Grundlagen der geometrischen Datenverarbeitung (2. Aufl.).
... Schließlich benötigen wir noch die Transformationsmatrizen zwischen unseren Koordinatensystemen. Die Zeilen dieser Matrizen erhält man bekanntlich (vgl. z.B. [11]) aus den Zeileneinheitsvektoren des neuen Systems im alten System. ...
... die Festlegung der Drehachse verwenden wir wieder temporär die Vektoren p = (x, y, z) T und u = (0, 0, 0) T mit Blickrichtung von p nach u und dem dadurch gegebenen positiven Drehsinn. Eine derartige Matrix hat die folgende Gestalt (siehe z.B.[11]) wenn noch zusätzlich für eine einfachere Schreibung r = 1 − c gesetzt wird 2 + c rxy − sz rxz + sy ryx + sz ry 2 + c ryz − sx rzx − sy rzy + sx rz 2 ...
Due to the continuing technical progress of the last decades, the process of capturing, analyzing and displaying spatial three-dimensional (3D) measurement data has fundamentally changed. Areabased sensors, such as laser scanners, provide a very detailed image of the reality. However, the captured huge 3D point clouds can contain a significant number of outliers that cannot be eliminated due to the lack of redundancy in the polar measurement procedures. In the field of displaying 3D measurement data, the user has at his disposal a large number of intuitively usable and easily understandable display options. Examples are point cloud viewer, computer-aided design software or virtual reality environments. The analysis, as the link between capture and representation, must be adapted to these changes. The scope of the analysis ranges, depending on the chosen capturing and presentation method, from the cleaning, segmentation and transformation of the point clouds to the modelling and further analyses, e.g. time series or deformation analysis. When modelling, i.e. interpolating or approximating point clouds, an effort is made to describe the point cloud using a mathematical function. On the one hand, this allows object properties, e.g. gradients, or statistically assured quality parameters, e.g. for accuracy and reliability, to be derived. On the other hand, the point cloud is transformed into a resource-saving format and it is available for further computer-aided processing. The piecewise polynomial functions and in particular the B-splines have proven to be particularly suitable mathematical functions for the approximation of arbitrary shapes. The approximation process for B-splines is divided into four steps: model selection, parameterization, knot vector selection and control point estimation. In the substep of the knot vector selection multimodal, multivariate,
continuous and nonlinear optimization problems must be solved. On the one hand, deterministic methods are used to determine the knot vector once from (derived) properties of the point cloud to be approximated. On the other hand, heuristic methods are used. Here a multiple random selection of the knot vector takes place, which is evaluated on the basis of a selected quality measure. Finally, the knot vector with the highest quality measure is selected. In this thesis three new or modified methods of knot vector selection process are presented. The residual-based iterative update algorithm is presented as a novel deterministic method. The knot vector is changed iteratively by moving individual knots according to the previously determined residuals. In addition, two modified heuristic methods are presented. On the one hand, a modified version of an evolutionary Monte Carlo method is shown. On the other hand, a modified version of
an elitist genetic algorithm is presented. In a simulation with normally distributed noise it is shown that this version of the elitist genetic algorithm leads to nearly optimal knot vectors compared to other methods. This is shown, among others, by the lowest proportion of rejected null hypothesis for the variance of the unit weight and the smallest distance between the estimated and true curve. In addition, this thesis systematically investigates the influence of outliers on the quality of B-spline curve approximation. For this purpose, data sets are generated using different noise models. In addition to the normal distribution, the noise models use the t-distribution, the Gaussian mixture
distribution and two variants of a biased noise. In control point estimation, three estimators, least
squares method, Huber and Hampel estimators are used and analyzed. For each noise model and
each estimator, the limit of resilience, i.e. the maximum possible proportion of outliers at which
acceptable results can still be achieved, is determined. The limit of resilience varies depending on
the noise model and estimator used. Basically, the most robust results are obtained with the Hampel
estimator.
