images as occlusions of textures a framework for segmentation we propose a new mathematical and algorithmic framework for unsupervised image segmentation which is a critical step in a wide variety of image processing applications we have found that most existing segmentation methods are not successful on histopathology images which prompted us to investigate segmentation of a broader class of images namely those without clear edges between the regions to be segmented we model these images as occlusions of random images which we call textures and show that local histograms are a useful tool for segmenting them based on our theoretical results we describe a flexible segmentation framework that draws on existing work on nonnegative matrix factorization and image deconvolution results on synthetic texture mosaics and real histology images show the promise of the method

computer model calibration using high dimensional output this work focuses on combining observations from field experiments with detailed computer simulations of a physical process to carry out statistical inference of particular interest here is determining uncertainty in resulting predictions this typically involves calibration of parameters in the computer simulator as well as accounting for inadequate physics in the simulator the problem is complicated by the fact that simulation code is sufficiently demanding that only a limited number of simulations can be carried out we consider applications in characterizing material properties for which the field data and the simulator output are highly multivariate for example the experimental data and simulation output may be an image or may describe the shape of a physical object we make use of the basic framework of kennedy and o hagan however the size and multivariate nature of the data lead to computational challenges in implementing the framework to overcome these challenges we make use of basis repre

complete discrete 2 d gabor transforms by neural networks for image analysis and compression a three layered neural network is described for transforming two dimensional discrete signals into generalized nonorthogonal 2 d gabor representations for image analysis segmentation and compression these transforms are conjoint spatial spectral representations which provide a complete image description in terms of locally windowed 2 d spectral coordinates embedded within global 2 d spatial coordinates in the present neural network approach based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights the network finds coefficients for complete conjoint 2 d gabor transforms without restrictive conditions in wavelet expansions based on a biologically inspired log polar ensemble of dilations rotations and translations of a single underlying 2 d gabor wavelet template image compression is illustrated with ratios up to 20 1 also demonstrated is image segmentation based on the clustering of coefficients in the complete 2 d gabor transform

rings quadratic forms and complete degeneracy for a subclass of highly overmoded rectangular waveguides in this article eigenvalue degeneracy formulas for a subclass of highly overmoded rectangular waveguides are determined this subclass consists of those rectangular geometries with side length ratio a b equal to the square root of a rational integer z and contains an unusual amount of degeneracy even when the side length ratio is irrational the complete degeneracy for members of this subclass 10 is determined by interpreting its normalized eigenvalue formula as a the norm of an associated ring of algebraic integers and b a special case of a positive definite binary integral quadratic form

solution of the schrodinger equation for a particle in an equilateral triangle the complete solution for the quantum mechanical problem of a particle in an equilateral triangle is derived by use of projection operators eigenfunctions belonging explicitly to each of the irreducible representations of the symmetry group c3v are constructed the most natural definition of the quantum numbers p and q includes not only integers but also nonintegers of the class 1 3 and 2 3 modulo 1 some relevant features relating to symmetry and degeneracy are also discussed

how many clusters are best an experiment this paper reports the results of a monte carlo study on the relative effectiveness of two internal indices in estimating the true number of clusters in multivariate data the two indices are the davies and bouldin 1 index and a new modification of the hubert statistic 2 data in d dimensions are clustered to create sequences of partitions estimates are based on plots of the indices as functions of the number of recovered clusters neither index uses a priori information the effects of sample size dimensionality cluster spread number of true clusters and sampling window are examined clustered data are generated to assure a given number of distinct clusters the degree of clustering in the data is verified by a separate monte carlo study based on the jaccard and corrected rand indices that exhibits the importance of correcting external indices for chance the modified hubert index proposed here for the first time is shown to perform better than the davies bouldin index under all experimental conditions recovery of the true number of clusters gets better as the number of true clusters decreases and as the number of dimensions increases no effect occurs due to sampling window the complete link clustering method and a square error clustering method recognize the true number of clusters consistently better than the single link method this study demonstrates the difficulty inherent in estimating the number of clusters

