The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favourably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing. / The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research.

Previous studies on Leafy reduced-stature (LRS) maize found that it had extremely early maturity and a higher harvest index (HI), leading to high yields for its maturity rating. Whether this apparent high HI is relaxed to its earliness, or can also exist among the medium or late maturity LRS maize has not been previously investigated. It was also of interest to know if the traits that produced the LRS canopy structure have pleiotropic effects on root architecture. Finally, field observations indicated that LRS maize had a lower incidence of common smut. It is not known whether this apparent resistance is specific to smut or includes other diseases. / Using a wide range of the most recently developed LRS hybrids and some conventional hybrids, a two-year field experiment was conducted to examine the HI and disease resistance of LRS maize. HI, yield, and yield components were compared between the two genotype groups (LRS and conventional) under different population densities. The resistance to the natural incidence of common smut and artificially inoculated Gibberella ear rot was also tested. Morphology and fractal dimension analyses of roots at an early development stage were conducted in indoor experiments. These analyses were performed with WinRHIZO (version 3.9), an interactive scanner-based image analysis system. / This work showed that: (1) There was no relationship between the HI and maturity; higher HIs can also exist among the medium and late maturity LRS hybrids. (2) While LRS maize hybrids have the potential for high yield this was not realized in the LRS hybrids used in this work. Further breeding and development of optimum management practices are needed to fully exploit this potential. (3) During early development LRS hybrids generally had more branching and more complex root systems than conventional hybrids. (4) Fractal dimension, as a comprehensive estimation of root complexity, was highly related to major root morphological variables, such as root total length, surface area, branching frequency and dry mass. (5) Of the hybrids tested the greatest resistance to both common smut and Gibberella ear rot, two major ear diseases, occurred in some of the LRS types.

Increase in technological advancements in fields of telecommunications, computers and television have prompted the need to exchange video, image and audio files between people. Transmission of such files finds numerous multimedia applications such as, internet multimedia, video conferencing, videophone, etc. However, the transmission and rece-ption of these files are limited by the available bandwidth as well as storage capacities of systems. Thus there is a need to develop compression systems, such that required multimedia applications can operate within these limited capacities. This dissertation presents two well established coding approaches that are used in modern' image and video compression systems. These are the wavelet and fractal methods. The wavelet based coder, which adopts the transform coding paradigm, performs the discrete wavelet transform on an image before any compression algorithms are implemented. The wavelet transform provides good energy compaction and decorrelating properties that make it suited for compression. Fractal compression systems on the other hand differ from the traditional transform coders. These algorithms are based on the theory of iterated function systems and take advantage of local self-similarities present in images. In this dissertation, we first review the theoretical foundations of both wavelet and fractal coders. Thereafter we evaluate different wavelet and fractal based compression algorithms, and assess the strengths and weakness in each case. Due to the short-comings of fractal based compression schemes, such as the tiling effect appearing in reconstructed images, a wavelet based analysis of fractal image compression is presented. This is the link that produces fractal coding in the wavelet domain, and presents a hybrid coding scheme called fractal-wavelet coders. We show that by using smooth wavelet basis in computing the wavelet transform, the tiling effect of fractal systems can be removed. The few wavelet-fractal coders that have been proposed in literature are discussed, showing advantages over the traditional fractal coders. This dissertation will present a new low-bit rate video compression system that is based on fractal coding in the wavelet domain. This coder makes use of the advantages of both the wavelet and fractal coders discussed in their review. The self-similarity property of fractal coders exploits the high spatial and temporal correlation between video frames. Thus the fractal coding step gives an approximate representation of the coded frame, while the wavelet technique adds detail to the frame. In this proposed scheme, each frame is decomposed using the pyramidal multi-resolution wavelet transform. Thereafter a motion detection operation is used in which the subtrees are partitioned into motion and non-motion subtrees. The nonmotion subtrees are easily coded by a binary decision, whereas the moving ones are coded using the combination of the wavelet SPIHT and fractal variable subtree size coding scheme. All intra-frame compression is performed using the SPIHT compression algorithm and inter-frame using the fractal-wavelet method described above. The proposed coder is then compared to current low bit-rate video coding standards such as the H.263+ and MPEG-4 coders through analysis and simulations. Results show that the proposed coder is competitive with the current standards, with a performance improvement been shown in video sequences that do not posses large global motion. Finally, a real-time implementation of the proposed algorithm is performed on a digital signal processor. This illustrates the suitability of the proposed coder being applied to numerous multimedia applications. / Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, Durban, 2004.

Migrating from a wired to a wireless implementation for communication system used in industrial applications is a logical move to avoid the many shortcomings associated with wires. When operated under harsh environments, those wires can break and could cause not only damage to the system but also endanger human lives. However, it is not well documented how well a wireless protocol can work under such harsh industrial environments. This thesis attempts to answer this research question in the point of view of gauging the performance of a wireless protocol under the influence of electromagnetic noise. More specifically, the type of noise signal that is the focus of this investigation is the random, pulsed type (e.g., discharges caused by sparking) that creates a hyperbolic broadband disturbance in the frequency domain. Consequently, a fractal noise model is used to study noise of this nature. The steps toward achieving this goal include: requirements gathering, wireless technology selection; noise modelling and synthesis; real noise capture and analysis to validate the chosen noise model; high-frequency fractal noise emulation in hardware; the use of a novel noise injection method for empirical work; and the conducting of a controlled synthetic noise-to-wireless node performance evaluation to obtain performance measure in the form of packet error rate (PER). Performance data in terms of PER versus signal-to-noise ratio (SNR) for various nodes separation have been collected. There were three significant findings: the obtained performance curves follow the standard 'S' trend; for a specific desired reliability (denoted by a certain PER), the SNR at the transmitter needs to be boosted as the correlation of the noise being present increases; and the maximum distance between nodes separation for a certain reliability to be achieved depends exponentially with the transmitter‟s SNR. The relationship in the third finding assists in placement of wireless nodes, which in turn can determine the minimum amount of wireless nodes required for an industrial system to reach the desired system reliability, thus boasting network cost saving.

Humans are experts at understanding what they see. Similarity and analogy play a significant role in making sense of the visual world by forming analogies to similar images encountered previously. Yet, while these acts of visual reasoning may be commonplace, the processes of visual analogy are not yet well understood. In this dissertation, I investigate the utility of representing visual information in a fractal manner for computing visual similarity and analogy. In particular, I develop a computational technique of fractal reasoning for addressing problems of visual similarity and novelty. I illustrate the effectiveness of fractal reasoning on problems of visual similarity and analogy on the Raven’s Progressive Matrices and Miller’s Analogies tests of intelligence, problems of visual novelty and oddity on the Odd One Out test of intelligence, and problems of visual similarity and oddity on the Dehaene test of core geometric reasoning. I show that the performance of my computational model on these various tests is comparable to human performance. Fractal reasoning provides a new method for computing answers to such problems. Specifically, I show that the choice of the level of abstraction of problem representation determines the degree to which an answer may be regarded as confident, and that that choice of abstraction may be controlled automatically by the algorithm as a means of seeking that confident answer. This emergence of ambiguity and its remedy via problem re-representation is afforded by the fractal representation. I also show how reasoning over sparse data (at coarse levels of abstraction) or homogeneous data (at finest levels of abstraction) could both drive the automatic exclusion of certain levels of abstraction, as well as provide a signal to shift the analogical reasoning from consideration of simple analogies (such as analogies between pairs of objects) to more complex analogies (such as analogies among triplets, or larger groups of objects). My dissertation also explores fractal reasoning in perception, including both biologically-inspired imprinting and bistable perception. In particular, it provides a computational explanation of bistable perception in the famous Necker cube problem that is directly tied to the process of determining a confident interpretation via re-representation. Thus, my research makes two primary contributions to AI theories of visual similarity and analogy. The first contribution is the Extended Analogy By Recall (ABR*) algorithm, the computational technique for visual reasoning that automatically adjusts fractal representations to an appropriate level of abstraction. The second contribution is the fractal representation itself, a knowledge representation that add the notion of self-similarity and re-representation to analogy making.

