Discrete curvature approximations and segmentation of polyhedral surfaces

The segmentation of digitized data to divide a free form surface into patches is one of the key steps required to perform a reverse engineering process of an object. To this end, discrete curvature approximations are introduced as the basis of a segmentation process that lead to a decomposition of digitized data into areas that will help the construction of parametric surface patches. The approach proposed relies on the use of a polyhedral representation of the object built from the digitized data input. Then, it is shown how noise reduction, edge swapping techniques and adapted remeshing schemes can participate to different preparation phases to provide a geometry that highlights useful characteristics for the segmentation process. The segmentation process is performed with various approximations of discrete curvatures evaluated on the polyhedron produced during the preparation phases. The segmentation process proposed involves two phases: the identification of characteristic polygonal lines and the identification of polyhedral areas useful for a patch construction process. Discrete curvature criteria are adapted to each phase and the concept of invariant evaluation of curvatures is introduced to generate criteria that are constant over equivalent meshes. A description of the segmentation procedure is provided together with examples of results for free form object surfaces

Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method

This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain. The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities. The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy. In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost. To reach this goal, we first need to answer the question: ''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?'' We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications. This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh. Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation. Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements. The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios

Texture-Based Segmentation and Finite Element Mesh Generation for Heterogeneous Biological Image Data

The design, analysis, and control of bio-systems remain an engineering challenge. This is mainly due to the material heterogeneity, boundary irregularity, and nonlinear dynamics associated with these systems. The recent developments in imaging techniques and stochastic upscaling methods provides a window of opportunity to more accurately assess these bio-systems than ever before. However, the use of image data directly in upscaled stochastic framework can only be realized by the development of certain intermediate steps. The goal of the research presented in this dissertation is to develop a texture-segmentation method and a unstructured mesh generation for heterogeneous image data. The following two new techniques are described and evaluated in this dissertation: 1. A new texture-based segmentation method, using the stochastic continuum concepts and wavelet multi-resolution analysis, is developed for characterization of heterogeneous materials in image data. The feature descriptors are developed to efficiently capture the micro-scale heterogeneity of macro-scale entities. The materials are then segmented at a representative elementary scale at which the statistics of the feature descriptor stabilize. 2. A new unstructured mesh generation technique for image data is developed using a hierarchical data structure. This representation allows for generating quality guaranteed finite element meshes. The framework for both the methods presented in this dissertation, as such, allows them for extending to higher dimensions. The experimental results using these methods conclude them to be promising tools for unifying data processing concepts within the upscaled stochastic framework across biological systems. These are targeted for inclusion in decision support systems where biological image data, simulation techniques and artificial intelligence will be used conjunctively and uniformly to assess bio-system quality and design effective and appropriate treatments that restore system health