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

ground state energy of a particle in a polygonal box the authors find the ground state energy of a particle enclosed in a regular polygon of n sides by perturbing about the equivalent circle and thereby setting up a n 1 expansion the first correction is o n 3 and hence a two term answer is very accurate for polygons with many sides

exact solution of the schrodinger equation for a particle in a tetrahedral box the authors obtain the exact solution of the schrodinger equation for a particle confined to i an equilateral triangle ii a tetrahedral box with corners pi square root 2 pi square root 2 pi square root 2 pi square root 2 pi square root 2 pi 2 pi square root 2 pi square root 2 pi 2 and pi square root 2 pi square root 2 pi 2 the energies are for i enl 8h cross 2 pi 2 9ml2 n2 l2 ln where l is the side of the triangle and l n are distinct non zero integers and for ii elmn h cross 2 8m 3 l2 m2 n2 2lm 2mn 2nl where l m and n are distinct non zero integers the wavefunctions have been classified according to the irreducible representation of the corresponding symmetry groups

segmentation of color textures this paper describes an approach to perceptual segmentation of color image textures a multiscale representation of the texture image generated by a multiband smoothing algorithm based on human psychophysical measurements of color appearance is used as the input initial segmentation is achieved by applying a clustering algorithm to the image at the coarsest level of smoothing the segmented clusters are then restructured in order to isolate core clusters i e patches in which the pixels are definitely associated with the same region the image pixels representing the core clusters are used to form 3d color histograms which are then used for probabilistic assignment of all other pixels to the core clusters to form larger clusters and categorise the rest of the image the process of setting up color histograms and probabilistic reassignment of the pixels to the clusters is then propagated through finer levels of smoothing until a full segmentation is achieved at the highest level of resolution

the exact eigenfunctions and eigenvalues of a particle in a box obtained using gaussian wavepacket dynamics exact eigenfunctions for a particle in a box are obtained using gaussian wavepacket dynamics the eigenfunctions are obtained by propagating without approximation an infinite set of gaussian wavepackets that collectively satisfy the boundary conditions of the problem being coherent states appropriate to this problem the method of images is applied to enforce these boundary conditions this technique may be applied to the quantum billiard problem whenever the particle is confined to any open or closed region that tessellates space regardless of the dimension of the region also it is shown that the use of frozen gaussians along with the de leon heller spectral quantisation method gives the exact solution for the one dimensional problem as well as for the above multi dimensional problems provided the components of the momentum of the wavepackets are chosen at random

choice of a pertinent color space for color texture characterization using parametric spectral analysis this article presents a comparison of different color spaces including rgb ihls and email protected a b for color texture characterization this comparison is based on the fusion of the independent spatial structure and color feature cues in ihls and l a b two channel complex color images are created from the luminance and the chrominance values for such images two dimensional complex multichannel linear prediction models are used to perform parametric power spectrum estimation and the structure feature cues are computed from this estimated power spectrum quantitative comparison of auto spectra of luminance and combined chrominance channels for different color spaces is done this comparison is based on the degree of decorrelation between luminance and chrominance information provided by different color space transformations three dimensional histograms are used as color feature cues then to classify color textures kullback leibler divergence based symmetric distance measures are calculated for pure color luminance structure and chrominance structure feature cues individual as well as combined effect of information from all feature cues on classification results is then compared for different color spaces and different color texture data sets the proposed color texture classification method performs better than the state of the art methods in certain cases the l a b color space gives us a better characterization of the chrominance spatial structure as well as the overall spatial structure for all of the chosen data sets experimental results on pixel classification of color textures are also presented and discussed

information cut for clustering using a gradient descent approach we introduce a new graph cut for clustering which we call the information cut it is derived using parzen windowing to estimate an information theoretic distance measure between probability density functions we propose to optimize the information cut using a gradient descent based approach our algorithm has several advantages compared to many other graph based methods in terms of determining an appropriate affinity measure computational complexity memory requirements and coping with different data scales we show that our method may produce clustering and image segmentation results comparable or better than the state of the art graph based methods

playing billiards in version space a ray tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a given set of linear separable examples for randomly distributed examples the billiard estimate of the single perceptron with best average generalization probability agrees with known analytic results while for real life classification problems the generalization probability is consistently enhanced when compared to the maximal stability perceptron

on the quantum particle in a polyhedral box the recently obtained solution of the schrodinger equation for a particle confined to a particular non regular tetrahedral box is rederived directly rather than through a transformation of a four body one dimensional problem its simple form is shown to be related to a space filling property of this particular tetrahedron the class of all such polyhedra is determined

blobworld image segmentation using expectation maximization and its application to image querying retrieving images from large and varied collections using image content as a key is a challenging and important problem we present a new image representation that provides a transformation from the raw pixel data to a small set of image regions that are coherent in color and texture this blobworld representation is created by clustering pixels in a joint color texture position feature space the segmentation algorithm is fully automatic and has been run on a collection of 10 000 natural images we describe a system that uses the blobworld representation to retrieve images from this collection an important aspect of the system is that the user is allowed to view the internal representation of the submitted image and the query results similar systems do not offer the user this view into the workings of the system consequently query results from these systems can be inexplicable despite the availability of knobs for adjusting the similarity metrics by finding image regions that roughly correspond to objects we allow querying at the level of objects rather than global image properties we present results indicating that querying for images using blobworld produces higher precision than does querying using color and texture histograms of the entire image in cases where the image contains distinctive objects

