A qualitative physics based on confluences

In this paper I will deal with the effect of external random perturbations, “noise”, on chemical systems and other open nonlinear systems. As a concrete example let us consider a CSTR. This is an open system and as such subject to external constraints, namely the concentrations of the chemical species in the feed streams, the flow rate, the stirring rate, the temperature, and the incident light intensity in the case of a photochemical reaction. These external constraints characterize the state of the environment of the open system and will, in general, fluctuate more or less strongly. Such environmental fluctuations are particularly important for natural systems; here random fluctuations are always present and their amplitude is not necessarily small as in laboratory systems. In the latter systems the experimenter will of course try to minimize the effect of random perturbations, though it is impossible to eliminate noise completely. Clearly, random external noise is ubiquitous in open systems, but this fact by itself would hardly warrant a systematic study of the effects of external fluctuations. The question is whether noise is more than a mere nuisance we have to live with. Is there any hope of finding interesting physics? The intuitive, and wrong, answer would be negative: The system averages out rapid fluctuations and the only trace of external noise would be a certain fuzziness in the state of the system. Of course, if the state of the system becomes unstable, the fluctuations initiate the departure from the unstable state. Then the dynamics of the system take over and the system evolves to a new stable state.

Noise-induced transitions

Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown real-world signals, then ...

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Scale-Space Theory : A Basic Tool for Analysing Structures at Different Scales

When computing descriptors of image data, the type of information that can be extracted may be strongly dependent on the scales at which the image operators are applied. This article presents a systematic methodology for addressing this problem. A mechanism is presented for automatic selection of scale levels when detecting one-dimensional image features, such as edges and ridges.

A novel concept of a scale-space edge is introduced, defined as a connected set of points in scale-space at which: (i) the gradient magnitude assumes a local maximum in the gradient direction, and (ii) a normalized measure of the strength of the edge response is locally maximal over scales. An important consequence of this definition is that it allows the scale levels to vary along the edge. Two specific measures of edge strength are analyzed in detail, the gradient magnitude and a differential expression derived from the third-order derivative in the gradient direction. For a certain way of normalizing these differential descriptors, by expressing them in terms of so-called γ-normalized derivatives, an immediate consequence of this definition is that the edge detector will adapt its scale levels to the local image structure. Specifically, sharp edges will be detected at fine scales so as to reduce the shape distortions due to scale-space smoothing, whereas sufficiently coarse scales will be selected at diffuse edges, such that an edge model is a valid abstraction of the intensity profile across the edge.

Since the scale-space edge is defined from the intersection of two zero-crossing surfaces in scale-space, the edges will by definition form closed curves. This simplifies selection of salient edges, and a novel significance measure is proposed, by integrating the edge strength along the edge. Moreover, the scale information associated with each edge provides useful clues to the physical nature of the edge.

With just slight modifications, similar ideas can be used for formulating ridge detectors with automatic selection, having the characteristic property that the selected scales on a scale-space ridge instead reflect the width of the ridge.

It is shown how the methodology can be implemented in terms of straightforward visual front-end operations, and the validity of the approach is supported by theoretical analysis as well as experiments on real-world and synthetic data.

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Edge Detection and Ridge Detection with Automatic Scale Selection

Catastrophe theory and its applications

0. Fundamental Not(at)ions.- I. Real Vector Spaces.- II. Affine Spaces.- III. Dual Spaces.- IV. Metric Vector Spaces.- V. Tensors and Multilinear Forms.- VI Topological Vector Spaces.- VII. Differentiation and Manifolds.- VIII. Connections and Covariant Differentiation.- IX. Geodesics.- X. Curvature.- XI. Special Relativity.- XII. General Relativity.- Index of Notations.

Tensor Geometry: The Geometric Viewpoint and Its Uses

Taylor expansions and catastrophes

Multistable figures show that the stimulus-percept relation is not a single valued function. We therefore propose a tentative nonlinear model on the hypothesis that the graph of this relation is the equilibrium set of a dynamic system. For simplicity and to obtain testable predictions, we consider a system whose bifurcations are gradient-like and thus generically described by the elementary catastrophes. We motivate this general model, and then show how, in conjunction with the principle of minimal singularity, it implies cusp catastrophe geometry in a specific perceptual example. Indeed, we argue for canonical cusp geometry in this case. The model incorporates naturally certain observed features of multistable perception, such as hysteresis and bias effects. Despite being a continuum model it is naturally compatible with the subjective dichotomy of bistable perception. The model makes testable predictions which may easily be extended to other specific examples of multistable perception.

Timothy Poston

Papers

Catastrophe theory and its applications

Tensor Geometry: The Geometric Viewpoint and Its Uses

Taylor expansions and catastrophes

Nonlinear modeling of multistable perception

Winding and Euler numbers for 2D and 3D digital images