graphical models for statistical inference and data assimilation in data assimilation for a system which evolves in time one combines past and current observations with a model of the dynamics of the system in order to improve the simulation of the system as well as any future predictions about it from a statistical point of view this process can be regarded as estimating many random variables which are related both spatially and temporally given observations of some of these variables typically corresponding to times past we require estimates of several others typically corresponding to future times graphical models have emerged as an efiective formalism for assisting in these types of inference tasks particularly for large numbers of random variables graphical models provide a means of representing dependency structure among the variables and can provide both intuition and e ciency in estimation and other inference computations we provide an overview and introduction to graphical models and describe how they can be used to represent statistical dependency and how the resulting structure can be used to organize computation the relation between statistical inference using graphical models and optimal sequential estimation algorithms such as kalman flltering is discussed we then give several additional examples of how graphical models can be applied to climate dynamics speciflcally estimation using multi resolution models of large scale data sets such as satellite imagery and learning hidden markov models to capture rainfall patterns in space and time

self similar set identification in the time scale domain we study the wavelet transform of deterministic self similar signals and derive their properties as well as a new algorithm for identification of the self similarity parameter we also include such applications as characterization and analysis of real chaotic signals in the presence of additive noise

a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

solution of inverse problems in image processing by wavelet expansion we describe a wavelet based approach to linear inverse problems in image processing in this approach both the images and the linear operator to be inverted are represented by wavelet expansions leading to a multiresolution sparse matrix representation of the inverse problem the constraints for a regularized solution are enforced through wavelet expansion coefficients a unique feature of the wavelet approach is a general and consistent scheme for representing an operator in different resolutions an important problem in multigrid multiresolution processing this and the sparseness of the representation induce a multigrid algorithm the proposed approach was tested on image restoration problems and produced good results

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

deconvolution filter design for fractal signal transmission systems a multiscale kalman filter bank approach a deconvolution filtering design is proposed for the 1 f fractal signal transmission systems with its design philosophy being based on multiscale kalman deconvolution filter bank equipped in the analysis synthesis wavelet filter bank the role of wavelet transformation for 1 f fractal signal process is exploited as a multiscale whitening filter for removing the properties of self similarity and long range dependence from the fractal signals

accurate estimation of the hurst parameter of long range dependent traffic using modified allan and hadamard variances internet traffic exhibits self similarity and longrange dependence lrd on various time scales in this paper we propose to use the modified allan variance mavar and a modified hadamard variance mhvar to estimate the hurst parameter h of the lrd traffic series or more generally the exponent alpha of data with 1 falpha alpha ges 0 power law spectrum mhvar generalizes the principle of mavar a time domain quantity widely used for frequency stability characterization to higher order differences of input data in our knowledge this mhvar has been mentioned in literature only few times and with little detail so far the behaviour of mavar and mhvar with power law random processes and some common deter ministic signals viz drifts sine waves steps is studied by analysis and simulation the mavar and mhvar accuracy in estimating h is evaluated and compared to that of wavelet logscale diagram ld extensive simulations show that mavar and mhvar achieve significantly better confidence and no bias in h estimation moreover mavar and mhvar feature a number of other advantages which make them valuable to complement other established techniques such as ld finally mhvar and ld are also applied to a real ip traffic trace

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

a wavelet based kl like expansion for wide sense stationary random processes we describe a wavelet based series expansion for wide sense stationary processes the expansion coefficients are uncorrelated random variables a property similar to that of a karhunen loeve kl expansion unlike the kl expansion however the wavelet based expansion does not require the solution of the eigen equation and does not require that the process be time limited this expansion also has advantages over fourier series which is often used as an approximation to the kl expansion in that it completely eliminates correlation and that the computation for its coefficients are more stable over large time intervals the basis functions of this expansion can be obtained easily from wavelets of the lemaire meyer 1990 type and the power spectral density of the process finally the expansion can be extended to some nonstationary processes such as those with wide sense stationary increments

regularization in tomographic reconstruction using thresholding estimators in tomographic medical devices such as single photon emission computed tomography or positron emission tomography cameras image reconstruction is an unstable inverse problem due to the presence of additive noise a new family of regularization methods for reconstruction based on a thresholding procedure in wavelet and wavelet packet wp decompositions is studied this approach is based on the fact that the decompositions provide a near diagonalization of the inverse radon transform and of prior information in medical images a wp decomposition is adaptively chosen for the specific image to be restored corresponding algorithms have been developed for both two dimensional and full three dimensional reconstruction these procedures are fast noniterative and flexible numerical results suggest that they outperform filtered back projection and iterative procedures such as ordered subset expectation maximization

multiresolution tomographic reconstruction using wavelets shows how the separable two dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm this algorithm is similar to the conventional filtered backprojection algorithm except that the filters are now angle dependent and the backprojection gives the wavelet coefficients of the reconstruction which are then used to synthesize the reconstruction at various resolution levels by reconstructing only a small localized region at high resolution the authors show how radiation exposure and computation can be significantly reduced compared to a standard reconstruction

a wavelet based method for multiscale tomographic reconstruction the authors represent the standard ramp filter operator of the filtered back projection fbp reconstruction in different bases composed of haar and daubechies compactly supported wavelets the resulting multiscale representation of the ramp filter matrix operator is approximately diagonal the accuracy of this diagonal approximation becomes better as wavelets with larger numbers of vanishing moments are used this wavelet based representation enables the authors to formulate a multiscale tomographic reconstruction technique in which the object is reconstructed at multiple scales or resolutions a complete reconstruction is obtained by combining the reconstructions at different scales the authors multiscale reconstruction technique has the same computational complexity as the fbp reconstruction method it differs from other multiscale reconstruction techniques in that 1 the object is defined through a one dimensional multiscale transformation of the projection domain and 2 the authors explicitly account for noise in the projection data by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients the computational complexity of this maximum a posteriori probability map solution is also the same as that of the fbp reconstruction this result is in contrast to commonly used methods of statistical regularization which result in computationally intensive optimization algorithms

computational methods for hidden markov tree models an application to wavelet trees the hidden markov tree models were introduced by crouse et al in 1998 for modeling nonindependent non gaussian wavelet transform coefficients in their paper they developed the equivalent of the forward backward algorithm for hidden markov tree models and called it the upward downward algorithm this algorithm is subject to the same numerical limitations as the forward backward algorithm for hidden markov chains hmcs in this paper adapting the ideas of devijver from 1985 we propose a new upward downward algorithm which is a true smoothing algorithm and is immune to numerical underflow furthermore we propose a viterbi like algorithm for global restoration of the hidden state tree the contribution of those algorithms as diagnosis tools is illustrated through the modeling of statistical dependencies between wavelet coefficients with a special emphasis on local regularity changes

estimation of 1 f noise several models have emerged for describing 1 f sup spl gamma noise processes based on these various techniques for estimating the properties of such processes have been developed this paper provides theoretical analysis of a new wavelet based approach which has the advantages of having low computational complexity and being able to handle the case where the 1 f sup spl gamma noise might be embedded in a further white noise process however the analysis conducted here shows that these advantages are balanced by the fact that the wavelet based scheme is only consistent for spectral exponents spl gamma in the range spl gamma spl isin 0 1 this is in contradiction to the results suggested in previous empirical studies when spl gamma spl isin 0 1 this paper also establishes that wavelet based maximum likelihood methods are asymptotically gaussian and efficient finally the asymptotic rate of mean square convergence of the parameter estimates is established and is shown to slow as spl gamma approaches one combined with a survey of non wavelet based methods these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1 f sup spl gamma noise processes

wavelet sampling and localization schemes for the radon transform in two dimensions two theorems are presented for wavelet decompositions of the two dimensional radon transform the rst theorem establishes an upper error bound in l 2 norm between the radon transform and its wavelet approximation whose coecients at dierent scales are estimated from radon data acquired at corresponding sampling rates the second theorem gives an estimate of the accuracy of a local image reconstructed from localized radon data at multiple levels these results show how to design a multilevel sampling and localization strategy for parallel beam scanning by using wavelet regularity and vanishing moment characteristics clarify the interaction between the wavelet structure and the essential bandwidth of an object image and provide guidelines for wavelet local tomography

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

signal detection in fractional gaussian noise several results related to the reproducing kernel hilbert space of fractional brownian motion are presented to facilitate the study of signal detection in additive fractional gaussian noise this hilbert space is completely characterized and an alternative characterization for the restriction of this class of functions to a compact interval 0 t is given infinite and finite interval whitening filters for fractional brownian motion are also developed the application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional gaussian noise a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

image reconstruction from projections under wavelet constraints first the authors discuss how the wavelet transform can be used to perform spatially varying filtering of an image suppressing noise locally in smooth regions of the image and discuss detection of such regions in a noise corrupted image second they show how to compute the minimum mean square estimate of an image given 1 noisy projections of the image 2 statistics of additive noise in the projections and 3 constraints on wavelet coefficients of the image examples illustrate the resulting procedure

fractal connectivity of long memory networks using the multivariate long memory lm model and taylor expansions we find the conditions for convergence of the wavelet correlations between two lm processes on an asymptotic value at low frequencies these mathematical results and a least squares estimator of lm parameters are validated in simulations and applied to neurophysiological human brain and financial market time series both brain and market systems had multivariate lm properties including a fractal connectivity regime of scales over which wavelet correlations were invariantly close to their asymptotic value this analysis provides efficient and unbiased estimation of long term correlations in diverse dynamic networks

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

platelets a multiscale approach for recovering edges and surfaces in photon limited medical imaging the nonparametric multiscale platelet algorithms presented in this paper unlike traditional wavelet based methods are both well suited to photon limited medical imaging applications involving poisson data and capable of better approximating edge contours this paper introduces platelets localized functions at various scales locations and orientations that produce piecewise linear image approximations and a new multiscale image decomposition based on these functions platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries for smoothness measured in certain holder classes it is shown that the error of m term platelet approximations can decay significantly faster than that of m term approximations in terms of sinusoids wavelets or wedgelets this suggests that platelets may outperform existing techniques for image denoising and reconstruction fast platelet based maximum penalized likelihood methods for photon limited image denoising deblurring and tomographic reconstruction problems are developed because platelet decompositions of poisson distributed images are tractable and computationally efficient existing image reconstruction methods based on expectation maximization type algorithms can be easily enhanced with platelet techniques experimental results suggest that platelet based methods can outperform standard reconstruction methods currently in use in confocal microscopy image restoration and emission tomography

efficient framework for model based tomographic image reconstruction using wavelet packets the use of model based algorithms in tomographic imaging offers many advantages over analytical inversion methods however the relatively high computational complexity of model based approaches often restricts their efficient implementation in practice many modern imaging modalities such as computed tomography positron emission tomography or optoacoustic tomography normally use a very large number of pixels voxels for image reconstruction consequently the size of the forward model matrix hinders the use of many inversion algorithms in this paper we present a new framework for model based tomographic reconstructions which is based on a wavelet packet representation of the imaged object and the acquired projection data the frequency localization property of the wavelet packet base leads to an approximately separable model matrix for which reconstruction at each spatial frequency band is independent and requires only a fraction of the projection data thus the large model matrix is effectively separated into a set of smaller matrices facilitating the use of inversion schemes whose complexity is highly nonlinear with respect to matrix size the performance of the new methodology is demonstrated for the case of 2 d optoacoustic tomography for both numerically generated and experimental data

fractional gaussian noise functional mri and alzheimer s disease fractional gaussian noise fgn provides a parsimonious model for stationary increments of a self similar process parameterised by the hurst exponent h and variance sigma2 fractional gaussian noise with h 0 5 demonstrates 1 f long memory or persistent behaviour and the special case of fgn with h 0 5 corresponds to classical gaussian white noise we comparatively evaluate four possible estimators of fgn parameters one method implemented in the time domain and three in the wavelet domain we show that a wavelet based maximum likelihood ml estimator yields the most efficient estimates of h and sigma2 in simulated fgn with 0 0 5 tend to be concentrated in cortex whereas voxels with h 0 5 are more frequently located in ventricles and sulcal csf the wavelet ml estimator can be generalised to estimate the parameter vector beta for general linear modelling glm of a physiological response to experimental stimulation and we demonstrate nominal type i error control in multiple testing of beta divided by its standard error in simulated and biological data under the null hypothesis beta 0 we illustrate these methods principally by showing that there are significant differences between patients with early alzheimer s disease ad and age matched comparison subjects in the persistence of fgn in the medial and lateral temporal lobes insula dorsal cingulate medial premotor cortex and left pre and postcentral gyrus patients with ad had greater persistence of resting fmri noise larger h in these regions comparable abnormalities in the ad patients were also identified by a permutation test of local differences in the first order autoregression ar 1 coefficient which was significantly more positive in patients however we found that the hurst exponent provided a more sensitive metric than the ar 1 coefficient to detect these differences perhaps because neurophysiological changes in early ad are naturally better described in terms of abnormal salience of long memory dynamics than a change in the strength of association between immediately consecutive time points we conclude that parsimonious mapping of fmri noise properties in terms of fgn parameters efficiently estimated in the wavelet domain is feasible and can enhance insight into the pathophysiology of alzheimer s disease

noise covariance properties in dual tree wavelet decompositions dual tree wavelet decompositions have recently gained much popularity mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy when the decomposition of a random process is performed which occurs in particular when an additive noise is corrupting the signal to be analyzed it is useful to characterize the statistical properties of the dual tree wavelet coefficients of this process as dual tree decompositions constitute over complete frame expansions correlation structures are introduced among the coefficients even when a white noise is analyzed in this paper we show that it is possible to provide an accurate description of the covariance properties of the dual tree coefficients of a wide sense stationary process the expressions of the cross covariance sequences of the coefficients are derived in the one and two dimensional cases asymptotic results are also provided allowing to predict the behavior of the second order moments for large lag values or at coarse resolution in addition the cross correlations between the primal and dual wavelets which play a primary role in our theoretical analysis are calculated for a number of classical wavelet families simulation results are finally provided to validate these results

analysis and synthesis of 1 f processes via shannon wavelets 1 f processes can be very useful in modeling processes with long term correlation we propose analysis and synthesis procedures to express these processes in terms of the shannon wavelet unlike previous techniques our analysis procedure generates uncorrelated decomposition coefficients for the 1 f process this is done by taking onto account and then removing the residual correlation between the wavelet components the analysis procedure is the major contribution of this work the proposed synthesis algorithm which is a byproduct of the proposed analysis algorithm is competitive with other techniques

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

symmetric singularity formation in lubrication type equations for interface motion fourth order degenerate diffusion equations arise in a lubrication approximation of a thin film or neck driven by surface tension numerical studies of the lubrication equation le h t h n h xxx x 0 with various boundary conditions indicate that singularity formation in which h x t t to 0 occurs for small enough n with anomalous or second type scaling inconsistent with usual dimensional analysis this paper considers locally symmetric or even singularities in the le and in the modified lubrication equation mle h t h n h xxxx 0 both equations have the property that entropy bounds forbid finite time singularities when n is sufficiently large power series expansions for local symmetric similarity solutions are proposed for equation le with n 1 and mle for all n in mathbb r in the latter case special boundary conditions that force singularity formation are required to produce singularities when n is large matching conditions at higher order terms in the

multiscale autoregressive processes ii lattice structures for whitening and modeling for pt i see ibid vol 40 no 8 p 1915 34 1992 lattice structures for the whitening and modeling of isotropic processes on trees are developed and a result relating the stability properties of these models to the reflection coefficient sequence introduced in pt i is presented this framework allows one to obtain a detailed analysis of the wold decomposition of processes on trees one interesting aspect of this is that there is a significantly larger class of singular processes on dyadic trees than on the integers

statistical mechanical systems on complete graphs infinite exchangeability finite extensions and a discrete finite moment problem we show that a large collection of statistical mechanical systems with quadratically represented hamiltonians on the complete graph can be extended to infinite exchangeable processes this includes all ferromagnetic ising potts and heisenberg models by de finetti s theorem this is equivalent to showing that these probability measures can be expressed as averages of product measures we provide examples showing that ferromagnetism is not however in itself sufficient and also study in some detail the ising model with an additional 3 body interaction finally we study the question of how much the antiferromagnetic ising model can be extended in this direction we obtain sharp asymptotic results via a solution to a new moment problem we also obtain a formula for the extension which is valid in many cases

gaussian non gaussian critical fluctuations in the curie weiss model it is known that at the critical temperature the curie weiss mean field model has non gaussian fluctuations and that internal fluctuations can be gaussian here we compute the distribution of theq mode magnetization fluctuations as a function of the temperature the wave vectorq and a fading out external field we obtain new classes of probability distributions generated by this external field as well as new critical behavior in terms of its rate of fading out we discuss also the susceptibility as the limitq tending to zero

anomalous exponents and dipole solutions for the thin film equation we investigate similarity solutions of the thin film equation in particular we look at solutions on the half line x ge0 with compact support and zero contact angle boundary conditions in x 0 such dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind using a combination of phase space analysis and numerical simulations we numerically construct trajectories representing these solutions at the same time obtaining broader insight into the nature of the four dimensional phase space additional asymptotic analysis provides further information concerning the evolution to self similarity

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

multiscale poisson intensity and density estimation the nonparametric poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness the methods and theory presented here share many of the desirable features associated with wavelet based estimators computational speed spatial adaptivity and the capability of detecting discontinuities and singularities with high resolution unlike traditional wavelet based approaches which impose an upper bound on the degree of smoothness to which they can adapt the estimators studied here guarantee nonnegativity and do not require any a priori knowledge of the underlying signal s smoothness to guarantee near optimal performance at the heart of these methods lie multiscale decompositions based on free knot free degree piecewise polynomial functions and penalized likelihood estimation the degrees as well as the locations of the polynomial pieces can be adapted to the observed data resulting in near minimax optimal convergence rates for piecewise analytic signals in particular the error of this estimator converges at nearly the parametric rate these methods can be further refined in two dimensions and it is demonstrated that platelet based estimators in two dimensions exhibit similar near optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries

nonnegative solutions of a fourth order nonlinear degenerate parabolic equation elena beretta michiel bertsch roberta dal passo communicated by m guatin o introduction in this paper we study nonnegative solutions of the fourth order equation 0 1 ut u uxxx x o where n is a real positive constant equation 0 1 is a nonlinear parabolic partial differential equation which is degenerate at points at which u vanishes it arises in several applications see the references in 2 4 5 6 in which u x t stands for a nonnegative quantity for example in the study of the dynamics of thin liquid films u denotes the thickness of the film or another relevant nonnegative quantity the case n 1 models the flow in a hele shaw cell while n 3 corresponds to viscous flow on a solid surface without slip driven by surface tension see also 6 7 8 for a related equation it is well known that fourth order equations do not satisfy a maximum principle in particular classical solutions of the linear equation ut u 0 may become negative in finite time bernis 8 friedman 2 have observed that if n is positive equation 0 1 has generalized solutions which remain nonnegative for all times in addition they show that the qualitative positivity of these solutions strongly depends on the value of n in more recent papers by kadanoff et al a combination of numerical calculations and matched asymptotics 4 and the analysis of travelling wave solutions 5 leads to results which suggest a very rich structure of the solutions of 0 1 it is the purpose of the present paper to clarify several aspects of the nonlinear equation 0 1 and its dependence on the parameter n our results lead to several conjectures and open problems as was pointed out in 4 for sufficiently small values of n solutions of 0 1 which are initially strictly positive may vanish at some point x 0 after a finite time to but to the best of our knowledge there is no rigorous proof of this phenomenon in the literature on the other hand it is of particular interest for applications for example if u denotes the thickness of a liquid film it implies that at time to the film breaks at xo

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

the linear limit of thin film flows as an obstacle type free boundary problem we study the limit as n to 0 of the nonnegative self similar source type solutions of the thin film equationut unuxxx x 0 we obtain a unique limiting function u which is a solution of an obstacle type free boundary problem with the constraint u geq 0 associated with the linear equationut uxxxx 0 the function u is c1 for t 0 and has finite speed of propagation the positivity set um 0 being bounded by two contact lines x pm a t 1 4 a constant the function u has a dirac mass as initial condition and satisfies the linear equation in the positivity set but not across the free boundaries or contact lines also known as moving boundaries we give an integral representation of u in the positivity set we set up a precise definition of the general non self similar obstacle type free boundary problem which is different from a standard parabolic variational inequality and compare it with the cauchy problem we also consider source type solutions for negative values of n which are so

a statistical multiscale framework for poisson inverse problems this paper describes a statistical multiscale modeling and analysis framework for linear inverse problems involving poisson data the framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity a corresponding multiscale factorization of the likelihood induced by this analysis and a choice of prior probability distribution made to match this factorization by modeling the splits in the underlying partition the class of priors used here has the interesting feature that the noninformative member yields the traditional maximum likelihood solution other choices are made to reflect prior belief as to the smoothness of the unknown intensity adopting the expectation maximization em algorithm for use in computing the maximum a posteriori map estimate corresponding to our model we find that our model permits remarkably simple closed form expressions for the em update equations the behavior of our em algorithm is examined and it is shown that convergence to the global map estimate can be guaranteed applications in emission computed tomography and astronomical energy spectral analysis demonstrate the potential of the new approach

hybrid diversification operator based evolutionary approach towards tomographic image reconstruction the proposed algorithm introduces a new and efficient hybrid diversification operator hdo in the evolution cycle to improve the tomographic image reconstruction and diversity in the population by using simulated annealing sa and the modified form of decreasing law of mutation probability this evolutionary approach has been used for parallel ray transmission tomography with the head and lung phantoms the algorithm is designed to address the observation that the convergence of a genetic algorithm slows down as it evolves the hdo is shown to yield a higher image quality as compared with the filtered back projection fbp the multiscale wavelet transform the sa and the hybrid continuous genetic algorithm hcga techniques various crossover operators including uniform block and image row crossover operators have also been analyzed and the latter has been generally found to give better image quality the hdo is shown to yield improvements of up to 92 and 120 when compared with fbp in terms of psnr for 128 t 128 head and lung phantoms respectively

tomographic traveltime inversion using natural pixels traditionally in the problem of tomographic traveltime inversion the model is divided into a number of rectangular cells of constant slowness inversion consists of finding these constant values using the measured traveltimes the inversion process can demand a large computational effort if a high resolution result is desired we show how to use a different kind of parameterization of the model based on beam propagation paths this parameterization is obtained within the framework of reconstruction in hilbert spaces by minimizing the error between the true model and the estimated model the traveltimes are interpreted as the projections of the slowness along the beampaths although the actual beampaths are described by complicated spatial functions we simplify the computations by approximating these functions with functions of constant width and height i e fat rays which collectively form a basis set of natural pixels with a simple numerical example we demonstrate that the main advantage of this

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

time varying system identification and model validation using wavelets parametric identification of time varying tv systems is possible if each tv coefficient can be expanded onto a finite set of basis sequences the problem then becomes time invariant with respect to the parameters of the expansion the authors address the question of selecting this set of basis sequences they advocate the use of a wavelet basis because of its flexibility in capturing the signal s characteristics at different scales and discuss how to choose the optimal wavelet basis for a given system trajectory they also develop statistical tests to keep only the basis sequences that significantly contribute to the description of the system s time variation by formulating the problem as a regressor selection problem they apply an p test and an aic based approach for multiresolution analysis of tv systems the resulting algorithm can estimate tv ar or armax models and determine their orders they apply this algorithm to both synthetic and real speech data and compare it with the kalman filtering tv parameter estimator

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

fractional brownian motion a maximum likelihood estimator and its application to image texture fractals have been shown to be useful in characterizing texture in a variety of contexts use of this methodology normally involves measurement of a parameter h which is directly related to fractal dimension in this work the basic theory of fractional brownian motion is extended to the discrete case it is shown that the power spectral density of such a discrete process is only approximately proportional to f a instead of in direct proportion as in the continuous case an asymptotic cramer rao bound is derived for the variance of an estimate of h subsequently a maximum likelihood estimator mle is developed to estimate h it is shown that the variance of this estimator nearly achieves the minimum bound a generation algorithm for discrete fractional motion is presented and used to demonstrate the capabilities of the mle when the discrete fractional brownian process is contaminated with additive gaussian noise the results show that even at signal to noise ratios of 30 db significant errors in estimation of h can result when noise is present the mle is then applied to x ray images of the human calcaneus to demonstrate how the line to line formulation can be applied to the two dimensional case these results indicate that it has strong potential for quantifying texture

fractal feature analysis and classification in medical imaging following b b mandelbrot s fractal theory 1982 it was found that the fractal dimension could be obtained in medical images by the concept of fractional brownian motion an estimation concept for determination of the fractal dimension based upon the concept of fractional brownian motion is discussed two applications are found 1 classification 2 edge enhancement and detection for the purpose of classification a normalized fractional brownian motion feature vector is defined from this estimation concept it represented the normalized average absolute intensity difference of pixel pairs on a surface of different scales the feature vector uses relatively few data items to represent the statistical characteristics of the medial image surface and is invariant to linear intensity transformation for edge enhancement and detection application a transformed image is obtained by calculating the fractal dimension of each pixel over the whole medical image the fractal dimension value of each pixel is obtained by calculating the fractal dimension of 7 7 pixel block centered on this pixel

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

a multi level preconditioned krylov method for the efficient solution of algebraic tomographic reconstruction problems classical iterative methods for tomographic reconstruction include the class of algebraic reconstruction techniques art convergence of these stationary linear iterative methods is however notably slow in this paper we propose the use of krylov solvers for tomographic linear inversion problems these advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration causing them to be generally uncompetitive without the inclusion of a suitable preconditioner combining elements from standard multigrid mg solvers and the theory of wavelets a novel wavelet based multi level wmg preconditioner is introduced which is shown to significantly speed up krylov convergence the performance of the wmg preconditioned krylov method is analyzed through a spectral analysis and the approach is compared to existing methods like the classical simultaneous iterative reconstruction technique sirt and unpreconditioned krylov methods on a 2d tomographic benchmark problem numerical experiments are promising showing the method to be competitive with the classical algebraic reconstruction techniques in terms of convergence speed and overall performance cpu time as well as precision of the reconstruction

new fast smoothers for multiscale systems we consider the smoothing problem for multiscale stochastic systems based on the wavelet transform these models involve processes indexed by the nodes of a dyadic tree each level of the dyadic tree represents one scale or resolution of the process therefore moving upward on the tree divides the resolution by 2 whereas moving downward multiplies it by 2 the processes are built according to a recursion in scale from coarse to fine to which random details are added to operate the change in scale one must perform an interpolation this is achieved using the qmf pair of operators attached to a wavelet transform these models have proved to be of great value to capture textures or fractal like processes as well as to perform multiresolution sensor fusion an example of which is given here up to now however only subclasses of multiscale systems were amenable to fast algorithms and through different formalisms those relying on haar s wavelet and those involving only one of the two wavelet interpolators we provide here a unifying framework that handles any system based on orthogonal wavelets a smoothing theory is presented to define the field of fast algorithms for markov random fields and give intuition on how to design them this theory reveals the difficulties arising with general multiscale systems we then prove that orthogonality properties of wavelets are the gate to fastness

wavelet transform inversion with prior scale information the concept of scale is of central importance in all problems related to the interpretation of earth data inversion of geophysical data does not at present however allow the incorporation of scale dependent information directly this article presents an application of the wavelet transform to traveltime inversion where we are specifically concerned with the selection and application of a priori scale information in the wavelet domain to reconstruct tomographic models because of the nature of the problem the wavelet basis chosen for this example is the haar basis the results are compared with the classic damped least squares dls method which is a special case of the wavelet transform approach when all the scales are treated equally we show that with scale dependent prior information most of the artifacts created by the dls technique are diminished considerably

estimation of 1 f signals on the basis of curve fitting an algorithm is proposed for the identification of fractal signals with wavelet based models from corrupted data this algorithm is based on curve fitting and wavelet transformation it is proved that whenever spl gamma is greater than 0 the algorithm provides an almost consistent estimation moreover the estimated parameters are asymptotically gaussian distributed a mean square asymptotic convergence rate of the estimated parameters has also been established simulation results verify the efficiency of the proposed algorithm

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

a statistical test for the time constancy of scaling exponents a statistical test is described for determining if scaling exponents vary over time it is applicable to diverse scaling phenomena including long range dependence and exactly self similar processes in a uniform framework without the need for prior knowledge of the type in question it is based on the special properties of wavelet based estimates of the scaling exponent strongly motivating an idealized inference problem the equality or otherwise of means of independent gaussian variables with known variances a uniformly most powerful invariant test exists for this problem and is described a separate uniformly most powerful invariant test is also given for when the scaling exponent undergoes a level change the power functions of both tests are given explicitly and compared using simulation the effect in practice of deviations from the idealizations made of the statistical properties of the wavelet detail coefficients are analyzed and found to be small the tests inherit the significant robustness and computational advantages of the underlying wavelet based estimator a detailed methodology is given describing its use in practical situations the use and benefits of the test are illustrated on the bellcore ethernet data sets

wavelet methods for inverting the radon transform with noisy data because the radon transform is a smoothing transform any noise in the radon data becomes magnified when the inverse radon transform is applied among the methods used to deal with this problem is the wavelet vaguelette decomposition wvd coupled with wavelet shrinkage as introduced by donoho 1995 we extend several results of donoho and others here first we introduce a new sufficient condition on wavelets to generate a wvd for a general homogeneous operator whose class includes the radon transform we show that a variant of donoho s method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a besov norm as the smoothing functional we give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise corrupted data we conduct tomographic reconstruction computations that support the hypothesis that near optimal shrinkage parameters can be derived if one can estimate only two besov space parameters about an image f both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than kolaczyk s 1996 variant of donoho s method and the classical filtered backprojection method

critical fluctuations of sums of weakly dependent random vectors letsn be sums of iid random vectors taking values in a banach space andf be a smooth function we study the fluctuations ofsn under the transformed measurepn given byd pn d p exp nf sn n zn if degeneracy occurs then the projection ofsn onto the degenerate subspace properly centered and scaled converges to a non gaussian probability measure with the degenerate subspace as its support the projection ofsn onto the non degenerate subspace scaled with the usual order sqrt n converges to a gaussian probability measure with the non degenerate subspace as its support the two projective limits are in general dependent we apply this theory to the critical mean field heisenberg model and prove a central limit type theorem for the empirical measure of this model

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

a wiener wintner theorem for 1 f power spectra abstract wiener s generalized harmonic analysis gha provides a theory of harmonic analysis for subspaces of tempered functions not accessible to the l 1 l 2 and fourier series theories and it does it in a way that is usually more quantitative than that provided by the theory of distributions on the other hand gha does not yield an adequate spectral analysis of large classes of functions including nonstationary processes and in particular 1 f noise in this paper we adapt gha to deal with 1 f noise by extending the wiener wintner theorem to the case of 1 f power spectra

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

limit theorems in the area of large deviations for some dependent random variables on considere des hamiltoniens de systemes de spins hn et on obtient des theoremes de limite

on center subspace behavior in thin film equations the large time behavior of weak nonnegative solutions of the thin film equation tfe with absorption u t nabla cdot u n nabla delta u u p 1 u with parameters n in 0 3 and p 1 is studied the standard free boundary problem fbp with zero height zero contact angle and zero flux conditions at the interface and bounded compactly supported initial data is considered it is shown that there exists the critical absorption exponent p 0 1 n frac 4 n such that for p p 0 the asymptotic behavior of solutions u x t for t gg1 is represented by the well known source type solution of the pure tfe absorption u s x t t beta n f y y x t beta with the exponent beta frac 1 4 nn which is perturbed by a couple of ln t factors for n 1 this behavior is associated with the center subspace for the rescaled linearized thin film operator and is given by u x t sim nobreak t ln t beta n f x t beta ln t beta n 4 with beta frac 1 4 n where f y fr

statistical analysis of dwt coefficients of fgn processes using arfima p d q models abstract fractional gaussian noise fgn provides an important parametric representation for the data recorded from long memory processes also it has been well established in literature that the orthogonal wavelet transforms prove to be the optimal bases to represent the data as fgn or fbm fractional brownian motion models this paper highlights the statistical properties of discrete wavelet transform dwt coefficients in the wavelet expansion of fgn statistical analysis was carried out by analyzing the inter scale and intra scale correlations of the dwt coefficients for wavelets with varying vanishing moments two types of auto regressive moving average arma models were fit to the wavelet coefficients of fgn namely i arma p q and ii arfima p d q models the latter represents the arma models with fractional differencing using the akaike information criteria aic and the bayesian information criteria bic it has been shown that arfima models best represent the wavelet coefficients of fgn the above observation holds good when wavelets with increasing number of vanishing moments are used for obtaining dwt coefficients after estimating the optimal model and its parameters different properties pertaining to the inter scale and intra scale correlations were verified using these models

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

on the gaussian fluctuations of the critical curie weiss model in statistical mechanics it has been known for some time that the fluctuations of the curie weiss mean field model of ferromagnetism are non gaussian at the critical temperature here we establish the presence of a substantial gaussian element in the critical model by showing that the internal fluctuations i e the fluctuations within the fluctuating field are gaussian we show for instance that as the number n of sites increases to infinity the system with the appropriate space and spin scalings behaves asymptotically as a brownian motion with randomised drift and that the distribution of the drift has the non gaussian form familiar in this theory with the more conventional space scaling 1 n it is known that the system degenerates asymptotically to a straight line with random slope we prove here that the error from the straight line when appropriately amplified converges in distribution to a brownian bridge

the linear limit of the dipole problem for the thin film equation we investigate self similar solutions of the dipole problem for the one dimensional thin film equation on the half line x ge 0 we study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem the similarity solutions are generally of the second kind given by the solution of a nonlinear eigenvalue problem although there are some notable cases where first kind solutions also arise we examine the conserved quantities connected to these first kind solutions difficulties associated with the lack of a maximum principle and the non self adjointness of the fundamental linear problem are also considered seeking similarity solutions that include sign changes yields a surprisingly rich set of coexisting stable solutions for the intermediate asymptotics of this problem our results include analysis of limiting cases and comparisons with numerical computations

wavelet based reconstruction for limited angle x ray tomography the aim of x ray tomography is to reconstruct an unknown physical body from a collection of projection images when the projection images are only available from a limited angle of view the reconstruction problem is a severely ill posed inverse problem statistical inversion allows stable solution of the limited angle tomography problem by complementing the measurement data by a priori information in this work the unknown attenuation distribution inside the body is represented as a wavelet expansion and a besov space prior distribution together with positivity constraint is used the wavelet expansion is thresholded before reconstruction to reduce the dimension of the computational problem feasibility of the method is demonstrated by numerical examples using in vitro data from mammography and dental radiology

tomographic reconstruction and estimation based on multiscale natural pixel bases we use a natural pixel type representation of an object originally developed for incomplete data tomography problems to construct nearly orthonormal multiscale basis functions the nearly orthonormal behavior of the multiscale basis functions results in a system matrix relating the input the object coefficients and the output the projection data which is extremely sparse in addition the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system we exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well conditioned and sparse matrix this enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions in case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability map multiscale reconstruction estimates based on a certain self similar prior on the multiscale object coefficients the framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete irregular and noisy and in object feature recognition directly from projection data

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling

nonlinear multigrid algorithms for bayesian optical diffusion tomography optical diffusion tomography is a technique for imaging a highly scattering medium using measurements of transmitted modulated light reconstruction of the spatial distribution of the optical properties of the medium from such data is a difficult nonlinear inverse problem bayesian approaches are effective but are computationally expensive especially for three dimensional 3 d imaging this paper presents a general nonlinear multigrid optimization technique suitable for reducing the computational burden in a range of nonquadratic optimization problems this multigrid method is applied to compute the maximum a posteriori map estimate of the reconstructed image in the optical diffusion tomography problem the proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality