Improving the Performance of the Prony Method Using a Wavelet Domain Filter for MRI Denoising

The Prony methods are used for exponential fitting. We use a variant of the Prony method for abnormal brain tissue detection in sequences of T2 weighted magnetic resonance images. Here, MR images are considered to be affected only by Rician noise, and a new wavelet domain bilateral filtering process is implemented to reduce the noise in the images. This filter is a modification of Kazubek’s algorithm and we use synthetic images to show the ability of the new procedure to suppress noise and compare its performance with respect to the original filter, using quantitative and qualitative criteria. The tissue classification process is illustrated using a real sequence of T2 MR images, and the filter is applied to each image before using the variant of the Prony method

Lemniscatas 3D

A 3D lemniscate is a surface consisting of all points whose product of distances toa (finite) set of points or foci is constant. We introduce 3D lemniscates in the contextof geometric modelling and consider its deformation, paying attention to disconnectednessissues. We deal mainly with lemniscates of three foci.Keywords: lemniscate, implicit modelling, closed surface, algebraic surface, interactivedeformationUna lemniscata 3D es una superficie que consiste en los puntos cuyo producto dedistancias a un conjunto (finito) de puntos fijos ´o focos es constante. En este trabajo seintroducen las lemniscatas 3D en el contexto de la modelaci´on geom´etrica, se exploraen particular, el caso de tres focos, especialmente en lo relativo a la deformaci´oncontrolada.Palabras clave: lemniscata, modelaci´on impl´?cita, superficie cerrada, superficie algebraica,deformaci´on interactiva

Métodos de Bézier y B-splines

Este libro provee una base sólida para la teoría de curvas de Bézier y B-spline, revelando su elegante estructura matemática. En el texto se hace énfasis en las nociones centrales del Diseño Geométrico Asistido por Computadora con la intención de dar un tratamiento analíticamente claro y geométricamente intuitivo de los principios básicos del área. También contiene material avanzado incluyendo splines multivariados, técnicas de subdivisión y la construcción a mano alzada de superficies con cualquier grado de suavidad. El libro está excelentemente bien ilustrado con diagramas y figuras que aluden directamente al material que se desarrolla en el texto y complementan su carácter constructivo. This book provides a solid and uniform derivation of the various properties Bezier and B-spline representations have, and shows the beauty of this underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric design with the intention to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness. The text is beautifully illustrated with many excellent fiugres to emphasize the geometric constructive approach of this book. In diesem Buch werden die grundlegenden Konzepte des Geometrischen Designs (CAGD) dargestellt. Die Eigenschaften von Bézier- und B-Spline Darstellungen werden mit Hilfe von Polarformen einheitlich und stringent hergeleitet. Darüber hinaus werden Konstruktionen von Freiformflächen beliebiger Glattheitsordnung, Unterteilungsalgorithmen, Boxsplines, Simplexsplines und multivariate Splines behandelt. Der Text ist mit vielen hervorragenden Abbildungen illustriert, die den geometrisch konstruktiven Zugang des Buches deutlich hervorheben

An intuitive way for constructing parametric quadric triangles

International audienceWe present an intuitive algorithm for providing quadric surface design elements with shape parameters. To this end, we construct rational parametric triangular quadratic patches which lie on quadrics. The input of the algorithm is three vertex data points in 3D and normals at these points. It emanates from a thorough analysis of two existing methods for the construction of rational parametric Bézier triangles on quadrics, that allows to establish an interesting geometric relation between them. The sufficient condition for a configuration of vertex and normal data to allow for the existence of a rational triangular quadratic patch lying on a quadric whose tangent planes at the vertices are those prescribed by the given normals is the concurrence of certain cevians. When these conditions are not met we offer an optimization procedure to tweak the normals, without varying the vertex data, so that for the new normals there is a rational triangular quadratic patch that lies on a quadric. The resulting quadric design element offers three free shape parameters