on the connection between irregular trajectories and the distribution of quantum level spacings for a model system the distribution of spacings between adjacent levels in specified intervals of the energy spectrum is compared with a semiclassical distribution this distribution berry and robnik 1984 depends on the size of the phase space volume filled with irregular trajectories good agreement is found between quantal and semiclassical spacing distributions in particular the transition from regularity to irregularity observed in the quantum calculation haller et al 1984 is well reproduced by the semiclassical results

spectral fluctuations and transport in phase space the authors extend the semiclassical theory of spectral fluctuations to include composite systems these systems are characterized by some number of weakly communicating highly chaotic subsystems they contain new and longer time scales depending on the connecting fluxes in addition to those associated with the rapid internal mixings the new time scales produce discernible modifications in the nature of the level fluctuations of the corresponding quantum systems the theory is applied to two coupled quartic oscillators

two dimensional critical point configuration graphs the configuration of the critical points of a smooth function of two variables is studied under the assumption that the function is morse that is that all of its critical points are nondegenerate a critical point configuration graph cpcg is derived from the critical points ridge lines and course lines of the function then a result from the theory of critical points of morse functions is applied to obtain several constraints on the number and type of critical points that appear on cycles of a cpcg these constraints yield a catalog of equivalent cpcg cycles containing four entries the slope districts induced by a critical point configuration graph appear useful for describing the behavior of smooth functions of two variables such as surfaces images and the radius function of three dimensional symmetric axes

level repulsion for band 3 3 random matrices in view of the great interest in band random matrices the authors investigate the level repulsion for the simplest case of 3 3 band matrices a non linear repulsion is found

reduced complexity rotation invariant texture classification using a blind deconvolution approach in this paper we present a texture classification procedure that makes use of a blind deconvolution approach specifically the texture is modeled as the output of a linear system driven by a binary excitation we show that features computed from one dimensional slices extracted from the two dimensional autocorrelation function acf of the binary excitation allows representing the texture for rotation invariant classification purposes the two dimensional classification problem is thus reconduced to a more simple one dimensional one which leads to a significant reduction of the classification procedure computational complexity

sorted consecutive local binary pattern for texture classification in this paper we propose a sorted consecutive local binary pattern sclbp for texture classification conventional methods encode only patterns whose spatial transitions are not more than two whereas sclbp encodes patterns regardless of their spatial transition conventional methods do not encode patterns on account of rotation invariant encoding on the other hand patterns with more than two spatial transitions have discriminative power the proposed sclbp encodes all patterns with any number of spatial transitions while maintaining their rotation invariant nature by sorting the consecutive patterns in addition we introduce dictionary learning of sclbp based on kd tree which separates data with a space partitioning strategy since the elements of sorted consecutive patterns lie in different space it can be generated to a discriminative code with kd tree finally we present a framework in which sclbps and the kd tree can be combined and utilized the results of experimental evaluation on five texture data sets outex curet uiuc umd and kth tips2 a indicate that our proposed framework achieves the best classification rate on the curet umd and kth tips2 a data sets compared with conventional methods the results additionally indicate that only a marginal difference exists between the best classification rate of conventional methods and that of the proposed framework on the uiuc and outex data sets

scaling properties of the eigenvalue spacing distribution for band random matrices the authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2 n where b is the band half width and n is the size of the matrices it is also shown that the eigenvalue s density obeys the semicircle law the found scaling behaviour suggests that the fluctuation properties in the intermediate regime between wigner dyson and poisson are universal

random antagonistic matrices the ensemble of antagonistic matrices is introduced and studied in antagonistic matrices the entries and are real and have opposite signs or are both zero and the diagonal is zero this generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

new approach to the statistical properties of energy levels the joint distribution of energy eigenvalues of a hamiltonian is derived by means of the usual statistical laws of classical many body systems it makes a transition from the poisson type to the gaussian type depending on the value of a single parameter characteristic of the hamiltonian

the statistics of eigenvector components of random band matrices analytical results by using a supersymmetric formalism the authors calculate analytically all higher moments pq sigma n mod psi n mod 2q generalizing the inverse participation ratio p2 where psi n 1 or n or n stands for the nth component of an eigenvector of a large random matrix with a band structure on this basis they reconstruct the whole probability distribution function of eigenvector components the relation with known numerical results is discussed

dyson s coulomb gas on a circle and intermediate eigenvalue statistics dyson s two dimensional coulomb gas on the unit circle with inverse temperature beta is investigated with the help of a metropolis algorithm theoretical predictions for energy and specific heat are verified the connection with the theory of ensembles of random unitary matrices with orthogonal unitary or symplectic symmetry corresponding to beta 1 2 4 is investigated in detail new approximation formulae for the spacing distribution and the so called delta statistics are proposed and found to be useful not only in the three cases mentioned but also for all beta between 0 and 4

