Fractals

Previously it has been shown that the branching pattern of pyramidal cells varies markedly between different cortical areas in simian primates. These differences are thought to influence the functional complexity of the cells. In particular, there is a progressive increase in the fractal dimension of pyramidal cells with anterior progression through cortical areas in the occipitotemporal (OT) visual stream, including the primary visual area (V1), the second visual area (V2), the dorsolateral area (DL, corresponding to the fourth visual area) and inferotemporal cortex (IT). However, there are as yet no data on the fractal dimension of these neurons in prosimian primates. Here we focused on the nocturnal prosimian galago (Otolemur garnetti). The fractal dimension (D), and aspect ratio (a measure of branching symmetry), was determined for 111 layer III pyramidal cells in V1, V2, DL and IT. We found, as in simian primates, that the fractal dimension of neurons increased with anterior progression from V1 through V2, DL, and IT. Two important conclusions can be drawn from these results: (1) the trend for increasing branching complexity with anterior progression through OT areas was likely to be present in a common primate ancestor, and (2) specialization in neuron structure more likely facilitates object recognition than spectral processing.

Self-similarity of plants has attracted the attention of biologists for at least 50 years, yet its formal treatment is rare, and no measure for quantifying the degree of self-similarity currently exists. We propose a formal definition and measures of self-similarity, tailored to branching plant structures. To evaluate self-similarity, we make use of an algorithm for computing topological distances between branching systems, developed in computer science. The formalism is illustrated using theoretical branching systems, and applied to analyze self-similarity in two sample plant structures: inflorescences of Syringa vulgaris (lilac) and shoots of Oryza sativa (rice).

We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability": at each magnification level any V-variable fractals has at most V key "forms" or "shapes". V-variable random fractals have the surprising property that they can be computed using a forward process. More precisely, a version of the usual Random Iteration Algorithm, operating on sets (or measures) rather than points, can be used to sample each family. To present this theory, we review relevant results on fractals (and fractal measures), both deterministic and random. Then our new results are obtained by constructing an iterated function system (a super IFS) from a collection of standard IFSs together with a corresponding set of probabilities. The attractor of the super IFS is called a superfractal; it is a collection of V-variable random fractals (sets or measures) together with an associated probability distribution on this collection. When the underlying space is for example ℝ², and the transformations are computationally straightforward (such as affine transformations), the superfractal can be sampled by means of the algorithm, which is highly efficient in terms of memory usage. The algorithm is illustrated by some computed examples. Some variants, special cases, generalizations of the framework, and potential applications are mentioned.

We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first- and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}_t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t).

We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents.

The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

We consider nonlinear generalizations of fractal interpolating functions applied to functions of one and two variables. The use of such interpolating functions in representing images by an iterated function system is illustrated, particularly as applied to the problem of resizing images, or portions thereof.