Construction of fractal interpolation surfaces on rectangular grids

We present a general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.

Construction of recurrent fractal interpolation surfaces(RFISs) on rectangular grids

A recurrent iterated function system (RIFS) is a genaralization of an IFS and provides nonself-affine fractal sets which are closer to natural objects. In general, it's attractor is not a continuous surface in R3. A recurrent fractal interpolation surface (RFIS) is an attractor of RIFS which is a graph of bivariate continuous interpolation function. We introduce a general method of generating recurrent interpolation surface which are at- tractors of RIFSs about any data set on a grid.

Actividades de geometría fractal en el aula de secundaria (II).

Análisis en torno al concepto de fractal. Se exponen diferentes formas de generar fractales y algunas de sus aplicaciones.

Actividades de geometría fractal en el aula de secundaria (I).

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Geometría fractal.

La geometría fractal permite estudiar de manera científica formas naturales como la de un arbol, romanesco o un copo de nieve en las que apreciamos irregularidades, estructura en todas las escalas y autosemenjanza. Algunos de los fractales más conocidos son la llamada curva de Koch o el triángulo de Sierpinski. Ambos se forman de una manera similar, se aplica una regla sencilla que se usa una y otra vez. Otra fuente de fractales es la iteración de funciones de variable compleja. El conjunto de Maldelbrot se crea a partir de este sistema. Para que cierta imagen sea un fractal no es suficiente con la autosemejanza, además hace falta una dimensión fractal, que se calcula con una serie de cuadrículas cada vez más finas que se superponen a la figura y se cuentan el número de cuadrados que tienen en común con la figura. A partir de los experimentos de Maldelbrot algunos artistas crearon el llamado arte fractal, obras de arte creadas mediante algoritmos matemáticos de generación de fractales y su posible manipulación posterior. También se usan para la composición musical que se crea a partir de una sucesión de números creados a partir de un algoritmo fractal. Esta música también se caracteriza por una estructura autosemejante.

Teaching as a fractal : from experience to model

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Artificial Cavities and Nest Site Selection by Puerto Rican Parrots: a Multiscale Assessment

We examined nest site selection by Puerto Rican Parrots, a secondary cavity nester, at several spatial scales using the nest entrance as the central focal point relative to 20 habitat and spatial variables. The Puerto Rican Parrot is unique in that, since 2001, all known nesting in the wild has occurred in artificial cavities, which also provided us with an opportunity to evaluate nest site selection without confounding effects of the actual nest cavity characteristics. Because of the data limitations imposed by the small population size of this critically endangered endemic species, we employed a distribution-free statistical simulation approach to assess site selection relative to characteristics of used and unused nesting sites. Nest sites selected by Puerto Rican Parrots were characterized by greater horizontal and vertical visibility from the nest entrance, greater density of mature sierra palms, and a more westerly and leeward orientation of nest entrances than unused sites. Our results suggest that nest site selection in this species is an adaptive response to predation pressure, to which the parrots respond by selecting nest sites offering advantages in predator detection and avoidance at all stages of the nesting cycle. We conclude that identifying and replicating the “nest gestalt” of successful nesting sites may facilitate conservation efforts for this and other endangered avian species.

Causes of the apparent scale independence of fractal indices associated with forest fragmentation in Bolivia

In a recent investigation, Landsat TM and ETM+ data were used to simulate different resolutions of remotely-sensed images (from 30 to 1100 m) and to analyze the effect of resolution on a range of landscape metrics associated with spatial patterns of forest fragmentation in Chapare, Bolivia since the mid-1980s. Whereas most metrics were found to be highly dependent on pixel size, several fractal metrics (DLFD, MPFD, and AWMPFD) were apparently independent of image resolution, in contradiction with a sizeable body of literature indicating that fractal dimensions of natural objects depend strongly on image characteristics. The present re-analysis of the Chapare images, using two alternative algorithms routinely used for the evaluation of fractal dimensions, shows that the values of the box-counting and information fractal dimensions are systematically larger, sometimes by as much as 85%, than the "fractal" indices DLFD, MPFD, and AWMFD for the same images. In addition, the geometrical fractal features of the forest and non-forest patches in the Chapare region strongly depend on the resolution of images used in the analysis. The largest dependency on resolution occurs for the box-counting fractal dimension in the case of the non-forest patches in 1993, where the difference between the 30 and I 100 m-resolution images corresponds to 24% of the full theoretical range (1.0 to 2.0) of the mass fractal dimension. The observation that the indices DLFD, MPFD, and AWMPFD, unlike the classical fractal dimensions, appear relatively unaffected by resolution in the case of the Chapare images seems due essentially to the fact that these indices are based on a heuristic, "non-geometric" approach to fractals. Because of their lack of a foundation in fractal geometry, nothing guarantees that these indices will be resolution-independent in general. (C) 2006 International Society for Photogrammetry and Remote Sensing, Inc. (ISPRS). Published by Elsevier B.V. All rights reserved.

Beurling zeta functions, generalised primes, and fractal membranes

We study generalised prime systems P (1 < p(1) <= p(2) <= ..., with p(j) is an element of R tending to infinity) and the associated Beurling zeta function zeta p(s) = Pi(infinity)(j=1)(1 - p(j)(-s))(-1). Under appropriate assumptions, we establish various analytic properties of zeta p(s), including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of zeta p(s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N-2. Some of the above results are relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.

Fractal quantization

A Fractal Quantizer is proposed that replaces the expensive division operation for the computation of scalar quantization by more modest and available multiplication, addition and shift operations. Although the proposed method is iterative in nature, simulations prove a virtually undetectable distortion to the naked eve for JPEG compressed images using a single iteration. The method requires a change to the usual tables used in JPEG algorithins but of similar size. For practical purposes, performing quantization is reduced to a multiplication plus addition operation easily programmed in either low-end embedded processors and suitable for efficient and very high speed implementation in ASIC or FPGA hardware. FPGA hardware implementation shows up to x15 area-time savingscompared to standars solutions for devices with dedicated multipliers. The method can be also immediately extended to perform adaptive quantization(1).

Characterization and classification of tumor lesions using computerized fractal-based texture analysis and support vector machines in digital mammograms

Objective: This paper presents a detailed study of fractal-based methods for texture characterization of mammographic mass lesions and architectural distortion. The purpose of this study is to explore the use of fractal and lacunarity analysis for the characterization and classification of both tumor lesions and normal breast parenchyma in mammography. Materials and methods: We conducted comparative evaluations of five popular fractal dimension estimation methods for the characterization of the texture of mass lesions and architectural distortion. We applied the concept of lacunarity to the description of the spatial distribution of the pixel intensities in mammographic images. These methods were tested with a set of 57 breast masses and 60 normal breast parenchyma (dataset1), and with another set of 19 architectural distortions and 41 normal breast parenchyma (dataset2). Support vector machines (SVM) were used as a pattern classification method for tumor classification. Results: Experimental results showed that the fractal dimension of region of interest (ROIs) depicting mass lesions and architectural distortion was statistically significantly lower than that of normal breast parenchyma for all five methods. Receiver operating characteristic (ROC) analysis showed that fractional Brownian motion (FBM) method generated the highest area under ROC curve (A z = 0.839 for dataset1, 0.828 for dataset2, respectively) among five methods for both datasets. Lacunarity analysis showed that the ROIs depicting mass lesions and architectural distortion had higher lacunarities than those of ROIs depicting normal breast parenchyma. The combination of FBM fractal dimension and lacunarity yielded the highest A z value (0.903 and 0.875, respectively) than those based on single feature alone for both given datasets. The application of the SVM improved the performance of the fractal-based features in differentiating tumor lesions from normal breast parenchyma by generating higher A z value. Conclusion: FBM texture model is the most appropriate model for characterizing mammographic images due to self-affinity assumption of the method being a better approximation. Lacunarity is an effective counterpart measure of the fractal dimension in texture feature extraction in mammographic images. The classification results obtained in this work suggest that the SVM is an effective method with great potential for classification in mammographic image analysis.

Integrating features for man-made target tracking from FLIR image sequence using particle filter

A new man-made target tracking algorithm integrating features from (Forward Looking InfraRed) image sequence is presented based on particle filter. Firstly, a multiscale fractal feature is used to enhance targets in FLIR images. Secondly, the gray space feature is defined by Bhattacharyya distance between intensity histograms of the reference target and a sample target from MFF (Multi-scale Fractal Feature) image. Thirdly, the motion feature is obtained by differencing between two MFF images. Fourthly, a fusion coefficient can be automatically obtained by online feature selection method for features integrating based on fuzzy logic. Finally, a particle filtering framework is developed to fulfill the target tracking. Experimental results have shown that the proposed algorithm can accurately track weak or small man-made target in FLIR images with complicated background. The algorithm is effective, robust and satisfied to real time tracking.

Multiscale evolving complex network model of functional connectivity in neuronal cultures

Cultures of cortical neurons grown on multielectrode arrays exhibit spontaneous, robust and recurrent patterns of highly synchronous activity called bursts. These bursts play a crucial role in the development and topological selforganization of neuronal networks. Thus, understanding the evolution of synchrony within these bursts could give insight into network growth and the functional processes involved in learning and memory. Functional connectivity networks can be constructed by observing patterns of synchrony that evolve during bursts. To capture this evolution, a modelling approach is adopted using a framework of emergent evolving complex networks and, through taking advantage of the multiple time scales of the system, aims to show the importance of sequential and ordered synchronization in network function.