a modified k means algorithm for circular invariant clustering several important pattern recognition applications are based on feature vector extraction and vector clustering directional patterns are commonly represented by rotation variant vectors f sub d formed from features uniformly extracted in m directions it is often desirable that pattern recognition algorithms are invariant under pattern rotation this paper introduces a distance measure and a k means based algorithm namely circular k means ck means to cluster vectors containing directional information such as f sub d in a circular shift invariant manner a circular shift of f sub d corresponds to pattern rotation thus the algorithm is rotation invariant an efficient fourier domain representation of the proposed measure is presented to reduce computational complexity a split and merge approach smck means suited to the proposed ck means technique is proposed to reduce the possibility of converging at local minima and to estimate the correct number of clusters experiments performed for textural images illustrate the superior performance of the proposed algorithm for clustering directional vectors f sub d compared to the alternative approach that uses the original k means and rotation invariant feature vectors transformed from f sub d

generalized wigner surmise for 2 2 random matrices we present new analytical results concerning the spectral distributions for 2 2 random real symmetric matrices which generalize the wigner surmise

nearest neighbour spacing distribution of energy levels in the region between integrability and chaos we derive an expression for the level spacing distribution for a quantal system displaying both regular and chaotic classical motion assuming that the level repulsion function for the mixed system is obtained by averaging the corresponding functions for the regular and chaotic regimes with weights given by their fractional phase space volumes the liouville measures of the regular region obtained from comparing this expression with the level spacing distributions of a hydrogen atom in a uniform magnetic field agree with the corresponding values obtained in the classical mechanical analysis of this system

poisson to wigner crossover transition in the nearest neighbor statistics of random points on fractals we show that the nearest neighbor spacing distribution for a model that consists of random points uniformly distributed on a self similar fractal is the brody distribution of random matrix theory in the usual context of hamiltonian systems the brody parameter does not have a definite physical meaning but in the model considered here the brody parameter is actually the fractal dimension exploiting this result we introduce a new model for a crossover transition between poisson and wigner statistics random points on a continuous family of self similar curves with fractal dimensions between 1 and 2 the implications to quantum chaos are discussed and a connection to conservative classical chaos is introduced

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

wigner surmise for mixed symmetry classes in random matrix theory we consider the nearest neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes or between integrable and nonintegrable systems we derive analytical formulas for the spacing distributions of 2 ifmmode times else texttimes fi 2 or 4 ifmmode times else texttimes fi 4 matrices and show numerically that they provide very good approximations for those of random matrices with large dimension this generalizes the wigner surmise which is valid for pure ensembles that are recovered as limits of the mixed ensembles we show how the coupling parameters of small and large matrices must be matched depending on the local eigenvalue density

quantum signatures of an integrable system with a chaotic scattering map a rather simple completely integrable hamiltonian with a chaotic classical scattering map is analysed in quantum mechanics after the analytical computation of the s matrix we investigate numerically the distribution of scattering phases giving the nearest neighbour distribution the number variance and the fourier transform of the spectrum we find fair agreement with the random matrix prediction for the circular orthogonal ensemble coe as appropriate for the time reversal invariant system with a chaotic scattering map this example confirms that coe properties of the s matrix do not necessarily indicate topological chaos in the flow generated by the classical hamiltonian

models of intermediate spectral statistics based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the anderson model at the transition point a simple theoretical model where such statistics can be obtained analytically is briefly discussed a few models with similar behavior are considered in particular we analytically found the eigenvalue statistics of a poisson distributed matrix perturbed by a rank one matrix which is a good model for spectral statistics of a singular billiard

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

saturation of number variance in embedded random matrix ensembles we study fluctuation properties of embedded random matrix ensembles of noninteracting particles for ensemble of two noninteracting particle systems we find that unlike the spectra of classical random matrices correlation functions are nonstationary in the locally stationary region of spectra we study the number variance and the spacing distributions the spacing distributions follow the poisson statistics which is a key behavior of uncorrelated spectra the number variance varies linearly as in the poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths and the number variance approaches saturation values these results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory we conjecture that the interacting particle cases which exhibit the characteristics of classical random matrices for short correlation lengths will also show saturation effects for large correlation lengths

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

texture synthesis textons revisited this paper introduces a technique for synthesizing natural textures with emphasis on quasiperiodic and structural textures textures are assumed to be composed of three components namely illumination structure and stochastic the contribution of this work is that in contrast to previous techniques it proposes a joint approach for handling the texture s global illumination irregular structure and stochastic component which may be correlated to the other two components furthermore the proposed technique does not produce verbatim copies in the synthesized texture more specifically a top down approach is used for extraction of texture elements textons in which in contrast to previous texton based approaches no assumptions regarding perfect periodicity are made the structure itself can be modeled as a stochastic process consequently textons are allowed to have irregular and nonidentical shapes in the synthesis stage a new nonregular textural structure is designed from the original one that defines the place holders for textons we call such place holders empty textons e textons the e textons are filled in by a representative texton since e textons do not have identical shapes a texton shape matching procedure is required after adding the illumination to the structural component a strictly localized version of a block sampling technique is applied to add the stochastic component the block sampling technique combined with the addition of the illumination component provides a significant improvement in the appearance of synthesized textures results show that the proposed method is successful in synthesizing structural textures visually indistinguishable to the original moreover the method is successful in synthesizing a variety of stochastic textures

density of states of sparse random matrices a supersymmetric formalism is used to derive a set of equations giving the density of states of any real symmetric sparse random matrix as a function of the distribution of non zero elements and the mean number of non zero elements per row p in the matrix where the non zero elements take the values or 1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method as e to 0 the density of states rho e behaves as 1 e ln e 2 the more general case where or 1 occur with unequal probabilities is also considered

segmentation by texture using correlation the correlation coefficients are used for segmentation according to texture they are first evaluated on a set of square regions forming two levels of the quadratic picture tree or pyramid if the coefficients of a square and its four children in the tree are similar then that region is considered to be of uniform texture if not it is replaced by its children in this way the split and merge algorithm is used to achieve a preliminary segmentation it is followed by a grouping algorithm using the correlation coefficients and the region adjacency graph plus a small region elimination step the latter regions are grouped according to their gray level because texture cannot be defined reliably on very small regions examples of implementation on four pictures are included

on invariant 2 2 β ensembles of random matrices abstract we introduce and solve exactly a family of invariant 2 2 random matrices depending on one parameter and we show that rotational invariance and real dyson index are not incompatible properties the probability density for the entries contains a weight function and a multiple trace trace interaction term which corresponds to the representation of the vandermonde squared coupling on the basis of power sums as a result the effective dyson index eff of the ensemble can take any real value in an interval two weight functions gaussian and non gaussian are explored in detail and the connections with ensembles of dumitriu edelman and the so called poisson wigner crossover for the level spacing are respectively highlighted a curious spectral twinning between ensembles of different symmetry classes is unveiled as a consequence the identification between symmetry group orthogonal unitary or symplectic and the exponent of the vandermonde 1 2 4 is shown to be potentially deceptive the proposed technical tool more generically allows for designing actual matrix models which i are rotationally invariant ii have a real dyson index eff iii have a pre assigned confining potential or alternatively level spacing profile the analytical results have been checked through numerical simulations with an excellent agreement eventually we discuss possible generalizations and further directions of research

steerable weighted median filters a filter is steerable if transformed i e rotated scaled etc versions of its impulse response can be expressed as linear combinations of a fixed set of basis functions steerability is important for numerous image processing applications however it is a property presently shared only by a specific class of linear filters on the other hand several classes of nonlinear filters such as weighted median filters wmfs may offer certain advantages over linear filters such as robustness and edge preserving capabilities in this paper the concept of steerability is extended to encompass wmfs it will be shown that in general a steerable wmf design technique needs to be capable of handling negative weights although methods that allow the design of wmfs admitting negative weights have already been proposed such methods do not necessarily produce filters that are steerable as opposed to the approach presented in this work experimental results illustrate the applicability of steerable wmfs in two applications namely edge detection and orientation analysis

distribution of the ratio of consecutive level spacings in random matrix ensembles we derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices this ratio distribution was recently introduced to study spectral properties of many body problems as contrary to the standard level spacing distributions it does not depend on the local density of states our wigner like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit quantitative improvements are found through a polynomial expansion examples from a quantum many body lattice model and from zeros of the riemann zeta function are presented

a note on the level spacings distribution of the hamiltonians in the transition region between integrability and chaos it is well known that there is no strict universality of the spectral fluctuations of quantum hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos the author discusses the level spacings distribution p s and explains why the semiclassical formulae of berry and robnik cannot be correct for small s there is no global universality of p s for nearly integrable systems but the approach to the integrability as the perturbation parameter in goes to zero can be universal this is reflected in the fact that the slope of dp ds at s 0 for small in is universally inversely proportional to in the author gives two models in terms of two dimensional random matrices one of them being based on maximum entropy considerations the author also points out the connection to the statistics of zeros of random functions and discusses the numerical evidence

higher order level correlations in integrable quantum systems the author studies the second third and fourth cumulants of the distribution of the number of energy levels in an interval containing levels on an average the distribution function which also depends on the distance n measured in units of average level spacings above the ground state can be obtained either from the exact quantum levels or from the semiclassical approximation to the level density either of the two methods yields the same values for the cumulants in two limits the cumulants have been obtained analytically first for small the functional dependence of the cumulants is obtained using generalisations of berry s semiclassical sum rule for the two point function one finds that in this limit the first four cumulants are equal to the average number of levels in the interval second larger values of the functional dependence can be obtained by taking into account only the slowly oscillating contributions to the cumulants of the semiclassical level density for the second cumulant he obtains an analytical expression when 1

eigenvalue spacing statistics of a four matrix model of some four by four random matrices a simple four matrix model consisting of some 4 4 real and imaginary random matrices and three parameters is introduced it is shown that the eigenvalue spacing statistics of an ensemble of such matrices can be used to describe transitions between all the wigner surmises of random matrix theory formulae for the nearest neighbour spacing distributions of the ensemble are given for various parameter values

an intensity texture model based level set method for image segmentation in this paper a novel level set segmentation model integrating the intensity and texture terms is proposed to segment complicated two phase nature images firstly an intensity term based on the global division algorithm is proposed which can better capture intensity information of image than the chan vese model cv particularly the cv model is a special case of the proposed intensity term under a certain condition secondly a texture term based on the adaptive scale local variation degree aslvd algorithm is proposed the aslvd algorithm adaptively incorporates the amplitude and frequency components of local intensity variation thus it can extract the non stationary texture feature accurately finally the intensity term and the texture term are jointly incorporated into level set and used to construct effective image segmentation model named as the intensity texture model since the intensity term and the texture term are complementary for image segmentation the intensity texture model has strong ability to accurately segment those complicated two phase nature images experimental results demonstrate the effectiveness of the proposed intensity texture model an intensity term based on the so called global division algorithm is proposed we extract the amplitude and frequency components of local intensity variation we propose the adaptive scale local variation degree algorithm as texture term the intensity and texture terms are integrated into level set energy functional

failure of semiclassical methods to predict individual energy levels the authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity by this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below this they show for the case of the integrable circular billiard and the one dimensional potential uo cos2 alpha x by comparing the torus quantized semiclassical eigenenergies with the exact results since all the various semiclassical methods such as gutzwiller s and bogomolny s are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid they have theoretical arguments and strong numerical evidence for the case of the circular billiard that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations it is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting poissonian behaviour so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory they discuss the validity of semiclassical methods in the light of their negative and positive findings in addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary i e it is independent of the energy

unsupervised texture segmentation using gabor filters abstract this paper presents a texture segmentation algorithm inspired by the multi channel filtering theory for visual information processing in the early stages of human visual system the channels are characterized by a bank of gabor filters that nearly uniformly covers the spatial frequency domain and a systematic filter selection scheme is proposed which is based on reconstruction of the input image from the filtered images texture features are obtained by subjecting each selected filtered image to a nonlinear transformation and computing a measure of energy in a window around each pixel a square error clustering algorithm is then used to integrate the feature images and produce a segmentation a simple procedure to incorporate spatial information in the clustering process is proposed a relative index is used to estimate the true number of texture categories

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

ridges crests and sub parabolic lines of evolving surfaces the ridge lines on a surface can be defined either via contact of the surface with spheres or via extrema of principal curvatures along lines of curvature certain subsets of ridge lines called crest lines have been singled out by some authors for medical imaging applications there is a related concept of sub parabolic line on a surface also defined via extrema of principal curvatures in this paper we study in detail the structure of the ridge lines crest lines and sub parabolic lines on a generic surface and on a surface which is evolving in a generic one parameter family the mathematical details of this study are in bruce et al 1994c

random matrix theory models of electric grid topology abstract the random matrix theory is useful in the study of large systems such as electric grids these transmission systems can be modeled as complex networks with high voltage lines the edges that connect nodes representing power plants and substations we draw upon established literature of complex systems theory and introduce methods from nuclear and statistical physics to identify new characteristics of these networks we show that most grids can be characterized by the gaussian orthogonal ensemble an indicator of chaos in many complex systems under certain circumstances however grids may be described by poisson statistics an indicator of regularity we use the random matrix formalism to describe the interconnection of multiple grids and construct a simple model of a distributed grid

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

long range stiffness of spectral fluctuations in integrable scale invariant systems the authors study the semiclassical limit for the delta 3 statistic of integrable systems which have a homogeneous polynomial as potential these systems possess a scale invariance which provides one with the energy dependence of the statistic using the general expressions given by berry 1985 they also obtain the functional dependence of the universal part of the delta 3 statistic both for integrable and ergodic systems they use a stationary phase approximation to evaluate the semiclassical limit of the level density when the dimension of the system is larger than one the investigation of its validity leads to the introduction of a generalised perimeter term in the average level density the fluctuating part of the semiclassical level density yields the semiclassical limit of the delta 3 statistic which is compared numerically to results obtained for actual spectra they find that the semiclassical approximation is excellent provided the perimeter term is taken into account exactly they also study the dimensional dependence of delta 3 for energy levels around the nth level above the ground state the position of the kink in the delta 3 statistic is essentially proportional to n d 1 d

radon transform orientation estimation for rotation invariant texture analysis this paper presents a new approach to rotation invariant texture classification the proposed approach benefits from the fact that most of the texture patterns either have directionality anisotropic textures or are not with a specific direction isotropic textures the wavelet energy features of the directional textures change significantly when the image is rotated however for the isotropic images the wavelet features are not sensitive to rotation therefore for the directional textures it is essential to calculate the wavelet features along a specific direction in the proposed approach the radon transform is first employed to detect the principal direction of the texture then the texture is rotated to place its principal direction at 0 degrees a wavelet transform is applied to the rotated image to extract texture features this approach provides a features space with small intraclass variability and therefore good separation between different classes the performance of the method is evaluated using three texture sets experimental results show the superiority of the proposed approach compared with some existing methods

spectral fluctuation properties of hamiltonian systems the transition region between order and chaos the authors study numerically the classical dynamical behaviour and the spectral fluctuation properties of a class of hamiltonian systems with two degrees of freedom the quantum mechanical properties of these systems are monotonic but non universal functions of the fraction of classical phase space filled by chaotic trajectories it is found that the observed spectral fluctuation measures can be reproduced by a random matrix model which depends on one parameter only

segmentation of textured images based on fractals and image filtering this paper describes a new approach to the segmentation of textured gray scale images based on image pre ltering and fractal features traditionally lter bank decomposition methods consider the energy in each band as the textural feature a parameter that is highly dependent on image intensity in this paper we use fractal based features which depend more on textural characteristics and not intensity information to reduce the total number of features used in the segmentation the signi cance of each feature is examined using a test similar to the f test and less signi cant features are not used in the clustering process the commonly used k means algorithm is extended to an iterative k means by using a variable window size that preserves boundary details the number of clusters is estimated using an improved hierarchical approach that ignores information extracted around region boundaries 2001 pattern recognition society published by elsevier science ltd all rights reserved

spectral fluctuations of billiards with mixed dynamics from time series to superstatistics a statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards one with mushroomlike shape and the other from the family of the limacon billiards is presented these billiards have mixed regular chaotic dynamics but different structures in their classical phase spaces the spectrum of each billiard is represented as a time series where the level order plays the role of time two most important findings follow from the time series analysis first the spectra can be characterized by two distinct relaxation lengths this is a prerequisite for the validity of the superstatistical approach which is based on the folding of two distribution functions second the shape of the resulting probability density function of the so called superstatistical parameter is reasonably approximated by an inverse chi2 distribution this distribution is used to compute nearest neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics the obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions including the one from berry and robnik however in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the porter thomas distribution indeed the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the limacon billiards within the framework of superstatistics

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

rotation invariant multiresolution texture analysis using radon and wavelet transforms a new rotation invariant texture analysis technique using radon and wavelet transforms is proposed this technique utilizes the radon transform to convert the rotation to translation and then applies a translation invariant wavelet transform to the result to extract texture features a k nearest neighbors classifier is employed to classify texture patterns a method to find the optimal number of projections for the radon transform is proposed it is shown that the extracted features generate an efficient orthogonal feature space it is also shown that the proposed features extract both of the local and directional information of the texture patterns the proposed method is robust to additive white noise as a result of summing pixel values to generate projections in the radon transform step to test and evaluate the method we employed several sets of textures along with different wavelet bases experimental results show the superiority of the proposed method and its robustness to additive white noise in comparison with some recent texture analysis methods

level repulsion near integrability a random matrix analogy using the analogy between the statistics of the levels of quantum hamiltonians and the eigenvalues of random matrices the authors use an appropriate choice of the latter in order to study the transition region near integrability they show that the nearest neighbour spacing distribution is linear for small spacings while the inverse of its slope is proportional to the amplitude of the integrability destroying perturbation

quantum surface of section method demonstration of semiclassical berry robnik energy level spacing distribution in a generic two dimensional hamiltonian system the recently developed quantum surface of section method is applied to a search for extremely high lying energy levels in a simple but generic hamiltonian system between integrability and chaos namely the semiseparable two dimensional oscillator using the stretch of 13445 consecutive levels with the sequential number around 1 8 107 eighteen million we have clearly demonstrated the validity of the semiclassical berry robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power law level repulsion which is globally very well described by the phenomenological brody distribution this is the first statistically significant numerical confirmation of the berry robnik surmise in an autonomous two dimensional hamiltonian system

semi poisson statistics and beyond semi poisson statistics are shown to be obtained by removing every other number from a random sequence retaining every r 1 mathrm th level we obtain a family of sequences which we call daisy models their statistical properties coincide with those of bogomolny s nearest neighbor interaction coulomb gas if the inverse temperature coincides with the integer r in particular the case r 2 reproduces closely the statistics of quasioptimal solutions of the traveling salesman problem

logarithmic terms in the spectral statistics of band random matrices the authors present a detailed analysis of the nearest neighbour spacing distribution nnsd for random matrices on which the non zero elements are confined to a band around the diagonal the vanishing of the remaining elements induces non analytic logarithmic behaviour for the nnsd at very small spacings a conjective for the precise behaviour of these logarithmic terms is presented and supported by extensive numerical computations

vector transport for shape from shading in this paper we describe a new shape from shading method we show how the parallel transport of surface normals can be used to impose curvature consistency and also to iteratively update surface normal directions so as to improve the brightness error we commence by showing how to make local estimates of the hessian matrix from surface normal information with the local hessian matrix to hand we develop an em like algorithm for updating the surface normal directions at each image location parallel transport is applied to the neighbouring surface normals to generate a sample of local surface orientation predictions from this sample a local weighted estimate of the image brightness is made the transported surface normal which gives the brightness prediction which is closest to this value is selected as the revised estimate of surface orientation the revised surface normals obtained in this way may in turn be used to re estimate the hessian matrix and the process iterated until stability is reached we experiment with the method on a variety of real world and synthetic data here we explore the properties of the fields of surface normals and the height data delivered by the method

inferring surface trace and differential structure from 3 d images early image understanding seeks to derive analytic representations from image intensities the authors present steps towards this goal by considering the inference of surfaces from three dimensional images only smooth surfaces are considered and the focus is on the coupled problems of inferring the trace points the points through which the surface passes and estimating the associated differential structure given by the principal curvature and direction fields over the estimated smooth surfaces computation of these fields is based on determining an atlas of local charts or parameterizations at estimated surface points algorithm robustness and the stability of results are essential for analyzing real images to this end the authors present a functional minimization algorithm utilizing overlapping local charts to refine surface points and curvature estimates and develop an implementation as an iterative constraint satisfaction procedure based on local surface smoothness properties examples of the recovery of local structure are presented for synthetic images degraded by noise and for clinical magnetic resonance images

eigenfunction distribution for the rosenzweig porter model the statistical distribution of eigenfunctions for the rosenzweig porter model is derived for the region where eigenfunctions have fractal behaviour the result is based on simple physical ideas and leads to transparent explicit formulas which agree very well with numerical calculations it constitutes a rare case where a non trivial eigenfunction distribution is obtained in a closed form

learning completed discriminative local features for texture classification we propose a novel feature representation method namely completed discriminative local features cdlf for texture classification the cdlf learn transformation matrices for texture images that maximize the mutual information between the local features and their category labels we propose an adaptive histogram accumulation aha algorithm which leverages the local contrast characteristic in the process of histogram accumulation the cdlf achieves higher accuracy than state of the art methods on three databases local binary patterns lbp and its variants have shown great potentials in texture classification tasks lbp like texture classification methods usually follow a two step feature extraction process in the first pattern encoding step the local structure information around each pixel is encoded into a binary string in the second histogram accumulation step the binary strings are accumulated into a histogram as the feature vector of a texture image the performances of these classification methods are closely related to the distinctiveness of the feature vectors in this paper we propose a novel feature representation method namely completed discriminative local features cdlf for texture classification the proposed cdlf improves the distinctiveness of lbp like feature vectors in two aspects in the pattern encoding stage we learn a transformation matrix using labeled data which significantly increases the discrimination power of the encoded binary strings in the histogram accumulation step we use an adaptive weight strategy to consider the contributions of pixels in different regions the experimental results on three challenging texture databases demonstrate that the proposed cdlf achieves significantly better results than previous lbp like feature representation methods for texture classification tasks

radon representation based feature descriptor for texture classification in this paper we aim to handle the intraclass variation resulting from the geometric transformation and the illumination change for more robust texture classification to this end we propose a novel feature descriptor called radon representation based feature descriptor rrfd rrfd converts the original pixel represented images into radon pixel images by using the radon transform the new radon pixel representation is more informative in geometry and has a much lower dimension subsequently rrfd efficiently achieves affine invariance by projecting an image or an image patch from the space of radon pixel pairs onto an invariant feature space by using a ratiogram i e the histogram of ratios between the areas of triangle pairs the illumination invariance is also achieved by defining an illumination invariant distance metric on the invariant feature space comparing to the existing radon transform based texture features which only achieve rotation and or scaling invariance rrfd achieves affine invariance the experimental results on curet show that rrfd is a powerful feature descriptor that is suitable for texture classification

random matrix theory by the supersymmetry method beyond the sigma model abstract the leading correction to the smoothed connected energy density density correlation function is obtained for the large energy difference within the context of the gaussian random matrix theory in order to achieve this result the supersymmetry method is extended beyond the sigma model to include small quadratic fluctuations around the saddle point special care is taken to avoid the potential divergence arising from the unbounded nature of the saddle point also in the small energy difference regime the leading correction to a two point correlation function is obtained

spacing distributions for real symmetric 2 2 generalized gaussian ensembles for ensembles of 2 2 real symmetric matrices the normalized spacing distributions p s form a family whose parameter space is the unit cube with coordinates determined by the means and variances of the diagonal and off diagonal elements the cube contains a variety of spacing distributions that are calculated analytically they include the wigner poisson transition distributions with singularities and gaussians unfolding is superfluous for 2 2 matrices but it can be implemented giving rise to a further variety of spacing distributions some surprising

texture analysis by fractal descriptors over the wavelet domain using a best basis decomposition abstract this work proposes the development and study of a novel set of fractal descriptors for texture analysis these descriptors are obtained by exploring the fractal like relation among the coefficients and magnitudes of a particular type of wavelet decomposition to know the best basis selection the proposed method is tested in the classification of three sets of textures from the literature brodatz vistex and usptex the method is also applied to a challenging real world problem which is the identification of species of plants from the brazilian flora the results are compared with other classical and state of the art texture descriptors and demonstrate the efficiency of the proposed technique in this task

the extremal mesh and the understanding of 3d surfaces this paper is about a new concept for the description of 3d smooth surfaces the extremal mesh in previous works we have shown how to extract the extremal lines from 3d images which are the lines where one of the two principal surface curvatures is locally extremal we have also shown how to extract the extremal points which are specific points where the two principal curvatures are both extremal the extremal mesh is the graph of the surface whose vertices are the extremal points and whose edges are the extremal lines it is invariant with respect to rigid transforms the good topological properties of this graph are ensured with a new local geometric invariant of 3d surfaces that we call the gaussian extremality and which allows to overcome orientation problems encountered with previous definitions of the extremal lines and points this paper presents also an algorithm to extract the extremal mesh from 3d images and experiments with synthetic and real 3d medical images show that this graph can be extremely precise and stable

iterated impact dynamics of n beads on a ring when n beads slide along a frictionless hoop their collision sequence gives rise to a dynamical system that can be studied via matrix products it is of general interest to understand the distribution of velocities and the corresponding eigenvalue spectrum that a given collision sequence can produce we formulate the problem for general n and state some basic theorems regarding the eigenvalues of the collision matrices and their products the case of three beads of masses m1 m2 m3 is studied in detail we exploit the fact that each collision sequence can be viewed as a billiard trajectory in a right triangle with non standard reflection rules existence of families of periodic orbits are proven and orbits that densely fill the triangle are computed eigenvalue distributions and position and velocity histograms are computed as a function of the restitution coefficient both periodic and dense collision sequences are discussed and a series of conjectures based on computational evidence are formulated comparisons are made between the eigenvalue distributions and autocorrelation matrices associated with dense trajectories generated from a chaotic collision sequence and spectra from matrix sequences generated from random orderings and we describe how the three bead system could be used as the basis for a random number generating algorithm that is computationally efficient

algorithmic complexity of the eigenvalue sequence of a nonintegrable hamiltonian system irregularity of the eigenvalue sequence in a quantum nonintegrable system is investigated from the viewpoint of algorithmic complexity theory an example of a real symmetric matrix whose eigenvalue sequence has null complexity but whose nearest neighbor level spacing shows the wigner type distribution is presented

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

spacing distributions for point processes on a regular fractal the homogeneous poisson point process in rd denoted by pd is a basic model of stochastic geometry and modern statistical physics using ideas from fractal geometry geometrical statistics and random matrix theory we introduce the model of random points on a self similar fractal as a model of intermediate statistics in the sense that the interpoint spacing statistics of the model are intermediate between those of p1 and p2 when the fractal dimension is in between 1 and 2 and intermediate between those of p2 and p3 when the fractal dimension is in between 2 and 3 and so on we also introduce the idea of using a continuous family of such models to interpolate between p1 and p2 and thereby effectuate crossover transitions between p1 statistics and p2 statistics we first derive the kth nearest neighbor spacing distribution for the general model and then study the interpoint spacing statistics of several realizations of the model involving sierpinski fractals in r2 and r3 we also study a realization of a continuous interpolation between p1 and p2 in particular a continuous interpolation between a point process on a line and a point process on a plane filling curve using the continuous family of self similar koch curves in r2 in the latter study we specifically analyze the second nearest neighbor interpoint spacing statistics which undergo a crossover transition between semi poisson and ginibre statistics

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

wavelet based rotational invariant roughness features for texture classification and segmentation we introduce a rotational invariant feature set for texture segmentation and classification based on an extension of fractal dimension fd features the fd extracts roughness information from images considering all available scales at once in this work a single scale is considered at a time so that textures with scale dependent properties are satisfactorily characterized single scale features are combined with multiple scale features for a more complete textural representation wavelets are employed for the computation of single and multiple scale roughness features because of their ability to extract information at different resolutions features are extracted in multiple directions using directional wavelets and the feature vector is finally transformed to a rotational invariant feature vector that retains the texture directional information an iterative k means scheme is used for segmentation and a simplified form of a bayesian classifier is used for classification the use of the roughness feature set results in high quality segmentation performance furthermore it is shown that the roughness feature set exhibits a higher classification rate than other feature vectors presented in this work the feature set retains the important properties of fd based features namely insensitivity to absolute illumination and contrast

singularities of principal direction fields from 3 d images generic singularities can provide position independent information about the qualitative shape of surfaces the authors determine the singularities of the principal direction fields of a surface its umbilic points from a computation of the index of the fields the authors present examples both for 3 d synthetic images to which noise has been added and for clinical magnetic resonance images

level spacing for band random matrices the authors examine the spacing distribution of the eigenvalues of tridiagonal real symmetric random matrices the elements of which are distributed according to a gaussian law they show explicitly that for 4 4 matrices the distribution at small spacings behaves as s log2 s they surmise that for n n matrices the behaviour is s logn 2s and they present numerical results which support this conjecture

texture segmentation using fractal dimension this paper deals with the problem of recognizing and segmenting textures in images for this purpose the authors employ a technique based on the fractal dimension fd and the multi fractal concept six fd features are based on the original image the above average high gray level image the below average low gray level image the horizontally smoothed image the vertically smoothed image and the multi fractal dimension of order two a modified box counting approach is proposed to estimate the fd in combination with feature smoothing in order to reduce spurious regions to segment a scene into the desired number of classes an unsupervised k means like clustering approach is used mosaics of various natural textures from the brodatz album as well as microphotographs of thin sections of natural rocks are considered and the segmentation results to show the efficiency of the technique supervised techniques such as minimum distance and k nearest neighbor classification are also considered the results are compared with other techniques

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

energy level statistics and localization in sparsed banded random matrix ensemble the matrix elements of the quantum hamiltonian which corresponds to the classical kam system h h0 epsilon v where h0 is integrable in an energy ordered eigenbasis of h0 are considered their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble sbrme in analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the brody distribution the numerical study of sbrme shows two interesting scaling laws i the connection between the scaling variable x alpha b3 2 alpha mean diagonal increment b bandwidth and the level repulsion parameter beta as deduced from the brody distribution and ii the connection between the same scaling variable x and an expression which contains two types of localization length entropy and geometric the universal aspects and the importance of sbrme for the hamiltonian systems in the transition region between integrability and chaos are discussed

an evaluation of the integral of the product of the error function and the normal probability density with application to the bivariate normal integral abstract this paper derives the value of the integral of the product of the error function and the normal probability density as a series of the hermite polynomial and the normalized incomplete gamma function this expression is beneficial and can be used for evaluating the bivariate normal integral as a series expansion this expansion is a good alternative to the well known tetrachoric series when the correlation coefficient is large in absolute value

on asymptotic expansions and scales of spectral universality in band random matrix ensembles we consider real random symmetric n n matrices h of the band type form with characteristic length b the matrix entries are independent gaussian random variables and have the variance proportional to where u t vanishes at infinity we study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace

comment on energy level statistics in the mixed regime we comment on the recent paper by abul magd 1996 29 1 concerning the energy level statistics in the mixed regime i e such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space we point out that his basic assumption on the additive property of the level repulsion function r s conditional probability density in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified since among other things it relies on the type of berry s ergodic assumption which however is only correct in a homogeneous ensemble of ergodic systems not in the neighbourhood of an integrable system thus his resulting distribution cannot be regarded as a theoretically well founded object we point out that the semiclassical limiting energy level spacing distribution must be of berry robnik 1984 type and explain what transitional behaviour of the brody type with fractional power law energy level repulsion we observe in the near semiclassical regime where effective is not yet small enough thus we refer to the derivation arguments and conclusions in our previous paper prosen and robnik 1994 26 8059 and explain again the behaviour in this double transition region

intermediate statistics of the quasi energy spectrum and quantum localisation of classical chaos the statistical properties of the quasi energy spectrum in a simple quantum model are investigated for the case when the corresponding classical system is fully chaotic while quantum chaos is restricted by the localisation effects it is shown that the level spacing distribution depends effectively on some parameter which is the ratio of the dimension of the eigenfunctions mean localisation length to the total number of the quasi energy levels numerical data for a wide range of parameters of the system are given