ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

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

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

coupled variational image decomposition and restoration model for blurred cartoon plus texture images with missing pixels in this paper we develop a decomposition model to restore blurred images with missing pixels our assumption is that the underlying image is the superposition of cartoon and texture components we use the total variation norm and its dual norm to regularize the cartoon and texture respectively we recommend an efficient numerical algorithm based on the splitting versions of augmented lagrangian method to solve the problem theoretically the existence of a minimizer to the energy function and the convergence of the algorithm are guaranteed in contrast to recently developed methods for deblurring images the proposed algorithm not only gives the restored image but also gives a decomposition of cartoon and texture parts these two parts can be further used in segmentation and inpainting problems numerical comparisons between this algorithm and some state of the art methods are also reported

fast cartoon texture image filters can images be decomposed into the sum of a geometric part and a textural part in a theoretical breakthrough y meyer oscillating patterns in image processing and nonlinear evolution equations providence ri american mathematical society 2001 proposed variational models that force the geometric part into the space of functions with bounded variation and the textural part into a space of oscillatory distributions meyer s models are simple minimization problems extending the famous total variation model however their numerical solution has proved challenging it is the object of a literature rich in variants and numerical attempts this paper starts with the linear model which reduces to a low pass high pass filter pair a simple conversion of the linear filter pair into a nonlinear filter pair involving the total variation is introduced this new proposed nonlinear filter pair retains both the essential features of meyer s models and the simplicity and rapidity of the linear model it depends upon only one transparent parameter the texture scale measured in pixel mesh comparative experiments show a better and faster separation of cartoon from texture one application is illustrated edge detection

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

a new barrier to the existence of moving kinks in frenkel kontorova lattices an explanation is offered for an observed lower bound on the wave speed of travelling kinks in frenkel kontorova lattices kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves phonons however they fail to exist even in this codimension one sense whenever there is more than one phonon branch in the dispersion relation inside such bands only quasi kinks with nondecaying oscillatory tails are possible the results are presented for a discrete sine gordon lattice with an onsite potential that has a tunable amount of anharmonicity novel numerical methods are used to trace kinks with topological charge q 1 and 2 in three parameters representing the propagation speed lattice discreteness and anharmonicity although none of the analysis is presented as rigorous mathematics numerical results suggest that the bound on allowable wavespeeds is sharp the results also explain why the vanishing at discrete values of anharmonicity of the peierls nabarro barrier between stationary kinks as discovered by savin et al does not lead to bifurcation of kinks with small wave speed 2003 elsevier b v all rights reserved

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

cartoon texture image decomposition using blockwise low rank texture characterization using a novel characterization of texture we propose an image decomposition technique that can effectively decomposes an image into its cartoon and texture components the characterization rests on our observation that the texture component enjoys a blockwise low rank nature with possible overlap and shear because texture in general is globally dissimilar but locally well patterned more specifically one can observe that any local block of the texture component consists of only a few individual patterns based on this premise we first introduce a new convex prior named the block nuclear norm bnn leading to a suitable characterization of the texture component we then formulate a cartoon texture decomposition model as a convex optimization problem where the simultaneous estimation of the cartoon and texture components from a given image or degraded observation is executed by minimizing the total variation and bnn in addition patterns of texture extending in different directions are extracted separately which is a special feature of the proposed model and of benefit to texture analysis and other applications furthermore the model can handle various types of degradation occurring in image processing including blur missing pixels with several types of noise by rewriting the problem via variable splitting the so called alternating direction method of multipliers becomes applicable resulting in an efficient algorithmic solution to the problem numerical examples illustrate that the proposed model is very selective to patterns of texture which makes it produce better results than state of the art decomposition models

desynchronization transitions in nonlinearly coupled phase oscillators abstract we consider the nonlinear extension of the kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter a bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed we demonstrate that for small ensembles it is typically mediated by stable cluster states that disappear with creation of heteroclinic cycles while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs

a novel variational model for image segmentation in this paper we propose a new variational model for image segmentation our model is inspired by the complex ginzburg landau model and the semi norm defined by us this new model can detect both the convex and concave parts of images moreover it can also detect non closed edges as well as quadruple junctions compared with other methods the initialization is completely automatic and the segmented images obtained by using our new model could keep fine structures and edges of the original images very effectively finally numerical results show the effectiveness of our model

encoding via conjugate symmetries of slow oscillations for globally coupled oscillators we study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators these slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise free dynamics in the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them moreover these transitions may occur between cluster states of the same symmetry or between cluster states with conjugate symmetries given by some rearrangement of the oscillators we consider the system of coupled phase oscillators studied by hansel et al phys rev e 48 3470 1993 in which one can observe slow noise driven oscillations that occur between two families of two cluster periodic states in the noise free case there is a robust attracting heteroclinic cycle connecting these families the two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry for n 5 oscillators one of the periodic orbits has one unstable direction and the other has two unstable directions examining the behavior on the unstable manifold for the two unstable directions we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits by applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors finally we show that similar behavior occurs in a system of phase energy oscillators that are a natural extension of the phase model to two dimensional oscillators we suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information

nonlinear wavelet image processing variational problems compression and noise removal through wavelet shrinkage this paper examines the relationship between wavelet based image processing algorithms and variational problems algorithms are derived as exact or approximate minimizers of variational problems in particular we show that wavelet shrinkage can be considered the exact minimizer of the following problem given an image f defined on a square i minimize over all g in the besov space b11 l1 i the functional f g l2 i 2 g b11 l1 i we use the theory of nonlinear wavelet image compression in l2 i to derive accurate error bounds for noise removal through wavelet shrinkage applied to images corrupted with i i d mean zero gaussian noise a new signal to noise ratio snr which we claim more accurately reflects the visual perception of noise in images arises in this derivation we present extensive computations that support the hypothesis that near optimal shrinkage parameters can be derived if one knows or can estimate only two parameters about an image f the largest for which f bq lq i 1 q 2 1 2 and the norm f bq lq i both theoretical and experimental results indicate that our choice of shrinkage parameters yields uniformly better results than donoho and johnstone s visushrink procedure an example suggests however that donoho and johnstone s 1994 1995 1996 sureshrink method which uses a different shrinkage parameter for each dyadic level achieves a lower error than our procedure

a total variation based approach to correcting surface coil magnetic resonance images abstract magnetic resonance images which are corrupted by noise and by smooth modulations are corrected using a variational formulation incorporating a total variation like penalty for the image and a high order penalty for the modulation the optimality system is derived and numerically discretized the cost functional used is non convex but it possesses a bilinear structure which allows the ambiguity among solutions to be resolved technically by regularization and practically by normalizing the maximum value of the modulation since the cost is convex in each single argument convex analysis is used to formulate the optimality condition for the image in terms of a primal dual system to solve the optimality system a nonlinear gauss seidel outer iteration is used in which the cost is minimized with respect to one variable after the other using an inner generalized newton iteration favorable computational results are shown for artificial phantoms as well as for realistic magnetic resonance images reported computational times demonstrate the feasibility of the approach in practice

shearlet based total variation diffusion for denoising we propose a shearlet formulation of the total variation tv method for denoising images shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization common approaches in combining wavelet like representations such as curvelets with tv or diffusion methods aim at reducing gibbs type artifacts after obtaining a nearly optimal estimate we show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain we also analyze the performance of a shearlet based diffusion method numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform furthermore the shearlet tv scheme requires far fewer iterations than similar competitors

numerical computation of travelling breathers in klein gordon chains abstract we numerically study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time t these solutions generalize the concept of solitary waves for which p 1 in the case of even on site potentials the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently g james y sire travelling breathers with exponentially small tails in a chain of nonlinear oscillators commun math phys 2005 in press available online at http www springerlink com the tail being exponentially small with respect to the central oscillation size in this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials we find that klein gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail we provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations in addition we numerically observe the existence of these solutions in the case of non even potentials

generalized flooding and multicue pde based image segmentation image segmentation remains an important but hard to solve problem since it appears to be application dependent with usually no a priori information available regarding the image structure moreover the increasing demands of image analysis tasks in terms of segmentation results quality introduce the necessity of employing multiple cues for improving image segmentation results in this paper we attempt to incorporate cues such as intensity contrast region size and texture in the segmentation procedure and derive improved results compared to using individual cues separately we emphasize on the overall segmentation procedure and we propose efficient simplification operators and feature extraction schemes capable of quantifying important characteristics like geometrical complexity rate of change in local contrast variations and orientation that eventually favor the final segmentation result based on the well known morphological paradigm of watershed transform segmentation which exploits intensity contrast and region size criteria we investigate its partial differential equation pde formulation and we extend it in order to satisfy various flooding criteria thus making it applicable to a wider range of images going a step further we introduce a segmentation scheme that couples contrast criteria in flooding with texture information the modeling of the proposed scheme is done via pdes and the efficient incorporation of the available contrast and texture information is done by selecting an appropriate cartoon texture image decomposition scheme the proposed coupled segmentation scheme is driven by two separate image components artoon u for contrast information and texture component the performance of the proposed segmentation scheme is demonstrated through a complete set of experimental results and substantiated using quantitative and qualitative criteria

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

moving kinks and nanopterons in the nonlinear klein gordon lattice we study moving topological solitons kinks and antikinks in the nonlinear klein gordon chain these solitons are shown to exist with both monotonic non oscillating and oscillating asymptotics tails using the pseudo spectral method the anti kink solutions with oscillating background so called nanopterons are found as travelling waves of permanent profile propagating with constant velocity each of these solutions may be considered as a bound state of an anti kink with a background nonlinear periodic wave so that the wave pushes the anti kink over the peierls nabarro barrier the stability of these bound states is confirmed numerically travelling wave solutions of permanent profile are shown to exist depending on the convexity of the on site substrate potential the set of velocities at which the anti kinks with monotonic asymptotics propagate freely is calculated we also find moving non oscillating anti kink profiles with higher topological charges each of which appears to be the bound state of anti kinks with lower topological charge jq jd 1 2000 elsevier science b v all rights reserved

image zoom completion we consider the problem of recovering a high resolution image from a pair consisting of a complete low resolution image and a high resolution but incomplete one we refer to this task as the image zoom completion problem after discussing possible contexts in which this setting may arise we introduce a nonlocal regularization strategy giving full details concerning the numerical optimization of the corresponding energy and discussing its benefits and shortcomings we also derive two total variation based algorithms and evaluate the performance of the proposed methods on a set of natural and textured images we compare the results and get with those obtained with two recent state of the art single image super resolution algorithms

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

moving discrete breathers abstract we give definitions for different types of moving spatially localized objects in discrete nonlinear lattices we derive general analytical relations connecting frequency velocity and localization length of moving discrete breathers and kinks in nonlinear one dimensional lattices then we propose a new numerical algorithm to find these solutions

weighted and extended total variation for image restoration and decomposition in various information processing tasks obtaining regularized versions of a noisy or corrupted image data is often a prerequisite for successful use of classical image analysis algorithms image restoration and decomposition methods need to be robust if they are to be useful in practice in particular this property has to be verified in engineering and scientific applications by robustness we mean that the performance of an algorithm should not be affected significantly by small deviations from the assumed model in image processing total variation tv is a powerful tool to increase robustness in this paper we define several concepts that are useful in robust restoration and robust decomposition we propose two extended total variation models weighted total variation wtv and extended total variation etv we state generic approaches the idea is to replace the tv penalty term with more general terms the motivation is to increase the robustness of rof rudin osher fatemi model and to prevent the staircasing effect due to this method moreover rewriting the non convex sublinear regularizing terms as wtv we provide a new approach to perform minimization via the well known chambolle s algorithm the implementation is then more straightforward than the half quadratic algorithm the behavior of image decomposition methods is also a challenging problem which is closely related to anisotropic diffusion etv leads to an anisotropic decomposition close to edges improving the robustness it allows to respect desired geometric properties during the restoration and to control more precisely the regularization process we also discuss why compression algorithms can be an objective method to evaluate the image decomposition quality

bifurcations in phase oscillator networks with a central element abstract a system of phase oscillators with identical natural frequencies and the star like architecture of connections is considered interaction functions are described by two terms of fourier expansion bifurcation analysis of small systems containing 3 or 4 oscillators has been performed the results are summarized in bifurcation diagrams that provide a full description of the boundaries between regions with different dynamics and the types of bifurcations that lead to the changes in the topology of phase space the bifurcations include changes of fixed point stability and formation destruction of limit and heteroclinic cycles for the system with 4 oscillators chaotic behaviour has been investigated the results can be useful to control system dynamics through an appropriate choice and variation of parameter values the generalization of the results to the systems with an arbitrary number of oscillators and application of the results in computational neuroscience are discussed

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

synchronization and multi frequency oscillations in the low dimensional chain of the self oscillators abstract the problem of growing complexity of the dynamics of the coupled phase oscillators as the number of oscillators in the chain increases is considered the organization of the parameter space parameter of the frequency detuning between the second and the first oscillators versus parameter of dissipative coupling is discussed the regions of complete synchronization quasi periodic regimes of different dimensions and chaos are identified we discuss transformation of the domains of different dynamics as the number of oscillators grows we use the method of charts of lyapunov exponents and modification of the method of the chart of dynamical regimes to visualize two frequency regimes of different type limits of applicability of the quasi harmonic approximation and the features of the dynamics of the original system which are not described by the approximate phase equations are discussed for the case of three coupled oscillators

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

dual norms and image decomposition models following a recent work by y meyer decomposition models into a geometrical component and a textured component have recently been proposed in image processing in such approaches negative sobolev norms have seemed to be useful to modelize oscillating patterns in this paper we compare the properties of various norms that are dual of sobolev or besov norms we then propose a decomposition model which splits an image into three components a first one containing the structure of the image a second one the texture of the image and a third one the noise our decomposition model relies on the use of three different semi norms the total variation for the geometrical component a negative sobolev norm for the texture and a negative besov norm for the noise we illustrate our study with numerical examples

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

a global sparse gradient based coupled system for image denoising abstract we consider a coupled system for image denoising which consists of an anisotropic diffusion equation and a global sparse gradient model gsg the global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly the anisotropic diffusion process is guided by the edge map furthermore we prove the existence and uniqueness of the solutions of the coupled system with dirichlet initial boundary value problem finally comparing with the edge indicators based on local methods experimental results demonstrate that our proposed method has the state of the art performance among the pde based anisotropic diffusion approaches both in objective measurement and visual evaluation

effective hamiltonian for travelling discrete breathers hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers however solutions which look like moving discrete breathers for some time are not difficult to observe in numerics in this paper we propose an abstract framework for the description of approximate travelling discrete breathers in hamiltonian chains of oscillators the method is based on the construction of an effective hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers error estimate on the approximate dynamics is also studied the concept of the peierls nabarro barrier can be made clear in this framework we illustrate the method with two simple examples namely the salerno model which interpolates between the ablowitz ladik lattice and the discrete nonlinear schrodinger system and the fermi pasta ulam chain

nonlinear regularized reaction diffusion filters for denoising of images with textures denoising is always a challenging problem in natural imaging and geophysical data processing in this paper we consider the denoising of texture images using a nonlinear reaction diffusion equation and directional wavelet frames in our model a curvelet shrinkage is used for regularization of the diffusion process to preserve important features in the diffusion smoothing and a wave atom shrinkage is used as the reaction in order to preserve and enhance interesting oriented textures we derive a digital reaction diffusion filter that lives on graphs and show convergence of the corresponding iteration process experimental results and comparisons show very good performance of the proposed model for texture preserving denoising

bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators abstract we study the bifurcation and dynamical behaviour of the system of n globally coupled identical phase oscillators introduced by hansel mato and meunier in the cases n 3 and n 4 this model has been found to exhibit robust slow switching oscillations that are caused by the presence of robust heteroclinic attractors this paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors we consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function these bifurcations preserve the permutation symmetry of the system we then investigate the implications of these bifurcations for the sensitivity to detuning i e the size of the smallest perturbations that give rise to loss of frequency locking for n 3 we find three types of heteroclinic bifurcation that are codimension one with symmetry on varying two parameters in the coupling function we find three curves giving a an s 3 transcritical homoclinic bifurcation b a saddle node heteroclinic bifurcation and c a z 3 heteroclinic bifurcation we also identify several global bifurcations with symmetry that organize the bifurcation diagram these are codimension two with symmetry for n 4 oscillators we determine many but not all codimension one bifurcations with symmetry including those that lead to a robust heteroclinic cycle a robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region furthermore we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system while for other parts of the phase plane it can coexist with stable limit cycles we finish with a discussion of bifurcations that appear for this coupling function and general n as well as for more general coupling functions

intermittent switching for three repulsively coupled oscillators we study intermittent switching behaviors in a system with three identical oscillators coupled diffusively and repulsively to clarify a bifurcation scenario which generates such intermittent switching behaviors we use the stuart landau oscillator which is a general form of hopf bifurcation and can describe both cases limit cycle and inactive i e non self oscillatory cases from a numerical study of the bifurcation structure two different routes to chaos which has s 3 symmetry were found one is the sudden appearance of chaos as pomeau manneville intermittency which is found for the inactive case in this case a trajectory shows switching among three mutually symmetric tori when a parameter exceeds critical value the other route which appears for the limit cycle case consists of two parts first chaos with lower symmetry appears through period doubling and after the two successive attractor merging crises chaos which has s 3 symmetry appears at each crisis the attractor changes its symmetry

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

minimizing the total variation under a general convex constraint for image restoration we present a general framework for image restoration despite its simplicity certain variational and certain wavelet approaches can be formulated within this framework this permits the construction of a natural model with only one parameter which has the advantages of both approaches we give a mathematical analysis of this model describe our algorithm and illustrate this by some experiments

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

polynomial normal forms with exponentially small remainder for analytic vector fields a key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms invented by poincare which gives simple forms to which a vector field can be reduced close to the equilibrium in the class of formal vector valued vector fields the problem can be easily solved whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order this paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in r m more precisely we prove that for any vector field in r m admitting the origin as a fixed point with a semi simple linearization the order of the normal form can be optimized so that the remainder is exponentially small we also give several examples of non semi simple linearization for which this result is still true

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

an effective method for solving nonlinear equations and its application the linearized partial differential equation from the nonlinear partial differential equation which was proposed by rudin osher and fatemi l i rudin s osher and e fatemi nonlinear total variation based noise removal algorithms for solving image decomposition was introduced by chambolle a chambolle an algorithm for total variation minimization and applications and r acar and c r vogel r acar and c r vogel analysis of bounded variation penalty methods for ill posed problems in this paper we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works

on a class of ill posed minimization problems in image processing in this paper we show that minimization problems involving sublinear regularizing terms are ill posed in general although numerical experiments in image processing give very good results the energies studied here are inspired by image restoration and image decomposition rewriting the nonconvex sublinear regularizing terms as weighted total variations we give a new approach to perform minimization via the well known chambolle s algorithm the approach developed here provides an alternative to the well known half quadratic minimization one

simultaneous higher order optical flow estimation and decomposition we study the estimation and decomposition of optical flows from highly nonrigid motions to this end recent methods from image decomposition into structural and textural parts are combined with variational optical flow estimation the approaches we suggest amount to minimizing discrete convex functionals using second order cone programming higher order regularization is necessary in order to accurately recover important flow structure like vortices and to incorporate key physical properties such as vanishing divergence for proper discretization we apply the finite mimetic difference method which preserves the identities fulfilled by the continuous differential operators numerical examples demonstrate the feasibility of the complex approaches

slow switching in a population of delayed pulse coupled oscillators we show that peculiar collective dynamics called slow switching arises in a population of leaky integrate and fire oscillators with delayed all to all pulse couplings by considering the stability of cluster states and symmetry possessed by our model we argue that saddle connections between a pair of the two cluster states are formed under general conditions slow switching appears as a result of the system s approach to the saddle connections it is also argued that such saddle connections are easy to arise near the bifurcation point where the state of perfect synchrony loses stability we develop an asymptotic theory to reduce the model into a simpler form with which an analytical study of the cluster states becomes possible

a multiscale image segmentation method this paper presents a novel image segmentation framework that combines image segmentation and feature extraction into a unified model the proposed model consists of two parts the segmentation part and the multiscale decomposition part in the model the segmentation part relies on the image intensities in the regions of interest while the multiscale decomposition part depends on the features in different scales the multiscale decomposition facilitates the process of segmentation since the region of interest can be easily detected from a proper scale the total variation projection regularization tvpr is used to preserve geometric shape of the segmented regions according to the geometric significance of tvpr parameters an adaptive tvpr parameters selection method is presented and edges of different region can be well preserved the proposed method is able to deal with intensity inhomogeneities and mixed noises often occurred in real world images which present challenges in image segmentation numerical examples on synthetic and real images are given to demonstrate the effectiveness of the proposed method this paper proposes a novel image segmentation framework a multiscale image segmentation method is presented within our framework total variation projection regularization tvpr is used to the proposed model we present an adaptive tvpr parameters selection method for image segmentation the experimental results show the effectiveness of the proposed method

variational models for image colorization via chromaticity and brightness decomposition colorization refers to an image processing task which recovers color in grayscale images when only small regions with color are given we propose a couple of variational models using chromaticity color components to colorize black and white images we first consider total variation minimizing tv colorization which is an extension from tv inpainting to color using chromaticity model second we further modify our model to weighted harmonic maps for colorization this model adds edge information from the brightness data while it reconstructs smooth color values for each homogeneous region we introduce penalized versions of the variational models we analyze their convergence properties and we present numerical results including extension to texture colorization

discrete breathers in two dimensional anisotropic nonlinear schrodinger lattices we study the structure and stability of discrete breathers both pinned and mobile in two dimensional nonlinear anisotropic schrodinger lattices starting from a set of identical one dimensional systems we develop the continuation of the localized pulses from the weakly coupled regime strongly anisotropic to the homogeneous one isotropic mobile discrete breathers are seen to be a superposition of a localized mobile core and an extended background of two dimensional nonlinear plane waves this structure is in agreement with previous results on one dimensional breather mobility the study of the stability of both pinned and mobile solutions is performed using standard floquet analysis regimes of quasi collapse are found for both types of solutions while another kind of instability responsible for the discrete breather fission is found for mobile solutions the development of such instabilities is studied examining typical trajectories on the unstable nonlinear manifold

a global minimization hybrid active contour model with applications to oil spill images active contour model is popularly and widely used in the field of image segmentation which is based on superior theoretical properties and efficient numerical methods nevertheless one of the prominent disadvantages of this kind of model is the existence of local minima in its functional energy in this paper we propose a novel global minimization hybrid active contour model this model effectively integrates the edge information the local region information and the global region information of the image which is relatively sufficient to extract the object boundaries furthermore we introduce an efficient and fast numerical approach to globally minimize the proposed model which is through a dual formulation of the minimization problem and easy to implement the proposed model is robust enough to the initial condition and does not need to initialize the contour in a distance function and re initialize it periodically during the evolution process specially we applied the proposed model to segment oil spill images in which there usually exist the noise blurry boundaries and intensity inhomogeneity compared with the state of the art models experiment results demonstrate the performance and effectiveness of the proposed model with applications to synthetical and real images especially for oil spill images

variational denoising of partly textured images by spatially varying constraints denoising algorithms based on gradient dependent regularizers such as nonlinear diffusion processes and total variation denoising modify images towards piecewise constant functions although edge sharpness and location is well preserved important information encoded in image features like textures or certain details is often compromised in the process of denoising we propose a mechanism that better preserves fine scale features in such denoising processes a basic pyramidal structure texture decomposition of images is presented and analyzed a first level of this pyramid is used to isolate the noise and the relevant texture components in order to compute spatially varying constraints based on local variance measures a variational formulation with a spatially varying fidelity term controls the extent of denoising over image regions our results show visual improvement as well as an increase in the signal to noise ratio over scalar fidelity term processes this type of processing can be used for a variety of tasks in partial differential equation based image processing and computer vision and is stable and meaningful from a mathematical viewpoint

constants of motion for superconducting josephson arrays abstract we show that series arrays of n identical overdamped josephson junctions have extremely degenerate dynamics in particular we prove that such arrays have n 3 constants of motion for all n 3 the analysis is based on a coordinate transformation that reduces the governing equations to an n 3 parameter family of low dimensional systems in the weak coupling limit the reduced equations can be analyzed completely either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations depending on a certain parameter value at the transitional value of this parameter the system becomes completely integrable then the phase space is foliated by invariant two dimensional tori for any n 3 the infinite n limit of the system is an integro partial differential equation with rigorously low dimensional dynamics it supports solitons in the integrable case and chaotic waves in the non integrable case

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