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

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

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

gaussian process models for computer experiments with qualitative and quantitative factors modeling experiments with qualitative and quantitative factors is an important issue in computer modeling we propose a framework for building gaussian process models that incorporate both types of factors the key to the development of these new models is an approach for constructing correlation functions with qualitative and quantitative factors an iterative estimation procedure is developed for the proposed models modern optimization techniques are used in the estimation to ensure the validity of the constructed correlation functions the proposed method is illustrated with an example involving a known function and a real example for modeling the thermal distribution of a data center

color image segmentation advances and prospects abstract image segmentation is very essential and critical to image processing and pattern recognition this survey provides a summary of color image segmentation techniques available now basically color segmentation approaches are based on monochrome segmentation approaches operating in different color spaces therefore we first discuss the major segmentation approaches for segmenting monochrome images histogram thresholding characteristic feature clustering edge detection region based methods fuzzy techniques neural networks etc then review some major color representation methods and their advantages disadvantages finally summarize the color image segmentation techniques using different color representations the usage of color models for image segmentation is also discussed some novel approaches such as fuzzy method and physics based method are investigated as well

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

ctex an adaptive unsupervised segmentation algorithm based on color texture coherence this paper presents the development of an unsupervised image segmentation framework referred to as ctex that is based on the adaptive inclusion of color and texture in the process of data partition an important contribution of this work consists of a new formulation for the extraction of color features that evaluates the input image in a multispace color representation to achieve this we have used the opponent characteristics of the rgb and yiq color spaces where the key component was the inclusion of the self organizing map som network in the computation of the dominant colors and estimation of the optimal number of clusters in the image the texture features are computed using a multichannel texture decomposition scheme based on gabor filtering the major contribution of this work resides in the adaptive integration of the color and texture features in a compound mathematical descriptor with the aim of identifying the homogenous regions in the image this integration is performed by a novel adaptive clustering algorithm that enforces the spatial continuity during the data assignment process a comprehensive qualitative and quantitative performance evaluation has been carried out and the experimental results indicate that the proposed technique is accurate in capturing the color and texture characteristics when applied to complex natural images

segmentation of textured images and gestalt organization using spatial spatial frequency representations the generic issue of clustering grouping is addressed recent research both in computer and human vision suggests the use of joint spatial spatial frequency s sf representations the spectrogram the difference of gaussians representation the gabor representation and the wigner distribution are discussed and compared it is noted that the wigner distribution gives superior joint resolution experimental results in the area of texture segmentation and gestalt grouping using the wigner distribution are presented proving the feasibility of using s sf representations for low level early preattentive vision

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

radial rescaling approach for the eigenvalue problem of a particle in an arbitrarily shaped box in the present work we introduce a methodology for solving a quantum billiard with dirichlet boundary conditions the procedure starts from the exactly known solutions for the particle in a circular disk which are subsequently radially rescaled in such a way that they obey the new boundary conditions in this way one constructs a complete basis set which can be used to obtain the eigenstates and eigenenergies of the corresponding quantum billiard to a high level of precision test calculations for several regular polygons show the efficiency of the method which often requires one or two basis functions to describe the lowest eigenstates with high accuracy

sums of laplace eigenvalues rotations and tight frames in higher dimensions the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among simplexes for the regular simplex the regular tetrahedron in three dimensions maximal among parallelepipeds for the hypercube and maximal among ellipsoids for the ball provided the volume and moment of inertia of an auxiliary body are suitably normalized this result holds for dirichlet robin and neumann eigenvalues additionally the cubical torus is shown to be maximal among flat tori the proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain among general convex domains the ball is conjectured to maximize sums of neumann eigenvalues provided the volume and moment of inertia of the polar dual are suitably normalized

voronoi region based adaptive unsupervised color image segmentation color image segmentation is a crucial step in many computer vision and pattern recognition applications this paper introduces an adaptive and unsupervised approach based on voronoi regions to solve the color image segmentation problem the proposed method uses a hybrid of spatial and feature space dirichlet tessellation followed by inter voronoi region proximal cluster merging to automatically find the number of clusters and cluster centroids in an image since the voronoi regions are much smaller compared to the whole image voronoi region wise clustering improves the efficiency and accuracy of the number of clusters and cluster centroid estimation process the proposed method was compared with four other adaptive unsupervised cluster based image segmentation algorithms on three image segmentation evaluation benchmarks the experimental results reported in this paper confirm that the proposed method outperforms the existing algorithms in terms of the image segmentation quality and results in much lower average execution time per image highlightswe propose a hybrid of dirichlet tessellation to automatically segment an image spatial dirichlet tessellation adaptively divides an image into voronoi regions feature space dirichlet tessellation adaptively clusters pixels in voronoi regions inter voronoi region proximal cluster merging automatically finds the final clusters

a simple approach to emulation for computer models with qualitative and quantitative factors we propose a flexible yet computationally efficient approach for building gaussian process models for computer experiments with both qualitative and quantitative factors this approach uses the hypersphere parameterization to model the correlations of the qualitative factors thus avoiding the need of directly solving optimization problems with positive definite constraints the effectiveness of the proposed method is successfully illustrated by several examples

the generalized gabor scheme of image representation in biological and machine vision a scheme suitable for visual information representation in a combined frequency position space is investigated through image decomposition into a finite set of two dimensional gabor elementary functions gef the scheme is generalized to account for the position dependent gabor sampling rate oversampling logarithmic frequency scaling and phase quantization characteristic of the visual system comparison of reconstructed signal highlights the advantages of the generalized gabor scheme in coding typical bandlimited images it is shown that there exists a tradeoff between the number of frequency components used per position and the number of such clusters sampling rate utilized along the spatial coordinate

filtering for texture classification a comparative study in this paper we review most major filtering approaches to texture feature extraction and perform a comparative study filtering approaches included are laws masks 1980 ring wedge filters dyadic gabor filter banks wavelet transforms wavelet packets and wavelet frames quadrature mirror filters discrete cosine transform eigenfilters optimized gabor filters linear predictors and optimized finite impulse response filters the features are computed as the local energy of the filter responses the effect of the filtering is highlighted keeping the local energy function and the classification algorithm identical for most approaches for reference comparisons with two classical nonfiltering approaches co occurrence statistical and autoregressive model based features are given we present a ranking of the tested approaches based on extensive experiments

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

new quartile based region merging algorithm for unsupervised image segmentation using color alone feature this study concerns the image segmentation problem and the use of a color alone feature for reducing the system complexity on the basis of the color based mathematical morphology method the similarity measure between neighboring regions can be obtained as the solution of a ranking problem to avoid the creation of a false color and false segmentation a hybrid ordering approach was used instead of vectorized and marginal ordering approaches ordering methods that use black as the reference color to sort pixels face a problem the scope of distance measurement is not optimal to avoid this problem we present a scheme for selecting a global reference color moreover for determining orders of color vectors the hue saturation intensity color distance was used instead of the euclidean distance the aforementioned scheme involves segmentation that is in accord with human visual perception quartile analysis indicated that threshold determination for region merging showed less sensitivity to context variations of images to evaluate the algorithm it was experimentally compared with two typical segmentation schemes on the basis of four quantitative indices

multilayer graph cuts based unsupervised color texture image segmentation using multivariate mixed student s t distribution and regional credibility merging this paper proposes an unsupervised color texture image segmentation method in order to enhance the effects of segmentation a new color texture descriptor is designed by integrating the compact multi scale structure tensor msst total variation tv flow and the color information due to the fact that msst does not work well for separating regions with large scale texture the total variation flow is used to auxiliarily describe the texture feature by extracting local scale information to segment the color texture image in an unsupervised and multi label way the multivariate mixed student s t distribution mmst is chosen for probability distribution modeling as mmst can describe the distribution of color texture features accurately since the valid class number is hard to adaptively determine in advance a component wise expectation maximization for mmst cem 3st algorithm is proposed which can effectively initialize the valid class number then we can build up the energy functional according to the valid class number and optimize it by multilayer graph cuts method however the problem of over error segmentation often happens to overcome this problem a strategy of regional credibility merging rcm is presented by integrating the regional adjacency relationship region size common edge between regions and regional color texture dissimilarity in order to terminate the whole segmentation process an adaptive iteration convergence criterion is designed which combines the negative logarithm of probability of all color texture features with the kullback leibler kl divergence for mmst experiments using a large number of synthesis color texture images and real natural scene images demonstrate the superiority of our proposed method such as the effective over error segmentation reduction high segmentation accuracy and outperforming visual entirety consistency

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

a study of gaussian mixture models of color and texture features for image classification and segmentation the aims of this paper are two fold to define gaussian mixture models gmms of colored texture on several feature spaces and to compare the performance of these models in various classification tasks both with each other and with other models popular in the literature we construct gmms over a variety of different color and texture feature spaces with a view to the retrieval of textured color images from databases we compare supervised classification results for different choices of color and texture features using the vistex database and explore the best set of features and the best gmm configuration for this task in addition we introduce several methods for combining the color and structure information in order to improve the classification performances we then apply the resulting models to the classification of texture databases and to the classification of man made and natural areas in aerial images we compare the gmm model with other models in the literature and show an overall improvement in performance

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed