Internet of Things (IoT) is commonly adopted in Industry 4.0/5.0, but generates an excessive amount of data that affects quality of service (QoS). To address this challenge, Fog-based architectures have emerged to enable faster and more efficient data processing. However, their deployment can be challenging, particularly when dealing with mobile connected devices like robots. To optimize the deployment of Fog-based solution in the presence of mobile connected devices, this article proposes an Integer Linear Programming (ILP) model to find the optimal network structure of hardware devices that will ensure complete coverage of the working area, good QoS, and cost minimization. The proposed model is tested in real and benchmark cases. Results show that the model is effective and robust in optimizing Fog-based solutions deployment while considering the dynamic aspect of mobile devices. Overall, the model offers a valuable approach for managing the increased data volume generated by IoT in industrial environments.

In this work, the main goal is to study the evolution of the coastline at a local scale, and to get a better understanding of the erosion process. We propose a methodology based on a new software (PyMERI) in order to build a 3D mesh of the subsoil from a Digital Elevation Model (DEM) on which a Partial Differential Equation-direct problem-allowing to recover the voltage (electrical field) in the ground is solved yielding numerical simulations of Electrical Resistivity Imaging (ERI). This study is complemented by the resolution of an inverse problem of ERI for which, unlike the direct problem, the conductivity is now the searched quantity, and by the introduction of a segmentation approach to segment faults on images.

We study a spline-based approximation of vector fields in the conservative case. Its motivation comes from for instance when approximating current or wind velocity fields, the data deriving in those cases from a potential (pressure for the wind, etc..). To model the problem, we introduce a minimization problem on a Hilbert space for which the existence and uniqueness of the solution are provided. A convergence result in Sobolev metric spaces is established using norm equivalences and compactness arguments, as well as an approximation error estimate of the involved smoothing Dmsplines. Possible applications include estimation of wind energy potential and prediction of wind power or marine current simulation.

This paper falls within the scope of joint segmentation-registration using nonlinear elasticity principles. Because Saint Venant--Kirchhoff materials are the simplest hyperelastic materials (hyperelasticity being a suitable framework when dealing with large and nonlinear deformations), we propose viewing the shapes to be matched as such materials. Then we introduce a variational model combining a measure of dissimilarity based on weighted total variation and a regularizer based on the stored energy function of a Saint Venant--Kirchhoff material. Adding a weighted total variation--based criterion enables us to align the edges of the objects even when the modalities are different. We derive a relaxed problem associated to the initial one for which we are able to provide a result of existence of minimizers. A description and analysis of a numerical method of resolution based on a decoupling principle is then provided including a theoretical result of $\Gamma$-convergence. Applications are illustrated in academic and biological images.

The problem of constructing a smooth surface from large datasets coming from different measurements (teledetection, topography, ship track data...) appears in many instances in oceanography, geophysics and geology. One can think for instance of the problem of reconstructing seafloor surfaces from SONAR ship track bathymetry data. Another example is the construction of a Digital Elevation Model from a given set of topography isolines,. Classical algorithms used to solve this class of problems usually select points on the curves to define a Lagrange dataset, and subsequently make use of classical spline functions, bivariate splines, or spline functions in Hilbert spaces. In this work, we propose an approximation method that honors the continuous aspect of the data. We use a fidelity criterion to the data, of integral type, which is based upon a L2-norm. In this respect, the method is related to the surface approximation technique introduced in the context of partial datasets by Gout et al.

The problem of vector field approximation emerges in a wide range of fields such as: motion control, computer vision, geometrical analysis, geometrical design, analysis of acoustic and/or electromagnetic waves, as well as in geophysics, medical imaging, fluid mechanics and so on... Many different approaches have been introduced to solve each specific problem occuring in the above fields of investigation. One can mention for instance: finite element approximation, PDE-based methods, spline and Rational Basis Function approximations.. The originality of this work consists: - in considering that the vector field derives from a potential (conservative vector field): it occurs for instance in meteorology (winds derive from temperature potentials), oceanography (currents derive from pression potentials), image processing... For in land wind velocity field, we also take into account the topography effects. - of a rigorous study of existence-uniqueness of the solution of the problem phrased as an energy minimization, - in establishing a convergence result (while many approaches only give algorithms without mathematical study) and providing an approximation error estimate. - in using a specific visualization tool.

In this paper, we investigate a new method to enforce topology preservation on deformation fields. The method is composed of two steps. The first one consists in correcting the gradient vector fields of the deformation at the discrete level, in order to fulfill a set of conditions ensuring topology preservation in the continuous domain after bilinear interpolation. This part, although related to prior works by Karaçali and Davatzikos, proposes a new approach based on interval analysis. The second one aims to reconstruct the deformation, given its full set of discrete gradient vectors. The problem is phrased as a functional minimization problem on the convex subset K of the Hilbert space V. The existence and uniqueness of the solution of the problem are established, and the use of Lagrange's multipliers allows to obtain the variational formulation of the problem on the Hilbert space V . Experimental results demonstrate the efficiency of the method.

In this paper, we propose an approximation method for surfaces with fault and /or large variation. We use image segmentation tools, meshing constraints, finite element methods and spline theory.

In this paper we propose a segmentation method to locate faults and/or large variations in order to approximate a surface from Lagrange dataset.

In this article, we introduce a surface approximation technique to address the problem of fitting a surface to a given set of curves. The problem of constructing a smooth surface from a given set of curves appears in many instances in geophysics and geology. The problem of reconstructing seafloor surfaces from SONAR ship track bathymetry data, as is studied in this article.

Addresses the problem from a mathematical point of view, and shows applications to geophysical data coming from surface patches from digital elevation models (DEM) of the Pyre/spl acute/ne/spl acute/es mountains at the border between France and Spain. The new method proposed here uses scale transformations, and is applied without any particular a priori knowledge of the data. Construction of these scale transformation families is proposed first and then the efficiency of this innovative approach is shown by applying it to real data.

Image segmentation is one of the most important steps leading to the analysis of processed image data; its main goal is to divide an image into parts that have a strong correlation with objects or areas of the real world contained in the image. We present a segmentation method which uses deformable models from a very complex image (with layers, faults...) having homogeneous grey levels. We first give a method to improve the contrast of the image before proposing a method to segment the obtained image. The originality of this segmentation method is that we have interpolation data and triple (and quadruple!) points that involve making some geometric constraints on the model. Numerical results are given.