symmetry breaking for δu 2δe u 0 on a disk with general boundary conditions we discuss the radially symmetric solutions and the symmetry breaking of the equation u 2 e u 0 in d and u b u n 0 on d where d is the unit disk in 2 0 and b is a constant we prove that for any b 0 such that there are exactly two radially symmetric solutions for 0 one for and none for b for where m is a positive integer there are b k 1 m such that the equation has symmetry breaking at k b on the lower branch of radially symmetric solutions

two dimensional spectral estimation in this paper effective methods for generating two dimensional quarter plane causal autoregressive ar and autoregressive moving average arma spectral estimation models are developed these procedures are found to provide super resolution capabilities when compared to other more classical methods such as the fourier transform the arma method involves manipulation of the model equation sum min k 0 max p 1 sum min k 0 max p 2 a km x n 1 k n 2 m sum min k 0 max q 1 sum min k 0 max q 2 b km epsilon n 1 k n 2 m and utilizes the given finite set of observations x n 1 n 2 for 1 leq n 1 leq n 1 1 leq n 2 leq n 2 in the above relationship the random excitation epsilon n 1 n 2 is taken to be white this arma model s autoregressive a km coefficients are selected to minimize a weighted least squares criterion composed of error elements while the moving average b km coefficients are obtained using an alternative approach the spectral estimation performance of the ar and arma methods will be empirically demonstrated by considering the problem of resolving two sinusoids embedded in noise

color spectral analysis for spatial structure characterization of textures in ihls color space in this article a linear prediction model based approach for color texture characterization and classification in the improved hue luminance and saturation color space is presented pure chrominance structure information is used in addition to the normally used luminance structure information for color texture classification hue and saturation channels of a color image in ihls color space are combined using a complex exponential to give a single channel which holds all the chrominance information of the image two dimensional complex multichannel versions of the non symmetric half plane autoregressive model the quarter plane autoregressive model and the gauss markov random field model are used to perform parametric power spectrum estimation of both luminance and the combined chrominance channels of the image the accuracy and precision of these spectral estimates are proven quantitatively by performing tests on a large number of images spectral distance measures are calculated for the spectral information of luminance and chrominance channels individually as well as combined through a combination coefficient using these distance measures color texture classification is done with k nearest neighbor algorithm experimental results verify that the ihls color space exhibits better performance than the rgb color space indicating the significance of using ihls for such analysis they also show that color texture characterization and percentage classification obtained by combined luminance and chrominance structure information is better than the color texture classification done using only the luminance structure information

an efficient algorithm for two dimensional autoregressive spectrum estimation an algorithm similar to the burg algorithm is given for autoregressive estimation in one dimension coefficients of the filter used to decorrelate the signal are found recursively from lower order filters the coefficients needed to get the larger sized filters can be found at each stage of the recursion from the correlations of lower order filter outputs thus the resulting estimate does not assume knowledge of ensemble averages it offers a computational saving of 25 over the commonly used method in which an estimate of the correlation matrix is first computed and then inverted

symmetry breaking at non positive solutions of semilinear elliptic equations we consider symmetry breaking bifurcations at non positive radially symmetric solutions of semilinear elliptic equations on a ball with dirichlet boundary conditions for nonlinearities which are asymptotically affine linear we find solutions at which the symmetry breaks the kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group o n for this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough thus we obtain by the equivariant branching lemma a large variety of isotropy subgroups of o n which occur as symmetries of the bifurcating solution branches

model identification of a noncausal 2 d ar process using a causal 2 d ar model on the nonsymmetric half plane if a noncausal two dimensional 2 d autoregressive ar process is bi causal there exists a causal 2 d ar process on the nonsymmetric half plane having the same autocorrelations as the noncausal 2 d ar process a formula is presented to relate the ar coefficients of the noncausal 2 d ar process with those of the causal 2 d ar process on the nonsymmetric half plane the 2 d yule walker equations are derived for causal 2 d ar models on the nonsymmetric half plane a computationally efficient order recursive algorithm is proposed to solve the 2 d yule walker equations using the autocorrelation equivalence relation and the order recursive algorithm we can easily identify a noncausal 2 d ar process from its autocorrelations

symmetry breaking for a class of semilinear elliptic problems abstract this document considers a nonlinear elliptic problem suppose this problem has a family of positive radial solutions parameterized by r i e ur x this paper studies the possibility of the existence of nonradial solutions of bifurcating from the radial solutions family answering a question posed by smoller and wasserman it is shown this happens if f satisfies suitable assumptions therefore the global structure of the nonradial solution set is investigated

extendability tests for multidimensional covariance sequences the problem of extendibility of multidimensional covariance sequences and the equivalent problem of the existence of maximum entropy me spectral estimates are analyzed using some recent results on the valid parameter space of gaussian markov random fields gmrfs for several nontrivial examples explicit conditions for extendability and using those conditions sketches of the space of extendible covariance sequences are obtained for the general case a cutting plane algorithm is proposed as an alternative to the two existing numerical procedures for ascertaining extendibility namely the linear programming procedure and the expanding hull algorithm the duality between the valid parameter space and the space of extendible covariances and the relationship between those two spaces and the space of admissible covariances for finite size data sequences are examined the connection between extendability and maximum likelihood ml estimation is established and some properties extendibility of covariances specified over increasing window sizes are presented

a spatial correlation based method for neighbor set selection in random field image models random field rf models have widespread application in image modeling and analysis the effectiveness of these models is largely dependent on the choice of neighbor sets which determine the spatial interactions that are representable by the model we consider the problem of selecting these neighbor sets for simultaneous autoregressive and gauss markov random field models based on the correlation structure of the image to be modeled a procedure for identifying appropriate neighbor sets is proposed and experimental results which demonstrate the viability of this method are presented

characterization and estimation of two dimensional arma models a class of finite order two dimensional autoregressive moving average arma is introduced that can represent any process with rational spectral density in this model the driving noise is correlated and need not be gaussian currently known classes of arma models or ar models are shown to be subsets of the above class the three definitions of markov property are discussed and the class of arma models are precisely stated which have the noncausal and semicausal markov property without imposing any specific boundary conditions next two approaches are considered to estimate the parameters of a model to fit a given image the first method uses only the empirical correlations and involves the solution of linear equations the second method is the likelihood approach since the exact likelihood function is difficult to compute we resort to approximations suggested by the toroidal models numerical experiments compare the quality of the two estimation schemes finally the problem of synthesizing a texture obeying an arma model is considered

feature extraction for texture discrimination via random field models with random spatial interaction in this paper we attack the problem of distinguishing textured images of real surfaces using small samples we first analyze experimental data that results from applying ordinary conditional markov fields in the face of the disappointing performance of these models we introduce a random field with spatial interaction that is itself a random variable usually referred to as a random field in a random environment for this class of models we establish the power spectrum and the autocorrelation function as well defined quantities and we devise a scheme for the estimation of related parameters the new set of features that resulted from this approach was applied to real images accurate discrimination was observed even for boxes of size 10 spl times 16

on two dimensional markov spectral estimation we give systematic methods for two dimensional 2 d spectral estimation from raw data using gaussian markov random field mrf models the mrf models considered in this paper characterize the statistical dependence of the observation y s on its neighbors in all directions due to the modeling assumption the 2 d spectrum is an explicit function of the parameters of the mrf model thus the spectral estimation problem reduces to that of estimating the appropriate structure and the parameters of the mrf model a tractable algorithm for maximum likelihood ml estimation of parameters is obtained by using doubly periodic boundary conditions for the given data asymptotically consistent decision rules are used to choose the appropriate neighbor set of the mrf model the mrf spectral estimate obtained using the doubly periodic boundary conditions has the property that its inverse discrete fourier transforms are in perfect agreement with the periodic sample correlation values of the given observations over a symmetric window identical to the structure of the mrf model using this result and the fact that the 2 d maximum entropy spectral mes estimate has a structure similar to the mrf power spectrum we show that the mrf spectral estimate developed in this paper is a good approximation to the mes estimate this result provides a reasonable basis i e maximum likelihood on which to justify the use of the mes estimate with correlation estimates obtained from a finite piece of data samples

passive error concealment for wavelet coded i frames with an inhomogeneous gauss markov random field model in video communication over lossy packet networks e g the internet packet loss errors can severely damage the transmitted video the damaged video can largely be repaired with passive error concealment where neighboring information is used to estimate missing information we address the problem of passive error concealment for wavelet coded data with dispersive packetization the reported techniques of this kind have many problems and usually fail in the reconstruction of high frequency content this paper presents a novel locally adaptive error concealment method for subband coded i frames based on an inhomogeneous gaussian markov random field model we estimate the parameters of this model from a local context of each lost coefficient and we interpolate the lost coefficients accordingly the results demonstrate a significant improvement over the reported related methods both in terms of objective performance measures and visually the biggest improvement of the proposed method compared to the state of the art in the field is the correct reconstruction of high frequency information such as textures and edges

two dimensional robust spectrum estimation robust estimation is studied of two dimensional 2 d power spectra of signals which are adequately represented by gaussian random field models but for which there are imperfect observations two situations of particular interest occur when the contaminating noise is additive and when the contaminating noise appears in the innovations in these cases the observed data are not gaussian and conventional procedures are no longer efficient to estimate the parameters of the signal model from the contaminated data two procedures are described which were originally proposed for estimation of scale and location from independent data and adapted to one dimensional autoregression parameter estimation by previous researchers the first algorithm is a robustification of least squares and equivalent to an iterated weighted least squares problem where the weights are data dependent the second algorithm is an iterative procedure known as a filter cleaner experiments using the robust procedures with synthetic data are reported and the results compared to a conventional method of model based spectrum estimation

existence uniqueness and nondegeneracy of positive solutions of semilinear elliptic equations we study positive solutions of the dirichlet problem a u x f u x 0 x d u x o x e ad where d is an n ball we find necessary and sufficient conditions for solutions to be nondegenerate we also give some new existence and uniqueness theorems in this paper we study positive solutions of the dirichlet problem au x f u x o x f2

an analytical and experimental study of the performance of markov random fields applied to textured images using small samples we investigate to what extent textures can be distinguished using conditional markov fields and small samples we establish that the least square ls estimator is the only reasonable choice for this task and we prove its asymptotic consistency and normality for a general class of random fields that includes gaussian markov fields as a special case the performance of this estimator when applied to textured images of real surfaces is poor if small boxes are used 20 spl times 20 or less we investigate the nature of this problem by comparing the behavior predicted by the rigorous theory to the one that has been experimentally observed our analysis reveals that 20 spl times 20 samples contain enough information to distinguish between the textures in our experiments and that the poor performance mentioned above should be attributed to the fact that conditional markov fields do not provide accurate models for textured images of many real surfaces a more general model that exploits more efficiently the information contained in small samples is also suggested

on resolving two dimensional sinusoids in white noise using different spectral estimates several well known spectrum estimates are examined in terms of their performance in resolving two closely located sinusoids in white noise closed form expressions for the estimates using the different approaches and models are derived and used to study the resolution question the notions of 3 db contour and single peak area spa are introduced and used to compare the different models the harmonic mean of two single quadrant linear prediction models is shown to be particularly promising as a high resolution estimate

mean field annealing using compound gauss markov random fields for edge detection and image estimation the authors consider the problem of edge detection and image estimation in nonstationary images corrupted by additive gaussian noise the noise free image is represented using the compound gauss markov random field developed by f c jeng and j w woods 1990 and the problem of image estimation and edge detection is posed as a maximum a posteriori estimation problem since the a posteriori probability function is nonconvex computationally intensive stochastic relaxation algorithms are normally required a deterministic relaxation method based on mean field annealing with a compound gauss markov random cgmrf field model is proposed the authors present a set of iterative equations for the mean values of the intensity and both horizontal and vertical line processes with or without taking into account some interaction between them the relationship between this technique and two other methods is considered edge detection and image estimation results on several noisy images are included

high resolution two dimensional arma spectral estimation the authors present a practical algorithm for estimating the power spectrum of a 2 d homogeneous random field based on 2 d autoregressive moving average arma modeling this algorithm is a two step approach first the ar parameters are estimated by solving a version of the 2 d modified yule walker equation for which some existing efficient algorithms are available then the ma spectrum parameters are obtained by simple computations the potential capability and the high resolution performance of the algorithm are demonstrated by using some numerical examples

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

a model based approach for estimation of two dimensional maximum entropy power spectra a stochastic model based approach is presented for estimation of the two dimensional maximum entropy power spectrum meps from given finite uniform array data the method consists of fitting an appropriate two dimensional noncausal gaussian markov random field gmrf model to the given data using the maximum likelihood ml technique for parameter estimation the nonlinear criterion function used for ml estimation is similar in structure to the function arising in the deterministic approach of lang and mcclellan the model based approach provides new insights into the two dimensional meps estimation problem for example using the asymptotic normality of ml estimates we derive simultaneous confidence bands for the estimated meps it turns out that when the true correlations are generated by a noncausal gmrf model the two dimensional meps can be obtained by solving linear equations this approach also suggests techniques for realizing two dimensional gmrf models from the given correlation data several numerical examples are given to illustrate the usefulness of the approach

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

on two dimensional maximum entropy spectral estimation a novel method of two dimensional spectral estimation which the authors introduced recently 1987 using the radon transform and a one dimensional autoregressive model ar led them to investigate the maximization of entropy subject to the correlation matching constraints in the radon space instead of solving the 2d maximum entropy me spectral estimation problem the authors convert it into a problem which is easier to solve it is shown that a radial slice of the 2d me spectrum can be obtained by 1d ar modeling of the projections radon transform of a stationary random field srf the advantages and limitations of using this new duality relation to estimate the complete 2d me spectra on a polar raster are discussed