L. Bodri

Fractal analysis of more than 100 borehole temperature logs indicates that fine scale fluctuations around the deterministic temperature field possess definite scale invariance from metres up to kilometres. We have used spectral technique to test scaling properties of the records. All obtained spectra show uniform power-law behaviour. The obtained spectral exponents seem to show environmental dependence. In sedimentary geologic environments they range between 2 to 3 indicating long-range persistence of data. In dense rocks the values of spectral exponent span in the interval of 1 to 2 characteristic for the antipersistent fractional noise. These differences are likely to be connected with the differences in diffusive behaviour of the physical processes generating the fractal structure of microtemperature fluctuations. A possible geophysical interpretation is proposed.

A. Saucier and J. Muller

We propose a systematic method to choose the scaling range for multifractal analysis, and we illustrate this method by examining the texture of paper formation. To clarify the statistical meaning of a texture characterization based on the moments of a measure (such as multifractal analysis), we examine the connection between these moments and the multipoint correlations of the underlying signal.

V.L. Aksenov, M.V. Avdeev, I.N. Serdyuk, A.A. Timchenko

The dependence of the protein molecular surface area (Am) on the volume (V) for different protein families is studied. Three protein families (globular, DNA- and tRNA-binding proteins) are considered. To construct the surface, techniques with different spatial resolutions are used. It is shown that the protein surface has two levels of organization - micro (<10 angstroms) and macro (>10 angstroms) levels. The fractal properties of these levels are studied. The fractal dimension of the microlevel for the families under consideration varies about 2.1. The dimension of the macrolevel cannot be determined in a standard way. However, we can evaluate it indirectly by using the dependence Am(V), which indicates that they differ greatly for the given families.

Z. Sokolowska, M. Hajnos, S. Sokolowski

We study changes of surface fractal dimension of soil taken from a former sewage farm, caused by leaching of dissolved organic carbon. The leaching at various pH is simulated in the laboratory using sodium hydroxide as the leaching agent. The removal of dissolved organic matter alters the surface properties of the soil. We measure water vapour adsorption and desorption isotherms and mercury intrusion pore size distributions. Experimental adsorption isotherms are analyzed by means of Frenkel-Hill-Halsey type equation. This analysis leads to the determination of the surface fractal dimension. The fractal dimensions obtained from adsorption isotherms are compared with those resulting from the analysis of mercury intrusion data.

Wenxia Li and Dongmei Xiao

The difference set E-F of Cantor sets E and F embedded in the real line is the set of real numbers t such that the intersection of E and F+t is nonemptyset . Problems in dynamical systems involving homoclinic tangencies, homoclinic bifurcations, and the creation of horseshoes have led to the problem of analyzing the difference set E-F. Now we take E=F=C where C is the middle-alpha Cantor set. The difference set C-C is analyzed by Roger Kraft (ref. Monthly of AMS, 640--650). In the present paper, we try to compute the myriad kinds of fractal dimensions of the difference set C-C, the intersection of C and C+t for t in the C-C, the set of those real numbers t which are such that the intersection sets of C and C+t have the same Hausdorff dimension.

T. G. Dewey

A thermodynamic formalism based on information theory is presented for investigating non-equilibrium growth structures. It is seen that the algorithmic complexity of such structures can be related to their configurational entropy. This approach allows one to develop Flory type arguments based on algorithmic complexity. This formalism is used to describe the entropy of random fractals. With this approach, it is seen that the scaling regime available to natural fractal structures is constrained by the thermodynamics of the system. A specific model is presented that relates the fractal scaling parameters to the surface energy of the system. The implication is that fractal scaling will only occur over a limited range in natural growth structures.

A. S. Sidorenko

The influence of fractal geometry on superconductivity has been analyzed for superconductors with different Euclidean dimensions. The complexity of fractal structures gives rise to ''multi-crossover'' behavior in one-dimensional fractal multilayers, unusual self-similar oscillations of the superconducting transition temperature in magnetic field in two-dimensional fractal networks ''Sierpinski gasket'', and temporal fractality of three-dimensional superconducting glasses. Artificial fractal structures serve as a suitable model object for simulations and for experimental studies of disordered superconductors and superconducting devices with complicated topology.

L. Nottale and G. Schumacher

The theory of scale relativity extends Einstein's principle of relativity to scale transformations of resolutions. It is based on the giving up of the axiom of differentiability of the space-time continuum. Three consequences arise from this withdrawal: (i) The geometry of space-time must be fractal, i.e., explicitly resolution-dependent. (ii) The geodesics of the non-differentiable space-time are themselves fractal and in infinite number. (iii) Time reversibility is broken at the infinitesimal level. These three effects are finally combined in terms of a new tool, the scale-covariant derivative, which transforms classical mechanics into a generalized, quantum-like mechanics. We describe virtual families of geodesics in terms of probability densities, interpreted as a tendency for the system to make structures. In the present contribution, we apply our scale-covariant procedure to the equations of motion of test-particles in a gravitational potential. A generalized Newton-Schrödinger equation is obtained, and its solutions are studied. Our theoretical predictions are then successfully checked by a comparison with observational data at various scales, ranging from planetary systems to large scale structures. A possible interpretation of these results is that the underlying fractal geometry of space-time plays the role of a universal structuring "field" that leads to self-organization and morphogenesis of matter in the Universe.

P.M. Iannaccone, S. Morley, T. Skimina, G. Landini

During fetal development in vertebrate species organogenesis precedes rapidly and faithfully, indicating robust mechanisms of regulation. An important part of that process is the generation of parenchyma which is then organized into functional tissues. The method by which the growth of organ parenchyma is regulated is not known but insight into this regulation has been obtained by studying mosaic tissues of experimental chimeras. These animals are produced by amalgamating the embryonic tissues of genetically distinguishable strains then observing the mosaicism microscopically. The patterns revealed by this procedure offer an indication of how the parenchyma was generated. It has been observed that patterns in tissues of chimeras are similar from animal to animal but differ from tissue to tissue. This suggests that the forces that give rise to the pattern are conserved and regulated. In the liver the pattern appears as islands of one cell type in a sea of the other cell type while in the adrenal cortex the pattern is one of alternating cords of one cell type adjacent to the other cell type. Recently we have established that the fractal dimensions of the patches in the liver are consistent with a growth model in which simple stereotypical division rules are applied to initial conditions with iterations until the parenchyma has grown. Here we report that the patches in adrenal cortex are also fractal and that the fractal dimension of the surface of the patches is lower than that in the liver. The analysis is consistent with a growth model in which parenchymal growth is constrained to patch edges forcing centripetal cord structures to form. Fractal analysis of the geometry of mosaic patches in tissues of experimental chimeras is helpful in constructing hypotheses of organ growth.

M Baldoni, C Baroglio, D Cavagnino, L Egidi

The full automation of object visual recognition is a hard
and computationally expensive task, mainly for two reasons:
on one hand, it is difficult to extract the necessary distinctive
information from the raw data, while, on the other, the obtained
representations must have characteristics that make them apt
to training adaptive classifiers to perform the recognition
task of interest.
In this paper we present a new method for representing 2-D
images, based on the extraction of a set of {*fractal features*},
which exploits the approximation of an image with
an { *Iterated Function System*}, a technique that is already at the basis
of many successful image compression tools. In particular, we show
that such features have a high discriminatory power, can be easily
extracted in an automatic way from the raw data and can effectively
be used to train adaptive classifiers to discriminate between
different kinds of objects.

O Stenflo

Time-dependent iterated function systems with time dependent probabilities are introduced, generalizing the concept of iterated function systems with probabilities introduced by Barnsley and Demko. A distributional ergodic theorem including rates of convergence and a law of large numbers are obtained under the assumptions that the system is asymptotically average-contractive.

M. Haase, B. Lehle

Wavelets provide new techniques which are well suited to extract microscopic information about the scaling properties of fractals. The wavelet transform modulus maxima lines allow a localization and characterization of the singularities of a signal. An algorithm is described which traces the maxima lines continuously by integrating the underlying differential equations for wavelets from the Gaussian family. The method is applied to determine the multifractal spectra arising at the quasiperiodic transition to chaos and for turbulent data.

M A Alber, S Lueck, C Renner, and J Peinke

The notion of self-similar energy cascades and multifractality has long since been connected with fully developed, homogeneous and isotropic turbulence. We introduce a number of amendments to the standard methods for analysing the multifractal properties of the energy dissipation field of a turbulent flow. We conjecture that the valid range for the scaling assumption for the moments of the energy dissipation rate coincides with the transition range to dissipation by Castaing et al. (Physica D 46, 177 (1990)). The multifractal spectral functions appear to be universal well within the error margins and exhibit some as yet undiscussed features. Furthermore, we are able to show that this universality extends to the neither homogeneous nor isotropic flows in the wake very close to a cylinder or the off-centre region of a free jet.

J. Kalda

A model of blood-vessel tree is derived. The basic assumptions are: a) the tree can be considered to be self-similar; b) variations of the blood consumption rate of the body cells are not significant. The similarity dimension of the model D_s is approx 3.4 and exceeds the topological dimension of the embedding space D = 3. The notion of self-overlapping exponent is introduced. The model is consistent with the processes governing the growth of the blood-vessels. The model is used to analyze the transport of passive component by blood.

E.F Mikhailov, S. S. Vlasenko, A. A. Kiselev

Structure features of atmospheric soot aerosol particles believed to be crucial for its optical properties due to its sparse flake-like fractal organization. In its turn, soot aerosol structure is known to be extremely sensible to the type of combusted fuel, flame mode, ageing effect of atmospheric environment. In this work scattering and absorbing properties of fractal soot aggregates are considered both experimentally and numerically. Soot particles were generated in the acetylene burner which operated in the diffusion flame mode, provided the fuel was not premixed with the air. The ageing of soot in atmosphere accompanied by structure changing was simulated by heating the aerosol flow up to the temperatures of ~1000 C. The values of fractal prefactor, fractal dimension, mean gyration radius and mean size of primary nuclei were controlled during restructuring by transmission electron microscope. Scattering cross-section angular behaviour and value of reduced extinction were confirmed to be sensitive to the structural parameters of soot aerosol. Theoretical approach, regarding soot particle as fractal aggregate of Mie spheres was used to simulate numerically the optical properties of soot aerosol. Some known models for the scattering on fractal objects (RDG-FA, Mie theory or mean-field approximation in Berry-Percival model) have been also applied, though none of them gave complete insight of the found effect..

K Daoudi

In this paper, we show that the Weakly Self-Affine functions, which are a generalization of self-similar functions, can be used to represent turbulent velocity signals with parsimony and accuracy. We also use this representation approach as a non-parametric method to estimate the multifractal spectrum of these signals.

S Dos Santos, M.P. Planat

New experiments performed on the whole set of modelockings in a voltage controlled electronic loop have revealed a rich arithmetical and fractal structure. It is interpreted from a balance between nonlinear and number theoretical concepts. Continuous fractions expansions of frequency ratios of signals at the input of the mixer govern the dynamics of harmonics. It is found for the first time that due to the finite resolution 1/f frequency noise occurs in the vicinity of modelockings with a magnitude scaled by the nonlinearity.

F.M. Dekking, P.v.d. Wal

We present a generalisation of fractal percolation, which admits neighbour interaction. In this model cells evolve as in a cellular automaton under influence of their neighbours, but contrary to cellular automata give rise to more than one new cell in the next generation. Rather surprisingly this makes the problem of describing their (rescaled) asymptotic behaviour much more tractable than in the case of ordinary cellular automata. One of our results is that many branching cellular automa seem rather fractal at finite time but turn out to be non-fractal in the limit.

E. Hocevar, W.G. Kropatsch

Global IFS (Iterated Function Systems) seem to be suited best for the compressed encoding of natural objects which are in most cases self similar even if not always exactly. An object has merely to be represented by the union of affine contractive transformed copies of itself. It's shown that the corners of a fractal hull (a hull bounded by log. spirals) can be related to those of a minimal set of the copies to calculate their transformations (the IFS-Codes). These copies within the object are identified by their intersection points (not expensively and affinely mappable corners of the fractal hull). This technique can be generalized to encode assemblies of arbitrary coloured objects. Even if they are not close to an IFS, compression ratios in the range of the standard encoding methods can be reached. For objects close to an IFS the compression is even far better.

J. M. Kowalski

A. N. Goltsov

B. Kerman

The observed Gibbs probability distribution of intensity differences between nearest neighbours in synthetic aperture imagery of sea ice leads to the formulation of a probability measure which quantifies how likely it is that two adjacent pixels belong to the same ice type. Also from the Gibbs nature of the observations it is possible to characterize the textural and structural information in the image. The correlation of these forms of information leads to distinct sub-ranges associated with ice type, separated by phase transition points, allowing for an initial segmentation. The fractal properties of the image are examined, particularly of trees, both those constituting the percolation cluster and those constrained about test points. It is also shown that both the partition function and a probability measure over extended separations are fractal. After an initial segmentation using the phase transition points, the extended fractal property along a local network leads to a voting procedure. Each member on a tree, after a correction for the distance to a test point, contributes its probability of being of the same ice type as the test point's ice type. A replacement is made at the test point of the most likely type as sampled along the tree. The process converges rapidly during successive iterations.

Y. Huang

Chinese acupuncture-moxibustion, a therapeutic method by needling in or burning moxa(herba) on the acupuncture points(acupoints) of the body surface, is used to diseases treatment. There exist over one thousand acupoints on the body surface. These acupoints(acupoints system) posses very complex functions. We have found that "fractal theory" is a good tool used to study the irregularity of acupoints distribution on the body surface and the complexity of the acpoint system functions. In this paper, we determine fractal dimensions of acupoint distribution on the body surface by "box counting method", describe the self-similar and nest structure (fractal structure) for the acupoint sytem function on the basis of acupuncture clinical researches. That is to say, the whole of the acuponit system can be divided into two large parts: the channel(Jingluo, macro-acupuncture)system and micro-acupuncture system. The micro-acupuncture system includes many various hierarchical classification "holographic units" similar to the human body. The channel also can be divided into many different hierarchical classification "holographic units", and there are functional nests among various "holographic units". The present work presents a clear answer to the complexity problem of the acupoint system function.

R.R. Filgueira, G.O. Sarli, A.I. Piro, L.L. Fournier

Specific surface of soil aggregates and crushed rock were investigated using physical adsorption of nitrogen at low temperature (78 K). From the variation of this parameter with diameter of particles the surface fractal dimensions were determined. In the case of soil aggregates, it was possible to observe two different zones. In the size range 30-4000 µm, the variation of the specific surface with the diameter showed a non-fractal behavior, with a relatively abrupt change around 30-100 µm. In the range 1-20 µm, the specific surface showed an increase with the decrease in diameter and it was possible to fit the data to a straight line in a log-log representation. For the crushed rock, the specific surface of particle size distributions in the range 40 to 700 (m was investigated. The plot of the specific surface values against particle size, in a log-log representation, could also be fitted to a straight line. The minimal range of self-similarity was calculated in all cases.

T. R. Thomas, N. Amini, B.-G. Rosen

The profiles of many natural and man-made surfaces can be represented over at least part of their structural range as self-affine fractals, characterised by two parameters, the fractal dimension and the topothesy. These parameters are conveniently determined experimentally by measuring the slope and intercept of a logarithmic plot of the profiles structure function. The problems of multifractality and anisotropy arising when this technique is generalised to three dimensions are discussed. A three-dimensional version of the structure function is presented, any section through which is equivalent to an ensemble average of profile structure functions. The angular variation of topothesy and fractal dimension obtained from such sections describes the anisotropy of the parent surface. Also, a bifractal structure function may be split into its component straight lines by fitting with a hyperbola. The intersection of the asymptotes then defines the so- called "corner frequency" between the two fractals. Examples are presented from 3D height measurements of several technical surfaces made with a scanning white-light interferometer and a stylus instrument, and the measured corner frequencies and fractal parameters are related to their respective finishing processes. All three parameters are found to be sensitive to anisotropy, the topothesy particularly so.

F. Roueff, J. Levy Vehel

We propose a new way of evaluating the regularity of a graph of a function f. Our approach is based on measuring the growth rate of the lengths of less and less regularized versions of f. This leads to a new index, that we call regularization dimension, dim_R. We derive some analytical properties of dim_R and compare it with other fractional dimensions. A statistical estimator is derived, and numerical experiments are performed, which suggest that dim_R may be computed in a robust way. Finally, we apply the regularization dimension to the study of Ethernet traffic.

C Canus, J Levy Vehel, C Tricot

The large deviation multifractal spectrum gives important statistical informations on irregular measures. However it is difficult to estimate. In this paper, we propose two new definitions of the large deviation spectrum better adapted to the design of various estimators. They rely on the computation of the Lebesgue measure of the reunion of all intervals of same size whose coarse grain Hoelder exponent is equal to a Hoelder exponent. In particular, we introduce the continuous large deviation spectrum for which we construct different estimators. We finally show some numerical results obtained on both deterministic and random synthetical signals.

M Meyer

The dynamics of heartbeat interval time series were studied by a modified random walk analysis recently introduced as Detrended Fluctuation Analysis. In this analysis, the intrinsic fractal long-range power-law correlation properties of beat-to-beat fluctuations generated by the dynamical system (i.e., cardiac rhythm generator), after decomposition from extrinsic uncorrelated sources, can be quantified by the scaling exponent which, in healthy subjects, is about 1.0. The finding of a scaling coefficient of 1.0, indicating scale-invariant long-range power-law correlations (1/f noise) of heartbeat fluctuations, would reflect a genuinely self-similar fractal process that typically generates fluctuations on a wide range of time scales. Lack of a characteristic time scale suggests that the neuroautonomic system underlying the control of heart rate dynamics helps prevent excessive mode-locking (error tolerance) that would restrict its functional responsiveness (plasticity) to environmental stimuli. The 1/f dynamics of heartbeat interval fluctuations are unaffected by exposure to chronic hypoxia at high altitude (> 5000 m) suggesting that the neuroautonomic cardiac control system is preadapted to hypoxia. Functional (hypothermia, cardiac diaease) and/or structural (cardiac transplantation, early cardiac development) inactivation of neuroautonomic control is associated with the breakdown or absence of fractal complexity reflected by anticorrelated random walk-like dynamics, indicating that in these conditions the heart is unadapted to its environment.

E. Faleiro and J.M.G. Gomez

The interaction of cosmic rays, mainly high energy gamma rays and protons with the Earth atmosphere produces secondary particles which again interact with the atmosphere, giving rise to multiplicative cascades of particles called Extensive Air Showers (EAS). The interplay of hadronic and electromagnetic processes and the multiplicative nature of the process make the study of the shower development non-trivial. The distribution of particles in the EAS can be measured at ground level with an array of detectors. Although detectors have steadily improved in the last years, the available experimental data at present are not sufficient for detailed statistical studies of the secondary particle distributions. Therefore we have used for this purpose a Monte Carlo simulation of the EAS by means of the CORSIKA code [1]. A large number of simulated EAS have been generated for primary cosmic protons and gamma rays at diferent energies ranging from 10 to 40 TeV. The density distribution of the secondary particles reaching the Earth surface at ground level has then been analysed in a ring placed between the circumferences of 50 and 100 m radii, as a function of the polar angle. It is found that the mean Power Spectrum Density P(k) is not constant, as would be the case for a signal whose fluctuations were uncorrelated. Instead, it exhibits a complex structure as a function of k. It is shown that the fluctuations of the data are composed of a white noise plus a 1/f noise. Finally the scaling properties of the 1/f noise statistical moments are studied, and it is found that the low momenta have a universal multifractal structure [2]. [1]- J.N. Capdevielle et al., KfK Report 4998, Kernforschungzentrum. Karlsruhe (1992). [2]- Schertzer, D. and Lovejoy, S. in Scaling, Fractals and Nonlinear Variability in Geophysics., eds. Schertzer, D. and Lovejoy (Kluwer, Holland, 1991)

L. Roters, S. Luebeck, K. D. Usadel

The Nagel-Schreckenberg traffic flow model shows a transition from a free flow regime to a jammed regime for increasing car density. The measurement of the dynamical structure factor offers the chance to observe the evolution of jams without the necessity to define a car to be jammed or not. Above the jamming transition the dynamical structure factor S(k,omega) exhibits for a given k-value two maxima corresponding to the separation of the system into the free flow phase and jammed phase. We obtain from a finite-size scaling analysis of the smallest jam mode that approaching the transition point long range correlations of the jams occur.

Z. R. Struzik

We present a promising technique which is capable of accessing the divergence free component of the partition function for the negative moments of the multi-fractal analysis of data using the wavelet transformation. It is based on implicitly bounding the local logarithmic slope of the wavelet maxima lines between the values of the Holder exponent of the singularities which are accessible for the wavelet used. The method delivers correct and stable results, illustrated using a test example of the Besicovich measure analysed with the Mexican hat wavelet.

James Kadtke and Michael Kremliovsky

Time series analysis methods derived from the theory of Nonlinear Dynamics have resulted in several new signal processing techniques in the last decade, with applications to prediction, noise reduction, and signal separation. Recently, many efforts have been directed towards applications to the detection and classification (D/C) of signals, or alternately, classification of the generating physical system. Although these techniques make the ansatz that the signal is generated by a dynamical system, and typically are interested in detecting determinism (esp. chaotic behavior), these techniques can be considered more general, in the sense that we quantify nonlinear correlations in the signal. These methods may be particularly relevant to the analysis of biologic, and geophysical signals, and machine diagnostics, with applications to automated classification being of fundamental interest. Recently, we have presented a method for the D/C of noisy signals which performs D/C by assuming a dynamical systems hypothesis [Kadtke,1995]. In practice, we develop a D/C processing chain which uses dynamical models ( DDEs ) to extract signal correlations. In this paper, we will discuss the extraction of the dynamical models from arbitrary data sets, the use of a coefficient space to provide a classification metric, and the generation of classification probabilities from the statistics of the coefficient ensembles. We indicate how non-autonomous models can be used for the classification of transient (pulse-like) data in a natural way, and demonstrate classification in noisy simulated data examples, even in high noise ( 1000% ). In particular, we show that this method may be valuable for the classification of biologic and medical data, which often presents significant problems because of noise levels, short observations, or the lack of any a priori system model. Since this method performs relative classification of short observations, and relies on the statistics of the observations in the coefficient feature space to estimate classification probabilities, many of the difficulties inherent in real-world data processing are reduced. Here, we will demonstrate the performance of these methods by presenting results of the analysis of several real-world data sets, including acoustic echolocation recordings of marine mammals, and medical observations of human patients. Eventual applications include voice/speech classification, remote sensing, and automated physiological monitoring.

H.H.J. Buijs and A.J. van Duyneveldt

Problem solving is, besides artistic creativity, a major creative act of human behaviour. Creative behaviour derives, as all human behaviour, from the brain. The brain, being a living system, has as one of its characteristics that it has an autopoietic, self making pattern of organisation, in which continual structural changes take place while the pattern of organisati-on itself does not change. These changes, new neural connections, appear as a consequence of environmental influences or of the internal dynamics of the system (structural coupling). They take place in the structure of each individual brain. The structure of the brain is dissipative. The processes in the brain, in organisation (structural coupling) and in structure (dissipative), bring about that the brain is a learning system. Structural coupling, the creative characteristic of the pattern of organisation of the brain, brings us on the one hand into many fields of psychology. On the other hand, application of this process to psychological and psychiatric therapies can bring helpful insights to those active with helping people in finding solutions for the problems in their lives. Thirdly, the dissipative structure of the brain leads us to the application of chaos theory and fractal geometry to brain processes, especially creative ones. The brain as a learning system causes that developing creative abilities, including problem solving, is possible for and ultimately characteristic of all of us, not only artists.

P. Parmananda

We report the synchronization of dynamics in a numerical model simulating electrochemical corrosion using external chaotic, periodic and finally forcing including a random component (random forcing). For all the three external forcings synchronization is achieved when the two response systems are at identical parameter conditions exhibiting similar behavior. However when the two response systems are at unequal parameter values exhibiting different dynamical behavior synchronization is achieved only for forcing including a random variable (random forcing). This ability of random perturbations to achieve generalized synchronization makes it a candidate worthy of consideration in problems involving synchronization of non-identical systems.

I. H. Parkinson and N. L. Fazzalari

Fractal analysis using image analyzers is influenced by device settings, initial magnification and specimen orientation. Analysis of the Richardson plot of trabecular bone profiles reveals more than one straight-line segment but an objective method for determining these segments has not been available to researchers. This study describes a standardised methodology for the fractal analysis of histological sections of trabecular bone. A modified box counting method has been developed for use on a PC based image analyzer (Quantimet 500MC, Leica Cambridge). The effect of image analyzer settings, such as initial magnification, image orientation and threshold levels, have been studied. Also, the range of scale over which trabecular bone is fractal was analyzed and a method formulated to objectively calculate more than one fractal dimension from the Richardson plot. The results show that initial magnification ranging from x2.6 to 13 increases the fractal dimension, but change in image orientation and threshold settings have little effect. Trabecular bone has a lower limit below which it is not fractal. There are 3 distinct fractal dimensions for trabecular bone, which relate directly to bone cell functional and structural entities.

Jizhong Zhang, Xiaoyan Ye, Xiaodong Yang

Morphology of tin iodide deposits grown during vapor-solid transformation was studied experimentally. The deposits included three kinds of forms: fractal grain-aggregate, tribranch-like structure of clusters, and flower-like pattern of thin film. The fractal grain-aggregate displayed random branched characteristic, and consisted of numerous tin iodide grains. The tribranch-like structure of clusters was composed of many tin iodide clusters, and displayed branched pattern. The flower- like pattern of tin iodide thin film was characterized by advancing front and random branching. The experimental results implied that morphology of tin iodide deposits was much sensitive to local non-equilibrium state of experimental system. Their growth process is discussed.

P. Arena, L. Bertucco, L. Fortuna, G. Nunnari, and D Porto

In this paper a new kind of Cellular Neural Network is introduced. Its mainly characteristic consists in a state representation using q-order derivatives being q a non integer quantity. This approach can be considered as a generalisation of the traditional CNN model, which is obtained from the one presented in the paper as a particular case setting q=1. It is shown that this more general CNN structure exhibits suitable performance in terms of speed-up. This is very important expecially for image processing, which is one of the most important applications of Cellular Neural Networks. In the paper various examples are reported to show the suitability of non-integer order CNNs .

P D Wakefield and D M Monro

The Implicit Fractal Transform (IFT) is a technique for image compression in which fractal content of the image is deduced by the decoder. This makes the option of fractal enhancement virtually code free. We use this technique in conjunction with a quad-tree decomposition which uses a fractal term for each block only when it improves the overall rate-distortion performance of the coder. Additionally we examine the multi-resolution properties of fractal coding and compare the results of zooming several fractal transforms. In the case of the IFT an enhancement technique is presented which visually improves results.

22 May 1998