Adérito Araújo

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.

Symplectic Methods Based on Decompositions

We consider methods that integrate systems of differential equations

Stability of Finite Difference Schemes for Complex Diffusion Processes

dy/dt=f(y)

Error propagation in the numerical integration of solitary waves. the regularized long wave equation

by taking advantage of a decomposition of the right-hand side

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

f=\sum f^{[\nu]}

Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

. We derive a general necessary and sufficient condition for those methods to be symplectic for Hamiltonian problems. Special attention is given to the case of additive Runge--Kutta methods.

Convergence of Finite Difference Schemes for Nonlinear Complex Reaction-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.)

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

Abstract We study the error propagation of time integrators of solitary wave solutions for the regularized long wave equation, u t +u x + 1 2 (u 2 ) x −u xxt =0 , by using a geometric interpretation of these waves as relative equilibria. We show that the error growth is linear for schemes that preserve invariant quantities of the problem and quadratic for ‘nonconservative’ methods. Numerical experiments are presented.

An alternating direction implicit method for a second-order hyperbolic diffusion equation with convection

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.

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

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and meansquare-displacement (covering both inertial and diffusive regimes) are presented.