Implicit method for nonlinear complex diffusion with applications to image denoising

In this paper we focus on the development and implementation of an implicit finite difference method for solving a complex diffusion differential equation with applications to noise filtering in images. We give a complete characterization of the implementation of the fully implicit method, including the characterization of the derivatives for Newton's method at each step. In order to show the feasibility of the method, we compare the implicit method with the semi-implicit and apply it to both in vivo images of the human retina by optical coherence tomography (OCT) and standard images (Lena and the Cameraman). We compute noise removal performance metrics to show the effect of several parameters in the performance of the method.info:eu-repo/semantics/publishedVersio

A hybrid method for inverse scattering for shape and impedance

We present a hybrid method to numerically solve the inverse scattering problem for shape and impedance, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting the advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. An optimization problem is presented as the theoretical background of the method and numerical results show its feasibility.FC

A hybrid method for inverse scattering for sound-soft obstacles in R3

We present a hybrid method to numerically solve the inverse acoustic soundsoft obstacle scattering problem in R3, given the far-field pattern for one incident direction. This method combines ideas of both iterative and decomposition methods, inheriting advantages of each of them, such as getting good reconstructions and not needing a forward solver at each step. A related Newton method is presented to show convergence of the method and numerical results show its feasibility

Inverse problems and medical imaging: lecture notes

Lecture notes for the curricular unit of Inverse Problems and Medical Imaging (23036 - https://guiadoscursos.uab.pt/en/ucs/problemas-inversos-e-imagiologia-medica/ ) of the Doctor’s Degree in Applied Mathematics and Modelling of Universidade Aberta.info:eu-repo/semantics/draf

Matemática aplicada ao estudo da visão

Neste texto pretendemos ilustrar algumas aplicações de matemática no campo da medicina, nomeadamente na imagiologia do fundo ocular. Vamos enumerar algumas técnicas, problemas actuais e em aberto nos quais a Matemática pode ser útil

On the use of quasi-equidistant source points over the sphere surface for the method of fundamental solutions

The method of fundamental solutions is broadly used in science and engineering to numerically solve the direct time-harmonic scattering problem. In 2D the choice of source points is usually made by considering an inner pseudo-boundary over which equidistant source points are placed. In 3D, however, this problem is much more challenging, since, in general, equidistant points over a closed surface do not exist. In this paper we discuss a method to obtain a quasi-equidistant point distribution over the unit sphere surface, giving rise to a Delaunay triangulation that might also be used for other boundary element methods. We give theoretical estimates for the expected distance between points and the expect area of each triangle. We illustrate the feasibility of the proposed method in terms of the comparison with the expected values for distance and area. We also provide numerical evidence that this point distribution leads to a good conditioning of the linear system associated with the direct scattering problem, being therefore an adequated choice of source points for the method of fundamental solutions.The first author acknowledges his work is partially supported by National Funding from FCT (Portugal) UID/Multi/04019/2013 and UID/MAT/04561/2019. The second author acknowledges his work is partially supported by National Funding from FCT (Portugal) UID/Multi/04621/2013 and by National Funding from FCT (Portugal) under the project PTDC/EMD-EMD/32162/2017, co-funded by FEDER through COMPETE 2020.info:eu-repo/semantics/publishedVersio

A hybrid method for sound-hard obstacle reconstruction

We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method.FC

Matemática aplicada e análise numérica: uma introdução com octave

Neste texto são as notas de base para algumas áreas da Análise Numérica, com ilustrações da sua implementação em Octave. O texto é composta por 10 capítulos: 1-Motivação; 2-Teoria do Erro; 3-Principios básicos de Análise Funcional; 4-Condicionamento; 5-Aproximação de funções; 6-Métodos Iterativos para Equações Não-lineares; 7-Sistema Lineares de Equações; 8-Integração Numérica; 9-Métodos Numéricos para Equações Diferenciais Ordinárias; 10-Métodos das Diferenças Finitas.N/

Huygens’ principle and iterative methods in inverse obstacle scattering

The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens’ principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens’ principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the traditional regularized Newton type iterations as applied to the boundary to far field map, including alternatives for the implementation of these Newton iterations.Fundação Calouste Gulbenkia

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.FCT PTDC/SAU-ENB/111139/200