verified solution of large systems and global optimization problems recent results on the solution of large banded or sparse systems and on global unconstrained optimization problems including verification of the correctness of the result will be discussed the computing time for the verification for large linear systems is the same as for the decomposition the total computational effort for verifying the global optimum value is for well known test examples competitive to those of pure floating point algorithms computational examples will be demonstrated

a numerical verification method for nonlinear functional equations based on infinite dimensional newton like iteration this paper describes a numerical verification of solutions for infinite dimensional functional equations based on residual forms and newton like iteration the method is based upon a verification method previously developed by the authors several computer assisted proofs for differential equations including nonlinear partial differential equations are presented

finite word length effects in implementation of distributions for time frequency signal analysis this correspondence presents an analysis of the finite register length influence on the accuracy of results obtained by the time frequency distributions tfds in order to measure quality of the obtained results the variance of the proposed model is found signal to quantization noise ratio snr is defined and appropriate expressions are derived floating and fixed point arithmetic are considered with the analysis of discrete random and discrete deterministic signals it is shown that commonly used reduced interference distributions rids exhibit similar performance with respect to the snr we have also derived the expressions establishing the relationship between the number of bits and the required quality of representation which is defined by the snr which may be used for register length design in hardware implementation of time frequency algorithms

the arithmetic of the digital computer a new approach a new approach to the arithmetic of the digital computer is surveyed the methodology for defining and implementing floating point arithmetic is described shortcomings of elementary floating point arithmetic are revealed through sample problems the development of automatic computation with emphasis on the user control of errors is reviewed the limitations of conventional rule of thumb procedures for error control in scientific computation are demonstrated by means of examples computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers a new fundamental computer operation the scalar product is introduced to develop this advanced computer arithmetic a process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described validation of computations for a large class of nu

time frequency distributions for a wide sense stationary random signal it is shown that a wide class of time frequency distributions gives the power spectrum for the case of a wide sense stationary random signal a simple criterion on the cohen 1966 kernel for determining when a bilinear time frequency distribution gives the power spectrum of the signal is derived

numerical verification of solutions for a simplified signorini problem abstract we proposed some numerical methods for the automatic proof of existence of solutions for variational inequalities i e obstacle problems and elastoplastic torsion problems it is based on nakao s method by using finite element approximation and its explicit error estimates for the problem in this paper we present a numerical verification algorithm of exact solutions for a simplified signorini problem in a computer a numerical example for the signorini problem is presented

a framework of verified eigenvalue bounds for self adjoint differential operators for eigenvalue problems of self adjoint differential operators a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues in the case of the laplacian operator by applying crouzeix raviart finite elements an efficient algorithm is developed to bound the eigenvalues for the laplacian defined in 1d 2d and 3d spaces moreover for nonconvex domains for which case there may exist singularities of eigenfunctions around re entrant corners the proposed algorithm can easily provide eigenvalue bounds by further adopting the interval arithmetic the explicit eigenvalue bounds from numerical computations can be mathematically correct

an enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems abstract we consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation this problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems we formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation we also consider the indices of eigenvalues especially the first eigenvalue of such a problem in these enclosing procedures the finite dimensional verified computations for linear and nonlinear system of equations play an essential role a numerical example is presented

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

bayesian inference for neighborhood filters with application in denoising range weighted neighborhood filters are useful and popular for their edge preserving property and simplicity but they are originally proposed as intuitive tools previous works needed to connect them to other tools or models for indirect property reasoning or parameter estimation in this paper we introduce a unified empirical bayesian framework to do both directly a neighborhood noise model is proposed to reason and infer the yaroslavsky bilateral and modified non local means filters by joint maximum a posteriori and maximum likelihood estimation then the essential parameter range variance can be estimated via model fitting to the empirical distribution of an observable chi scale mixture variable an algorithm based on expectation maximization and quasi newton optimization is devised to perform the model fitting efficiently finally we apply this framework to the problem of color image denoising a recursive fitting and filtering scheme is proposed to improve the image quality extensive experiments are performed for a variety of configurations including different kernel functions filter types and support sizes color channel numbers and noise types the results show that the proposed framework can fit noisy images well and the range variance can be estimated successfully and efficiently

wigner distribution of noisy signals the effect of noise on the wigner distribution is analyzed the mean and variance of the wigner distribution of signals contaminated by additive noise are derived it is shown that in the case of white noise the wigner distribution calculated by definition calculated by definition cannot be used for estimation for estimation windowing in the time domain is not sufficient smoothing in the time domain is necessary very simple expressions for variance and bias estimation are obtained if the signal and noise are analytic only one window is sufficient to make the variance finite the results are demonstrated by numerical examples with linear and sinusoidal chirp pulses

evaluation of the least constant in sobolev s inequality for h 1 0 s variational methods are applied to evaluate the least constant c in sobolev s inequality for the imbedding of the sobolev space h 1 0 s where 0 s is a real interval into the space c 0 s of bounded continuous functions on 0 s the method yields the exact solution explicitly it is shown that c tanh 1 2 s cong 1 1459 text for s 1 and that equality holds for the hyperbolic cosine function

numerical verification of solutions for some unilateral boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some unilateral boundary value problems using the finite element approximations and explicit a priori error estimates we construct in a computer a set of solutions which satisfies the hypothesis of schauder s fixed point theorem for a compact map on a certain sobolev space further the conditions of verifiability by this method are considered and some numerical examples are presented

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

an extension of ortiz recursive formulation of the tau method to certain linear systems of ordinary differential equations ortiz step by step recursive formulation of the lanczos tau method is extended to the numerical solution of linear systems of differential equations with polynomial coefficients numerical comparisons are made with gear s and enright s methods

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

numerical verification of solutions for obstacle problems this article is an extension of the previous paper numer math 81 1998 305 by the same authors we propose a method to prove the existence of solutions for more general obstacle problems by numerical computations the main task in this paper consists of the numerical determination of some constants which appear in a priori error estimations for the finite element approximation of a simple variational inequality we present a numerical example for verification

a computer assisted proof for the kolmogorov flows of incompressible viscous fluid a computer assisted proof of non trivial steady state solutions for the kolmogorov flows is described the method is based on the infinite dimensional fixed point theorem using newton like operator this paper also proposes a numerical verification algorithm which generates automatically on a computer a set including the exact non trivial solution with local uniqueness all discussed numerical results take into account the effects of rounding errors in the floating point computations

wavelet thresholding of multivalued images in this paper a denoising technique for multivalued images exploiting interband correlations is proposed a redundant wavelet transform is applied and denoising is applied by thresholding wavelet coefficients specific functions of the wavelet coefficients are defined that exploit interscale and or interband correlation of the signal three functions are studied the square of the wavelet coefficients products of coefficients at adjacent scales and products of coefficients from different bands for these functions the signal and noise probability density functions pdf become more separated the high signal correlation between bands is exploited by summing these products over all bands in this way separating noise and signal pdfs even more the noise pdf of the proposed quantities is derived analytically and from this a wavelet threshold is derived the technique is demonstrated to outperform single band wavelet thresholding on multispectral remote sensing images and on multimodal mri images

a nonlinear stein based estimator for multichannel image denoising the use of multicomponent images has become widespread with the improvement of multisensor systems having increased spatial and spectral resolutions however the observed images are often corrupted by an additive gaussian noise in this paper we are interested in multichannel image denoising based on a multiscale representation of the images a multivariate statistical approach is adopted to take into account both the spatial and the intercomponent correlations existing between the different wavelet subbands more precisely we propose a new parametric nonlinear estimator which generalizes many reported denoising methods the derivation of the optimal parameters is achieved by applying stein s principle in the multivariate case experiments performed on multispectral remote sensing images clearly indicate that our method outperforms conventional wavelet denoising techniques

occlusion models for natural images a statistical study of a scale invariant dead leaves model we develop a scale invariant version of matheron s dead leaves model for the statistics of natural images the model takes occlusions into account and resembles the image formation process by randomly adding independent elementary shapes such as disks in layers we compare the empirical statistics of two large databases of natural images with the statistics of the occlusion model and find an excellent qualitative and good quantitative agreement at this point this is the only image model which comes close to duplicating the simplest elementary statistics of natural images such as the scale invariance property of marginal distributions of filter responses the full co occurrence statistics of two pixels and the joint statistics of pairs of haar wavelet responses

numerical verification of solutions for obstacle problems using a newton like method abstract this paper is a continuation of the preceding study 1 in which we described a method which automatically proves the existence of solutions for variational inequalities by computer we newly formulate a verification method using a newton like method this approach enables us to remove the restriction in the previous paper to the retraction property of the operator in a neighborhood of the solution we show some numerical examples which confirm that the method is really applicable to problems which have no retraction property

on computational proofs of the existence of solutions to nonlinear parabolic problems abstract this paper is an extension of the preceding study nakao this journal 1991 in which we described a numerical verification method of the solution for one space dimensional parabolic problems to the several space dimensional case here numerical verification means the automatic proof of the existence of solutions to the problems by some numerical techniques on a computer we reformulate the verification condition for nonlinear parabolic initial boundary value problems using the fixed point problem of a compact operator on certain function spaces as in the preceding study based upon a simple c 0 finite element approximation and its constructive a priori error estimates a numerical verification procedure is presented with some numerical examples

space spatial frequency analysis based filtering space invariant filtering of signals that overlap with noise in both space and frequency can be inefficient however the signal and noise may be well separated in the joint space spatial frequency domain then it is possible to benefit from the application of space spatial frequency approaches processing based on these approaches can outperform space or frequency invariant based methods to this aim the concept of nonstationary space varying filtering is introduced in this paper as an extension of the time varying filtering concept the filtering definitions are based on statistical averages although the filtering should commonly be applied knowing only a single noisy signal realization the procedures that can produce good estimates of quantities crucial for efficient filtering based on a single noisy signal realization are considered special attention has been paid to the region of support estimation and cross term effects removal the efficiency of the proposed space spatial frequency filtering concept is tested on the signal forms inspired by the interferograms in optics including real images as disturbances examples demonstrate the superiority of the proposed filtering over the space invariant one for the considered type of signals and noise

generalized sure for exponential families applications to regularization stein s unbiased risk estimate sure was proposed by stein for the independent identically distributed i i d gaussian model in order to derive estimates that dominate least squares ls recently the sure criterion has been employed in a variety of denoising problems for choosing regularization parameters that minimize an estimate of the mean squared error mse however its use has been limited to the i i d case which precludes many important applications in this paper we begin by deriving a sure counterpart for general not necessarily i i d distributions from the exponential family this enables extending the sure design technique to a much broader class of problems based on this generalization we suggest a new method for choosing regularization parameters in penalized ls estimators we then demonstrate its superior performance over the conventional generalized cross validation and discrepancy approaches in the context of image deblurring and deconvolution the sure technique can also be used to design estimates without predefining their structure however allowing for too many free parameters impairs the estimate s performance to address this inherent tradeoff we propose a regularized sure objective and demonstrate its use in the context of wavelet denoising

nonstationary spectral analysis based on time frequency operator symbols and underspread approximations we present a unified framework for time varying or time frequency tf spectra of nonstationary random processes in terms of tf operator symbols we provide axiomatic definitions and tf operator symbol formulations for two broad classes of tf spectra one of which is new these classes contain all major existing tf spectra such as the wigner ville evolutionary instantaneous power and physical spectrum our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high lag tf correlations so called underspread processes we demonstrate that for underspread processes all tf spectra yield effectively identical results and satisfy several desirable properties at least approximately we also show that gabor frames provide approximate karhunen loeve kl functions of underspread processes and tf spectra provide a corresponding approximate kl spectrum finally we formulate simple approximate input output relations for the tf spectra of underspread processes that are passed through underspread linear time varying systems all approximations are substantiated mathematically by upper bounds on the associated approximation errors our results establish a tf calculus for the second order analysis and time varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes

kernel design for reduced interference distributions the authors present a class of time frequency signal representations tfrs called the reduced interference distribution rid an overview of commonly used tfrs is given and desirable distribution properties are introduced particular attention is paid to the interpretation of cohen s class of time frequency distributions of tfrs in the ambiguity temporal correlation spectral correlation and time frequency domains based on the desirable kernel requirements the rid is discussed and further defined a systematic procedure to create rid kernels or equivalently compute rids is proposed some aspects and properties of the rid are discussed the authors estimate design considerations for rids and compare various selections of the primitive window some experimental results demonstrating the performance of the rid are presented

some lower bound estimates for resolvents of a compact operator on an infinite dimensional hilbert space abstract this paper presents some lower estimates for resolvents of a compact operator a on an infinite dimensional hilbert space usually upper estimates for resolvents are known under suitable conditions but lower estimates are not known these estimates enable us to calculate suitability results for approximation of a resolvent

norm bound computation for inverses of linear operators in hilbert spaces abstract this paper presents a computer assisted procedure to prove the invertibility of a linear operator which is the sum of an unbounded bijective and a bounded operator in a hilbert space and to compute a bound for the norm of its inverse by using some projection and constructive a priori error estimates the invertibility condition together with the norm computation is formulated as an inequality based upon a method originally developed by the authors for obtaining existence and enclosure results for nonlinear partial differential equations several examples which confirm the actual effectiveness of the procedure are reported

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

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

numerical verification of existence and inclusion of solutions for nonlinear operator equations abstract nonlinear operator equations of the type f u lu nu 0 u d l are considered where l is a closed linear operator from a banach space x to another banach space y and n a nonlinear operator from x to y a method is presented for numerical verification and inclusion of solutions for the equations as an example the existence of a periodic solution is proved for the duffing equation

an efficient numerical verification method for the kolmogorov problem of incompressible viscous fluid some computer assisted proofs of nontrivial steady state solutions for the kolmogorov flows are presented the method is based on the infinite dimensional fixed point theorem using a newton like operator with a numerical verification algorithm that automatically generates a set that includes the exact nontrivial solution when discussing the numerical results we consider the effects of rounding errors in the floating point computations this is a continuation of our study that was presented in watanabe 2009

robust wigner distribution with application to the instantaneous frequency estimation the wigner distribution wd produces highly concentrated time frequency tf representation of nonstationary signals it may be used as an efficient signal analysis tool including the cases of frequency modulated signals corrupted with the gaussian noise in some applications a significant amount of impulse noise is present then the wd fails to produce satisfactory results the robust periodogram has been introduced for spectral estimation of this kind of noisy signals it can produce good concentration for pure harmonic signals however it is not so efficient in the cases of signals with rapidly varying frequency this is the motivation for introducing the robust wd it is a reliable tf representation tool for wide class of nonstationary signals corrupted with impulse noise this distribution produces good accuracy of the instantaneous frequency if estimation using the huber 1981 loss function a generalization of the wd is presented it includes both the standard and the robust wd as special cases this distribution can be used for tf analysis of signals corrupted with a mixture of impulse and gaussian noise the presented theory is illustrated on examples including applications on the if estimation and time varying filtering of signals corrupted with a mixture of the gaussian and impulse noise the case study analysis of the if estimators accuracy based on the standard and the robust wd forms is performed in order to improve the if estimation a median filter is applied on the obtained if estimate

numerical verifications of solutions for elliptic equations in nonconvex polygonal domains in this paper methods for numerical verifications of solutions for elliptic equations in nonconvex polygonal domains are studied in order to verify solutions using computer it is necessary to determine some constants which appear in a priori error estimations we propose some methods for determination of these constants in numerical examples calculating these constants for anl shaped domain we verify the solution of a nonlinear elliptic equation

a new formulation using the schur complement for the numerical existence proof of solutions to elliptic problems without direct estimation for an inverse of the linearized operator infinite dimensional newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations pdes in computer assisted proofs of pdes the original problem is transformed into the infinite dimensional newton type fixed point equation w mathcal l 1 mathcal f hat u mathcal l 1 mathcal g w where mathcal l is a linearized operator mathcal f hat u is a residual and mathcal g w is a nonlinear term therefore the estimations of vert mathcal l 1 mathcal f hat u vert and vert mathcal l 1 mathcal g w vert play major roles in the verification procedures in this paper using a similar concept to block gaussian elimination and its corresponding schur complement for matrix problems we represent the inverse operator mathcal l 1 as an infinite dimensional operator matrix that can be decomposed into two parts finite dimensional and infinite dimensional this operator matrix yields a new effective realization of the infinite dimensional newton method which enables a more efficient verification procedure compared with existing nakao s methods for the solution of elliptic pdes we present some numerical examples that confirm the usefulness of the proposed method related results obtained from the representation of the operator matrix as mathcal l 1 are presented in the appendix

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

solving nonlinear parabolic problems with result verification part i one space dimensional case abstract we propose some numerical methods for the automatic proof of existence of weak solutions for parabolic initial boundary value problems with one space dimension it also means that one can obtain a posteriori error bounds for the approximate solutions of the problems based upon schauder s fixed point theorem a verification condition is formulated and by the use of finite element approximation and its error estimates for a simple parabolic problem we present a numerical verification algorithm of exact solutions in a computer some numerical examples which are verified by the method are illustrated

