A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Geometric Surface Processing and Virtual Modeling

In this work we focus on two main topics "Geometric Surface Processing" and "Virtual Modeling". The inspiration and coordination for most of the research work contained in the thesis has been driven by the project New Interactive and Innovative Technologies for CAD (NIIT4CAD), funded by the European Eurostars Programme. NIIT4CAD has the ambitious aim of overcoming the limitations of the traditional approach to surface modeling of current 3D CAD systems by introducing new methodologies and technologies based on subdivision surfaces in a new virtual modeling framework. These innovations will allow designers and engineers to transform quickly and intuitively an idea of shape in a high-quality geometrical model suited for engineering and manufacturing purposes. One of the objective of the thesis is indeed the reconstruction and modeling of surfaces, representing arbitrary topology objects, starting from 3D irregular curve networks acquired through an ad-hoc smart-pen device. The thesis is organized in two main parts: "Geometric Surface Processing" and "Virtual Modeling". During the development of the geometric pipeline in our Virtual Modeling system, we faced many challenges that captured our interest and opened new areas of research and experimentation. In the first part, we present these theories and some applications to Geometric Surface Processing. This allowed us to better formalize and give a broader understanding on some of the techniques used in our latest advancements on virtual modeling and surface reconstruction. The research on both topics led to important results that have been published and presented in articles and conferences of international relevance

08221 Abstracts Collection -- Geometric Modeling

From May 26 to May 30 2008 the Dagstuhl Seminar 08221 ``Geometric Modeling\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Accurate correction of surface noises of polygonal meshes

In this paper we propose a new algorithm for accurate correction of surface noises of polygonal meshes. It consists of three basic components: (a) feature-preserving pre-smoothing; (b) partitioning of feature and non-feature regions; (c) second-order predictor for non-feature regions and median filter for feature regions. The unique contributions of our approach include (a) an idea of partitioning an input surface into feature and non-feature regions so that different smoothing algorithms, which are best suited for either feature or non-feature regions can be, respectively, applied; (b) a second-order predictor that provides higher smoothing accuracy and better convergence on smoothly curved surfaces. In comparison with several existing algorithms, our algorithm is evaluated quantitatively in terms of surface normal and vertex distance error metrics. Numerical experiments indicate the effectiveness of our approach in the aspects of convergence and accuracy. Copyright © 2005 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/48758/1/1441_ftp.pd

Geometry Processing of Conventionally Produced Mouse Brain Slice Images

Brain mapping research in most neuroanatomical laboratories relies on conventional processing techniques, which often introduce histological artifacts such as tissue tears and tissue loss. In this paper we present techniques and algorithms for automatic registration and 3D reconstruction of conventionally produced mouse brain slices in a standardized atlas space. This is achieved first by constructing a virtual 3D mouse brain model from annotated slices of Allen Reference Atlas (ARA). Virtual re-slicing of the reconstructed model generates ARA-based slice images corresponding to the microscopic images of histological brain sections. These image pairs are aligned using a geometric approach through contour images. Histological artifacts in the microscopic images are detected and removed using Constrained Delaunay Triangulation before performing global alignment. Finally, non-linear registration is performed by solving Laplace's equation with Dirichlet boundary conditions. Our methods provide significant improvements over previously reported registration techniques for the tested slices in 3D space, especially on slices with significant histological artifacts. Further, as an application we count the number of neurons in various anatomical regions using a dataset of 51 microscopic slices from a single mouse brain. This work represents a significant contribution to this subfield of neuroscience as it provides tools to neuroanatomist for analyzing and processing histological data.Comment: 14 pages, 11 figure

A Concept For Surface Reconstruction From Digitised Data

Reverse engineering and in particular the reconstruction of surfaces from digitized data is an important task in industry. With the development of new digitizing technologies such as laser or photogrammetry, real objects can be measured or digitized quickly and cost effectively. The result of the digitizing process is a set of discrete 3D sample points. These sample points have to be converted into a mathematical, continuous surface description, which can be further processed in different computer applications. The main goal of this work is to develop a concept for such a computer aided surface generation tool, that supports the new scanning technologies and meets the requirements in industry towards such a product. Therefore first, the requirements to be met by a surface reconstruction tool are determined. This marketing study has been done by analysing different departments of several companies. As a result, a catalogue of requirements is developed. The number of tasks and applications shows the importance of a fast and precise computer aided reconstruction tool in industry. The main result from the analysis is, that many important applications such as stereolithographie, copy milling etc. are based on triangular meshes or they are able to handle these polygonal surfaces. Secondly the digitizer, currently available on the market and used in industry are analysed. Any scanning system has its strength and weaknesses. A typical problem in digitizing is, that some areas of a model cannot be digitized due to occlusion or obstruction. The systems are also different in terms of accuracy, flexibility etc. The analysis of the systems leads to a second catalogue of requirements and tasks, which have to be solved in order to provide a complete and effective software tool. The analysis also shows, that the reconstruction problem cannot be solved fully automatically due to many limitations of the scanning technologies. Based on the two requirements, a concept for a software tool in order to process digitized data is developed and presented. The concept is restricted to the generation of polygonal surfaces. It combines automatic processes, such as the generation of triangular meshes from digitized data, as well as user interactive tools such as the reconstruction of sharp corners or the compensation of the scanning probe radius in tactile measured data. The most difficult problem in this reconstruction process is the automatic generation of a surface from discrete measured sample points. Hence, an algorithm for generating triangular meshes from digitized data has been developed. The algorithm is based on the principle of multiple view combination. The proposed approach is able to handle large numbers of data points (examples with up to 20 million data points were processed). Two pre-processing algorithm for triangle decimation and surface smoothing are also presented and part of the mesh generation process. Several practical examples, which show the effectiveness, robustness and reliability of the algorithm are presented