Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

1 Graphical modeling using L-systems.- 1.1 Rewriting systems.- 1.2 DOL-systems.- 1.3 Turtle interpretation of strings.- 1.4 Synthesis of DOL-systems.- 1.4.1 Edge rewriting.- 1.4.2 Node rewriting.- 1.4.3 Relationship between edge and node rewriting.- 1.5 Modeling in three dimensions.- 1.6 Branching structures.- 1.6.1 Axial trees.- 1.6.2 Tree OL-systems.- 1.6.3 Bracketed OL-systems.- 1.7 Stochastic L-systems.- 1.8 Context-sensitive L-systems.- 1.9 Growth functions.- 1.10 Parametric L-systems.- 1.10.1 Parametric OL-systems.- 1.10.2 Parametric 2L-systems.- 1.10.3 Turtle interpretation of parametric words.- 2 Modeling of trees.- 3 Developmental models of herbaceous plants.- 3.1 Levels of model specification.- 3.1.1 Partial L-systems.- 3.1.2 Control mechanisms in plants.- 3.1.3 Complete models.- 3.2 Branching patterns.- 3.3 Models of inflorescences.- 3.3.1 Monopodial inflorescences.- 3.3.2 Sympodial inflorescences.- 3.3.3 Polypodial inflorescences.- 3.3.4 Modified racemes.- 4 Phyllotaxis.- 4.1 The planar model.- 4.2 The cylindrical model.- 5 Models of plant organs.- 5.1 Predefined surfaces.- 5.2 Developmental surface models.- 5.3 Models of compound leaves.- 6 Animation of plant development.- 6.1 Timed DOL-systems.- 6.2 Selection of growth functions.- 6.2.1 Development of nonbranching filaments.- 6.2.2 Development of branching structures.- 7 Modeling of cellular layers.- 7.1 Map L-systems.- 7.2 Graphical interpretation of maps.- 7.3 Microsorium linguaeforme.- 7.4 Dryopteris thelypteris.- 7.5 Modeling spherical cell layers.- 7.6 Modeling 3D cellular structures.- 8 Fractal properties of plants.- 8.1 Symmetry and self-similarity.- 8.2 Plant models and iterated function systems.- Epilogue.- Appendix A Software environment for plant modeling.- A.1 A virtual laboratory in botany.- A.2 List of laboratory programs.- Appendix B About the figures.- Turtle interpretation of symbols.

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The Algorithmic Beauty of Plants

An intelligent electronic appliance preferably includes a user interface, data input and/or output port, and an intelligent processor. A preferred embodiment comprises a set top box for interacting with broadband media streams, with an adaptive user interface, content-based media processing and/or media metadata processing, and telecommunications integration. An adaptive user interface models the user, by observation, feedback, and/or explicit input, and presents a user interface and/or executes functions based on the user model. A content-based media processing system analyzes media content, for example audio and video, to understand the content, for example to generate content-descriptive metadata. A media metadata processing system operates on locally or remotely generated metadata to process the media in accordance with the metadata, which may be, for example, an electronic program guide, MPEG 7 data, and/or automatically generated format. A set top box preferably includes digital trick play effects, and incorporated digital rights management features.

Intelligent electronic appliance system and method

An adaptive interface for a programmable system, for predicting a desired user function, based on user history, as well as machine internal status and context. The apparatus receives an input from the user and other data. A predicted input is presented for confirmation by the user, and the predictive mechanism is updated based on this feedback. Also provided is a pattern recognition system for a multimedia device, wherein a user input is matched to a video stream on a conceptual basis, allowing inexact programming of a multimedia device. The system analyzes a data stream for correspondence with a data pattern for processing and storage. The data stream is subjected to adaptive pattern recognition to extract features of interest to provide a highly compressed representation which may be efficiently processed to determine correspondence. Applications of the interface and system include a VCR, medical device, vehicle control system, audio device, environmental control system, securities trading terminal, and smart house. The system optionally includes an actuator for effecting the environment of operation, allowing closed-loop feedback operation and automated learning.

Ergonomic man-machine interface incorporating adaptive pattern recognition based control system

The author proposes an independent and novel approach to image coding, based on a fractal theory of iterated transformations. The main characteristics of this approach are that (i) it relies on the assumption that image redundancy can be efficiently exploited through self-transformability on a block-wise basis, and (ii) it approximates an original image by a fractal image. The author refers to the approach as fractal block coding. The coding-decoding system is based on the construction, for an original image to encode, of a specific image transformation-a fractal code-which, when iterated on any initial image, produces a sequence of images that converges to a fractal approximation of the original. It is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0.5-1.0 b/pixel. The fractal block coder has performance comparable to state-of-the-art vector quantizers. >

Image coding based on a fractal theory of iterated contractive image transformations

Recurrent iterated function systems generalize iterated function systems as introduced by Barnsley and Demko [BD] in that a Markov chain (typically with some zeros in the transition probability matrix) is used to drive a system of maps w j : K→ K, j = 1, 2,…, N, where K is a complete metric space. It is proved that under “average contractivity,” a convergence and ergodic theorem obtains, which extends the results of Barnsley and Elton [BE]. It is also proved that a Collage Theorem is true, which generalizes the main result of Barnsley et al. [BEHL] and which broadens the class of images which can be encoded using iterated map techniques. The theory of fractal interpolation functions [B] is extended, and the fractal dimensions for certain attractors is derived, extending the technique of Hardin and Massopust [HM]. Applications to Julia set theory and to the study of the boundary of IFS attractors are presented.

Recurrent iterated function systems

Consider a Markov process on a locally compact metric space arising from iteratively applying maps chosen randomly from a finite set of Lipschitz maps which, on the average, contract between any two points (no map need be a global contraction). The distribution of the maps is allowed to depend on current position, with mild restrictions. Such processes have unique stationary initial distribution [BE], [BDEG].We show that, starting at any point, time averages along trajectories of the process converge almost surely to a constant independent of the starting point. This has applications to computer graphics.

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An ergodic theorem for iterated maps

On considere un processus de Markov en temps discret sur un espace metrique localement compact obtenu par iteration aleatoire des cartes de Lipschitz w 1 , w 2 , , w n

/pdf/invariant-measures-for-markov-processes-arising-from-2gxen5n75f.pdf

Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities

Interpolation functions $f:[0,1] \to \mathbb{R}$ of the following nature are constructed. Given data \[ \left\{ {\left( {t_n ,x_n } \right) \in [0,1] \times \mathbb{R}:n = 0,1,2, \cdots ,N} \right\}\] with $0 = t_0 < t_1 < \cdots <' t_N = 1$, f obeys \[ f(t_n ) = x_n ,\qquad n = 0,1,2, \cdots ,N.\] Furthermore, the graph of f is the projection of a set G in $\mathbb{R}^M $ (M an integer greater than or equal to 2) that is homeomorphic to $[0,1]$ and is the attractor for an iterated function system consisting of afflne maps in $\mathbb{R}^M $. The latter characterization ensures that f can be computed rapidly while possessing many ”hidden” variables, on which its values continuously depend, which allow great flexibility and diversity in the interpolant, making it potentially useful in approximation theory. Estimtes and exact values for the fractal dimensions of G and the graph of f are obtained.

Hidden variable fractal interpolation functions

A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained. Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

John H. Elton

Papers

Recurrent iterated function systems

An ergodic theorem for iterated maps

Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities

Hidden variable fractal interpolation functions

A new class of markov processes for image encoding