Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation

In order to reduce the data size and simplify the process of creating characters’ 3D models, a new and interactive ordinary differential equation (ODE)-based C2 continuous surface creation algorithm is introduced in this paper. With this approach, the creation of a three-dimensional surface is transformed into generating two boundary curves plus four control curves and solving a vector-valued sixth order ordinary differential equation subjected to boundary constraints consisting of boundary curves, and first and second partial derivatives at the boundary curves. Unlike the existing patch modeling approaches which require tedious and time-consuming manual operations to stitch two separate patches together to achieve continuity between two stitched patches, the proposed technique maintains the C2 continuity between adjacent surface patches naturally, which avoids manual stitching operations. Besides, compared with polygon surface modeling, our ODE C2 surface creation method can significantly reduce and compress the data size, deform the surface easily by simply changing the first and second partial derivatives, and shape control parameters instead of manipulating loads of polygon points

Modelado geométrico personalizado de la córnea humana y su aplicación a la detección de ectasias corneales

[SPA] La córnea es una estructura biológica viva cuya arquitectura presenta una morfología singular, ya sea en un estado natural o patológico. Esta singularidad ha sido caracterizada a lo largo de toda la historia en el campo de la oftalmología y la óptica a través de la generación de modelos genéricos o de modelos personalizados de la córnea humana. Hoy en día, el desarrollo de nuevas tecnologías permite caracterizar la morfología corneal a partir de los denominados equipos topográficos; estos equipos aportan una caracterización personalizada de índole cualitativa y cuantitativa al médico oftalmólogo. Sin embargo, los sistemas de diagnóstico de las patologías corneales están basados en unos índices de valoración de la irregularidad de las superficies corneales que son calculados a partir de algoritmos específicos internos para cada topógrafo corneal y de los cuales se desconoce su programación. Por este motivo en esta tesis doctoral se establece un nuevo procedimiento fundamentado en la geometría computacional para obtener un modelo sólido 3D personalizado in vivo de la córnea humana utilizando herramientas de Diseño Geométrico Asistido por Ordenador. Este modelo virtual reconstruye fidedignamente las superficies de la cara anterior y posterior de la córnea, a partir de unos datos aportados por los topógrafos corneales denominados datos en bruto (sin ningún trato mediante algoritmo) tanto para los ojos de pacientes sanos como para los ojos de pacientes diagnosticados con la patología ectásica más común, el queratocono. A partir del nuevo modelo sólido obtenido, se definen unos índices de caracterización de la morfología corneal basados en variables geométricas, los cuales pueden ser utilizados como unos nuevos índices de diagnóstico de la patología ectásica objeto de estudio debido a que presentan una elevada sensibilidad y especificidad para su diagnóstico. [ENG] The cornea is a living biological structure whose architecture has a unique morphology, either in a natural or diseased condition. This uniqueness has been characterized throughout all history in the field of ophthalmology and optics through the generation of generic or customized models of human cornea. Today, the development of new technologies leads to characterize the corneal morphology from the so‐called topographic devices; these devices provide a personalized qualitative and quantitative characterization of its nature for the ophthalmologist. However corneal pathological diagnosis systems are based on indicators of the irregularity of the corneal surfaces, which are calculated from specific internal algorithms for each corneal topographer and whose programming is unknown. For that reason, this doctoral thesis establishes a new procedure based on computational geometry to obtain a 3D solid model, personalized and in vivo of the human cornea by using Computer Aided Geometrical Design tools. This virtual model represents accurately both the anterior and posterior corneal surfaces from a set of raw data (without any algorithm treatment) provided by the corneal topographers for both healthy corneas and corneas with the most common ectasic disease, the keratoconus. The new solid model obtained is later analyzed to define a set of indices that enable the characterization of the corneal morphology and that are based on geometric variables. These indices can be used as new indicators for the diagnosis of the keratoconus disease due to their high sensibility and specificity.[ENG] The cornea is a living biological structure whose architecture has a unique morphology, either in a natural or diseased condition. This uniqueness has been characterized throughout all history in the field of ophthalmology and optics through the generation of generic or customized models of human cornea. Today, the development of new technologies leads to characterize the corneal morphology from the so‐called topographic devices; these devices provide a personalized qualitative and quantitative characterization of its nature for the ophthalmologist. However corneal pathological diagnosis systems are based on indicators of the irregularity of the corneal surfaces, which are calculated from specific internal algorithms for each corneal topographer and whose programming is unknown. For that reason, this doctoral thesis establishes a new procedure based on computational geometry to obtain a 3D solid model, personalized and in vivo of the human cornea by using Computer Aided Geometrical Design tools. This virtual model represents accurately both the anterior and posterior corneal surfaces from a set of raw data (without any algorithm treatment) provided by the corneal topographers for both healthy corneas and corneas with the most common ectasic disease, the keratoconus. The new solid model obtained is later analyzed to define a set of indices that enable the characterization of the corneal morphology and that are based on geometric variables. These indices can be used as new indicators for the diagnosis of the keratoconus disease due to their high sensibility and specificity.Esta tesis se ha realizado en parte gracias a la financiación del proyecto del Fondo Europeo de Desarrollo Regional (FEDER) y del Ministerio Español de Economía y Competitividad, Instituto Carlos III, Red Temática de Investigación Cooperativa en Salud (RETICS) «Prevención, detección precoz y tratamiento de la patología ocular prevalente, degenerativa y crónica». Subprograma «dioptrio ocular y patologías frecuentes» (RD12/0034/0007).Escuela Internacional de DoctoradoUniversidad Politécnica de CartagenaPrograma Oficial de Doctorado en Tecnologías Industriale

New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context