Surface reconstruction of anatomical structures is an integral part of medical modeling. Contour information is extracted from serial cross-sections of tissue data and is stored as "slice" files. Although there are several reasonably efficient triangulation algorithms that reconstruct surfaces from slice data, the models generated from them have a jagged or faceted appearance due to the large inter-slice distance created by the sectioning process. Moreover, inconsistencies in user input aggravate the problem. So, we created a method that reduces inter-slice distance, as well as ignores the inconsistencies in the user input. Our method called the piecewise weighted implicit functions, is based on the approach of weighting smaller implicit functions. It takes only a few slices at a time to construct the implicit function. This method is based on a technique called variational interpolation. <p> Other approaches based on variational interpolation have the disadvantage of becoming unstable when the model is quite large with more than a few thousand constraint points. Furthermore, tracing the intermediate contours becomes expensive for large models. Even though some fast fitting methods handle such instability problems, there is no apparent improvement in contour tracing time, because, the value of each data point on the contour boundary is evaluated using a single large implicit function that essentially uses all constraint points. Our method handles both these problems using a sliding window approach. As our method uses only a local domain to construct each implicit function, it achieves a considerable run-time saving over the other methods. The resulting software produces interpolated models from large data sets in a few minutes on an ordinary desktop computer.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 95-98). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 95-98). Also available on the Internet.

Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Tobin A. Driscoll, Dept. of Mathematical Sciences. Includes bibliographical references.

Thesis (M. Eng.(Mechanical Engineering))--University of Pretoria, 2004. / Includes bibliographical references.

Thesis (M.S.)--Florida State University, 2005. / Advisor: Dr. Anke Meyer Baese, Florida State University, College of Engineering, Dept. of Electrical and Computer Engineering. Title and description from dissertation home page (viewed June 7, 2005). Document formatted into pages; contains ix, 39 pages. Includes bibliographical references.

MESQUITA, Diego Parente Paiva. Machine Learning for incomplete data. 2017. 55 f. Dissertação (Mestrado em Ciência da Computação)-Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Jonatas Martins (jonatasmartins@lia.ufc.br) on 2017-08-29T14:42:43Z No. of bitstreams: 1 2017_dis_dppmesquita.pdf: 673221 bytes, checksum: eec550f75e2965d1120185327465a595 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-08-29T16:04:36Z (GMT) No. of bitstreams: 1 2017_dis_dppmesquita.pdf: 673221 bytes, checksum: eec550f75e2965d1120185327465a595 (MD5) / Made available in DSpace on 2017-08-29T16:04:36Z (GMT). No. of bitstreams: 1 2017_dis_dppmesquita.pdf: 673221 bytes, checksum: eec550f75e2965d1120185327465a595 (MD5) Previous issue date: 2017 / Methods based on basis functions (such as the sigmoid and q-Gaussian functions) and similarity measures (such as distances or kernel functions) are widely used in machine learning and related fields. These methods often take for granted that data is fully observed and are not equipped to handle incomplete data in an organic manner. This assumption is often flawed, as incomplete data is a fact in various domains such as medical diagnosis and sensor analytics. Therefore, one might find it useful to be able to estimate the value of these functions in the presence of partially observed data. We propose methodologies to estimate the Gaussian Kernel, the Euclidean Distance, the Epanechnikov kernel and arbitrary basis functions in the presence of possibly incomplete feature vectors. To obtain such estimates, the incomplete feature vectors are treated as continuous random variables and, based on that, we take the expected value of the transforms of interest. / Métodos baseados em funções de base (como as funções sigmoid e a q-Gaussian) e medidas de similaridade (como distâncias ou funções de kernel) são comuns em Aprendizado de Máquina e áreas correlatas. Comumente, no entanto, esses métodos não são equipados para utilizar dados incompletos de maneira orgânica. Isso pode ser visto como um impedimento, uma vez que dados parcialmente observados são comuns em vários domínios, como aplicações médicas e dados provenientes de sensores. Nesta dissertação, propomos metodologias para estimar o valor do kernel Gaussiano, da distância Euclidiana, do kernel Epanechnikov e de funções de base arbitrárias na presença de vetores possivelmente parcialmente observados. Para obter tais estimativas, os vetores incompletos são tratados como variáveis aleatórias contínuas e, baseado nisso, tomamos o valor esperado da transformada de interesse.