Methods for surface design based on certain desired properties will be presented. Working in this way enable CAD-users to entirely concentrate on prescribing properties, leaving the task to fulfil them to the computer. In this way, the operator may almost instantaneously attain design solutions than would takes days or weeks with traditional means — if reached at all.
A user may, however, prescribe properties for which there does not exist any physically realisable surface having them, as will be illustrated. From a mathematical point, this is to be expected, since most styling properties gives rise to nonlinear partial differential equations that in general, does not have smooth solutions. It is shown that for several visual- and curvature properties, there are unique perturbation solutions admitting a user, adhering to a few basic principles, to work rather freely.
Mathematik ist die Wissenschaft vom Formalisieren; sie widmet sich der Untersuchung und Charakterisierung der hierdurch definierbaren Kalküle, Räume und Strukturen, entwickelt Beweis- und Rechenmethoden und leitet gültige Eigenschaften her. Hierbei haben sich prinzipielle Grenzen herausgestellt: Die Widerspruchsfreiheit läßt sich i. allg. nicht innerhalb von Kalkülen beweisen, viele Probleme sind mit Rechenmethoden unlösbar oder nicht effizient lösbar, die meisten nichtlinearen Systeme besitzen keine Lösung in knapper geschlossener mathematischer Darstellung usw. Solche Grenzen gelten auch für die Informatik, die sich mit maschinell bearbeitbaren Lösungsverfahren (also mit Algorithmen), ihren Darstellungen, ihren Eigenschaften, ihrer Realisierung und ihren Anwendungen befaßt. Bis in die 60er Jahre wurde die Informatik noch als ein Bereich aufgefaßt, der sich in die Mathematik eingliedern läßt. In der Tat entwickelt die Informatik (wie die Mathematik) grundlegende, meist formale Methoden und Techniken, die in anderen Wissenschaften benötigt und in immer stärkerem Maße dort eingesetzt werden. Doch ebenso, wie die Ingenieurwissenschaften nicht mehr als Teil der Naturwissenschaften angesehen werden, haben die Probleme bei der Realisierung und beim Einsatz und die in großen Mengen entwickelten Systeme imd Werkzeuge aus der Informatik eine eigenständige, seit 1970 vorwiegend ingenieurwissenschaftlich arbeitende Disziplin gemacht, die sich aber nicht mit den Grundstoffen „Materie“ und „Energie“, sondern mit der „Information“ beschäftigt. Da auf der Information persönliche und gesellschaftliche Entscheidungs- und Lernprozesse beruhen, wirkt die Informatik nachhaltig in fast alle Bereiche des menschlichen Lebens und Zusammenlebens hinein.
Mathematik ist die Wissenschaft vom Formalisieren; sie widmet sich der Untersuchung und Charakterisierung der hierdurch definierbaren Kalküle, Räume und Strukturen, entwickelt Beweis- und Rechenmethoden und leitet gültige Eigenschaften her. Hierbei haben sich prinzipielle Grenzen herausgestellt: Die Widerspruchsfreiheit läßt sich i.allg. nicht innerhalb von Kalkülen beweisen, viele Probleme sind mit Rechenmethoden unlösbar oder nicht effizient lösbar, die meisten nichtlinearen Systeme besitzen keine Lösung in knapper geschlossener mathematischer Darstellung usw. Solche Grenzen gelten auch für die Informatik, die sich mit maschinell bearbeitbaren Lösungsverfahren (also mit Algorithmen), ihren Darstellungen, ihren Eigenschaften, ihrer Realisierung und ihren Anwendungen befaßt. Bis in die 60er Jahre wurde die Informatik noch als ein Bereich aufgefaßt, der sich in die Mathematik eingliedern läßt. In der Tat entwickelt die Informatik (wie die Mathematik) grundlegende, meist formale Methoden und Techniken, die in anderen Wissenschaften benötigt und in immer stärkerem Maße dort eingesetzt werden. Doch ebenso, wie die Ingenieurwissenschaften nicht mehr als Teil der Naturwissenschaften angesehen werden, haben die Probleme bei der Realisierung und beim Einsatz und die in großen Mengen entwickelten Systeme und Werkzeuge aus der Informatik eine eigenständige, seit 1970 vorwiegend ingenieurwissenschaftlich arbeitende Disziplin gemacht, die sich aber nicht mit den Grundstoffen „Materie“ und „Energie“, sondern mit der „Information“ beschäftigt. Da auf der Information persönliche und gesellschaftliche Entscheidungs- und Lernprozesse beruhen, wirkt die Informatik nachhaltig in fast alle Bereiche des menschlichen Lebens und Zusammenlebens hinein.
An algorithm for approximation of arbitrary clouds of points with integral tensor product B-spline surfaces is presented. The clouds may be scattered may have holes and may have arbitrary boundaries. The usual methods in Reverse Engineering subdivide the given cloud into rectangular parts and approximate these parts individually. In the presented paper an overall algorithm for tensor product B-spline approximation with free boundary curves is introduced.
In this paper we will consider surface design through interactive improvements of image intensities of a current surface. Systems supporting this kind of design are immensely efficient tools, enabling operators quickly achieve surface shapes that may be unattainable with other means.
The problem that must be solved in this process is that of finding a surface from its image intensity and some boundary conditions. It is a non-linear boundary value problem that, unfortunately, may very well lack a smooth solution. The main issue of the paper is to find conditions that can be used in practice, ensuring useful solutions
Given an extended Tchebycheff space of dimension m + 1, we associate with it a normal curve C in real projective m—space P
m
. The curve C has the properties that any hyperplane intersects it in at most m points and that through any point of P
m
there pass at most m of its osculating hyperplanes. This allows us to study extended Tchebycheff spaces and Tchebycheffian splines in a geometric way. We introduce a generalized blossom which directly leads to generalizations of Bézier und B—spline curves. Furthermore, we obtain connections to isotropic curve theory and to recent results of Carnicer and Peña on B—bases and extensibility of Tchebycheff systems. The paper both surveys recent research and presents new results.
Physics and geometry based variational techniques for surface construction have been shown to be advanced methods for designing high quality surfaces in the fields of CAD and CAGD. In this paper, we derive an Euler–Lagrange equation from a geometric invariant curvature integral functional—the integral about the mean curvature gradient. Using this Euler–Lagrange equation, we construct a sixth-order geometric flow, which is solved numerically by a divided-difference-like method. We apply our equation to solving several surface modeling problems, including surface blending, N-sided hole filling and point interpolating, with G2 continuity. The illustrative examples provided show that this sixth-order flow yields high quality surfaces.
Trimming of surfaces and volumes, curve and surface modeling via Bézier’s idea of destortion, segmentation, reparametrization, geometric continuity are examples of applications of functional composition. This paper shows how to compose polynomial and rational tensor product Bézier representations. The problem of composing Bézier splines and B-spline representations will also be addressed in this paper.
Based upon the Loewner ellipsoid an affine invariant norm will be presented. This norm will be compared with the norm established by Nielson [10] using results of scattered data interpolation.
A high quality surface representation is indispensable for a hydrodynamic numerical model, which is related to nature. Standard algorithms for meshing are not able to guarantee these requirements. Applied methods approximate the surface by patches. As a result an implementation for a b-spline surface was realised and tested. The used method “tensor product b-spline surface” is defined by an array of control points, called de Boor points. The associated b-spline functions are defined in every local segment. Changes at one point of the control point mesh result only in local changes of the b-spline surface.
We propose a method for determining the accurate local 3-D motion and shape of objects in image sequences from multiple views. The shape of the considered object is calculated at each time step using image sequences, synchronously acquired by two or more calibrated cameras. Starting from an initial shape model of the object a stepwise refinement process is performed in which the model is adapted to the actual shape according to a cost function. This function maximizes the similarity between the respective model projections of the object onto the images. In order to obtain an appropriate initial shape model at the consecutive time step a motion estimation technique for monocular image sequences is applied. The proposed method is applicable for rigid and deformable objects as well. Selected results for real-world image sequences are presented.
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