a class of elliptic partial differential equations with exponential nonlinearities on demontre l existence d au moins une infinite denombrable de familles a 1 parametre de solutions non triviales continues de holder pour une classe de problemes aux valeurs limites de dirichlet d ordre 2 fortement non lineaires a 2 dimensions et plus

the discrete zak transform application to time frequency analysis and synthesis of nonstationary signals algorithms to compute coefficients of the finite double sum expansion of time varying nonstationary signals and to synthesize them from a finite number of expansion coefficients are presented the algorithms are based on the computation of the discrete zak transform dzt fast algorithms to compute dzt are presented modifications to the algorithms which increase robustness are given

the undersampled discrete gabor transform conventional studies on discrete gabor transforms have generally been confined to the cases of critical sampling and oversampling in which the gabor families span the whole signal space in this paper we investigate undersampled discrete gabor transforms for an undersampled gabor triple g a b i e a spl middot b n we show that the associated generalized dual gabor window gdgw function is the same as the one associated with the oversampled g n b n a except for the constant factor ab n computations of undersampled gabor transforms are made possible by applying the methods algorithms developed in oversampled settings the undersampled gdgw is determined then we are able to obtain the best approximation of a signal x by linear combinations of vectors in the gabor family

complex multipole beam approach to acoustic scattering problems a novel approach to reducing the matrix size associated with the boundary element moment method solution of the problem of acoustic scattering from arbitrarily shaped closed bodies is presented in this paper the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure the accuracy of the proposed method is verified by using internal accuracy checks and by comparing the numerical results with the analytic solutions for spherical scatterers

variational nonlinear elliptic equations in nonreflexive musielak spaces abstract we investigate the existence and uniqueness of variational solutions for the dirichlet problem associated with variational nonlinear elliptic equations in musielak spaces we assume neither 2 nor 2 condition on the functions but suppose that they satisfy some structural assumptions that allow to define the variational framework and to obtain poincare type inequalities within the approach we use the galerkin method is coupled with an abstract existence result by f e browder 1966

signal representation in the combined phase spatial space reconstruction and criteria for uniqueness motivated by the major roles of the spectral phase in signal structure and by previous results on the importance of localized representation in images we develop conditions for unique representation of a signal by a combination of its spectral fourier phase and spatial samples methods for signal reconstruction from various combinations of spatial and phase information are introduced it is concluded that most of the previous results in the area of phase only representation and reconstruction are special cases of the general approach presented here

efficient computation of gabor filter based multiresolution responses abstract multiresolution representation in machine vision systems provides a means to detect analyze and interpret the information content of an image at multiple resolutions the pyramidal representation of images suggested by burt and crowley and the hierarchical construction of orientation and velocity tuned filters developed by fleet and jepson are some of the examples for multiresolution representation of image details in recent years gabor filter based methods have been suggested by several researchers for machine vision applications such as edge detection texture classification and optical flow estimation these applications require generating a family of gabor filters tuned to several resolutions and orientations and computing their responses the computation of gabor filter responses at multiple resolutions and orientations is a very computationally intensive task two efficient schemes for computing gabor filter based multiresolution responses are proposed the first scheme namely recursive filtering method generates a family of gabor filters and their responses starting from a high resolution and proceeding towards the lowest resolution in steps of an octave the second approach namely successive sampling method generates the same responses in the reverse order in both these methods the symmetric asymmetric and wavelet nature of gabor filters are exploited in order to speed up the computation the applicability and usefulness of the schemes are established through experimental results special purpose architectures are proposed to implement the proposed techniques in hardware in order to further speed up the computation

optimal reconstruction of images from localized phase the importance of localized phase in signal representation is investigated the convergence rate of the pocs algorithm projection onto convex sets used for image reconstruction from spectral phase is defined and analyzed and the characteristics of images optimally reconstructed from phase only information are presented it is concluded that images of geometric form are most efficiently reconstructed from their spectral phase whereas images of symmetric form have the poorest convergence characteristics the transition between the two extremes is shown to be continuous the results provide a new approach and analysis of the previously reported advantages of the localized phase representation over the global approach and suggest possible compression schemes

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

long term behavior of polynomial chaos in stochastic flow simulations in this paper we focus on the long term behavior of generalized polynomial chaos gpc and multi element generalized polynomial chaos me gpc for partial differential equations with stochastic coefficients first we consider the one dimensional advection equation with a uniform random transport velocity and derive error estimates for gpc and me gpc discretizations subsequently we extend these results to other random distributions and high dimensional random inputs with numerical verification using the algebraic convergence rate of me gpc finally we apply our results to noisy flow past a stationary circular cylinder simulation results demonstrate that me gpc is effective in improving the accuracy of gpc for a long term integration whereas high order gpc cannot capture the correct asymptotic behavior

a generalized wavelet transform for fourier analysis the multiresolution fourier transform and its application to image and audio signal analysis a wavelet transform specifically designed for fourier analysis at multiple scales is described and shown to be capable of providing a local representation which is particularly well suited to segmentation problems it is shown that by an appropriate choice of analysis window and sampling intervals it is possible to obtain a fourier representation which can be computed efficiently and overcomes the limitations of using a fixed scale of window yet by virtue of its symmetry properties allows simple estimation of such fundamental signal parameters as instantaneous frequency and onset time position the transform is applied to the segmentation of both image and audio signals demonstrating its power to deal with signal events which are localized in either time space or frequency feature extraction and segmentation are performed through the introduction of a class of multiresolution markov models whose parameters represent the signal events underlying the segmentation

the canonical coordinates method for pattern deformation theoretical and computational considerations a method for the analysis of deformed patterns is presented and analyzed the image is transformed into a new set of coordinates in which the deformation has a particular simple form a number of deformations are considered the practical implementation of the method is discussed similar aspects of biological vision are also considered

nonuniform sampling and antialiasing in image representation a unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented a nonuniform sampling theorem clark et al 1985 for functions belonging to this space is extended and a class of nonstationary stochastic processes is considered the space of locally bandlimited signals is shown to be a reproducing kernel space a generalized projection theorem can therefore be applied yielding either a continuous or a discrete projection filter the former can be used for image conditioning prior to nonuniform sampling while the latter provides a general tool for image representation by nonuniform sampling schemes the problem of finding the local bandwidth of a given signal in order to generate an optimal sampling scheme is addressed in the context of signal representation in the combined position frequency space the stochastic estimation of parameters which characterize the local bandwidth is discussed bounds on the error resulting from the utilization of nonexact position varying signal parameters are derived

on the regularization of zeno hybrid automata fundamental properties of hybrid automata such as existence and uniqueness of executions are studied particular attention is devoted to zeno hybrid automata which are hybrid automata that take infinitely many discrete transitions in finite time it is shown that regularization techniques can be used to extend the zeno executions of these automata to times beyond the zeno time different types of regularization may however lead to different extensions a water tank control problem and a bouncing ball system are used to illustrate the results

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

rotation invariant texture classification using a complete space frequency model a method of rotation invariant texture classification based on a complete space frequency model is introduced a polar analytic form of a two dimensional 2 d gabor wavelet is developed and a multiresolution family of these wavelets is used to compute information conserving microfeatures from these microfeatures a micromodel which characterizes spatially localized amplitude frequency and directional behavior of the texture is formed the essential characteristics of a texture sample its macrofeatures are derived from the estimated selected parameters of the micromodel classification of texture samples is based on the macromodel derived from a rotation invariant subset of macrofeatures in experiments comparatively high correct classification rates were obtained using large sample sets

perceptually uniform color spaces for color texture analysis an empirical evaluation rgb a nonuniform color space is almost universally accepted by the image processing community as the means for representing color on the other hand perceptually uniform spaces such as l a b as well as approximately uniform color spaces such as hsv exist in which measured color differences are proportional to the human perception of such differences this paper compares rgb with l a b and hsv in terms of their effectiveness in color texture analysis there has been a limited but increasing amount of work on the color aspects of textured images the results have shown that incorporating color into a texture analysis and recognition scheme can be very important and beneficial the presented methodology uses a family of gabor filters specially tuned to measure specific orientations and sizes within each color texture effectiveness is measured by the classification performance of each color space as well as by classifier independent measures experimental results are obtained with a variety of color texture images perceptually uniform spaces are shown to outperform rgb in many cases

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations a representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed it is shown that by appropriate grouping of terms feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings it is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks for the purpose of analysis the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class spatio spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task two synthesis procedures based on spatio spectral localization that reduce the training problem to one of convex optimization are outlined

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

isotropic polyharmonic b splines scaling functions and wavelets in this paper we use polyharmonic b splines to build multidimensional wavelet bases these functions are nonseparable multidimensional basis functions that are localized versions of radial basis functions we show that rabut s elementary polyharmonic b splines do not converge to a gaussian as the order parameter increases as opposed to their separable b spline counterparts therefore we introduce a more isotropic localization operator that guarantees this convergence resulting into the isotropic polyharmonic b splines next we focus on the two dimensional quincunx subsampling scheme this configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach however up until now the design of appropriate filters for the quincunx scheme has mainly been done using the mcclellan transform in our approach we start from the scaling functions which are the polyharmonic b splines and as such explicitly known and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi orthogonal wavelet transform e g orthonormal b spline and dual the filters are automatically specified by the scaling relations satisfied by these functions we prove that the isotropic polyharmonic b spline wavelet converges to a combination of four gabor atoms which are well separated in the frequency domain we also show that these wavelets are nearly isotropic and that they behave as an iterated laplacian operator at low frequencies we describe an efficient fast fourier transform based implementation of the discrete wavelet transform based on polyharmonic b splines

a fast learning algorithm for gabor transformation an adaptive learning approach for the computation of the coefficients of the generalized nonorthogonal 2 d gabor 1946 transform representation is introduced the algorithm uses a recursive least squares rls type algorithm the aim is to achieve minimum mean squared error for the reconstructed image from the set of the gabor coefficients the proposed rls learning offers better accuracy and faster convergence behavior when compared with the least mean squares lms based algorithms applications of this scheme in image data reduction are also demonstrated

structure texture image decomposition modeling algorithms and parameter selection this paper explores various aspects of the image decomposition problem using modern variational techniques we aim at splitting an original image f into two components u and where u holds the geometrical information and holds the textural information the focus of this paper is to study different energy terms and functional spaces that suit various types of textures our modeling uses the total variation energy for extracting the structural part and one of four of the following norms for the textural part l2 g l1 and a new tunable norm suggested here for the first time based on gabor functions apart from the broad perspective and our suggestions when each model should be used the paper contains three specific novelties first we show that the correlation graph between u and may serve as an efficient tool to select the splitting parameter second we propose a new fast algorithm to solve the tv l1 minimization problem and third we introduce the theory and design tools for the tv gabor model

defect detection in periodically patterned surfaces using independent component analysis in this paper we propose a fast self comparison scheme for defect detection in structural surfaces containing periodic complicated patterns it works directly on a one dimensional line image instead of a two dimensional array image that contains a periodic pattern in the line the proposed self comparison scheme is simply carried out by dividing a sensed line image into two segments of equal length since the line image contains a periodic pattern the two divided segments are only translated versions to each other in this study an independent component analysis ica model is proposed to obtain the de mixing matrix that can recover the translation between the two divided segments the proposed ica model directly measures the independency of signals by minimizing the difference between the joint probability density function pdf and the product of marginal pdfs in which the pdfs are estimated by relative frequency distributions the particle swarm optimization pso algorithm is used to search for the de mixing matrix the proposed ica model can effectively separate highly correlated signals and is well suited for translation recovery between two signals with the same periodic pattern in the detection stage each line image is first divided into two segments and the de mixing matrix learned off line from a defect free line image is used to recover the signals with well aligned translation the normalized cross correlation is adopted to measure the similarity between two compared segments since the de mixing matrix is only of a small size of 2x2 the proposed method in the detection stage is very computationally efficient the performance of the proposed method is demonstrated with test samples of tft lcd panels and color filters found in lcd manufacturing experimental results have shown that the proposed self comparison scheme can effectively and efficiently detect the presence of defects in periodically patterned surfaces

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

uncertainty quantification in kinematic wave models we develop a probabilistic approach to quantify parametric uncertainty in first order hyperbolic conservation laws kinematic wave equations the approach relies on the derivation of a deterministic equation for the cumulative density function cdf of a system state in which probabilistic descriptions probability density functions or pdfs of system parameters and or initial and boundary conditions serve as inputs in contrast to pdf equations which are often used in other contexts cdf equations allow for straightforward and unambiguous determination of boundary conditions with respect to sample variables the accuracy and robustness of solutions of the cdf equation for one such system the saint venant equations of river flows are investigated via comparison with monte carlo simulations

motion estimation in the 3 d gabor domain motion estimation methods can be broadly classified as being spatiotemporal or frequency domain in nature the gabor representation is an analysis framework providing localized frequency information when applied to image sequences the 3 d gabor representation displays spatiotemporal spatiotemporal frequency st stf information enabling the application of robust frequency domain methods with adjustable spatiotemporal resolution in this work the 3 d gabor representation is applied to motion analysis we demonstrate that piecewise uniform translational motion can be estimated by using a uniform translation motion model in the st stf domain the resulting motion estimation method exhibits both good spatiotemporal resolution and substantial noise resistance compared to existing spatiotemporal methods to form the basis of this model we derive the signature of the translational motion in the 3 d gabor domain finally to obtain higher spatiotemporal resolution for more complex motions a dense motion field estimation method is developed to find a motion estimate for every pixel in the sequence

complete gabor transformation for signal representation properties of the gabor transformation used for image representation are discussed the properties can be expressed in matrix notation and the complete gabor coefficients can be found by multiplying the inverse of the gabor 1946 matrix and the signal vector the gabor matrix can be decomposed into the product of a sparse constant complex matrix and another sparse matrix that depends only on the window function a fast algorithm is suggested to compute the inverse of the window function matrix enabling discrete signals to be transformed into generalized nonorthogonal gabor representations efficiently a comparison is made between this method and the analytical method the relation between the window function matrix and the biorthogonal functions is demonstrated a numerical computation method for the biorthogonal functions is proposed

a theory for invariant object recognition in the frontoparallel plane we suggest a new computational theory for position size and orientation invariant recognition of planar gray scale forms in frontoparallel view the theory employs a spatial frequency representation which approximates the complex logarithmic conformally mapped wigner distribution of a gray scale image our discussion emphasizes the theoretical significance method of computation and practical utility of this powerful new representation for image pattern recognition

experiments on estimating egomotion and structure parameters using long monocular image sequences this paper presents a simple but robust model based approach to estimating the kinematics of a moving camera and the structure of the objects in a stationary environment using long noisy monocular image sequences both batch and recursive algorithms are presented and the problem due to occlusion is addressed the approach is based on representing the constant translational velocity and constant angular velocity of the camera motion using nine rectilinear motion parameters which are 3 d vectors of the position of the rotation center linear and angular velocities the structure parameters are 3 d coordinates of the salient feature points in the inertial coordinate system due to redundancies in parameterization the total number of independent parameters to be estimated is 3m 7 wherem is the number of feature points the image plane coordinates of these feature points in each frame are first detected and matched over the frames these noisy image coordinates serve as the input to our algorithms due to the nonlinear nature of perspective projection a nonlinear least squares method is formulated for the batch algorithm and a conjugate gradient method is then applied to find the solution a recursive method using an iterated extended kalman filter iekf for incremental estimation of motion and structure is also presented since the plant model is simple in our formulation closed form solutions for the state and covariance transition equations are easily derived experimental results for simulated imagery as well as several real image sequences are included

pseudo monotone mappings in sobolev orlicz spaces and nonlinear boundary value problems on unbounded domains where r is a domain not necessarily bounded in r in the case where the a have at most polynomial growth in u and its derivatives this problem can be studied in the context of the mappings of monotone type from a sobolev space into its conjugate when the functions a even the top order terms do not obey any polynomial growth it is customary to use a sobolev orlicz space rather than a sobolev space although at the cost of some degree of complexity in the arguments caused mainly by the fact that orlicz spaces are not reflexive in general we shall be dealing here with the latter case the concept of a pseudo monotone operator was introduced to take into account the difference of the properties of the highest order functions a in their top order variables from the lower order derivatives when the domain r is bounded and a polynomial growth condition is satisfied it is a classical result to derive the pseudo monotonicity of the mapping t induced by the operator a in a natural manner from a relatively simple set of hypotheses leray lions conditions on the functions a 3 131 an analogous conclusion for the case of sobolev orlicz spaces was also shown to be true 161 for unbounded domains r other methods were developed to obtain some

an unsupervised texture segmentation algorithm with feature space reduction and knowledge feedback this paper presents an unsupervised texture segmentation algorithm based on feature extraction using multichannel gabor filtering it is shown that feature contrast a criterion derived for gabor filter parameter selection is well suited for feature coordinate weighting in order to reduce the feature space dimension the central idea of the proposed segmentation algorithm is to decompose the actual segmented image into disjunct areas called scrap images and use them after lowpass filtering as additional features for repeated k means clustering and minimum distance classification this yields a classification of texture regions with an improved degree of homogeneity while preserving precise texture boundaries

automatic surface inspection using wavelet reconstruction abstract in this paper we present a multiresolution approach for the inspection of local defects embedded in homogeneous textured surfaces the proposed method does not rely on the extraction of local textural features in a pixel by pixel basis it is based on an efficient image restoration scheme using the wavelet transforms by properly selecting the smooth subimage or the combination of detail subimages in different decomposition levels for backward wavelet transform the reconstructed image will remove regular repetitive texture patterns and enhance only local anomalies then a simple thresholding can be used to discriminate between defective regions and homogeneous regions in the reconstructed image this converts the difficult defect detection in complicated textured images into a simple binary thresholding in nontextured images effects of different wavelet bases the number of multiresolution levels the decomposed subimages used for reconstruction and the change in image rotation on defect detection are thoroughly evaluated and discussed based on the experiment on a variety of real textures including machined surfaces natural wood textile fabrics sandpaper and leather

unsupervised texture segmentation of images using tuned matched gabor filters recent studies have confirmed that the multichannel gabor decomposition represents an excellent tool for image segmentation and boundary detection unfortunately this approach when used for unsupervised image analysis tasks imposes excessive storage requirements due to the nonorthogonality of the basis functions and is computationally highly demanding in this correspondence we propose a novel method for efficient image analysis that uses tuned matched gabor filters the algorithmic determination of the parameters of the gabor filters is based on the analysis of spectral feature contrasts obtained from iterative computation of pyramidal gabor transforms with progressive dyadic decrease of elementary cell sizes the method requires no a priori knowledge of the analyzed image so that the analysis is unsupervised computer simulations applied to different classes of textures illustrate the matching property of the tuned gabor filters derived using our determination algorithm also their capability to extract significant image information and thus enable an easy and efficient low level image analysis will be demonstrated

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

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

leray lions operators with logarithmic growth abstract in this paper we prove existence uniqueness and regularity results for solutions of nonlinear elliptic equations in which the differential operator has logarithmic growth with respect to the gradient the solution will belong to the sobolev space w 0 1 1 to the orlicz sobolev space generated by the function t log 1 t but not to any sobolev space w 0 1 p with p 1

recognition of distorted patterns by invariance kernels abstract a novel method for the recognition of deformed patterns is presented it is shown that for planar deformations generated by lie transformation groups there exist kernels such that the associated integral transforms are invariant under the deformations many examples of deformations and the corresponding kernels are given

texture segmentation using 2 d gabor elementary functions many texture segmentation schemes use an elaborate bank of filters to decompose a textured image into a joint space spatial frequency representation although these schemes show promise and although some analytical work has been done the relationship between texture differences and the filter configurations required to distinguish them remain largely unknown this paper examines the issue of designing individual filters using a 2 d texture model we show analytically that applying a properly configured bandpass filter to a textured image produces distinct output discontinuities at texture boundaries the analysis is based on gabor elementary functions but it is the bandpass nature of the filter that is essential depending on the type of texture difference these discontinuities form one of four characteristic signatures a step ridge valley or a step change in average local output variation accompanying experimental evidence indicates that these signatures are useful for segmenting an image the analysis indicates those texture characteristics that are responsible for each signature type detailed criteria are provided for designing filters that can produce quality output signatures we also illustrate occasions when asymmetric filters are beneficial an issue not previously addressed

localized measurement of emergent image frequencies by gabor wavelets the authors derive implement and demonstrate a computational approach for the measurement of emergent image frequencies measuring emergent signal frequencies requires spectral measurements accurate in both frequency and time or space conflicting requirements that are shown to be balanced by a generalized uncertainty relationship such spectral measurements can be obtained from the responses of multiple wavelet like channel filters that sample the signal spectrum and that yield a locus of possible solutions for each locally emergent frequency it is shown analytically that this locus of solutions is maximally localized in both space and frequency if the channel filters used are gabor wavelets a constrained solution is obtained by imposing a stabilizing term that develops naturally from the assumptions on the signal the measurement of frequencies is then cast as an ill posed extremum problem regularized by the stabilizing term leading to an iterative constraint propagation algorithm the technique is demonstrated by application to a variety of 2 d textured images

adaptive gabor transformation for image processing an algorithm that computes the gabor coefficients of an image is presented an adaptive filter that uses the complex least mean square algorithm for their computation is proposed and its numerical characteristics are discussed it is shown that the filter is stable under certain conditions because the gabor transformation seems to be an excellent tool for image compression the efficiency of information coding using the gabor coefficients is investigated and compared with coding that uses the coefficients of the discrete cosine transformation dct properties of both transformations are discussed

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

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

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

iterative methods for solving the gabor expansion considerations of convergence j g daugman s 1988 neural network solution to the gabor expansion of an image is reformulated as a steepest descent implementation nonlinear optimization theory is then applied to select an appropriate convergence factor two quasi newton based nonlinear optimization techniques are applied to improve the convergence for certain types of lattice

computationally attractive real gabor transforms we present a gabor transform for real discrete signals and present a computationally attractive method for computing the transform for the critically sampled case we derive a biorthogonal function which is very localized in the time domain therefore truncation of this biorthogonal function allows us to compute approximate expansion coefficients with significantly reduced computational requirements further truncation does not degrade the numerical stability of the transform we present a tight upper bound on the reconstruction error incurred due to use of a truncated biorthogonal function and summarize computational savings for example the expense of transforming a length 2048 signal using length 16 blocks is reduced by a factor of 26 over similar fft based methods with at most 0 04 squared error in the reconstruction

gabor filter based edge detection abstract it is common practice to utilize evidence from biological and psychological vision experiments to develop computational models for low level feature extraction the receptive profiles of simple cells in mammalian visual systems have been found to closely resemble gabor filters daugman proved that gabor filters achieve joint minimal joint uncertainty these results led researchers to develop computational models based on gabor filters for several low level vision applications such as edge detection texture classification optical flow estimation and data compression in this paper the performance of a gabor odd filter based edge detector is investigated using the measures proposed by canny based on this performance analysis a design criterion for one dimensional 1d gabor filter based edge detector is derived it is shown that this design criterion also holds good for a two dimensional 2d gabor filter based edge detector experimental results are presented to demonstrate the performance of the gabor filter based edge detector

efficient gabor filter design for texture segmentation gabor filters have been successfully applied to a broad range of image processing tasks the present paper considers the design of a single filter to segment a two texture image a new efficient algorithm for gabor filter design is presented along with methods for estimating filter output statistics the algorithm draws upon previous results that showed that the output of a gabor filtered texture is modeled well by a rician distribution a measure of the total output power is used to select the center frequency of the filter and is used to estimate the rician statistics of the gabor filtered image the method is further generalized to include the statistics of postfiltered outputs that are generated by a gaussian filtering operation following the gabor filter the new method typically requires an order of magnitude less computation to design a filter than a previously proposed method experimental results demonstrate the efficacy of the method

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

variational inequalities in orlicz sobolev spaces on etudie l existence de solutions d inegalites variationnelles associees a des operateurs differentiels elliptiques

can one evaluate the gabor expansion using gabor s iterative algorithm d gabor s original 1946 approach for calculation of expansion coefficients was examined and it was found that convergence of the suggested algorithm is conditional on the window function chosen the method of analysis enables a process of synthesis whereby eligible windows can be defined ensuring convergence of the algorithm for every represented signal for windows having a finite support of width d the coefficients calculated according to the algorithm are obtained after a single iteration it is proved that in the special case of a gaussian window the algorithm converges only for specific signals on the basis of the conclusions drawn conditions are formulated which ensure convergence of the gabor algorithm permitting examination of alternative windows

the berkeley wavelet transform a biologically inspired orthogonal wavelet transform we describe the berkeley wavelet transform bwt a two dimensional triadic wavelet transform the bwt comprises four pairs of mother wavelets at four orientations within each pair one wavelet has odd symmetry and the other has even symmetry by translation and scaling of the whole set plus a single constant term the wavelets form a complete orthonormal basis in two dimensions the bwt shares many characteristics with the receptive fields of neurons in mammalian primary visual cortex v1 like these receptive fields bwt wavelets are localized in space tuned in spatial frequency and orientation and form a set that is approximately scale invariant the wavelets also have spatial frequency and orientation bandwidths that are comparable with biological values although the classical gabor wavelet model is a more accurate description of the receptive fields of individual v1 neurons the bwt has some interesting advantages it is a complete orthonormal basis and is therefore inexpensive to compute manipulate and invert these properties make the bwt useful in situations where computational power or experimental data are limited such as estimation of the spatiotemporal receptive fields of neurons

texture classification using windowed fourier filters we define a distance between textures for texture classification from texture features based on windowed fourier filters the definition of the distance relies on an interpretation of our texture attributes in terms of spectral density when the texture can be considered as a gaussian random field the distance between textures is then defined as a symmetrized kullback distance which is a simple function of the attributes and does not require any normalization an experimental analysis using gabor filters and in particular a comparison to quadratic distances shows the efficiency and robustness of the method

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

a unified approach to boundary perception edges textures and illusory contours a model consisting of a multistage system which extracts and groups salient features in the image at different spatial scales or frequencies is used in the first stage a gabor wavelet decomposition provides a representation of the image which is orientation selective and has optimal localization properties in space and frequency this decomposition is useful in detecting significant features such as step and line edges at different scales and orientations in the image following the wavelet transformation local competitive interactions are introduced to reduce the effects of noise and changes in illumination interscale interactions help in localizing the line ends and corners and play a crucial role in boundary perception the final stage groups similar features aiding in boundary completion the different stages can be identified with processing by simple complex and hypercomplex cells in the visual cortex of mammals experimental results demonstrate the performance of this model in detecting boundaries both real and illusory in real and synthetic images

wave propagation and sampling theory part i complex signal and scattering in multilayered media from experimental studies in digital processing of seismic reflection data geophysicists know that a seismic signal does vary in amplitude shape frequency and phase versus propagation time to enhance the resolution of the seismic reflection method we must investigate these variations in more detail we present quantitative results of theoretical studies on propagation of plane waves for normal incidence through perfectly elastic multilayered media as wavelet shapes we use zero phase cosine wavelets modulated by a gaussian envelope and the corresponding complex wavelets a finite set of such wavelets for an appropriate sampling of the frequency domain may be taken as the basic wavelets for a gabor expansion of any signal or trace in a two dimensional 2 d domain time and frequency we can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals these results appear as complex 2 d functions of time and frequency i e as instantaneous frequency spectra choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces for this purpose we developed a gabor expansion involving basic wavelets with a constant time duration mean period ratio for layered media as found in sedimentary basins

a joint frequency position domain structure identification of nonlinear discrete time systems by neural networks a new technique is proposed to identify the structure and the parameters of nonlinear discrete time system models the structure is represented in a frequency position domain of gabor basis functions gbfs a simplification to the gbf is also presented where the spatial gaussian envelope of gbf is replaced with a triangular one a modification to the gbf has also been introduced in order to suppress the effects of noise on the procedure a three layered neural network augmented with nonuniform sampling is described for solving the system identification problem

integrated active contours for texture segmentation we address the issue of textured image segmentation in the context of the gabor feature space of images gabor filters tuned to a set of orientations scales and frequencies are applied to the images to create the gabor feature space a two dimensional riemannian manifold of local features is extracted via the beltrami framework the metric of this surface provides a good indicator of texture changes and is used therefore in a beltrami based diffusion mechanism and in a geodesic active contours algorithm for texture segmentation the performance of the proposed algorithm is compared with that of the edgeless active contours algorithm applied for texture segmentation moreover an integrated approach extending the geodesic and edgeless active contours approaches to texture segmentation is presented we show that combining boundary and region information yields more robust and accurate texture segmentation results

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

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

a two component model of texture for analysis and synthesis a model of natural texture based on a structural component that uses affine coordinate transformations and a stochastic residual component is presented it is argued that the selection of an appropriate analysis scale can be formulated in terms of a tradeoff between the rate at which parameters are generated and the distortion resulting from the approximation by the structural component an efficient algorithm for identifying the parameters of the structural model is described and its utility demonstrated on a number of synthetic and natural textures

gabor wavelet representation for 3 d object recognition this paper presents a model based object recognition approach that uses a gabor wavelet representation the key idea is to use magnitude phase and frequency measures of the gabor wavelet representation in an innovative flexible matching approach that can provide robust recognition the gabor grid a topology preserving map efficiently encodes both signal energy and structural information of an object in a sparse multiresolution representation the gabor grid subsamples the gabor wavelet decomposition of an object model and is deformed to allow the indexed object model match with similar representation obtained using image data flexible matching between the model and the image minimizes a cost function based on local similarity and geometric distortion of the gabor grid grid erosion and repairing is performed whenever a collapsed grid due to object occlusion is detected the results on infrared imagery are presented where objects undergo rotation translation scale occlusion and aspect variations under changing environmental conditions

orthogonalization of circular stationary vector sequences and its application to the gabor decomposition certain vector sequences in hermitian or in hilbert spaces can be orthogonalized by a fourier transform in the finite dimensional case the discrete fourier transform dft accomplishes the orthogonalization the property of a vector sequence which allows the orthogonalization of the sequence by the dft called circular stationarity cs is discussed in this paper applying the dft to a given cs vector sequence results in an orthogonal vector sequence which has the same span as the original one in order to obtain coefficients of the decomposition of a vector upon a particular nonorthogonal cs vector sequence the decomposition is first found upon the equivalent dft orthogonalized one and then the required coefficients are found through the dft it is shown that the sequence of discrete gabor 1946 basis functions with periodic kernel and with a certain inner product on the space of n periodic discrete functions satisfies the cs condition the theory of decomposition upon cs vector sequences is then applied to the gabor basis functions to produce a fast algorithm for calculation of the gabor coefficients

the farthest point strategy for progressive image sampling a new method of farthest point strategy fps for progressive image acquisition an acquisition process that enables an approximation of the whole image at each sampling stage is presented its main advantage is in retaining its uniformity with the increased density providing efficient means for sparse image sampling and display in contrast to previously presented stochastic approaches the fps guarantees the uniformity in a deterministic min max sense within this uniformity criterion the sampling points are irregularly spaced exhibiting anti aliasing properties comparable to those characteristic of the best available method poisson disk a straightforward modification of the fps yields an image dependent adaptive sampling scheme an efficient o n log n algorithm for both versions is introduced and several applications of the fps are discussed

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

n folded symmetries by complex moments in gabor space and their application to unsupervised texture segmentation complex moments of the gabor power spectrum yield estimates of the n folded symmetry of the local image content at different frequency scales that is they allow to detect linear rectangular hexagonal triangular and so on structures with very fine to very coarse resolutions results from experiments on the unsupervised segmentation of real textures indicate their importance for image processing applications real geometric moments computed in gabor space also provide for very powerful texture features but lack the clear geometrical interpretation of complex moments

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

frame analysis of the discrete gabor scheme the properties of the discrete gabor scheme are considered in the context of oversampling the approach is based on the concept of frames and utilizes the piecewise finite zak transform pfzt the frame operator is represented as a matrix valued function in the pfzt domain and its properties are examined in relation to this function the frame bounds are calculated by means of the eigenvalues of the matrix valued function and the dual frame which is used in calculation of the expansion coefficients is expressed by means of the inverse matrix dft based algorithms for computation of the expansion coefficients and for the reconstruction of signals from these coefficients are generalized for the case of oversampling of the gabor space the algorithms are implemented in an example of representation of a nonstationary signal

an image representation algorithm compatible with neural associative processor based hardware recognition systems a robust image representation algorithm compatible with the vlsi matching engine based image recognition system has been developed the spatial distributions of four principal direction edges in a 64 spl times 64 pels gray scale image are coded to form a 64 dimension feature vector since the 2d edge information is reduced to a feature vector by projecting edge flags to the principal directions it is named the projected principal edge distribution pped representation the pped vectors very well preserve the human perception of similarity among images in the vector space while achieving a substantial dimensionality reduction in the image data the pped algorithm has been applied to medical radiograph analysis which was taken as a test vehicle for algorithm optimization the robust nature of the pped representation has been confirmed by the recognition results comparable to the diagnosis by experts having several years of experience in a university hospital dedicated digital vlsi circuits have been developed for pped vector generation in order to expedite the processing a test hardware recognition system was constructed using the vector generation circuits where the analog neural associative processor chip developed in a separate project was employed as a vector matching engine as a result a successful medical radiograph analysis has been experimentally demonstrated using the hardware system feasibility of a very low power operation of the system has been also demonstrated

uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

nonuniform image representation in area of interest systems this paper is concerned with image representation by data distributed nonuniformly and in particular with a representation scheme suitable for area of interest imaging the class of signals under consideration whose information density varies with position can be represented according to a nonuniform sampling scheme position varying projection operators are presented as simple low pass filtering operations in a fourier like domain sequential projections are used for pyramidal representation of nonuniformly sampled images it is shown that irregular random sampling prevents under certain mild restrictions aliasing effects

image representation using 2d gabor wavelets this paper extends to two dimensions the frame criterion developed by daubechies for one dimensional wavelets and it computes the frame bounds for the particular case of 2d gabor wavelets completeness criteria for 2d gabor image representations are important because of their increasing role in many computer vision applications and also in modeling biological vision since recent neurophysiological evidence from the visual cortex of mammalian brains suggests that the filter response profiles of the main class of linearly responding cortical neurons called simple cells are best modeled as a family of self similar 2d gabor wavelets we therefore derive the conditions under which a set of continuous 2d gabor wavelets will provide a complete representation of any image and we also find self similar wavelet parametrization which allow stable reconstruction by summation as though the wavelets formed an orthonormal basis approximating a tight frame generates redundancy which allows low resolution neural responses to represent high resolution images