Constructing new control points for Bézier interpolating polynomials using new geometrical approach

Interpolation is a mathematical technique employed for estimating the value of missing data between data points. This technique assures that the resulting polynomial passes through all data points. One of the most useful interpolating polynomials is the parametric interpolating polynomial. Bézier interpolating curves and surfaces are parametric interpolating polynomials for two-dimensional (2D) and three-dimensional (3D) datasets, respectively, that produce smooth, flexible, and accurate functions. According to the previous studies, the most crucial component in deriving Bézier interpolating polynomials is the construction of control points. However, most of the existing strategies constructed control points that produce partial smooth functions. As a result, the approximate values of the missing data are not accurate. In this study, nine new strategies of geometrical approach for constructing new 2D and 3D Bézier control points are proposed. The obtained control points from each new strategies are substituted in the relevant Bézier curve and surface equations to derive Bézier piecewise and non-piecewise interpolating polynomials which leads to the development of nine new methods. The proposed methods are proven to preserve the stability and smoothness of the generated Bézier interpolating curves and surfaces. In addition, the numerical results show that most of the resulting polynomials are able to approximate the missing values more accurately compared to those derived by the existing methods. The Bézier interpolating surfaces derived by the proposed method with highest accuracy for 3D datasets are then applied to upscale grey and colour images by the factors of two and three. Not only does the proposed method produces higher quality upscaled images, the numerical results also show that it outperforms the existing methods in terms of accuracy. Therefore, this study has successfully proposed new strategies for constructing new 2D and 3D control points for deriving Bézier interpolating polynomials that are capable of approximating the missing values accurately. In terms of application, the derived Bézier interpolating surfaces have a great potential to be employed in image upscaling

Arbitrary topology meshes in geometric design and vector graphics

Meshes are a powerful means to represent objects and shapes both in 2D and 3D, but the techniques based on meshes can only be used in certain regular settings and restrict their usage. Meshes with an arbitrary topology have many interesting applications in geometric design and (vector) graphics, and can give designers more freedom in designing complex objects. In the first part of the thesis we look at how these meshes can be used in computer aided design to represent objects that consist of multiple regular meshes that are constructed together. Then we extend the B-spline surface technique from the regular setting to work on extraordinary regions in meshes so that multisided B-spline patches are created. In addition, we show how to render multisided objects efficiently, through using the GPU and tessellation. In the second part of the thesis we look at how the gradient mesh vector graphics primitives can be combined with procedural noise functions to create expressive but sparsely defined vector graphic images. We also look at how the gradient mesh can be extended to arbitrary topology variants. Here, we compare existing work with two new formulations of a polygonal gradient mesh. Finally we show how we can turn any image into a vector graphics image in an efficient manner. This vectorisation process automatically extracts important image features and constructs a mesh around it. This automatic pipeline is very efficient and even facilitates interactive image vectorisation

Information Extraction and Modeling from Remote Sensing Images: Application to the Enhancement of Digital Elevation Models

To deal with high complexity data such as remote sensing images presenting metric resolution over large areas, an innovative, fast and robust image processing system is presented. The modeling of increasing level of information is used to extract, represent and link image features to semantic content. The potential of the proposed techniques is demonstrated with an application to enhance and regularize digital elevation models based on information collected from RS images

Developing a flexible and expressive realtime polyphonic wave terrain synthesis instrument based on a visual and multidimensional methodology

The Jitter extended library for Max/MSP is distributed with a gamut of tools for the generation, processing, storage, and visual display of multidimensional data structures. With additional support for a wide range of media types, and the interaction between these mediums, the environment presents a perfect working ground for Wave Terrain Synthesis. This research details the practical development of a realtime Wave Terrain Synthesis instrument within the Max/MSP programming environment utilizing the Jitter extended library. Various graphical processing routines are explored in relation to their potential use for Wave Terrain Synthesis

Motion patterns of subviral particles: Digital tracking, image data processing and analysis

At the Institute of Virology, Philipps-University, Marburg, Germany, currently research on the understanding of the transport mechanisms of Ebola- and Marburgvirus nucleocapsids is carried out. This research demands a profound knowledge about the various motion characteristics of the nucleocapids. The analysis of large amounts of samples by conventional manual evaluation is a laborious task and does not always lead to reproducible and comparable results. In a cooperation between the Institute of Virology, Marburg, and the Institute for Biomedical Engineering, University of Applied Sciences, Giessen, Germany, algorithms are developed and programmed that enable an automatic tracking of subviral particles in fluorescence microscopic image sequences. The algorithms form an interface between the biologic and the algorithmic domain. Furthermore, methods to automatically parameterize and classify subviral particle motions are created. Geometric and mathematical approaches, like curvature-, fractal dimension- and mean squared displacement-determination are applied. Statistical methods are used to compare the measured subviral particle motion parameters between different biological samples. In this thesis, the biological, mathematical and algorithmic basics are described and the state of the art methods of other research groups are presented and compared. The algorithms to track, parameterize, classify and statistically analyze subviral particle tracks are presented in the Methods section. All methods are evaluated with simulated data and/or compared to data validated by a virologist. The methods are applied to a set of real fluorescence microscopic image sequences of Marburgvirus infected live-cells. The Results chapter shows that subviral particle motion can be successfully analyzed using the presented tracking and analysis methods. Furthermore, differences between the subviral particle motions in the analyzed groups could be detected. However, further optimization with manually evaluated data can improve the results. The methods developed in this project enhance the knowledge about nucleocapsid transport and may be valuable for the development of effective antiviral agents to cure Ebola- and Marburgvirus diseases. The thesis concludes with a chapter Discussion and Conclusions

Colour depth-from-defocus incorporating experimental point spread function measurements

Depth-From-Defocus (DFD) is a monocular computer vision technique for creating depth maps from two images taken on the same optical axis with different intrinsic camera parameters. A pre-processing stage for optimally converting colour images to monochrome using a linear combination of the colour planes has been shown to improve the accuracy of the depth map. It was found that the first component formed using Principal Component Analysis (PCA) and a technique to maximise the signal-to-noise ratio (SNR) performed better than using an equal weighting of the colour planes with an additive noise model. When the noise is non-isotropic the Mean Square Error (MSE) of the depth map by maximising the SNR was improved by 7.8 times compared to an equal weighting and 1.9 compared to PCA. The fractal dimension (FD) of a monochrome image gives a measure of its roughness and an algorithm was devised to maximise its FD through colour mixing. The formulation using a fractional Brownian motion (mm) model reduced the SNR and thus produced depth maps that were less accurate than using PCA or an equal weighting. An active DFD algorithm to reduce the image overlap problem has been developed, called Localisation through Colour Mixing (LCM), that uses a projected colour pattern. Simulation results showed that LCM produces a MSE 9.4 times lower than equal weighting and 2.2 times lower than PCA. The Point Spread Function (PSF) of a camera system models how a point source of light is imaged. For depth maps to be accurately created using DFD a high-precision PSF must be known. Improvements to a sub-sampled, knife-edge based technique are presented that account for non-uniform illumination of the light box and this reduced the MSE by 25%. The Generalised Gaussian is presented as a model of the PSF and shown to be up to 16 times better than the conventional models of the Gaussian and pillbox

Blending techniques in Curve and Surface constructions

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