Sílvia Barbeiro

Improved adaptive complex diffusion despeckling filter

Despeckling optical coherence tomograms from the human retina is a fundamental step to a better diagnosis or as a preprocessing stage for retinal layer segmentation. Both of these applications are particularly important in monitoring the progression of retinal disorders. In this study we propose a new formulation for a well-known nonlinear complex diffusion filter. A regularization factor is now made to be dependent on data, and the process itself is now an adaptive one. Experimental results making use of synthetic data show the good performance of the proposed formulation by achieving better quantitative results and increasing computation speed.

Stability of Finite Difference Schemes for Complex Diffusion Processes

In this paper we present a rigorous proof for the stability of a class of finite difference schemes applied to nonlinear complex diffusion equations. Complex diffusion is a common and broadly used denoising procedure in image processing. To illustrate the theoretical results we present some numerical examples based on an explicit scheme applied to a nonlinear equation in the context of image denoising. (A correction is attached.)

Integro-differential models for percutaneous drug absorption

In this paper we propose new mathematical models for percutaneous absorption of a drug. The new models are established by introducing, in the classical Ficks law, a memory term being the advection–diffusion equations of the classical models replaced by integro-differential equations. The well-posedness of the models is studied with Dirichlet, Neumann and natural boundary conditions. Methods for the computation of numerical solutions are proposed. Stability and convergence of the introduced methods are studied. Finally, numerical simulations illustrating the behaviour of the model are included.

Simulation of cellular changes on Optical Coherence Tomography of human retina.

We present a methodology to assess cell level alterations on the human retina responsible for functional changes observable in the Optical Coherence Tomography data in healthy ageing and in disease conditions, in the absence of structural alterations. The methodology is based in a 3D multilayer Monte Carlo computational model of the human retina. The optical properties of each layer are obtained by solving the Maxwells equations for 3D domains representative of small regions of those layers, using a Discontinuous Galerkin Finite Element Method (DG-FEM). Here we present the DG-FEM Maxwell 3D model and its validation against Mies theory for spherical scatterers. We also present an application of our methodology to the assessment of cell level alterations responsible for the OCT data in Diabetic Macular Edema. It was possible to identify which alterations are responsible for the changes observed in the OCT scans of the diseased groups.

Convergence of Finite Difference Schemes for Nonlinear Complex Reaction-Diffusion Processes

This paper is devoted to the proof of the convergence properties of a class of finite difference schemes applied to nonlinear complex reaction-diffusion equations. We investigate the accuracy of the numerical solution considering implicit and semi-implicit discretizations. To illustrate the theoretical results we present some numerical examples computed with a semi-implicit scheme applied to a nonlinear equation.

Monte Carlo simulation of diabetic macular edema changes on optical coherence tomography data

Optical coherence tomography (OCT) scans were acquired from healthy controls and patients with diabetic macular edema (DME), a common complication of diabetes characterized by increased retinal thickness due to fluid accumulation. The collected OCT data was divided into three distinct groups: healthy subjects, DME patients with significantly increased outer nuclear layer (ONL) thickness and DME patients without visible changes in the ONL. For each group, the ONL was segmented and processed, yielding a representative A-scan. Using reference values for the physical and optical characteristics of the healthy human retina, we used a Monte Carlo method with a model for the ONL to simulate an A-scan for each group and compare it to the real OCT data. This allowed to identify which alterations in the cellular characteristics are responsible for the changes observed in the OCT scans of the diseased groups.

Maxwell's equations based 3D model of light scattering in the retina

The goal of this work is to develop a computational model of the human retina and simulate light scattering through its structure aiming to shed light on data obtained by optical coherence tomography in human retinas. Currently, light propagation in scattering media is often described by Mies solution to Maxwells equations, which only describes the scattering patterns for homogeneous spheres, thus limiting its application for scatterers of more complex shapes. In this work, we propose a discontinuous Galerkin method combined with a low-storage Runge-Kutta method as an accurate and efficient way to numerically solve the time-dependent Maxwells equations. In this work, we report on the validation of the proposed methodology by comparison with Mies solution, a mandatory step before further elaborating the numerical scheme towards the propagation of electromagnetic waves through the human retina.

Laboratory for Computational Mathematics: An Interface Between Academia and Industry

“If Europe is to achieve its goal of becoming the leading knowledge-based economy in the world, mathematics has a vital role to play” (MACSI-net 2004). In spite of the fact that in many industrial sectors the value of mathematics is already proven, there is a need for positive action to promote the use of mathematics by European industry. A dynamic mathematical community interacting actively with industry and commerce, on the one hand and the science base, on the other, has been pointed out as important ingredients for competing in the global market of the future where innovation will be the key to success.

Discretely Compact Embeddings of Spaces of Cell-Centered Grid Functions

Compactness of embeddings in discrete counterparts of Sobolev spaces is considered. We study the embeddings in spaces of cell-centered grid functions, in one- and two-dimensional domains. No restrictions are made on the mesh-ratios of the underlying meshes.

A nonstandard linear finite element method for a planar elasticity problem

The aim of this work is to present a nonstandard linear finite element method for a planar elasticity problem. The error for the solution computed with this method is estimated with respect to H1×H1-norm and second-order convergence is shown.