This dissertation addresses problems that arise in a diverse group of fields including cosmology, electromagnetism, and graphic design. While these topics may seem disparate, they share a commonality in their need for fast and accurate algorithms which can handle large datasets collected on irregular domains. An important issue in cosmology is the calculation of the angular power spectrum of the cosmic microwave background (CMB) radiation. CMB photons offer a direct insight into the early stages of the universe's development and give the strongest evidence for the Big Bang theory to date. The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) grid is used by cosmologists to collect CMB data and store it as points on the sphere. HEALPix also refers to the software package that analyzes CMB maps and calculates their angular power spectrums. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the curvature of the universe, dark matter density, and the nature of dark energy. In the first paper, we present a new method for performing spherical harmonic analysis for HEALPix data, which is a vital component for computing the CMB angular power spectrum. Using numerical experiments, we demonstrate that the new method provides better accuracy and a higher convergence rate when compared to the current methods on synthetic data. This paper is presented in Chapter 2.
The problem of constructing smooth approximants to divergence-free (div-free) and curl-free vector fields and/or their potentials based only on discrete samples arises in science applications like fluid dynamics and electromagnetism. It is often necessary that the vector approximants preserve the div-free or curl-free properties of the field. Div/curl-free radial basis functions (RBFs) have traditionally been utilized for constructing these vector approximants, but their global nature can make them computationally expensive and impractical. In the second paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity (PUM) framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. This method can be used to approximate vector fields and their scalar potentials on the sphere and in irregular domains in ℝ2 and ℝ3. We present error estimates and demonstrate the effectiveness of the method on several test problems. This paper is presented in Chapter 3.
The issue of reconstructing implicit surfaces from oriented point clouds has applications in computer aided design, medical imaging, and remote sensing. Utilizing the technique from the second paper, we introduce a novel approach to this problem by exploiting a fundamental result from vector calculus. In our method, deemed CFPU, we interpolate the normal vectors of the point cloud with a curl-free RBF-PUM interpolant and extract a potential of the reconstructed vector field. The zero-level surface of this potential approximates the implicit surface of the point cloud. Benefits of this method include its ability to represent local sharp features, handle noise in the normal vectors, and even exactly interpolate a point cloud. We demonstrate in the third paper that our method converges for known surfaces and also show how it performs on various surfaces found in the literature. This paper is presented in Chapter 4.
A Novel Vegetation Point Cloud Density Tree-Segmentation Model for Overlapping Crowns Using UAV LiDAR