Alan D. Sloan

Harnessing chaos for image synthesis

Chaotic dynamics can be used to model shapes and render textures in digital images. This paper addresses the problem of how to model geometrically shapes and textures of two dimensional images using iterated function systems. The successful solution to this problem is demonstrated by the production and processing of synthetic images encoded from color photographs. The solution is achieved using two algorithms: (1) an interactive geometric modeling algorithm for finding iterated function system codes; and (2) a random iteration algorithm for computing the geometry and texture of images defined by iterated function system codes. Also, the underlying mathematical framework, where these two algorithms have their roots, is outlined. The algorithms are illustrated by showing how they can be used to produce images of clouds, mist and surf, seascapes and landscapes and even faces, all modeled from original photographs. The reasons for developing iterated function systems algorithms include their ability to produce complicated images and textures from small databases, and their potential for highly parallel implementation.

Fractal modeling of biological structures.

The purpose of this paper is to introduce to a general audience of scientists some recent mathematical results and to show how they can be applied to the construction of geometrical models for physical structures. Euclidean geometry works well to describe the conformation of elements in a building. However, it is an inefficient tool for modeling the placement of a quarter of a million pine needles on a pine tree. The basic tools of Euclidean geometry are readily available; they are straightedge and compass, and include some knowledge of how to write down equations for lines and circles in the Cartesian plane. Here, in equally simple terms, we present a basic tool for working with fractal geometry. Possible applications of the technique include the construction of geometrical models for features of plants; the spread of a virus on the surface of a human lung; the blood system; tissue masses; dynamical processes, such as growth of plants or networks; and functions on biological structures, such as temperatures on ferns. Some precisions are in order: ( 1 ) By a fractal, we mean here any subset of R” (typically n = 2, 3, and 4) that possesses features that are not simplified by magnification (observation at successively higher visual resolution). In two dimensions, a location on a set is simplified by magnification if it reveals a straight-line segment or isolated point in the asymptotic limit of infinite magnification. This definition is more general than the usual one that states that the Hausdorff-Besicovitch dimension of the set exceeds its topological dimension. (2) A set, such as a Sierpinski Triangle or Classical Cantor Set, which is made exactly of “little copies of itself,” is likely to be a fractal; however, in the sense and spirit with which we use the word, it would be a very special case. The fractal geometrical models that we describe here are, in general, much more complicated. Features that are apparent a t one location may not be present a t other locations nor be retrieved upon closer inspection. (3) We are concerned with deterministic geometry. Thus, any model produced will always be the same subset of R” irrespective of how many times it is regenerated. We are not concerned with random fractal geometries. Interest in the latter resides in their statistical properties; deterministic fractals may be used to model the exact structure of a specific object over a range of scales. (4) All geometrical models for physical entities are inevitably wrong at some high enough magnification. The architect’s drawing of a straight line representing the edge of a roof breaks down as a model if it is examined closely enough. On fine enough scales, the edge of the roof is wriggly, while the (intended) drawing remains endlessly flat. Fractal geometry can provide a better model for the edge of the roof: the model may appear as straight a t one scale of

The polaron without cutoffs in two space dimensions

Hamiltonians for the polaron of fixed total momentum are defined using momentum cutoffs. A renormalized Hamiltonian of fixed total momentum is defined in two space dimensions by proving the strong convergence of the resolvents of the cutoff Hamiltonians. The Hamiltonian for the physical polaron is defined as the direct integral of the fixed momentum Hamiltonians.

A nonperturbative approach to nondegeneracy of ground states in quantum field theory: Polaron models

Abstract An operator on an L2 space is said to maximize support if it takes every function not identically zero into a function which differs from zero almost everywhere. It will be shown that the semigroups generated by certain quantum field theory Hamiltonians maximize support. A simple consequence of the semigroups maximizing support is the nondegeneracy of ground states for the Hamiltonian.

Characteristic methods for multidimensional evolution equations

Abstract The stochastic method of characteristics is generalized to analyze multidimensional evolution equations of arbitrary order with constant coefficients. A linear space of abstract characteristics is introduced along with algorithms for associating characteristics with evolutions. A projection from a space of functions of characteristics to solutions of evolution equations plays the role of a generalized expectation. The solution is then viewed as an “average” of initial data. Examples are presented which indicate possible extensions of the theory presented here to equations with variable coefficients.

An Algebraic Generalization of Stochastic Integration

The authors use a generalization of stochastic integration to analyze real constant coefficient elliptic operators of order n which generate strongly continuous semigroups on L 2 ( k ). Let p n (x, t) be the fundamental solution to u t = Au where for n even. For real polynomials q l ,…,q k on m let q = (q l ,…,q k ) and define integral operators F n,q (t), for each t > 0 on functions on k by The authors determine when e tQ = lim F(t/j) J strongly on L 2 ( k ) as j →, for some q on some m .

Product formulas for semigroups with elliptic generators

Abstract Real constant coefficient n th order elliptic operators, Q , which generate strongly continuous semigroups on L 2 ( R k ) are analyzed in terms of the elementary generator, A = (−n) ( n 2 − 1 )(n!) −1 ∑ k j = 1 ∂ n ∂x j n , for n even. Integral operators are defined using the fundamental solutions p n ( x , t ) to u t = Au and using real polynomials q l ,…, q k on R m by the formula, for q = ( q l ,…, q k ), (F(t)ƒ)(x) = ∫ R m ƒ(x + q(z)) P n (z, t)dz . It is determined when, strongly on L 2 ( R k ), e tQ = lim j → ∞ F t j j . If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.

An application of the nonstandard Trotter product formula

The nonstandard Trotter product formula is used to extend the Feynman integral interpretation of solutions to the Schrodinger equation in the presence of a highly singular potential.

Fractal Image Encoding

Increasingly, the output of physical and numerical experiments is presented as two dimensional images, instead of as tables and graphs of observed real variables. Instances include pictures of diffusion limited aggregates, fractal fingering boundaries between fluids, and images of turbulent flows. One approach to understanding the intricate geometries of such experimental data is traditional: try to isolate physically meaningful real parameters, such as fractal dimension (Mandelbrot, 1982), by passing straight lines through data points derived from the pictures. Another approach is to attempt to approximate the image with a geometrical entity whose structure is, despite appearances, quite simple. Straight lines, squares, circles and other fundamentally Euclidean models will not suffice for this purpose when the data possesses structure which cannot be fully resolved or simplified by rescaling (i.e., magnification). The first purpose of this paper is to describe in simple terms how fractal geometries may be succinctly understood in terms of iterated function systems and to explain how one can go about finding a fractal model to fit given two dimensional data. The second purpose is to relate attractors for iterated function systems to attractors for cellular automata (Wolfram, 1983). The latter is motivated by the following question: Once one has found an iterated function system encoding of an image, can one deduce a set of cellular automata rules which would allow one to construct the image of a lattice? The answer is “Yes”, which delights us for this reason. For some time, via the Collage Theorem (Barnsley, et al., 1984), one has known how to make iterated function system models for ferns and leaves; now one may also model the manner in which these forms both grow and stop growing.

Product formulas for solutions of initial value partial differential equations, II

Abstract Families of operators which approximate semi-groups or evolution systems generated by partial differential operators are constructed. Product formulas are used to recover these semi-groups or evolution systems through product integrals. Conditions on generators are provided under which its semi-group or evolution system can be approximated in this way by families of specific types of operators.