the transient spectrum of a random system the frequency spectrum is a fundamental quantity for the analysis and design of a system when a system is turned on or off the frequency spectrum of its output changes with time we define the transient spectrum as the time frequency spectrum of the system output during the transient phase we obtain the exact transient spectrum for a wide class of random systems and we formulate it with respect to the classical power spectrum which is reached for large times we apply the developed method to the case of a harmonic oscillator and of a system with two resonances

multispectral image denoising with optimized vector bilateral filter vector bilateral filtering has been shown to provide good tradeoff between noise removal and edge degradation when applied to multispectral hyperspectral image denoising it has also been demonstrated to provide dynamic range enhancement of bands that have impaired signal to noise ratios snrs typical vector bilateral filtering described in the literature does not use parameters satisfying optimality criteria we introduce an approach for selection of the parameters of a vector bilateral filter through an optimization procedure rather than by ad hoc means the approach is based on posing the filtering problem as one of nonlinear estimation and minimization of the stein s unbiased risk estimate of this nonlinear estimator along the way we provide a plausibility argument through an analytical example as to why vector bilateral filtering outperforms band wise 2d bilateral filtering in enhancing snr experimental results show that the optimized vector bilateral filter provides improved denoising performance on multispectral images when compared with several other approaches

on the smoothing property of the galerkin method for parabolic equations we analyze the galerkin approximation of the general second order parabolic initial boundary value problem for the second order continuous time method with initial data only in l 2 we prove an l 2 error estimate of order o h 2 t analogous results are shown to hold for the error in negative sobolev norms and for the time derivative of the error our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward euler time discretization

numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element the verifications of solutions to weakly nonlinear elliptic equations by the method described e g by nakao 1988 1989 etc are sometimes hardly accomplished when the right hand sides of the equations are very large to overcome such difficulties a residual iteration technique with approximate solution was introduced by nakao 1993 in the present paper we propose an a posteriori method for the residual iteration and show that a remarkable improvement in efficiency and in accuracy of the verification can be obtained when we use a higher order finite element

a posteriori error bounds for two point boundary value problems consider a general two point boundary value problem tpbvp begin gathered y t f t y hfill b 1 y a b 2 y b w quad hfill end gathered a leqq t leqq b where f r n 1 to r n f in c 2 b 1 and b 2 are n times n matrices and w in r n it is shown how one can bound a posteriori the error made in the numerical solution of the tpbvp the error bounds obtained are rigorous and include the truncation and the roundoff error in addition the computations establish the existence of solutions to the tpbvp numerical schemes are developed for the case where f t y is a polynomial in t and y examples are given of computational existence proofs for problems where analytical existence proofs are not known

on very accurate enclosure of the optimal constant in the a priori error estimates for h02 projection we present constructive a priori error estimates for h 0 2 projection into a space of polynomials on a one dimensional interval here constructive indicates that we can obtain the error bounds in which all constants are explicitly given or are represented in a numerically computable form using the properties of legendre polynomials we consider a method by which to determine these constants to be as small as possible using the proposed technique the optimal constant could be enclosed in a very narrow interval with result verification furthermore constructive error estimates for finite element h 0 2 projection in one dimension are presented these types of estimates will play an important role in the numerical verification of solutions for nonlinear fourth order elliptic problems as well as in the guaranteed a posteriori error analysis for the finite element method or the spectral method e g hashimoto et al 2006 2 nakao et al 2008 3 watanabe et al 2009 11

minimum variance time frequency distribution kernels for signals in additive noise we derive the minimum variance time frequency distribution kernel for signals in additive circular complex gaussian white noise processes it is shown that the kernel that minimizes the average variance and simultaneously satisfies the time frequency constraints for the noise only ease remains optimum in the presence of frequency modulated fm signals

image denoising via bandwise adaptive modeling and regularization exploiting nonlocal similarity this paper proposes a new image denoising algorithm based on adaptive signal modeling and regularization it improves the quality of images by regularizing each image patch using bandwise distribution modeling in transform domain instead of using a global model for all the patches in an image it employs content dependent adaptive models to address the non stationarity of image signals and also the diversity among different transform bands the distribution model is adaptively estimated for each patch individually it varies from one patch location to another and also varies for different bands in particular we consider the estimated distribution to have non zero expectation to estimate the expectation and variance parameters for every band of a particular patch we exploit the nonlocal correlation in image to collect a set of highly similar patches as the data samples to form the distribution irrelevant patches are excluded so that such adaptively learned model is more accurate than a global one the image is ultimately restored via bandwise adaptive soft thresholding based on a laplacian approximation of the distribution of similar patch group transform coefficients experimental results demonstrate that the proposed scheme outperforms several state of the art denoising methods in both the objective and the perceptual qualities

nonparametric detection of fm signals using time frequency ridge energy in many practical applications signals to be detected are unknown nonlinear frequency modulated fm and are corrupted by strong noise the phase histories of signals are assumed to be unknown smooth functions of time and these functions are poorly modeled or unmodeled by a small number of parameters thus the conventional parametric based detection methods are invalid in these cases this paper proposes a nonparametric detection method using the ridge energy of observations the detection process consists of three steps tf ridge detection ridge energy extraction and decision first the directionally smoothed pseudo wigner ville distribution dspwvd is introduced to highlight the instantaneous frequency if points along a special direction on the if curve of a signal from noise further an angular maximal distribution amad is constructed from a set of dspwvds to highlight the entire if curve as a result the tf ridge of an observation can be estimated well from its amad by the maxima position detector second the ridge energy the total energy along the tf ridge on the pseudo wigner ville distribution pwvd is extracted a noisy signal has larger ridge energy than a pure noise does with a large probability because pure noise energy is randomly distributed throughout the tf plane while the signal energy in a noisy signal is concentrated along the estimated tf ridge third the ridge energy of an observation is used as the test statistic to decide whether or not a signal of interest is present in the observation where the decision threshold is determined by a large number of monte carlo simulations using pure noise finally the simulation experiments to two test signals are made to verify the effectiveness of the proposed method

the use of cone shaped kernels for generalized time frequency representations of nonstationary signals generalized time frequency representations gtfrs which use cone shaped kernels for nonstationary signal analysis are presented the cone shaped kernels are formulated for the gtfrs to produce good resolution simultaneously in time and frequency specifically for a gtfr with a cone shaped kernel finite time support is maintained in the time dimension along with an enhanced spectrum in the frequency dimension and the cross terms are smoothed out experimental results on simulated data and real speech show the advantages of the gtfrs with the cone shaped kernels through comparisons to the spectrogram and the pseudo wigner distribution

on the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems we consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial boundary conditions our estimation technique uses a full discrete numerical scheme which is based on the galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space in our technique the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role combining these estimates with an argument for the discretized inverse operator and a contraction property of the newton type formulation we derive an a posteriori estimate of the norm for the infinite dimensional operator in numerical examples we show that the proposed method should be more efficient than the existing method moreover as an application we give some prototype results for numerical verification of solutions of nonlinear parabolic problems which confirm the actual usefulness of our technique

nonlinear multivariate image filtering techniques in this paper nonlinear multivariate image filtering techniques are proposed to handle color images corrupted by noise first we briefly review the principle of reduced ordering r ordering and then define three r orderings by selecting different central locations considering noise attenuation edge preservation and detail retention r ordering based multivariate filters are designed by combining the r ordering schemes to implement color image filtering more effectively we develop them into a locally adaptive version the output of the adaptive filter is the closest sample to a central location that is a weighted linear combination of the mean the marginal median and the center sample as a result we study an adaptive hybrid multivariate ahm filter consisting of the mean filter the marginal median filter and the identity filter the performance of the two adaptive filtering techniques is compared with that of some nonadaptive ones the examples of color image filtering show that the adaptive multivariate image filtering gives a rather good performance improvement

the wigner distribution of noisy signals with adaptive time frequency varying window time frequency representations using the wigner distribution wd may be significantly obscured by the noise in the observations the analysis performed for the wd of discrete time noisy signals shows that this time frequency representation can be optimized by the appropriate choice of the window length however the practical value of this analysis is not significant because the optimization requires knowledge of the bias which depends on the unknown derivatives of the wd a simple adaptive algorithm for the efficient time frequency representation of noisy signals is developed in this paper the algorithm uses only the noisy estimate of the wd and the analytical formula for the variance of this estimate the quality of this adaptive algorithm is close to the one that could be achieved by the algorithm with the optimal window length provided that the wd derivatives were known in advance the proposed algorithm is based on the idea that has been developed in our previous work for the instantaneous frequency if estimation here a direct addressing to the wd itself rather than to the instantaneous frequency resulted in a time and frequency varying window length and showed that the assumption of small noise and bias is no longer necessary a simplified version of the algorithm using only two different window lengths is presented it is shown that the procedure developed for the adaptive window length selection can be generalized for application on multicomponent signals with any distribution from the cohen 1989 1990 1992 class simulations show that the developed algorithms are efficient even for a very low value of the signal to noise ratio

estimating the probability of the presence of a signal of interest in multiresolution single and multiband image denoising we develop three novel wavelet domain denoising methods for subband adaptive spatially adaptive and multivalued image denoising the core of our approach is the estimation of the probability that a given coefficient contains a significant noise free component which we call signal of interest in this respect we analyze cases where the probability of signal presence is 1 fixed per subband 2 conditioned on a local spatial context and 3 conditioned on information from multiple image bands all the probabilities are estimated assuming a generalized laplacian prior for noise free subband data and additive white gaussian noise the results demonstrate that the new subband adaptive shrinkage function outperforms bayesian thresholding approaches in terms of mean squared error the spatially adaptive version of the proposed method yields better results than the existing spatially adaptive ones of similar and higher complexity the performance on color and on multispectral images is superior with respect to recent multiband wavelet thresholding

a numerical approach to the proof of existence of solutions for some generalized obstacle problems in 8 we proposed some numerical verification methods for automatic proof of the existence of solution for obstacle problems in this paper we propose a new iterative algorithm to automatically prove the existence of solutions for some generalized obstacle problems

on necessary and sufficient conditions for numerical verification of double turning points this paper describes numerical verification of a double turning point of a nonlinear system using an extended system to verify the existence of a double turning point we need to prove that one of the solutions of the extended system corresponds to the double turning point for that we propose an extended system with an additional condition as an example for a finite dimensional problem we verify the existence and local uniqueness of a double turning point numerically using the extended system and a verification method based on the banach fixed point theorem

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

numerical inclusion of exact periodic solutions for time delay duffing equation abstract numerical inclusion results for exact periodic solutions are presented for the time delay autonomous duffing equations constructive implicit function theorem is used for including one dimensional solution manifolds consisting of exact periodic solutions a conjecture of a lower bound for a number of periodic solutions is given as a function of the time delay if the delay time is less than 30 we have proved this conjecture using verified numerical computations theory for proving the existence of periodic solutions of the forced delay duffing equation is proposed based on the verified numerical computations the forced term is sinusoidal waves stress is on a study of the bifurcation of periodic solutions synchronizing to the external forces a rich bifurcation phenomena of periodic solutions are reported taking the delay time as parameters especially a kind of fractal structure is observed concerning resonance peaks

performance of quadratic time frequency distributions as instantaneous frequency estimators general performance analysis of the shift covariant class of quadratic time frequency distributions tfds as instantaneous frequency if estimators for an arbitrary frequency modulated fm signal is presented expressions for the estimation bias and variance are derived this class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear if however when the if is not a linear function of time then the estimate is biased cases of white stationary and white nonstationary additive noises are considered the well known results for the wigner distribution wd and linear fm signal and the spectrogram of signals whose if may be considered as a constant within the lag window are presented as special cases in addition we have derived the variance expression for the spectrogram of a linear fm signal that is quite simple but highly signal dependent this signal is considered in the cases of other commonly used distributions such as the born jordan and the choi williams distributions it has been shown that the reduced interference distributions outperform the wd but only in the case when the if is constant or its variations are small analysis is extended to the if estimation of signal components in the case of multicomponent signals all theoretical results are statistically confirmed

verified eigenvalue evaluation for the laplacian over polygonal domains of arbitrary shape the finite element method fem is applied to bound leading eigenvalues of the laplace operator over polygonal domains compared with classical numerical methods most of which can only give concrete eigenvalue bounds over special domains of symmetry our proposed algorithm can provide concrete eigenvalue bounds for domains of arbitrary shape even when the eigenfunction has a singularity the problem of eigenvalue estimation is solved in two steps first we construct a computable a priori error estimation for the fem solution of poisson s problem which holds even for nonconvex domains with reentrant corners second new computable lower bounds are developed for the eigenvalues because the interval arithmetic is implemented throughout the computation the desired eigenvalue bounds are expected to be mathematically correct we illustrate several computation examples such as the cases of an l shaped domain and a crack domain to demonstrate the efficiency and flexibility of the proposed method

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

numerical verification of solutions for nonlinear elliptic problems using anl residual method abstract we consider a numerical enclosure method with guaranteedl error bounds for the solution of nonlinear elliptic problems of second order by using an a posteriori error estimate for the approximate solution of the problem with a higher orderc0 finite element it is shown that we can obtain the guaranteedl error bounds with high accuracy a particular emphasis is that our method needs no assumption of the existence of the solution of the original nonlinear equation but it follows as the result of computation itself a numerical example that confirms the effectiveness of the method is presented

further results on the minimum variance time frequency distribution kernels results for the minimum variance kernel presented by hearon and amin see ibid vol 43 p 1258 1995 for the complex gaussian white noise with independent real and imaginary parts remain valid for real noise and approximately valid for analytic noise these results are extended to the real and analytic noisy signals cases

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

building robust wavelet estimators for multicomponent images using stein s principle multichannel imaging systems provide several observations of the same scene which are often corrupted by noise in this paper we are interested in multispectral image denoising in the wavelet domain we adopt a multivariate statistical approach in order to exploit the correlations existing between the different spectral components our main contribution is the application of stein s principle to build a new estimator for arbitrary multichannel images embedded in additive gaussian noise simulation tests carried out on optical satellite images show that the proposed method outperforms conventional wavelet shrinkage techniques

sure let multichannel image denoising interscale orthonormal wavelet thresholding we propose a vector matrix extension of our denoising algorithm initially developed for grayscale images in order to efficiently process multichannel e g color images this work follows our recently published sure let approach where the denoising algorithm is parameterized as a linear expansion of thresholds let and optimized using stein s unbiased risk estimate sure the proposed wavelet thresholding function is pointwise and depends on the coefficients of same location in the other channels as well as on their parents in the coarser wavelet subband a nonredundant orthonormal wavelet transform is first applied to the noisy data followed by the subband dependent vector valued thresholding of individual multichannel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform extensive comparisons with the state of the art multiresolution image denoising algorithms indicate that despite being nonredundant our algorithm matches the quality of the best redundant approaches while maintaining a high computational efficiency and a low cpu memory consumption an online java demo illustrates these assertions

numerical verification of solutions for variational inequalities of the second kind abstract the purpose of this paper is to present an approach to the numerical proof of existence of solutions for the problem of the flow of a viscous plastic fluid in a pipe using the finite element approximations and the explicit a priori error estimates for the problem of the flow of a viscous plastic fluid in a pipe we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example is presented

multiple window time varying spectral analysis a multiwindow method for generating a time varying spectrum of nonstationary signals is presented the time varying spectrum is computed from an optimally weighted average of multiple orthogonal windowed spectrograms the weights are determined using linear least squares estimation with respect to a reference time frequency distribution examples are provided with performance criteria measures to demonstrate and quantify the effectiveness of the method

minimum variance time frequency distribution kernels when dealing with random processes reduced spectrum estimate variance becomes an important property that augments the list of desirable time frequency t f distribution properties in this correspondence we derive the t f kernel that satisfies the t f constraints and provides the minimum variance for the power spectrum estimate for gaussian white noise processes

numerical solution of stiff and singularly perturbed boundary value problems with a segmented adaptive formulation of the tau method this paper concerns the application of ortiz recursive formulation of the tau method to the construction of piecewise polynomial approximations to the solution of linear and nonlinear boundary value problems for ordinary differential equations a practical error estimation technique related to the concept of correction in zadunaisky s sense is considered and used in the design of an adaptive approach to the tau method it proves efficient in the numerical treatment of problems with rapid functional variations stiff and singularly per turbed problems a technique of increased accuracy at matching points of segmented tau approximants is also discussed and successfully applied to several problems numerical examples show that for a given degree of approximation our segmented tau approximant gives an accuracy comparable to that of the best segmented approximation of the exact solution by means of algebraic polynomials 1 introduction we discuss the use of ortiz recursive formulation of the tau method 23 25 in the numerical solution of boundary value problems for linear and nonlinear differential equations defined over an interval a x b we con sider global approximations over a b with a single polynomial expression and segmented forms based on a step by step formulation of the tau method considered by ortiz in 26 the tau approximate solution of a differential problem defined by a differential operator d is represented in terms of the elements of a sequence q of canonical polynomials such a sequence is uniquely determined by d it is independent of the specific boundary conditions of the problem and of the particular interval a b in which the solution is required these properties make possible the use of segmenta tion within the framework and with the software 32 designed for the recursive formulation of the tau method the concept of correction in zadunaisky s sense 37 see also stetter 36 is discussed in the context of the tau method and related to a practical error estimation technique this technique based on tau estimators introduced here is systematically applied to all examples linear or nonlinear it is