The Fourier transform (spectrum) of a signal is a complex function and is characterized by the magnitude and phase spectra. Phase retrieval is the reconstruction of the phase spectrum from the measurements of the magnitude spectrum. Such problems are encountered in imaging modalities such as X-ray crystallography, frequency-domain optical coherence tomography (FDOCT), quantitative phase microscopy, digital holography, etc., where only the magnitudes of the wavefront are detected by the sensors. The phase retrieval problem is ill-posed in general, since an in nite number of signals can have the same magnitude spectrum. Typical phase retrieval techniques rely on certain prior knowledge about the signal, such as its support or sparsity, to reconstruct the signal. A classical result in phase retrieval is that minimum-phase signals have log-magnitude and phase spectra that satisfy the Hilbert integral equations, thus facilitating exact phase retrieval. In this thesis, we demonstrate that there exist larger classes of signals beyond minimum-phase signals, for which exact phase retrieval is possible. We generalize Hilbert integral equations to 2-D, and also introduce a variant that we call the composite Hilbert transform in the context of 2-D periodic signals. Our first extension pertains to a particular type of parametric modelling of 2-D signals. While 1-D minimum-phase signals have a parametric representation, in terms of poles and zeros, there exists no such 2-D counterpart. We introduce a new class of parametric 2-D signals that possess the exact phase retrieval property, that is, their magnitude spectrum completely characterizes the signal. Starting from the magnitude spectrum, a sequence of non-linear operations lead us to a sum-of-exponentials signal, from which the parameters are computed employing concepts from high-resolution spectral estimation such as the annihilating filter and algebraically coupled matrix-pencil methods. We demonstrate that, for this new class of signals, our method outperforms existing techniques even in the presence of noise. Our second extension is to continuous-domain signals that lie in a principal shift-invariant space spanned by a known basis. Such signals are characterized by the basis combining coefficients. These signals need not be minimum-phase, but certain conditions on the coefficients lead to exact phase retrieval of the continuous-domain signal. In particular, we introduce the concept of causal, delta dominant (CDD) sequences, and show that such signals are characterized by their magnitude spectra. This condition pertains to the time/spatial-domain description of the signal, in contrast to the minimum-phase condition, which is described in the spectral domain. We show that there exist CDD sequences that are not minimum-phase, and vice versa. However, finite-length CDD sequences are always minimum-phase. Our method reconstructs the signal from the magnitude spectrum up to ma-chine precision. We thus have a class of continuous-domain signals that are neither causal nor minimum phase, and yet allow for exact phase retrieval. The shift-invariant structure is applicable to modelling signals encountered in imaging modalities such as FDOCT. We next present an application of 2-D phase retrieval to continuous-domain CDD signals in the context of quantiative phase microscopy. We develop sufficient conditions on the interfering reference wave for exact phase retrieval from magnitude measurements. In particular, we show that when the reference wave is a plane wave with magnitude greater that the intensity of the object wave, and when the carrier frequency is larger than the band-width of the object wave, we can reconstruct the object wave exactly. We demonstrate high-resolution reconstruction of our method on USAF target images. Our final and perhaps the most unifying contribution is in developing Hilbert integral equations for 2-D first-quadrant signals and in introducing the notion of generalized minimum-phase signals for both 1-D and 2-D signals. For 2-D continuous-domain, first-quadrant signals, we establish partial Hilbert transform relations between the real and imaginary parts of the spectrum. In the context of 2-D discrete-domain signals, we show that the partial Hilbert transform does not suffice and introduce the notion of composite Hilbert transform and establish the integral equations. We then introduce four classes of signals (combinations of 1-D/2-D and continuous/discrete-domain) that we call generalized minimum-phase signals, which satisfy corresponding Hilbert integral equations between log-magnitude and phase spectra, hence facilitating exact phase retrieval. This class of generalized minimum-phase signals subsumes the well known class of minimum-phase signals. We further show that, akin to minimum-phase signals, these signals also have stable inverses, which are also generalized minimum-phase signals.

Typescript (photocopy). / Digitized by Kansas Correctional Industries

One of the most important parameters of speech is the fundamental frequency of vibration of voiced sounds. The audio sensation of the fundamental frequency is known as the pitch. Depending on the tonal/non-tonal category of language, the fundamental frequency conveys intonation, pragmatics and meaning. In addition the fundamental frequency and intonation carry speaker gender, age, identity, speaking style and emotional state. Accurate estimation of the fundamental frequency is critically important for functioning of speech processing applications such as speech coding, speech recognition, speech synthesis and voice morphing. This thesis makes contributions to the development of accurate pitch estimation research in three distinct ways: (1) an investigation of the impact of the window length on pitch estimation error, (2) an investigation of the use of the higher order moments and (3) an investigation of an analysis-synthesis method for selection of the best pitch value among N proposed candidates. Experimental evaluations show that the length of the speech window has a major impact on the accuracy of pitch estimation. Depending on the similarity criteria and the order of the statistical moment a window length of 37 to 80 ms gives the least error. In order to avoid excessive delay as a consequence of using a longer window, a method is proposed ii where the current short window is concatenated with the previous frames to form a longer signal window for pitch extraction. The use of second order and higher order moments, and the magnitude difference function, as the similarity criteria were explored and compared. A novel method of calculation of moments is introduced where the signal is split, i.e. rectified, into positive and negative valued samples. The moments for the positive and negative parts of the signal are computed separately and combined. The new method of calculation of moments from positive and negative parts and the higher order criteria provide competitive results. A challenging issue in pitch estimation is the determination of the best candidate from N extrema of the similarity criteria. The analysis-synthesis method proposed in this thesis selects the pitch candidate that provides the best reproduction (synthesis) of the harmonic spectrum of the original speech. The synthesis method must be such that the distortion increases with the increasing error in the estimate of the fundamental frequency. To this end a new method of spectral synthesis is proposed using an estimate of the spectral envelop and harmonically spaced asymmetric Gaussian pulses as excitation. The N-best method provides consistent reduction in pitch estimation error. The methods described in this thesis result in a significant improvement in the pitch accuracy and outperform the benchmark YIN method.

Thesis (MScEng)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Capacity waste happens when right-turn vehicles have right-of-way during a separate right-turn phase and lanes (e.g., through lanes) of the same approach of the intersection cannot discharge vehicles during that green phase. Right-turn traffic consumes the capacity which otherwise could be provided to through traffic movements at an at-grade signalized intersection. Therefore, it is widely considered that it would lower intersection capacity and increase total delay (Lin, Machemehl, Lee & Herman, 1984). The pre-signal strategy proposed in this research is specifically designed to improve this problem. The following aspects of this strategy were studied in this research, - Capacity benefits of this strategy, - Relationships between the capacity and the length of sorting area (the area between the two signals), - Signal timing of both main signal and pre-signal, - Clearance time of the sorting area, - Main signal phasing options - Signal coordination between the main signal and the pre-signal, - Utilization of the sorting area, and - Pre-signal strategy performance in a simulated environment. The results of this study showed that right-turn movement benefits significantly from this pre-signal strategy. For example, right-turn capacity can be doubled if a presignal is installed on one through lane of an approach with one right-turn lane. It was also found, the maximum approach capacity benefit is not affected significantly by the length of the sorting area for a given green period. The optimal green time and the available pre-signal green time for right-turn movement were also derived in this research. Different main signal phasing options were studied and compared. Phasing options which fit the proposed pre-signal strategy were found. Recommended values for right-turn green time of both signals were given based on different lengths of sorting area. The case study, which compared the performance of some critical movements at the intersection with and without the proposed pre-signal system, confirms the results concluded in this study. A potential problem with this strategy when applied at a real intersection is that it may confuse drivers. Drivers need to be educated and will need time to get familiar with this signal control method. / AFRIKAANSE OPSOMMING: Beskermde regsdraaifases vir verkeer by gelykvlak seinbeheerde kruisings gebruik die kapasiteit wat benut kon word deur deurbewegings. Dit verlaag interseksie kapasiteit en totale oponthoud verhoog. Die voorseinstrategie wat in die navorsing studie voorgestel word is spesifiek ontwikkel om die probleem op te los of te verminder. Die volgende aspekte van die strategie is ondersoek in die navorsingsstudie: - Kapasiteitsvoordele van die strategie. - Die verhouding tussen die kapasiteit en die lengte van die sorteringsarea (die area tussen die twee seine). - Seintydstoedeling van beide die hoofseinfase en die voorseinfase. - Ontruimingstyd van die stoorarea. - Hoofseinfaseopsies. - Seinkoordenasie tussen die hoofsein en die voorsein. - Benutting van die sorterings area, en - Voorseinstrategieprestasie in ‘n gesimuleerde omgewing. Die resultate bewys dat die regsdraaibeweging grootliks bevoordeel word nadat die voorseinstrategie ingestel is. Byvoorbeeld, regsdraaikapasiteit kan verdubbel word as ‘n voorseinfase ingestel word op een van die deurlane tesame met ‘n enkele regsdraailaan. Daar is ook gevind dat die kapasiteit nie grootliks beinvloed word deur die lengte van die stoorgebied nie. Die optimale groentyd en die beskikbare voorsein groen tyd vir die regsdraaibeweging is ook afgelei in die navorsing. Verskillende hoofseinfaseopsies is bestudeer en vergelyk. Faseringsopsies vir die voorgestelde voorseinstrategie is gevind. Voorgestelde waardes vir regsdraaigroentyd van voorseine en hoofseine is bereken om kapasiteit te verbeter, gebasseer op verskillende lengtes van die stoorarea. Die gevallestudie wat die prestasie op ‘n aanloop met en sonder die voogestelde voorseinstrategie vergelyk, bewys resultate wat ooreenstem met die bevindinge in die studie. Die verwagte probleem met die voorseinstrategie, wanneer dit ingestel word by ‘n werklike interseksie, is verwarring van die bestuurders. Bestuurders sal opgevoed moet word en sal tyd nodig hê om gewoond te raak aan die voorseinmetode.

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

Current rates of technological advancement continue to translate into changes on our battlefields. Aerial robots capable of gathering reconnaissance along with unmanned underwater vehicles capable of defusing enemy minefields provide evidence that machines are playing key roles once played by humans within our military. This thesis explores one of the major problems facing both commercial and military UUVs to date. Successfully navigating in unfamiliar environments and maneuvering autonomously to avoid obstacles is a problem that has yet to be fully solved. Using a simulated 2-D ocean environment, the work of this thesis provides results of numerous REMUS simulations that model the vehicle's flight path over selected sea bottoms. Relying on a combination of sliding mode control and feedforward preview control, REMUS is able to locate obstacles such as seawalls using processed forward look sonar images. Once recognized, REMUS maneuvers to avoid the obstacle according to a Gaussian potential function. In summary, the integration of feedforward preview control and sliding mode control results in an obstacle avoidance controller that is not only robust, but also autonomous.

In this thesis, we investigate the application of compressive sensing and sparse signal processing techniques to image compression and inpainting problems. Considering that many signals are sparse in certain transformation domain, a natural question to ask is: can an image be represented by as few coefficients as possible? In this thesis, we propose a new model for image compression/decompression based on sparse representation. We suggest constructing an overcomplete dictionary by combining two compression matrices, the discrete cosine transform (DCT) matrix and Hadamard-Walsh transform (HWT) matrix, instead of using only one transformation matrix that has been used by the common compression techniques such as JPEG and JPEG2000. We analyze the Structural Similarity Index (SSIM) versus the number of coefficients, measured by the Normalized Sparse Coefficient Rate (NSCR) for our approach. We observe that using the same NSCR, SSIM for images compressed using the proposed approach is between 4%-17% higher than when using JPEG. Several algorithms have been used for sparse coding. Based on experimental results, Orthogonal Matching Pursuit (OMP) is proved to be the most efficient algorithm in terms of computational time and the quality of the decompressed image. In addition, based on compressive sensing techniques, we propose an image inpainting approach, which could be used to fill missing pixels and reconstruct damaged images. In this approach, we use the Gradient Projection for Sparse Reconstruction (GPSR) algorithm and wavelet transformation with Daubechies filters to reconstruct the damaged images based on the information available in the original image. Experimental results show that our approach outperforms existing image inpainting techniques in terms of computational time with reasonably good image reconstruction performance.

‘Tipping points’ in the climate system are characterised by a nonlinear response to gradual forcing, and may have severe and wide-ranging impacts. One of the best ways to identify and potentially predict threshold behaviour in the climate system is through analysis of palaeoclimate records. It has been suggested that early warning signals occur on the approach to a tipping point, generated from characteristic fluctuations in a time series as a system loses stability. Although early warning signals have been found in climate models and high-resolution marine and ice core palaeodata, studies from terrestrial records are lacking. In this study, a number of Pleistocene terrestrial records were selected to represent a range of regions strongly influenced by different climate modes which are thought to be capable of displaying threshold behaviour. These records included lake sediments from the North Atlantic, tree-rings from the South Pacific, a Chinese speleothem and were complemented by a new Greenland ice core chronology. Recently developed methods to detect signals of ‘critical slowing down’, ‘flickering’, and stability changes on the approach to a tipping point were utilised. Specific methodological issues arising from analysing palaeoclimate data were also investigated using a simple bifurcation model. A number of key criteria were found to be necessary for the reliable identification of early warning signals in palaeoclimate records, most crucially, the need for a low-noise record of sufficient data length, resolution and accuracy. Analysis of a Chinese speleothem identified the East Asian Summer Monsoon as an important climate ‘tipping element’, which may display a cascade of impacts. However, in some cases where early warning signals may fail, a deeper understanding of the underlying system dynamics is required to inform the development of more robust system-specific indicators. This was exemplified by the analysis of an abrupt, centennial-duration shutdown recorded during the Younger Dryas Chronozone in New Zealand, which demonstrated no slowing down, consistent with a freshwater pulse into the Southern Ocean. This study demonstrates that time series precursors from palaeoclimate archives provide a means of useful forewarning of many potential climate tipping points.

M.Ing. (Transportation) / Consecutive green phases at signalized intersections are separated by an amber and sometimes all-red period to allow a safe changeover from flow in one direction to another. Al though this signal change interval consists of only a few seconds, existing practice in setting this signal change interval varies considerably. Research on this subj ect has focused on understanding how individual drivers react when they are confronted by a change signal. Identifying the factors that describe the need for different change intervals can most easily be determined by measuring the aggregated behaviour of motorists under conditions where these factors vary. Evidence suggests that the South African driver overseas counterpart regarding his attitude towards interval differs from his the signal change Field observations of drivers' aggregated behaviour when confronted by a change interval were made at 17 approaches to 9 different intersections. The linear relationship between the signal change interval and various factors which apparently have an influence on the need for such a period were determined. The regression models were also structured to allow evaluation of existing models such as the well-known ITE-formula. The results were evaluated statistically. It was not possible to calibrate the ITE-formula for South African conditions. The approach adopted here conforms to tiling designs that use a constant (for a specific approach) amber interval. This constant is probably a summation of the influence of all the factors which influence the need for a signal change interval. The use of the clearance time for a crossing, as the all-red period, is also ascertained.

Recent advances in the optimization of fixed time traffic signals have demonstrated a move towards the use of genetic algorithm optimization with traffic network performance evaluated via stochastic microscopic simulation models. This dissertation examines methods for improved optimization. Several modified versions of the genetic algorithm and alternative genetic operators were evaluated on test networks. A traffic simulation model was developed for assessment purposes. Application of the CHC search algorithm with real crossover and mutation operators were found to offer improved optimization efficiency over the standard genetic algorithm with binary genetic operators. Computing resources are best utilized by using a single replication of the traffic simulation model with common random numbers for fitness evaluations. Combining the improvements, delay reductions between 13%-32% were obtained over the standard approaches. A coding scheme allowing for complete optimization of signal phasing is proposed and a statistical model for comparing genetic algorithm optimization efficiency on stochastic functions is also introduced. Alternative delay measurements, amendments to genetic operators and modifications to the CHC algorithm are also suggested.