scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

tsallis and renyi entropies in fractional diffusion and entropy production the entropy production rate for fractional diffusion processes using shannon entropy was calculated previously which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation renyi and tsallis entropies which have an additional parameter q generalizing the shannon case q 1 are shown here to have similar counter intuitive behaviours however the issue can be successfully treated in exactly the same manner as with shannon entropy for q being not too large i e generalizations near the shannon case whereas for larger q the renyi and tsallis entropies behave in a different way

generalized spatio chromatic diffusion a framework for diffusion of color images is presented the method is based on the theory of thermodynamics of irreversible transformations which provides a suitable basis for designing correlations between the different color channels more precisely we derive an equation for color evolution which comprises a purely spatial diffusive term and a nonlinear term that depends on the interactions among color channels over space we apply the proposed equation to images represented in several color spaces such as rgb cielab opponent colors and ihs

a new scheme for robust gradient vector estimation in color images gradient estimators are mostly designed to yield accurate and robust estimates of the gradient magnitude not the gradient direction this paper proposes a method for the accurate and robust estimation of both the gradient magnitude and direction it robustly estimates the gradient in the x and y directions the robustness against noise is achieved by prefiltering and postfiltering of the gradient in each direction to reduce edge blurring effects introduced by these filters the gradient in a certain direction is obtained by applying the prefilter and postfilter in the perpendicular direction the basic elements employed in each window are highpass lowpass and aggregation operators the highpass operator is used as a gradient estimator the lowpass operator is for prefiltering and postfiltering and the aggregation operator is for aggregating the prefiltered and postfiltered gradients four different combinations of highpass lowpass and aggregation operators are proposed mvd median mean mvd median max rcmg median mean and rcmg median max experimental results show that the rcmg median mean has the best performance in estimating the gradient and detecting the edges in noisy color images it is computationally more efficient than the state of the art gradient estimators and is able to accurately estimate the gradient direction as well as the gradient magnitude computer simulation results show that the proposed method outperforms other recently proposed color gradient estimators and edge detectors

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

the entropy production paradox for fractional master equations abstract time fractional evolution equations for probability distributions provide a means to describe an important class of stochastic processes their solutions show features which are essential in modeling a variety of phenomena in real world applications one aspect which has been observed in time fractional diffusion equations shows a surprising and unexpected behavior of the entropy production rate induced by these equations the entropy production rate increases as one moves away from the fully irreversible case corresponding to classical diffusion this rate is analyzed for a new class of systems with state spaces that are finite and denumerable we find that the entropy production paradox reemerges nonetheless but in a new and unexpected form

fuzzy peer groups for reducing mixed gaussian impulse noise from color images the peer group of an image pixel is a pixel similarity based concept which has been successfully used to devise image denoising methods however since it is difficult to define the pixel similarity in a crisp way we propose to represent this similarity in fuzzy terms in this paper we introduce the fuzzy peer group concept which extends the peer group concept in the fuzzy setting a fuzzy peer group will be defined as a fuzzy set that takes a peer group as support set and where the membership degree of each peer group member will be given by its fuzzy similarity with respect to the pixel under processing the fuzzy peer group of each image pixel will be determined by means of a novel fuzzy logic based procedure we use the fuzzy peer group concept to design a two step color image filter cascading a fuzzy rule based switching impulse noise filter by a fuzzy average filtering over the fuzzy peer group both steps use the same fuzzy peer group which leads to computational savings the proposed filter is able to efficiently suppress both gaussian noise and impulse noise as well as mixed gaussian impulse noise experimental results are provided to show that the proposed filter achieves a promising performance

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

defining probability density for a distribution of random functions the notion of probability density for a random function is not as straightforward as in finite dimensional cases while a probability density function generally does not exist for functional data we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis this leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension this density approximation is estimable readily from data it accurately represents in a monotone way key features of small ball approximations to density our results on estimators of the densities of principal component scores are also of independent interest they reveal interesting shape differences that have not previously been considered the statistical implications of these results and properties are identified and discussed and practical ramifications are illustrated in numerical work

