Fractals

Survey is time-consuming and expensive. Therefore, it needs to be both effective and efficient. Some archaeologists have argued that current survey techniques are not effective (Shott 1985, 1989), but most archaeologists continue to employ these methods and therefore must believe they are effective. If our survey techniques are effective, why do simulations suggest otherwise? If they are ineffective, can we improve them? The answers to these practical questions depend on the topological characteristics of archaeological site distributions. In this study I analyze archaeological site distributions in the Valley of Oaxaca, Mexico, using lacunarity and fractal dimension. Fractal dimension is a parameter of fractal patterns, which are complex, space-filling designs exhibiting self-similarity and power-law scaling. Lacunarity is a statistical measure that describes the texture of a spatial dispersion. It is useful in understanding how archaeological tests should be spaced during surveys. Between these two measures, I accurately describe the regional topology and suggest new considerations for archaeological survey design.

A coastline is an example of a statistically self-similar fractal. A standard characterization walks a ruler of fixed size along the coast and estimates fractal dimension from the power-law relationship between length and ruler size. Multiple intersection can lead to ambiguity in choosing the next step. The standard method always chooses the first intersection along the curve. Variations were developed to choose intersections which highlight geographic properties. The land method measures accessibility to the coast from land at each size while the water method probes water access. Measurements on sections of the East and West Coasts of the United States typically showed the land length exceeding water. Jumps in water length as step size decreased were often caused by narrow rivers or bays which have few corresponding land features. Simple recursive constructions were inadequate to model this asymmetry.

Application of fractal concepts to geographic problems is reviewed. The basics of fractals are presented. Scale, self-similarity and their effects on natural phenomena are discussed. Eight sites (six wetland and two non-wetland) are studied. Using Landsat Multi-Spectral Scanner data, the fractal dimensions of the surface reflectance of the sites are calculated by the contour method. Values obtained are related to the type of wetland and the structure and processes that characterize that type of wetland. Forested wetlands were found to have the lowest fractal dimension. Mixed forested and non-forested had the highest. Marshy non-forested sites were higher than expected. Conclusions drawn are that physical characteristics of different wetlands result in measurable differences in fractal dimension. Fractals will be valuable in computer aided wetland identification in the future.

Rapid Eye Movement (REM) sleep in adult and neonatal mammals is characterized by episodes of high variability and bursting in brainstem sites associated with spontaneous tonic and phasic behavioral events such as REMs, nuchal inactivity and twitches of the body. REM sleep is the principal behavioral state during fetal and neonatal life and as has been demonstrated by various REM deprivation procedures to be indispensable during this period and to lead to long lasting behavioral defects in adult life. The guiding hypothesis throughout this dissertation is that the variability of REM-associated nuchal atonia episodes and other spontaneous motor events reflects the fractal time patterns of a global fetal REM sleep state over multiple timescales serving as a transient behavioral ontogenetic adaptation to changing developmental environments. Further, spontaneous activity over many levels of organization, including phasic REM motor activity during ontogeny, could play a fundamental role in the development of appetitive behavioral processes (e.g., searching and orienting) and other forms of neuroplasticity (e.g., learning and dynamic regulation of receptor fields and maps). The nature of this variability was investigated by measuring the durations of nuchal atonia over extended periods in fetal sheep and neonatal rats, species which are in a REM sleep-like state >50% of the time. Hurst's rescaled range analysis, which affords comparisons between natural time series with short- and long-term correlated fluctuations, indicated that variability in both species over short time scales is statistically similar to longer time scales (i.e., is fractal in time) and remarkably stable over the developmental periods examined. Spontaneous nuchal events in both species were also found to be described by convolutionally stable self-similar Levy distributions, suggesting that activity associated with fetal REM sleep could provide a stable, scale invariant source of correlated stimulation, facilitating integration of new neural changes into developing motor and cortical networks over gestation. These fractal time descriptions of spontaneous prenatal behaviors have implications for conceptualizing the evolutionary mechanisms underlying heterochrony (shifting self-affine relationships between the timing of gene expression and behavioral activity) and the plasticity essential to the genesis of behavioral neophenotypes.

The main objective of the research is to develop computationally efficient hybrid coding schemes for the low bit implementations of image frames and image sequences. The basic fractal block coding can compress a relatively low resolution image efficiently without blocky artifacts, but it does not converge well at the high frequency edges. This research proposes a hybrid multi-resolution scheme which combines the advantages of fractal and DCT coding schemes. The fractal coding is applied to get a lower resolution, quarter size output image and DCT is then used to encode the error residual between original full bandwidth image signal and the fractal decoded image signal. At the decoder side, the full resolution, full size reproduced image is generated by adding decoded error image to the decoded fractal image. Also, the lower resolution, quarter size output image is automatically given by the iteration function scheme without having to spend extra effort. Other advantages of the scheme are that the high resolution layer is generated by error image which covers the bandwidth loss of the lower resolution layer as well as the coding error of the lower resolution layer, and that it does not need a sophisticated classification procedure. A series of computer simulation experiments are conducted and their results are presented to illustrate the merit of the scheme. The hybrid fractal coding method is then extended to process motion sequences as well. A new scheme is proposed for motion vector detection and motion compensation, by judiciously combining the techniques of fractal compression and block matching. The advantage of this scheme is that it improves the performance of the motion compensation, while keeping the overall computational complexity low for each frame. The simulation results on realistic video conference image sequences support the superiority of the proposed method in terms of reproduced picture quality and compression ratio.