on the equilibrium state of random walkers in random environments analytical results we study equilibrium properties of random walkers in one dimensional random environments of finite length l from an exact expression for the quenched average of the free energy we derive analytical results for all cumulants and all renyi entropies of the equilibrium distribution in contrast to the finite variance of a typical non equilibrium distribution in the unbiased situation we find that in equilibrium the disorder averaged variance diverges with the size of the system as

zero beta function for a model of diffusion in potential random field a rigorous perturbative proof is given of both the renormalisability of the field theoretic model of diffusion in potential random field and the triviality of the renormalisation group beta function

spatial reasoning and similarity retrieval of images using 2d c string knowledge representation abstract the data structure called 2d string to represent symbolic pictures was proposed by chang et al it allows a natural way to construct iconic indexes for pictures a new spatial knowledge representation of a 2d c string with accompanying cutting mechanism and a set of spatial operators are proposed 2d c string is more characteristic of spatial knowledge in images and is more efficient in representation and manipulation of images since a spatial relationship is a fuzzy concept the capability of similarity retrieval is essential in image database systems the spatial query and spatial reasoning based on a 2d c string representation are presented the similarity measure is defined and the algorithm for similarity retrieval of iconic images is also proposed

on toy ageing we consider the dynamics of a simple one dimensional model and we discuss the phenomenon of ageing i e the strong dependence of the dynamical correlation functions over the waiting time our model is the so called random random walk the toy model of a directed polymer evolving in a random medium

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

a review of the fractal image coding literature fractal image compression is a technique based on the representation of an image by a contractive transform on the space of images for which the fixed point is close to the original image this broad principle encompasses a very wide variety of coding schemes many of which have been explored in the rapidly growing body of published research while certain theoretical aspects of this representation are well established relatively little attention has been given to the construction of a coherent underlying image model that would justify its use most purely fractal based schemes are not competitive with the current state of the art but hybrid schemes incorporating fractal compression and alternative techniques have achieved considerably greater success this review represents a survey of the most significant advances both practical and theoretical since the publication of jacquin s 1990 original fractal coding scheme

fast fractal image block coding based on local variances in fractal image block coding most of the time is spent on finding a close match between a range block and a large pool of domain blocks for a large image this effect becomes aggravated as the domain pool increases exponentially we propose using the local variances of domain blocks to reduce the search space by sorting the contracted domain pool according to their local variances and defining an acceptance criterion for a close match we can confine all the potential close matches to a relatively small sized window to limit the search space the encoding time can hence be shortened with the decoded image quality as good as that using the full search method the speedup can be over ten times depending on the complexity of encoded images

huber fractal image coding based on a fitting plane recently there has been significant interest in robust fractal image coding for the purpose of robustness against outliers however the known robust fractal coding methods hfic and lad fic etc are not optimal since besides the high computational cost they use the corrupted domain block as the independent variable in the robust regression model which may adversely affect the robust estimator to calculate the fractal parameters depending on the noise level this paper presents a huber fitting plane based fractal image coding hfpfic method this method builds huber fitting planes hfps for the domain and range blocks respectively ensuring the use of an uncorrupted independent variable in the robust model on this basis a new matching error function is introduced to robustly evaluate the best scaling factor meanwhile a median absolute deviation mad about the median decomposition criterion is proposed to achieve fast adaptive quadtree partitioning for the image corrupted by salt pepper noise in order to reduce computational cost the no search method is applied to speedup the encoding process experimental results show that the proposed hfpfic can yield superior performance over conventional robust fractal image coding methods in encoding speed and the quality of the restored image furthermore the no search method can significantly reduce encoding time and achieve less than 2 0 s for the hfpfic with acceptable image quality degradation in addition we show that combined with the mad decomposition scheme the hfp technique used as a robust method can further reduce the encoding time while maintaining image quality

speed up in fractal image coding comparison of methods fractal image compression has received much attention from the research community because of some desirable properties like resolution independence fast decoding and very competitive rate distortion curves despite the advances made the long computing times in the encoding phase still remain the main drawback of this technique so far several methods have been proposed in order to speed up fractal image coding we address the problem of choosing the best speed up techniques for fractal image coding comparing some of the most effective classification and feature vector methods namely fisher 1994 hurtgen 1993 and saupe 1995 1996 and a new feature vector coding scheme based on the block s mass center furthermore we introduce two new coding schemes combining saupe with fisher and saupe with mass center coding scheme experimental results demonstrate both the superiority of feature vector techniques on classification and the effectiveness of combining saupe and the mass center coding scheme an approach that exhibits the best time distortion curves

iconic indexing by 2 d strings in this paper we describe a new way of representing a symbolic picture by a two dimensional string a picture query can also be specified as a 2 d string the problem of pictorial information retrieval then becomes a problem of 2 d subsequence matching we present algorithms for encoding a symbolic picture into its 2 d string representation reconstructing a picture from its 2 d string representation and matching a 2 d string with another 2 d string we also prove the necessary and sufficient conditions to characterize ambiguous pictures for reduced 2 d strings as well as normal 2 d strings this approach thus allows an efficient and natural way to construct iconic indexes for pictures

a nonlinear model for fractal image coding after a very promising start progress in fractal image coding has been relatively slow recently most improvements have been concentrating on better adaptive coding algorithms and on search strategies to reduce the encoding time very little has been done to challenge the linear model of the fractal transformations used so far in practical applications in this paper we explain why effective nonlinear transformations are not easy to find and propose a model based on conformal mappings in the geometric domain that are a natural extension of the affine model our compression results show improvements over the linear model and support the hope that a deeper understanding of the notion of self similarity would further advance fractal image coding

image coding by block prediction of multiresolution subimages the redundancy of the multiresolution representation has been clearly demonstrated in the case of fractal images but it has not been fully recognized and exploited for general images fractal block coders have exploited the self similarity among blocks in images we devise an image coder in which the causal similarity among blocks of different subbands in a multiresolution decomposition of the image is exploited in a pyramid subband decomposition the image is decomposed into a set of subbands that are localized in scale orientation and space the proposed coding scheme consists of predicting blocks in one subimage from blocks in lower resolution subbands with the same orientation although our prediction maps are of the same kind of those used in fractal block coders which are based on an iterative mapping scheme our coding technique does not impose any contractivity constraint on the block maps this makes the decoding procedure very simple and allows a direct evaluation of the mean squared error mse between the original and the reconstructed image at coding time more importantly we show that the subband pyramid acts as an automatic block classifier thus making the block search simpler and the block matching more effective these advantages are confirmed by the experimental results which show that the performance of our scheme is superior for both visual quality and mse to that obtainable with standard fractal block coders and also to that of other popular image coders such as jpeg

generalised hypergeometric series nf x1 xn arising in physical and quantum chemical applications functions nf x1 xn which are a straightforward generalisation of standard hypergeometric functions of n variables are introduced a convenient operator representation is established for these functions which permits one to consider in a uniform and simple way standard hypergeometric functions generalised hypergeometric series of one variable appell kampe de feriet and lauricella functions etc as well as many other nameless hypergeometric series arising in physical and quantum chemical applications generating functions generalisations of bailey 1935 and chaundy 1943 expansions recurrence relations reduction rules and some other topics are discussed

on the interpretation of 1 f noise we propose a model of 1 f noise based on a random walk in a random potential numerical support for the model is given and physical applicability discussed

a fast fractal image coding based on kick out and zero contrast conditions a fast algorithm for fractal image coding based on a single kick out condition and the zero contrast prediction is proposed in this paper the single kick out condition can avoid a large number of range domain block matches when finding the best matched domain block an efficient method for zero contrast prediction is also proposed which can determine whether the contrast factor for a domain block is zero or not and compute the corresponding difference between the range block and the transformed domain block efficiently and exactly the proposed algorithm can achieve the same reconstructed image quality as the exhaustive search and can greatly reduce the required computation or runtime in addition this algorithm does not need any pre processing step or additional memory for its implementation and can combine with other fast fractal algorithms to further improve the speed experimental results show that the runtime is reduced by about 50 of that of the exhaustive search method when combined with the dct inner product algorithm the required runtime for the algorithm can be further reduced by about 50 the proposed algorithm was also compared to two other fast fractal algorithms experimental results also show that our algorithm achieves a better efficiency and requires a much smaller amount of memory for implementation

a non wiener random walk in a two dimensional bernoulli environment for a degenerate random walk in a 2d bernoulli environment without local traps computer results show a non wiener behavior for a better exploitation of the memory the analysis is based on the statistics of the first exit time from a square

rigorous renormalisation group and disordered systems abstract we consider random walks in a non symmetric random environment we report a recent result based on a renormalisation group approach showing that for d 62 2 these walks are diffusive when the disorder is weak

generating differential operators for the basis functions in the variational lcao type methods the analytical expressions for operators which allow us to generate the basis functions from the simplest ground states by means of differentiating by parameters are obtained some important properties of these operators are indicated this approach allows us to simplify the structure of molecular matrix elements whose calculation presents considerable difficulties in the variational lcao type methods

a fuzzy image metric with application to fractal coding image quality assessment is an important issue addressed in various image processing applications such as image video compression and image reconstruction the peak signal to noise ratio psnr with the l sup 2 metric is commonly used in objective image quality assessment however the measure does not agree very well with the human visual perception in many cases a fuzzy image metric fim is defined based on sugeno s 1977 fuzzy integral this new objective image metric which is to some extent a proper evaluation from the viewpoint of the judgment procedure is closely approximates the subjective mean opinion score mos with a correlation coefficient of about 0 94 as compared to 0 82 obtained using the psnr compared to the l sup 2 metric we demonstrate that a better performance can be achieved in fractal coding by using the proposed fim

fast fractal image encoding based on adaptive search this paper presents a new adaptive search approach to reduce the computational complexity of fractal encoding a simple but very efficient adaptive necessary condition is introduced to exclude a large number of unqualified domain blocks so as to speed up fractal image compression furthermore we analyzed an unconventional affine parameter that has better properties than the conventional luminance offset specifically we formulated an optimal bit allocation scheme for the simultaneous quantizations of the usual scaling and the aforementioned unconventional affine parameter experiments on standard images showed that our adaptive search method yields superior performance over conventional fractal encoding

addition theorems for b functions and other exponentially declining functions in this paper addition theorems are derived for a special class of exponentially declining functions the so called b functions bmn l r e filter and e o steinborn phys rev a 18 1 1978 although these b functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems also all the commonly occurring exponentially declining functions like for instance bound state hydrogen eigenfunctions and slater type functions can be expressed by simple finite sums of b functions consequently addition theorems for these functions can also be written down immediately the various addition theorems for b functions are derived by applying suitable generating differential operators to the well known addition theorem of the special b function bm l l which is that solution of the modified helmholtz equation that is irregular at the origin and regular at infinity e j weniger and e o

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

localization of random walks in one dimensional random environments we consider a random walk on the one dimensional semi lattice 0 1 2 we prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment the size of this neighbourhood is estimated the limit parameters of the walks are also determined

a simple derivation of the addition theorems of the irregular solid harmonics the helmholtz harmonics and the modified helmholtz harmonics in this article a simple derivation of the addition theorems of the irregular solid harmonics the helmholtz harmonics and the modified helmholtz harmonics is presented our derivation is based upon properties of the differential operator yml which is obtained from the regular solid harmonic yml r by replacing the cartesian components of r by the cartesian components of with the help of this differential operator yml which is an irreducible spherical tensor of rank l the addition theorems of the anisotropic functions are obtained by differentiating the addition theorems of the isotropic functions the performance of the necessary differentiations is greatly facilitated by a systematic exploitation of the tensorial nature of the differential operator yml

a fast and efficient hybrid fractal wavelet image coder the excellent visual quality and compression rate of fractal image coding have limited applications due to exhaustive inherent encoding time this paper presents a new fast and efficient image coder that applies the speed of the wavelet transform to the image quality of the fractal compression fast fractal encoding using fisher s domain classification is applied to the lowpass subband of wavelet transformed image and a modified set partitioning in hierarchical trees spiht coding on the remaining coefficients furthermore image details and wavelet progressive transmission characteristics are maintained no blocking effects from fractal techniques are introduced and the encoding fidelity problem common in fractal wavelet hybrid coders is solved the proposed scheme promotes an average of 94 reduction in encoding decoding time comparing to the pure accelerated fractal coding results the simulations also compare the results to the spiht wavelet coding in both cases the new scheme improves the subjective quality of pictures for high medium low bitrates

region based fractal image compression using heuristic search presents work carried out on fractal or attractor image compression the approach relies on the assumption that image redundancy can be efficiently exploited through self transformability the algorithms described utilize a novel region based partition of the image that greatly increases the compression ratios achieved over traditional block based partitionings due to the large search spaces involved heuristic algorithms are used to construct these region based transformations results for three different heuristic algorithms are given the results show that the region based system achieves almost double the compression ratio of the simple block based system at a similar decompressed image quality for the lena image compression ratios of 41 1 can be achieved at a psnr of 26 56 db

exact calculation of the anomalous dimension of the diffusion coefficient for a model of a random walk in a random potential renormalisation of a model of classical diffusion in a random potential is analysed it is shown that at two dimensions the one loop expression for the anomalous dimension of the diffusion coefficient is perturbatively exact leading to the coupling dependent value 1 nu 2 g 4 pi of the exponent nu

sinai disorder intermittency for random maps abstract the properties of the first passage probability for a random walk with sinai disorder are shown to be similar to those observed for deterministic intermittent chaos

multidimensional random walks in random environments with subclassical limiting behavior in this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them our results make rigorous a heuristic argument of marinari et al 1983 and show that in anyd we can have a xn is recurrent and b xn logn 2

a fast encoding algorithm for fractal image compression using the dct inner product in this paper a fast encoding algorithm is developed for fractal image compression at each search entry in the domain pool the mean square error mse calculations of the given range block and the eight dihedral symmetries of the domain block are obtained simultaneously in the frequency domain in which the redundant computations are all eliminated in the new encoding algorithm it is shown in software simulation that the encoding time is about six times faster than that of the baseline method with almost the same psnr for the retrieved image the fast algorithm is performed to deal with the eight dihedral symmetries at each search entry therefore it can be applied to various enhanced algorithms which are equipped with quadtree classification and other mechanisms

adaptive approximate nearest neighbor search for fractal image compression fractal image encoding is a computationally intensive method of compression due to its need to find the best match between image subblocks by repeatedly searching a large virtual codebook constructed from the image under compression one of the most innovative and promising approaches to speed up the encoding is to convert the range domain block matching problem to a nearest neighbor search problem this paper presents an improved formulation of approximate nearest neighbor search based on orthogonal projection and pre quantization of the fractal transform parameters furthermore an optimal adaptive scheme is derived for the approximate search parameter to further enhance the performance of the new algorithm experimental results showed that our new technique is able to improve both the fidelity and compression ratio while significantly reduce memory requirement and encoding time

statistical and dynamical properties of the discrete sinai model at finite times we study the sinai model for the diffusion of a particle in a one dimensional quenched random energy landscape we consider the particular case of discrete energy landscapes made of random jumps on the semi infinite line with a reflecting wall at the origin we compare the statistical distribution of the successive local minima of the energy landscapes which we derive explicitly with the dynamical distribution of the position of the diffusing particle which we obtain numerically at high temperature the two distributions match only in the large time asymptotic regime at low temperature however we find even at finite times a clear correspondence between the statistical and dynamical distributions with additional interesting oscillatory behaviours

a novel fractal image compression scheme with block classification and sorting based on pearson s correlation coefficient fractal image compression fic is an image coding technology based on the local similarity of image structure it is widely used in many fields such as image retrieval image denoising image authentication and encryption fic however suffers from the high computational complexity in encoding although many schemes are published to speed up encoding they do not easily satisfy the encoding time or the reconstructed image quality requirements in this paper a new fic scheme is proposed based on the fact that the affine similarity between two blocks in fic is equivalent to the absolute value of pearson s correlation coefficient apcc between them first all blocks in the range and domain pools are chosen and classified using an apcc based block classification method to increase the matching probability second by sorting the domain blocks with respect to apccs between these domain blocks and a preset block in each class the matching domain block for a range block can be searched in the selected domain set in which these apccs are closer to apcc between the range block and the preset block experimental results show that the proposed scheme can significantly speed up the encoding process in fic while preserving the reconstructed image quality well

three loop calculation of the random walk problem an application of dimensional transformation and the uniqueness method the method and details of the calculation of the three loop contribution to the anomalous dimension of the diffusion coefficient of the model of a random walk in a potential random field with long range correlations are presented contrary to earlier conjectures this contribution does not vanish identically a new method of calculation of multi loop feynman graphs with complicated numerator structure is suggested it leads to simpler integrals in a space of higher dimensionality which are computed using the recursion relations of the uniqueness method

a wavelet based analysis of fractal image compression why does fractal image compression work what is the implicit image model underlying fractal block coding how can we characterize the types of images for which fractal block coders will work well these are the central issues we address we introduce a new wavelet based framework for analyzing block based fractal compression schemes within this framework we are able to draw upon insights from the well established transform coder paradigm in order to address the issue of why fractal block coders work we show that fractal block coders of the form introduced by jacquin 1992 are haar wavelet subtree quantization schemes we examine a generalization of the schemes to smooth wavelets with additional vanishing moments the performance of our generalized coder is comparable to the best results in the literature for a jacquin style coding scheme our wavelet framework gives new insight into the convergence properties of fractal block coders and it leads us to develop an unconditionally convergent scheme with a fast decoding algorithm our experiments with this new algorithm indicate that fractal coders derive much of their effectiveness from their ability to efficiently represent wavelet zero trees finally our framework reveals some of the fundamental limitations of current fractal compression schemes

image compression with a hybrid wavelet fractal coder a hybrid wavelet fractal coder wfc for image compression is proposed the wfc uses the fractal contractive mapping to predict the wavelet coefficients of the higher resolution from those of the lower resolution and then encode the prediction residue with a bitplane wavelet coder the fractal prediction is adaptively applied only to regions where the rate saving offered by fractal prediction justifies its overhead a rate distortion criterion is derived to evaluate the fractal rate saving and used to select the optimal fractal parameter set for wfc the superior performance of the wfc is demonstrated with extensive experimental results

a range domain approximation error based approach for fractal image compression fractals can be an effective approach for several applications other than image coding and transmission database indexing texture mapping and even pattern recognition problems such as writer authentication however fractal based algorithms are strongly asymmetric because in spite of the linearity of the decoding phase the coding process is much more time consuming many different solutions have been proposed for this problem but there is not yet a standard for fractal coding this paper proposes a method to reduce the complexity of the image coding phase by classifying the blocks according to an approximation error measure it is formally shown that postponing range spl bsol slash domain comparisons with respect to a preset block it is possible to reduce drastically the amount of operations needed to encode each range the proposed method has been compared with three other fractal coding methods showing under which circumstances it performs better in terms of both bit rate and or computing time

fast fractal compression of greyscale images a new algorithm for fractal compression of greyscale images is presented it uses some previous results allowing the compression process to be reduced to a nearest neighbors problem and is essentially based on a geometrical partition of the image block feature space experimental comparisons with previously published methods show a significant improvement in speed with no quality loss

fire fractal indexing with robust extensions for image databases as already documented in the literature fractal image encoding is a family of techniques that achieves a good compromise between compression and perceived quality by exploiting the self similarities present in an image furthermore because of its compactness and stability the fractal approach can be used to produce a unique signature thus obtaining a practical image indexing system since fractal based indexing systems are able to deal with the images in compressed form they are suitable for use with large databases we propose a system called fire which is then proven to be invariant under three classes of pixel intensity transformations and under geometrical isometries such as rotations by multiples of spl pi 2 and reflections this property makes the system robust with respect to a large class of image transformations that can happen in practical applications the images can be retrieved even in the presence of illumination and or color alterations additionally the experimental results show the effectiveness of fire in terms of both compression and retrieval accuracy

study on huber fractal image compression in this paper a new similarity measure for fractal image compression fic is introduced in the proposed huber fractal image compression hfic the linear huber regression technique from robust statistics is embedded into the encoding procedure of the fractal image compression when the original image is corrupted by noises we argue that the fractal image compression scheme should be insensitive to those noises presented in the corrupted image this leads to a new concept of robust fractal image compression the proposed hfic is one of our attempts toward the design of robust fractal image compression the main disadvantage of hfic is the high computational cost to overcome this drawback particle swarm optimization pso technique is utilized to reduce the searching time simulation results show that the proposed hfic is robust against outliers in the image also the pso method can effectively reduce the encoding time while retaining the quality of the retrieved image

multifractal scaling in sinai diffusion abstract we consider the mean first passage time of random walks to go from one end of a segment of a sinai lattice to the other we show that the distribution of the mean first passage time over sinai disorder obeys a finite size multiscaling form

equilibrium distributions for random walkers in random media the authors study the renyi entropies of the probability distribution of a random walker in a random medium of finite total volume in one dimension of space it is shown that they tend for large lattices to constants independent of the lattice size thus the distribution is not multifractal in contrast to a recent claim the authors conjecture that the same is true for any dimension if the random force is derived from a potential in the more interesting general case unconstrained forces they cannot make a definite statement

region based fractal image compression a fractal coder partitions an image into blocks that are coded via self references to other parts of the image itself we present a fractal coder that derives highly image adaptive partitions and corresponding fractal codes in a time efficient manner using a region merging approach the proposed merging strategy leads to improved rate distortion performance compared to previously reported pure fractal coders and it is faster than other state of the art fractal coding methods