Cerebral cortex, classified as gray matter, is the superficial layer of the cerebrum. In recent years, many studies have shown the abnormality of cortical thickness is possibly correlated to the disease or disorder in central nervous system, such as Alzheimer¡¦s disease and lissencephaly. Therefore, this purpose of this work is to implement the measurement of the cortical thickness. In general, two approaches, surface-based and voxel-based methods, have been proposed to measure the cortical thickness. In this thesis, a procedure of the voxel-based method using Laplace¡¦s equation was developed on the basis of a 2008 publication reported by Chloe Hutton et al to obtain voxel-based cortical thickness (VBCT) map. The result of our home-made program was further compared with those calculated by Hutton¡¦s program, whic h was generously provided by the author. The difference between two implementations was consisted of four main parts. First of all, different strategies of the tissue classification were used to define boundary condition of Laplace¡¦s equation. When grey matter, white matter, and cerebrospinal fluid were classified by maximizing the tissue probability, Hutton¡¦s program tends to search more voxels of cerebrospinal fluid in sulci by skeletonizing the non-parenchyma area. Second, the algorithm of layer growing also differs. The single layer obtained by the 26-neighborhood algorithm in our program would be obviously thicker than that provided by Hutton¡¦s program using 6-neighborhood. Third, compared with a fixed step size (usually 0.5 mm) porposed in the main reference to track cortical streamline, we designed a variable step size, reducing the underestimation of cortical thickness. The last but not the least, the connecting points of the cortical streamline usually are not grid points, thus requiring interpolation to estimate the stepping gradient. We adapted the linear interpolation for better accuracy when Hutton et al searched for the closest grid point for replacement to achieve faster computation.

[pt] Invisibilidade sempre mexeu com a imaginação de crianças e adultos. Quem nunca imaginou ser capaz de tornar-se invisível em certas ocasiões? Recentemente essa ideia da ficção científica tomou forma na vida real, e um dos objetivos do presente texto é explicar de uma maneira acessível as principais ideias físicas e matemáticas por trás do conceito de invisibilidade. Pedimos do leitor somente uma modesta familiaridade com Cálculo Vetorial, Séries de Fourier e Álgebra Linear. O objetivo da capa de invisibilidade é tornar um objeto não detectável por meio de energia eletromagnética. A capa é fisicamente realizada por um metamaterial especialmente projetado para redirecionar certas ondas eletromagnéticas irradiadas sobre o objeto. Nesta exposição, usaremos como exemplo a tomografia de impedância elétrica (TIE) como método de detecção e explicaremos como criar uma capa invisível à TIE. Cabe ressaltar que o processamento de imagem através da TIE diz respeito a um problema inverso e, no contexto das equações diferenciais, esse problema envolve, a partir de determinadas simplificações, a equação de Laplace com condições de contorno. Despretensiosamente, optamos pelo caso bidimensional para facilitar a exposição das idéias principais, embora todos os nossos resultados possam ser generalizados em 3 dimensões. / [en] Invisibility has always instigated children and adult s imagination. Who never thought of occasionally being able to turn yourself invisible? Recently, this science fiction idea has taken shape in real life, and one of the objectives of this text is explain the main physical and mathematical ideas behind the invisibility concept, on a comprehensible way. We only require the reader has a modest familiarity with Vectorial Calculus, Fourier Series and Linear Algebra. Invisibility cloak aims to turn an object imperceptible to electromagnetic energy detection. The cloak is made of an especially projected metamaterial that redirects certain electromagnetic waves irradiated over the object. Here we will take as an example electrical impedance tomography (EIT) as a detection method and we will explain how to create an invisible cloak for EIT. It is worth mentioning that image processing through EIT is an inverse problem. Thereby, in the context of differential equations, this problem involves a few simplifications in the Laplace s problem with contours conditions. Unpretentiously, we chose the two-dimensional case to simplify the exposition of the main ideas, although all of our results may be generalized in three-dimensional case.