Migrating from a wired to a wireless implementation for communication system used in industrial applications is a logical move to avoid the many shortcomings associated with wires. When operated under harsh environments, those wires can break and could cause not only damage to the system but also endanger human lives. However, it is not well documented how well a wireless protocol can work under such harsh industrial environments. This thesis attempts to answer this research question in the point of view of gauging the performance of a wireless protocol under the influence of electromagnetic noise. More specifically, the type of noise signal that is the focus of this investigation is the random, pulsed type (e.g., discharges caused by sparking) that creates a hyperbolic broadband disturbance in the frequency domain. Consequently, a fractal noise model is used to study noise of this nature. The steps toward achieving this goal include: requirements gathering, wireless technology selection; noise modelling and synthesis; real noise capture and analysis to validate the chosen noise model; high-frequency fractal noise emulation in hardware; the use of a novel noise injection method for empirical work; and the conducting of a controlled synthetic noise-to-wireless node performance evaluation to obtain performance measure in the form of packet error rate (PER). Performance data in terms of PER versus signal-to-noise ratio (SNR) for various nodes separation have been collected. There were three significant findings: the obtained performance curves follow the standard 'S' trend; for a specific desired reliability (denoted by a certain PER), the SNR at the transmitter needs to be boosted as the correlation of the noise being present increases; and the maximum distance between nodes separation for a certain reliability to be achieved depends exponentially with the transmitter‟s SNR. The relationship in the third finding assists in placement of wireless nodes, which in turn can determine the minimum amount of wireless nodes required for an industrial system to reach the desired system reliability, thus boasting network cost saving.

Thesis (Ph. D.)--University of Hawaii at Manoa, 1994. / Includes bibliographical references (leaves 222-247). / Microfiche. / xvii, 246 p. ill. (some col.), maps 29 cm

El objetivo de esta investigación es descubrir la existencia de patrones gráficos fractales en el comportamiento colectivo de los españoles, en relación a su identidad ideológica. La hipótesis principal es la presencia de patrones fractales en la serie temporal de la identidad ideológica en España. Asimismo, estas series deberían mostrar ciclos característicos y su comportamiento no debería ser aleatorio.En cuanto al diseño y metodología, se utiliza una serie temporal de 251 puntos recogidos de la escala ideológica de los Barómetros del Centro de Investigaciones Sociológicas (CIS) de España, a nivel nacional. Esta serie se extiende desde junio de 1983 hasta abril de 2006. La metodología utilizada fue a través del Rescaled Range Analysis o Hurst Exponent. Este análisis detecta la presencia de características fractales en series temporales, permitiendo descubrir persistencia o antipersistencia en el comportamiento de la serie, además de ciclos de volatilidad, tal como se realiza en valores de acciones en bolsa.En relación a los resultados se observa que la serie general presenta un valor Hurst H, de 0,091, indicando la presencia de características fractales. La antipersistencia de la serie sugiere la presencia de mecanismos regulatorios de feed-back negativos para favorecer la estabilidad dinámica del sistema. La serie tiene un valor H de 0,398 en ventanas de tiempo menores a 45 meses y de H 0,115 en ventanas mayores, lo que indica la presencia de ciclos de cambio de alrededor de esa cantidad de meses. Por su parte, las series de los distintos sectores políticos desde la extrema izquierda a la extrema derecha presentan también características antipersistentes y ciclos de cambio.Dentro de las conclusiones, se puede decir que, al igual que en economía, la aplicación del Exponente de Hust en el análisis de comportamientos colectivos en política, permiten dar cuenta de su volatilidad y ciclos característicos. Pareciera que la fractalidad se da en muchos sistemas naturales y comportamientos humanos, abarcando también a la política. Este estudio ha podido determinar la fractalidad del comportamiento político español, de utilidad para el pronóstico de comportamiento de voto y, principalmente, de ayuda para comprender el surgimiento de la opinión pública. / The aim of this study is to demonstrate the existence of fractal graphical patterns in the collective behaviour of the Spanish people about their ideological identity. The main hypothesis is that the time series of the ideological identity (Left-Right Scale) in Spain will present fractal patterns of functioning. Also, these series should show characteristic cycles and its behaviour would not be a random walk.DESIGN AND METHODS: A time series of 251 points gathered of the ideological scale of the Barometers of the Sociologic Research Centre (CIS) of Spain (national level) is analyzed. This series extends from June of 1983 to April of 2006. The methodology used was Rescaled Range Analysis (Hurst Exponent). This analysis detects the presence of fractal features in time series, allowing to discover persistence or antipersistence in the behaviour of series, in addition to cycles of volatility, as it occurs in stock market prices.RESULTS: The general series displays a Hurst H = 0.091, indicating presence of fractal features. The antipersistence of the series suggests an action of regulatory mechanisms of negative feed-back to get dynamic stability of the system. The series has a value H of 0.398 in time lag smaller than 45 months and H 0.115 in greater lag, which indicates the existence of cycles of change, around this time lag. The series of the political sectors (from the extreme left to the extreme right) also displays antipersistence and cycles of change.CONCLUSION: Like an econometric study, the application of Hurst Exponent in the analysis of collective behaviours in policy, allows to report their volatility and characteristic cycles. It seems to be that the fractality occurs in many natural systems and human behaviours, also including to the politic issues. This study has been able to determine the fractality of the Spanish political behaviour, utility to forecast voting behaviour and, mainly, aid to understand the emergence of public opinion.

Fractals are popular because of their ability to create complex images using only several simple codes. This is possible by capturing image redundancy and presenting the image in compressed form using the self similarity feature. For many years fractals were used for image compression. In the last few years they have also been used for face recognition. In this research we present new fractal methods for recognition, especially human face recognition. This research introduces three new methods for using fractals for face recognition, the use of fractal codes directly as features, Fractal image-set coding and Subfractals. In the first part, the mathematical principle behind the application of fractal image codes for recognition is investigated. An image Xf can be represented as Xf = A x Xf + B which A and B are fractal parameters of image Xf . Different fractal codes can be presented for any arbitrary image. With the defnition of a fractal transformation, T(X) = A(X - Xf ) + Xf , we can define the relationship between any image produced in the fractal decoding process starting with any arbitrary image X0 as Xn = Tn(X) = An(X - Xf ) + Xf . We show that some choices for A or B lead to faster convergence to the final image. Fractal image-set coding is based on the fact that a fractal code of an arbitrary gray-scale image can be divided in two parts - geometrical parameters and luminance parameters. Because the fractal codes for an image are not unique, we can change the set of fractal parameters without significant change in the quality of the reconstructed image. Fractal image-set coding keeps geometrical parameters the same for all images in the database. Differences between images are captured in the non-geometrical or luminance parameters - which are faster to compute. For recognition purposes, the fractal code of a query image is applied to all the images in the training set for one iteration. The distance between an image and the result after one iteration is used to define a similarity measure between this image and the query image. The fractal code of an image is a set of contractive mappings each of which transfer a domain block to its corresponding range block. The distribution of selected domain blocks for range blocks in an image depends on the content of image and the fractal encoding algorithm used for coding. A small variation in a part of the input image may change the contents of the range and domain blocks in the fractal encoding process, resulting in a change in the transformation parameters in the same part or even other parts of the image. A subfractal is a set of fractal codes related to range blocks of a part of the image. These codes are calculated to be independent of other codes of the other parts of the same image. In this case the domain blocks nominated for each range block must be located in the same part of the image which the range blocks come from. The proposed fractal techniques were applied to face recognition using the MIT and XM2VTS face databases. Accuracies of 95% were obtained with up to 156 images.

Thesis (Honors)--College of William and Mary, 2009. / Includes bibliographical references (leaves 56-57). Also available via the World Wide Web.