algorithm while maintaining accuracy. Finally, the experimental results show that our method significantly outperforms existing ones for high-resolution images in paint stains.

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In this work we propose a hybrid model of cell population dynamics, where cells are considered as discrete elements whose dynamics depending on the intracellular and extracellular regulation. The model takes into account different cell types which include undifferentiated cells and two types of differentiated cells. We use a simulation algorithm based on the dynamical systems approach on the one hand, and the multi-agent approach on the other hand. Both approaches have been implemented using NetLogo and Python. We discuss cell choice between two types of differentiated cells and analyze the coexistence of cell lineages.

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