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

the similarity group and anomalous diffusion equations a number of distinct differential equations known as generalized diffusion equations have been proposed to describe the phenomenon of anomalous diffusion on fractal objects although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon using similarity methods it becomes very clear that this is far from sufficient to confirm their validity the similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals it also reduces the differential equations to extra ordinary ones which are presented here for the first time it becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data suggesting that a new approach is needed

anisotropic diffusion of multivalued images with applications to color filtering a general framework for anisotropic diffusion of multivalued images is presented we propose an evolution equation where at each point in time the directions and magnitudes of the maximal and minimal rate of change in the vector image are first evaluated these are given by eigenvectors and eigenvalues of the first fundamental form in the given image metric then the image diffuses via a system of coupled differential equations in the direction of minimal change the diffusion strength is controlled by a function that measures the degree of dissimilarity between the eigenvalues we apply the proposed framework to the filtering of color images represented in cie l a b space

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

componentwise classification and clustering of functional data the infinite dimension of functional data can challenge conventional methods for classification and clustering a variety of techniques have been introduced to address this problem particularly in the case of prediction but the structural models that they involve can be too inaccurate or too abstract or too difficult to interpret for practitioners in this paper we develop approaches to adaptively choose components enabling classification and clustering to be reduced to finite dimensional problems we explore and discuss properties of these methodologies our techniques involve methods for estimating classifier error rate and cluster tightness and for choosing both the number of components and their locations to optimize these quantities a major attraction of this approach is that it allows identification of parts of the function domain that convey important information for classification and clustering it also permits us to determine regions that are relevant to one of these analyses but not the other copyright 2012 oxford university press

diffusion of general data on non flat manifolds viaharmonic maps theory the direction diffusion case in a number of disciplines directional data provides a fundamental source of information a novel framework for isotropic and anisotropic diffusion of directions is presented in this paper the framework can be applied both to denoise directional data and to obtain multiscale representations of it the basic idea is to apply and extend results from the theory of harmonic maps and in particular harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data we show the corresponding variational and partial differential equations formulations for isotropic diffusion obtained from an l 2 norm and edge preserving diffusion obtained from an l norm in general and an l 1 norm in particular in contrast with previous approaches the framework is valid for directions in any dimensions supports non smooth data and gives both isotropic and anisotropic formulations in addition the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non flat manifolds that is functions between two general manifolds we present a number of theoretical results open questions and examples for gradient vectors optical flow and color images

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

color image enhancement via chromaticity diffusion a novel approach for color image denoising is proposed in this paper the algorithm is based on separating the color data into chromaticity and brightness and then processing each one of these components with partial differential equations or diffusion flows in the proposed algorithm each color pixel is considered as an n dimensional vector the vectors direction a unit vector gives the chromaticity while the magnitude represents the pixel brightness the chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals this theory deals with the regularization of vectorial data while satisfying the intrinsic unit norm constraint of directional data such as chromaticity both isotropic and anisotropic diffusion flows are presented for this n dimensional chromaticity diffusion flow the brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement we present the underlying theory a number of examples and briefly compare with the current literature

entropy production and nonlinear fokker planck equations the entropy time rate of systems described by nonlinear fokker planck equations which are directly related to generalized entropic forms is analyzed both entropy production associated with irreversible processes and entropy flux from the system to its surroundings are studied some examples of known generalized entropic forms are considered and particularly the flux and production of the boltzmann gibbs entropy obtained from the linear fokker planck equation are recovered as particular cases since nonlinear fokker planck equations are appropriate for the dynamical behavior of several physical phenomena in nature like many within the realm of complex systems the present analysis should be applicable to irreversible processes in a large class of nonlinear systems such as those described by tsallis and kaniadakis entropies

nonequilibrium statistical operator method in renyi statistics abstract the generalization of the zubarev nonequilibrium statistical operator nso method for the case of renyi statistics is proposed when the relevant statistical operator or distribution function is obtained based on the maximum entropy principle for the renyi entropy the nonequilibrium statistical operator and corresponding generalized transport equations for the reduced description parameters are obtained a consistent description of the kinetic and hydrodynamic processes in a system of interacting particles is considered as an example

a model based on local graphs for colour images and its application for gaussian noise smoothing in this paper a new model for processing colour images is presented a graph is built for each image pixel taking into account some constraints on links each pixel is characterized depending on the features of its related graph which allows to process it appropriately as an example we provide a characterization of each pixel based on the link cardinality of its connected component this feature enables us to properly distinguish flat image regions respect to edge and detail regions according to this we have designed a hybrid filter for colour image smoothing it combines a filter able to properly process flat image regions with another one that is more appropriate for details and texture experimental results show that our model performs appropriately we also see that our proposed filter is competitive with respect to state of the art methods it is close closer to the corresponding optimal switching filter respect to other analogous hybrid method

a new class of chromatic filters for color image processing theory and applications this paper advances a new framework for chromatic filtering of color images the chromatic content of a color image is encoded in the cie u v chromaticity coordinates whereas the achromatic content is encoded as cie y tristimulus value within the u v chromaticity diagram colors are added according to the well known center of gravity law of additive color mixtures which is generalized here into a nonlinear filtering scheme for processing the two chromatic signals u and v the achromatic channel y can be processed with traditional filtering schemes either linear or nonlinear depending on the specific task at hand the most interesting characteristics of the new filtering scheme are 1 the elimination of color smearing effects along edges between bright and dark areas 2 the possibility of processing chromatic components in a noniterative fashion through linear convolution operations and 3 the consequent amenability to computationally efficient implementations with fast fourier transform the paper includes several examples with both synthetic and real images where the performance of the new filtering method is compared with that of other color image processing algorithms

unbalanced haar technique for nonparametric function estimation the discrete unbalanced haar uh transform is a decomposition of one dimensional data with respect to an orthonormal haar like basis where jumps in the basis vectors do not necessarily occur in the middle of their support we introduce a multiscale procedure for estimation in gaussian noise that consists of three steps a uh transform thresholding of the decomposition coefficients and the inverse uh transform we show that our estimator is mean squared consistent with near optimal rates for a wide range of functions uniformly over uh bases that are not too unbalanced a vital ingredient of our approach is basis selection we choose each basis vector so that it best matches the data at a specific scale and location where the latter parameters are determined by the parent basis vector our estimator is computable in o n log n operations a simulation study demonstrates the good performance of our estimator compared with state of the art competitors we apply our method to the estimation of the mea

generalized multichannel image filtering structures recent works in multispectral image processing advocate the employment of vector approaches for this class of signals vector processing operators that involve the minimization of a suitable error criterion have been proposed and shown appropriate for this task in this framework two main classes of vector processing filters have been reported in the literature astola et al 1990 introduce the well known class of vector median filters vmf which are derived as maximum likelihood ml estimates from exponential distributions trahanias et al see ibid vol 2 no 4 p 528 34 1993 and vol 5 no 6 p 868 80 1996 study the processing of color image data using directional information considering the class of vector directional filters vdf we introduce a new filter structure the directional distance filters ddf which combine both vdf and vmf in a novel way we show that ddf are robust signal estimators under various noise distributions they have the property of chromaticity preservation and finally compare favorably to other multichannel image processing filters

generalized diffusion equation with fractional derivatives within renyi statistics by using the zubarev nonequilibrium statistical operator method and the liouville equation with fractional derivatives a generalized diffusion equation with fractional derivatives is obtained within the renyi statistics averaging in generalized diffusion coefficient is performed with a power distribution with the renyi parameter q

geometric features based filtering for suppression of impulse noise in color images a geometric features based filtering technique named as the adaptive geometric features based filtering technique agff is presented for removal of impulse noise in corrupted color images in contrast with the traditional noise detection techniques where only 1d statistical information is used for noise detection and estimation a novel noise detection method is proposed based on geometric characteristics and features i e the 2 d information of the corrupted pixel or the pixel region leading to effective and efficient noise detection and estimation outcomes a progressive restoration mechanism is devised using multipass nonlinear operations which adapt to the intensity and the types of the noise extensive experiments conducted using a wide range of test color images have shown that the agff is superior to a number of existing well known benchmark techniques in terms of standard image restoration performance criteria including objective measurements the visual image quality and the computational complexity

cats clustering after transformation and smoothing cats clustering after transformation and smoothing is a technique for nonparametrically estimating and clustering a large number of curves our motivating example is a genetic microarray experiment but the method is very general the method includes transformation and smoothing multiple curves multiple nonparametric testing for screening out flat curves clustering curves with similar shape and nonparametrically inferring the clustering estimation error rate

the superdiffusion entropy production paradox in the space fractional case for extended entropies abstract contrary to intuition entropy production rates grow as reversible wave like behavior is approached this paradox was discovered in time fractional diffusion equations it was found to persist for extended entropies and for space fractional diffusion as well this paper completes the possibilities by showing that the paradox persists for tsallis and renyi entropies in the space fractional case complications arising due to the heavy tail solutions of space fractional diffusion equations are discussed in detail

