Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

This paper presents a method for resampling a sequence of high dynamic range light probe images into a representation of Incident Light Field (ILF) illumination which enables realtime rendering. The light probe sequences are captured at varying positions in a real world environment using a high dynamic range video camera pointed at a mirror sphere. The sequences are then resampled to a set of radiance maps in a regular three dimensional grid before projection onto spherical harmonics. The capture locations and amount of samples in the original data make it inconvenient for direct use in rendering and resampling is necessary to produce an efficient data structure. Each light probe represents a large set of incident radiance samples from different directions around the capture location. Under the assumption that the spatial volume in which the capture was performed has no internal occlusion, the radiance samples are projected through the volume along their corresponding direction in order to build a new set of radiance maps at selected locations, in this case a three dimensional grid. The resampled data is projected onto a spherical harmonic basis to allow for realtime lighting of synthetic objects inside the incident light field.

Modeling of the ionosphere has been a highly interesting subject within the scientific community due to its effects on the propagation of electromagnetic waves. The development of the Global Positioning System (GPS) and creation of extensive ground-based GPS networks started a new period in observation of the ionosphere, which resulted in several studies on GPS-based modeling of the ionosphere. However, software studies on the subject that are open to the scientific community have not progressed in a similar manner and the options for the research community to reach ionospheric modeling results are still limited. Being aware of this need, a new MATLAB&reg / based ionosphere modeling software, i.e. TECmapper is developed within the study. The software uses three different algorithms for the modeling of the Vertical Total Electron Content (VTEC) of the ionosphere, namely, 2D B-spline, 3D B-spline and spherical harmonic models. The study includes modifications for the original forms of the B-spline and the spherical harmonic approaches. In order to decrease the effect of outliers in the data a robust regression algorithm is utilized as an alternative to the least squares estimation. Besides, two regularization methods are employed to stabilize the ill-conditioned problems in parameter estimation stage. The software and models are tested on a real data set from ground-based GPS receivers over Turkey. Results indicate that the B-spline models are more successful for the local or regional modeling of the VTEC. However, spherical harmonics should be preferred for global applications since the B-spline approach is based on Euclidean theory.

Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

Shadows provide an important cue in computer graphics. In this thesis we focus on real-time soft shadow algorithms. Two new techniques are presented, both run entirely on modern graphics hardware. "Soft Shadows Using Precomputed Visibility Distance Functions" renders fake soft shadows in static scenes using precomputed visibility information. The technique handles dynamic local light sources and contains special computation steps to generate smooth shadows from hard visibility functions. The resulting images are not physically accurate, nevertheless the method renders plausible images that imitate global illumination. "Soft Projected Shadows" is a simple method for simulating natural shadow penumbra for projected grayscale shadow textures. Shadow blurring is performed entirely in image space and needs only a couple of special blurring passes on pixel shader 2.0 hardware. The technique treats shadow receivers as nearly planar surfaces and doesn’t handle self shadowing, but executes very fast and renders plausible soft shadows. Multiple overlapping shadow casters in a single shadow map are natively supported without any performance overhead.

An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differential equations. A Fourier-Bessel expansion of the solution vector in terms of Bessel functions with real orders is employed, resulting in a generalized matrix eigenvalue problem. The method is applied to two particular examples. The first example is a prolate spheroidal billiard which is also treated by using an alternative method. The numerical results obtained by using both the methods are compared. The second exampleis a billiard family depending on a parameter. Numerical results concerning the second example include the statistical analysis of the eigenvalues.

Thesis (M.S.)--Nuclear and Radiological Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Stacey, Weston M.; Committee Co-Chair: de Oliveira, Cassiano R.E.; Committee Member: Hertel, Nolan; Committee Member: van Rooijen, Wilfred F.G.

Animals are often able to solve complex navigational tasks in very challenging terrain, despite using low resolution sensors and minimal computational power, providing inspiration for robots. In particular, many species of insect are known to solve complex navigation problems, often combining an array of different behaviours (Wehner et al., 1996; Collett, 1996). Their nervous system is also comparatively simple, relative to that of mammals and other vertebrates. In the first part of this thesis, the visual input of a navigating desert ant, Cataglyphis velox, was mimicked by capturing images in ultraviolet (UV) at similar wavelengths to the ant’s compound eye. The natural segmentation of ground and sky lead to the hypothesis that skyline contours could be used by ants as features for navigation. As proof of concept, sky-segmented binary images were used as input for an established localisation algorithm SeqSLAM (Milford and Wyeth, 2012), validating the plausibility of this claim (Stone et al., 2014). A follow-up investigation sought to determine whether using the sky as a feature would help overcome image matching problems that the ant often faced, such as variance in tilt and yaw rotation. A robotic localisation study showed that using spherical harmonics (SH), a representation in the frequency domain, combined with extracted sky can greatly help robots localise on uneven terrain. Results showed improved performance to state of the art point feature localisation methods on fast bumpy tracks (Stone et al., 2016a). In the second part, an approach to understand how insects perform a navigational task called path integration was attempted by modelling part of the brain of the sweat bee Megalopta genalis. A recent discovery that two populations of cells act as a celestial compass and visual odometer, respectively, led to the hypothesis that circuitry at their point of convergence in the central complex (CX) could give rise to path integration. A firing rate-based model was developed with connectivity derived from the overlap of observed neural arborisations of individual cells and successfully used to build up a home vector and steer an agent back to the nest (Stone et al., 2016b). This approach has the appeal that neural circuitry is highly conserved across insects, so findings here could have wide implications for insect navigation in general. The developed model is the first functioning path integrator that is based on individual cellular connections.

Halo nuclei are formed when the last protons or neutrons are weakly bound to a tightly bound core. This allows the halo nucleons to tunnel far away from the core, resulting in a large r.m.s radius and therefore a large reaction cross section. Usually, halo nuclei possess only one bound state, the ground state, with all excited states being more or less unbound. When a nuclear potential is too weak to form a bound ground or excited state, the state can nevertheless be manifest physically as a positive energy resonance. Experimentally, low energy resonance like structures have been observed in the three-body continuum of certain halo nuclei eg. 6He → alpha + n + n. However, from a strict theoretical point of view, a resonance corresponds to a pole in the scattering amplitude at a complex energy. Halo nuclei have been successfully modelled as three-body systems in the hyper-spherical harmonic calculation scheme. Here the R-matrix method is used in solving the coupled hyperradial equations. It is critical that the long-range nature of the couplings in this system are incorporated correctly when evaluating the S-matrix. This is achieved through the use of coupled asymptotic solutions to the radial equation. These procedures have enabled a number of resonance-like S-matrix poles to be located for the 2+, 0+ and 1- spin-parity states in the low energy continuum of 6He.

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