New Models for High-Quality Surface Reconstruction and Rendering

The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface

New Models for High-Quality Surface Reconstruction and Rendering

The efficient reconstruction and artifact-free visualization of surfaces from measured real-world data is an important issue in various applications, such as medical and scientific visualization, quality control, and the media-related industry. The main contribution of this thesis is the development of the first efficient GPU-based reconstruction and visualization methods using trivariate splines, i.e., splines defined on tetrahedral partitions. Our methods show that these models are very well-suited for real-time reconstruction and high-quality visualizations of surfaces from volume data. We create a new quasi-interpolating operator which for the first time solves the problem of finding a globally C1-smooth quadratic spline approximating data and where no tetrahedra need to be further subdivided. In addition, we devise a new projection method for point sets arising from a sufficiently dense sampling of objects. Compared with existing approaches, high-quality surface triangulations can be generated with guaranteed numerical stability. Keywords. Piecewise polynomials; trivariate splines; quasi-interpolation; volume data; GPU ray casting; surface reconstruction; point set surface

Trivariate C1-Splines auf gleichmäßigen Partitionen

In der vorliegenden Dissertation werden Splines auf gleichmäßigen Partitionen untersucht. Ziel der Arbeit ist die Analyse von multivariaten Splineräumen und die Entwicklung von neuen Methoden zur Lösung von Interpolations- und Approximationsproblemen mit trivariaten C1-Splines. Die entwickelten Methoden werden in Hinblick auf Lokalität, Stabilität und Approximationsordnung untersucht und die Ergebnisse dem Stand der Technik gegenübergestellt. Erstmalig kann dabei eine Quasi-Interpolationsmethode für trivariate C1-Splines vom totalen Grad zwei entwickelt werden und zur interaktiven Volumenvisualisierung mit Raycasting Techniken effizient eingesetzt werden

Lagrange interpolation and quasi-interpolation using trivariate splines on a uniform partition

We develop quasi-interpolation methods and a Lagrange interpolation method for trivariate splines on a regular tetrahedral partition, based on the Bernstein-Bézier representation of polynomials. The partition is based on the bodycentered cubic grid. Our quasi-interpolation operators use quintic C2 splines and are defined by giving explicit formulae for each coefficient. One operator satisfies a certain convexity condition, but has sub-optimal approximation order. A second operator has optimal approximation order, while a third operator interpolates the provided data values. The first two operators are defined by a small set of computation rules which can be applied independently to all tetrahedra of the underlying partition. The interpolating operator is more complex while maintaining the best-possible approximation order for the spline space. It relies on a decomposition of the partition into four classes, for each of which a set of computation rules is provided. Moreover, we develop algorithms that construct blending operators which are based on two quasi-interpolation operators defined for the same spline space, one of which is convex. The resulting blending operator satisfies the convexity condition for a given data set. The local Lagrange interpolation method is based on cubic C1 splines and focuses on low locality. Our method is 2-local, while comparable methods are at least 4-local. We provide numerical tests which confirm the results, and high-quality visualizations of both artificial and real-world data sets

New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes

With the advent of powerful 3D acquisition technology, there is a growing demand for the modeling, processing, and visualization of surfaces and volumes. The proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression. This thesis presents several novel solutions to these problems for surfaces (Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples

Lokale Lagrange-Interpolation mit Splineoberflächen

Wir entwickeln lokale Lagrange-Interpolationsverfahren für bivariate Splines und 3D-Splineoberflächen. In Zusammenhang mit Interpolationsalgorithmen für bivariate Splines auf Quadrangulierung entwickeln wir einen Färbungsalgorithmus, bei welchem gleichfarbige benachbarte Vierecke erstmals stets in nicht geschlossenen Ketten auftreten. Darüber hinaus ergeben sich große Klassen von Quadrangulierungen, sodass die maximale Kettenlänge festgelegt ist. Die entwickelten bivariaten Verfahren sind lokal, stabil und besitzen optimale Approximationsordnung. Hinsichtlich 3D-Splineoberflächen wird erstmals ein Verfahren entwickelt, bei welchem die geglätteten interpolierenden Oberflächen für den Großteil der Kanten der 3D-Triangulierungen differenzierbar an jedem Punkt sind. Weiterhin entwickeln wir ein Verfahren für 3D-Splineoberflächen auf Quadrangulierungen. Beide Verfahren sind lokal und stabil

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement

Critical Thinking Skills Profile of High School Students In Learning Science-Physics

This study aims to describe Critical Thinking Skills high school students in the city of Makassar. To achieve this goal, the researchers conducted an analysis of student test results of 200 people scattered in six schools in the city of Makassar. The results of the quantitative descriptive analysis of the data found that the average value of students doing the interpretation, analysis, and inference in a row by 1.53, 1.15, and 1.52. This value is still very low when compared with the maximum value that may be obtained by students, that is equal to 10.00. This shows that the critical thinking skills of high school students are still very low. One fact Competency Standards science subjects-Physics is demonstrating the ability to think logically, critically, and creatively with the guidance of teachers and demonstrate the ability to solve simple problems in daily life. In fact, according to Michael Scriven stated that the main task of education is to train students and or students to think critically because of the demands of work in the global economy, the survival of a democratic and personal decisions and decisions in an increasingly complex society needs people who can think well and make judgments good. Therefore, the need for teachers in the learning device scenario such as: driving question or problem, authentic Investigation: Science Processes