directional processing of color images theory and experimental results the processing of color image data using directional information is studied the class of vector directional filters vdf which was introduced by the authors in a previous work is further considered the analogy of vdf to the spherical median is shown and their relation to the spatial median is examined moreover their statistical and deterministic properties are studied which demonstrate their appropriateness in image processing vdf result in optimal estimates of the image vectors in the directional sense this is very important in the case of color images where the vectors direction signifies the chromaticity of a given color issues regarding the practical implementation of vdf are also considered in addition efficient filtering schemes based on vdf are proposed which include adaptive and or double window structures experimental and comparative results in image filtering show very good performance measures when the error is measured in the l a b space l a b is known as a space where equal color differences result in equal distances and therefore it is very close to the human perception of colors moreover an indication of the chromaticity error is obtained by measuring the error on the maxwell triangle the results demonstrate that vdf are very accurate chromaticity estimators

achieving near perfect classification for functional data summary we show that in functional data classification problems perfect asymptotic classification is often possible making use of the intrinsic very high dimensional nature of functional data this performance is often achieved by linear methods which are optimal in important cases these results point to a marked contrast between classification for functional data and its counterpart in conventional multivariate analysis where the dimension is kept fixed as the sample size diverges in the latter setting linear methods can sometimes be quite inefficient and there are no prospects for asymptotically perfect classification except in pathological cases where for example a variance vanishes by way of contrast in finite samples of functional data good performance can be achieved by truncated versions of linear methods truncation can be implemented by partial least squares or projection onto a finite number of principal components using in both cases cross validation to determine the truncation point we establish consistency of the cross validation procedure

information gain within nonextensive thermostatistics we discuss the information theoretical foundations of the kullback information gain recently generalized within a nonextensive thermostatistical formalism general properties are studied and in particular a consistent test for measuring the degree of correlation between random variables is proposed in addition minimum entropy distributions are discussed and the h theorem is proved within the generalized context

a comparison of heuristic procedures for minimum within cluster sums of squares partitioning perhaps the most common criterion for partitioning a data set is the minimization of the within cluster sums of squared deviation from cluster centroids although optimal solution procedures for within cluster sums of squares wcss partitioning are computationally feasible for small data sets heuristic procedures are required for most practical applications in the behavioral sciences we compared the performances of nine prominent heuristic procedures for wcss partitioning across 324 simulated data sets representative of a broad spectrum of test conditions performance comparisons focused on both percentage deviation from the best found wcss values as well as recovery of true cluster structure a real coded genetic algorithm and variable neighborhood search heuristic were the most effective methods however a straightforward two stage heuristic algorithm hk means also yielded exceptional performance a follow up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data our findings have important implications for behavioral science researchers whose theoretical conclusions could be adversely affected by poor algorithmic performances

functional clustering and identifying substructures of longitudinal data summary a functional clustering fc method k centres fc for longitudinal data is proposed the k centres fc approach accounts for both the means and the modes of variation differentials between clusters by predicting cluster membership with a reclassification step the cluster membership predictions are based on a non parametric random effect model of the truncated karhunen loeve expansion coupled with a non parametric iterative mean and covariance updating scheme we show that under the identifiability conditions derived the k centres fc method proposed can greatly improve cluster quality as compared with conventional clustering algorithms moreover by exploring the mean and covariance functions of each cluster thek centres fc method provides an additional insight into cluster structures which facilitates functional cluster analysis practical performance of the k centres fc method is demonstrated through simulation studies and data applications including growth curve and gene expression profile data

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

random walks of oriented particles on fractals random walks of point particles on fractals exhibit subdiffusive behavior where the anomalous diffusion exponent is smaller than one and the corresponding random walk dimension is larger than two this is due to the limited space available in fractal structures here we endow the particles with an orientation and analyze their dynamics on fractal structures in particular we focus on the dynamical consequences of the interactions between the local surrounding fractal structure and the particle orientation which are modeled using an appropriate move class these interactions can lead to particles becoming temporarily or permanently stuck in parts of the structure a surprising finding is that the random walk dimension is not affected by the orientation while the diffusion constant shows a variety of interesting and surprising features

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments