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The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an approximating function for a surface given some samples (x[sub i], y[sub j], z[sub ij]) of the surface.

In Reverse Engineering a physical object is digitally reconstructed from a set of boundary points. In the segmentation phase these points are grouped into subsets to facilitate consecutive steps as surface fitting. In this thesis we present a segmentation method with subsequent classification of simple algebraic surfaces. Our method is direct in the sense that it operates directly on the point set in contrast to other approaches that are based on a triangulation of the data set. The reconstruction process involves a fast algorithm for $k$-nearest neighbors search and an estimation of first and second order surface properties. The first order segmentation, that is based on normal vectors, provides an initial subdivision of the surface and detects sharp edges as well as flat or highly curved areas. One of the main features of our method is to proceed by alternating the steps of segmentation and normal vector estimation. The second order segmentation subdivides the surface according to principal curvatures and provides a sufficient foundation for the classification of simple algebraic surfaces. If the boundary of the original object contains such surfaces the segmentation is optimized based on the result of a surface fitting procedure. / Im Reverse Engineering wird ein existierendes Objekt aus einer Menge von Oberflächenpunkten digital rekonstruiert. Während der Segmentierungsphase werden diese Punkte in Teilmengen zusammengefügt, um die nachfolgenden Schritte wie Flächenerkennung (surface fitting) zu vereinfachen. Wir präsentieren in dieser Arbeit eine Methode zur Segmentierung der Punkte und die anschließende Klassifikation einfacher algebraischen Flächen. Unser Verfahren ist direkt in dem Sinne, dass es direkt an den Punkten arbeitet, im Gegensatz zu anderen Verfahren, die auf einer Triangulierung der Punktmenge basieren. Der Rekonstruktionsprozess schließt einen neuen Algorithmus zur Berechnung der k-nächsten Nachbarn eines Oberflächenpunktes und Verfahren zur Schätzung der Flächeneigenschaften ersten und zweiten Grades ein. Die normalenbasierte Segmentierung (Segmentierung ersten Grades) liefert eine Aufteilung des Objektes und detekiert scharfe Kanten, sowie flache oder stark gekrümmte Gebiete des Objektes. Ein zentrales Element unserer Methode ist die Wiederholung der Schritte der Segmentierung und der Schätzung der Normalen. Erst die Iteration ermöglicht die Schätzung der Normalen in der benötigten Genauigkeit und die Generierung einer zufriedenstellender Segmentierung. Die Segmentierung zweiten Grades teilt die Oberfläche nach den Hauptkrümmungen auf und bietet eine zuverlässige Grundlage für die Klassifizierung einfacher algebraischen Flächen. Falls der Rand des Ausgangsobjektes solche Flächen enthält, wird der Segmentierungsprozess auf der Grundlage des Ergebnisses der Flächenerkennungsprozedur optimiert.

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No description available.

Surface fitting methods play an important role in many scientific fields as well as in computer aided geometric design. The problem treated here is that of constructing a smooth surface that interpolates data values associated with scattered nodes in the plane. The data is said to be convex if there exists a convex interpolant. The problem of convexity-preserving interpolation is to determine if the data is convex, and construct a convex interpolant if it exists.

Context: Uncompressed Images contain redundancy of image data which can be reduced by image compression in order to store or transmit image data in an economic way. There are many techniques being used for this purpose but the rapid growth in digital media requires more research to make more efficient use of resources. Objectives: In this study we implement Polynomial curve fitting using 1st and 2nd curve orders with non-overlapping 4x4 and 8x8 block sizes. We investigate a selective quantization where each parameter is assigned a priority. The 1st parameter is assigned high priority compared to the other parameters. At the end Huffman coding is applied. Three standard grayscale images of LENA, PEPPER and BOAT are used in our experiment. Methods: We did a literature review, where we selected articles from known libraries i.e. IEEE, ACM Digital Library, ScienceDirect and SpringerLink etc. We have also performed two experiments, one experiment with 1st curve order using 4x4 and 8x8 block sizes and second experiment with 2nd curve order using same block sizes. Results: A comparison using 4x4 and 8x8 block sizes at 1st and 2nd curve orders shows that there is a large difference in compression ratio for the same value of Mean Square Error. Using 4x4 block size gives better quality of an image as compare to 8x8 block size at same curve order but less compression. JPEG gives higher value of PSNR at low and high compression. Conclusions: A selective quantization is good idea to use to get better subjective quality of an image. A comparison shows that to get good compression ratio, 8x8 block size at 1st curve order should be used but for good objective and subjective quality of an image 4x4 block size at 2nd order should be used. JPEG involves a lot of research and it outperforms in PSNR and CR as compare to our proposed scheme at low and high compression ratio. Our proposed scheme gives comparable objective quality (PSNR) of an image at high compression ratio as compare to the previous curve fitting techniques implemented by Salah and Ameer but we are unable to achieve subjective quality of an image.

The author developed a computational framework for the study of the correlation between airway morphology and aerosol deposition based on a population of human subjects. The major improvement on the previous framework, which consists of a geometric airway model, a computational fluid dynamics (CFD) model, and a particle tracking algorithm, lies in automatic geometry construction and mesh generation of airways, which is essential for a population-based study. The new geometric model overcomes the shortcomings of both centerline (CL)-based cylindrical models, which are based on the skeleton and average branch diameters of airways called one-dimensional (1-D) trees, and computed tomography (CT)-based models. CL-based models are efficient in terms of pre- and post-processing, but fail to represent trifurcations and local morphology. In contrast, in spite of the accuracy of CT-based models, it is time-consuming to build these models manually, and non-trivial to match 1-D trees and three-dimensional (3-D) geometry. The new model, also known as a hybrid CL-CT-based model, is able to construct a physiologically-consistent laryngeal geometry, represent trifurcations, fit cylindrical branches to CT data, and create the optimal CFD mesh in an automatic fashion. The hybrid airway geometries constructed for 8 healthy and 16 severe asthmatic (SA) subjects agreed well with their CT-based counterparts. Furthermore, the prediction of aerosol deposition in a healthy subject by the hybrid model agreed well with that by the CT-based model. To demonstrate the potential application of the hybrid model to investigating the correlation between skeleton structure and aerosol deposition, the author applied the large eddy simulation (LES)-based CFD model that accounts for the turbulent laryngeal jet to three hybrid models of SA subjects. The correlation between diseased branch and aerosol deposition was significant in one of the three SA subjects. However, whether skeleton structure contributes to airway abnormality requires further investigation.

In Reverse Engineering a physical object is digitally reconstructed from a set of boundary points. In the segmentation phase these points are grouped into subsets to facilitate consecutive steps as surface fitting. In this thesis we present a segmentation method with subsequent classification of simple algebraic surfaces. Our method is direct in the sense that it operates directly on the point set in contrast to other approaches that are based on a triangulation of the data set. The reconstruction process involves a fast algorithm for $k$-nearest neighbors search and an estimation of first and second order surface properties. The first order segmentation, that is based on normal vectors, provides an initial subdivision of the surface and detects sharp edges as well as flat or highly curved areas. One of the main features of our method is to proceed by alternating the steps of segmentation and normal vector estimation. The second order segmentation subdivides the surface according to principal curvatures and provides a sufficient foundation for the classification of simple algebraic surfaces. If the boundary of the original object contains such surfaces the segmentation is optimized based on the result of a surface fitting procedure. / Im Reverse Engineering wird ein existierendes Objekt aus einer Menge von Oberflächenpunkten digital rekonstruiert. Während der Segmentierungsphase werden diese Punkte in Teilmengen zusammengefügt, um die nachfolgenden Schritte wie Flächenerkennung (surface fitting) zu vereinfachen. Wir präsentieren in dieser Arbeit eine Methode zur Segmentierung der Punkte und die anschließende Klassifikation einfacher algebraischen Flächen. Unser Verfahren ist direkt in dem Sinne, dass es direkt an den Punkten arbeitet, im Gegensatz zu anderen Verfahren, die auf einer Triangulierung der Punktmenge basieren. Der Rekonstruktionsprozess schließt einen neuen Algorithmus zur Berechnung der k-nächsten Nachbarn eines Oberflächenpunktes und Verfahren zur Schätzung der Flächeneigenschaften ersten und zweiten Grades ein. Die normalenbasierte Segmentierung (Segmentierung ersten Grades) liefert eine Aufteilung des Objektes und detekiert scharfe Kanten, sowie flache oder stark gekrümmte Gebiete des Objektes. Ein zentrales Element unserer Methode ist die Wiederholung der Schritte der Segmentierung und der Schätzung der Normalen. Erst die Iteration ermöglicht die Schätzung der Normalen in der benötigten Genauigkeit und die Generierung einer zufriedenstellender Segmentierung. Die Segmentierung zweiten Grades teilt die Oberfläche nach den Hauptkrümmungen auf und bietet eine zuverlässige Grundlage für die Klassifizierung einfacher algebraischen Flächen. Falls der Rand des Ausgangsobjektes solche Flächen enthält, wird der Segmentierungsprozess auf der Grundlage des Ergebnisses der Flächenerkennungsprozedur optimiert.

No description available.