aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

conditional scores and optimal scores for generalized linear measurement error models abstract this paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error for the functional case i e when the explanatory vectors are fixed constants unbiased score functions are obtained by conditioning on certain sufficient statistics this work generalizes results obtained for logistic regression in the case that the explanatory vectors are independent and identically distributed with unknown distribtuion efficient score functions are obtained using the theory developed in begun et al 1983 keywords conditional score function efficient score function functional model generalized linear model measurement error structural model

understanding optimising and extending data compression with anisotropic diffusion galic et al journal of mathematical imaging and vision 31 255 269 2008 have shown that compression based on edge enhancing anisotropic diffusion eed can outperform the quality of jpeg for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation however the reasons for the good performance of eed remained unclear and they could not outperform the more advanced jpeg 2000 the goals of the present paper are threefold firstly we investigate the compression qualities of various partial differential equations this sheds light on the favourable properties of eed in the context of image compression secondly we demonstrate that it is even possible to beat the quality of jpeg 2000 with eed if one uses specific subdivisions on rectangles and several important optimisations these amendments include improved entropy coding brightness and diffusivity optimisation and interpolation swapping thirdly we demonstrate how to extend our approach to 3 d and shape data experiments on classical test images and 3 d medical data illustrate the high potential of our approach

existence of heteroclinic and homoclinic orbits in two different chaotic dynamical systems abstract this paper presents the existence of si lnikov orbits in two different chaotic systems belong to the class of lorenz systems more exactly in the lu system and in the zhou s system both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z axis by using the undetermined coefficient method the existence of the homoclinic orbit for the zhou s system has been proven also by using the undetermined coefficient method as a result the si lnikov criterion along with some technical conditions guarantees that lu and zhou s systems have both smale horseshoes and horseshoe type of chaos moreover the geometric structures of attractors are determined by these heteroclinic orbits

a deterministic control based approach motion by curvature the level set formulation of motion by mean curvature is a degenerate parabolic equation we show that its solution can be interpreted as the value function of a deterministic two person game more precisely we give a family of discrete time two person games whose value functions converge in the continuous time limit to the solution of the motion by curvature pde for a convex domain the boundary s first arrival time solves a degenerate elliptic equation this corresponds in our game theoretic setting to a minimum exit time problem for a nonconvex domain the two person game still makes sense we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the positive part of the curvature these results are unexpected because the value function of a deterministic control problem is normally the solution of a first order hamilton jacobi equation our situation is different because the usual first order calculation is singular 2005 wiley periodicals inc

solving integral and differential equations by the aid of non uniform haar wavelets a modification of the haar wavelet method for which the stepsize of the argument is variable is proposed to establish the efficiency of the method three test problems for which exact solution is known are considered computer simulations show clear preference of the suggested method compared with the haar wavelet method of a constant stepsize

on an empirical bayes test for a normal mean we exhibit an empirical bayes test n for the normal mean testing problem using a linear error loss under the condition that the critical point of a bayes test is within some known compact interval n is shown to be asymptotically optimal and its associated regret bayes risk converges to zero at a rate o n 1 ln n 1 5 where n is the number of past experiences available when the current component decision problem is considered under the same condition this rate is faster than the optimal rate of convergence claimed by karunamuni

nonlinear adaptive wavelet control using constructive wavelet networks in this paper an adaptive wavelet network based control approach is proposed for highly nonlinear uncertain dynamical systems wavelet network as a kind of universal approximator has two novel properties orthonormality and multiresolution the orthonormal property ensures that adding a new resolution new wavelets does not affect the existing wavelet network that may have been well tuned in the sequel the online adjustment of the structure of the nonlinear adaptive wavelet controller awc can be done in a constructive manner by gradually increasing the network resolution the multiresolution property on the other hand assures a guaranteed improvement of the approximation precision when a new resolution is added in real life problems we are unable to know the adequate size of a network either a neural network nn or a wavelet network to produce the required approximation precision by virtue of the novel wavelet network properties a coarse or very simple structure can be selected first if the system fails to converge after the elapse of a dwell time a new wavelet resolution is considered to be necessary and added directly in this manner the awc can be easily constructed and tuned from the coarse to finer levels until the performance requirement is satisfied the trial and error way of selecting the network structure which may lead to either an inadequate or a highly redundant structure can be avoided in this paper the proposed adaptive wavelet network is applied first to a class of nonlinear dynamical systems with a partially known model and an affine in input structure then the adaptive wavelet network is applied to a class of nonlinear nonaffine dynamical systems

structurally dynamic wavelet networks for adaptive control of robotic systems the practical applicability of recently developed adaptive neurocontrol algorithms for poorly modelled robotic systems depends crucially upon the accuracy and efficiency of the neural network used

a deterministic control based approach to fully nonlinear parabolic and elliptic equations we show that a broad class of fully nonlinear second order parabolic or elliptic pde s can be realized as the hamilton jacobi bellman equations of deterministic twoperson games more precisely given the pde we identify a deterministic discretetime two person game whose value function converges in the continuous time limit to the viscosity solution of the desired equation our game is roughly speaking a deterministic analogue of the stochastic representation recently introduced by cheridito soner touzi and victoir comm pure appl math 60 2006 1081 1110 in the parabolic setting with no u dependence it amounts to a semidiscrete numerical scheme whose timestep is a min max our result is interesting because the usual control based interpretations of second order pde s involve stochastic rather than deterministic control

a morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets this paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach an axiomatic approach of image processing and the mean curvature partial differential equation pde are briefly presented then the properties of the proposed scheme are studied in particular consistency and convergence are proved the applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented

enhancement of morphological snake based segmentation by imparting image attachment through scale space continuity in this paper we propose a new multi scale morphological approach to curve evolution useful for object extraction through segmentation the homogenous image structures that characterize the segmentation process are edges and terminations normally the conventional morphological snake ms technique employs morphological binary level set operators for realizing forces these operations handle definite components of the pde partial differential equation used for modeling the dynamic system the proposed model can segment with reasonably high level of accuracy and efficiency while ensuring smooth segmentation at object boundaries with scale space continuity application of discrete image force in ms is a per pixel decision based on the sign of image force pde component in the continuous domain however the intensity of the image force pde component is the primary factor for snake evolution in our model we embed scale space continuity into the morphological operators dictated by ms in order to realize the image force both in intensity and direction thus our model confirms to the speed agility and robustness of morphological snakes with regard to segmentation while ensuring enhanced efficiency of segmentation under noise we have rated the performance both on qualitative and quantitative basis against benchmark results on a set of 2d gray scale real images both in absence and presence of noise a comparative study has also been carried among our method ms geodesic active contour gac and distance regularized level set evolution drlse highlightsmulti scale morphological approach to curve evolution is suggested for segmentation we embed scale space continuity in morphological snakes morphological operators ensure fast curve evolution scale space continuity ensures smoother segmentation near edges extraction improvement near noisy pixels is observed with scale space continuity

a morphological approach to curvature based evolution of curves and surfaces we introduce new results connecting differential and morphological operators that provide a formal and theoretically grounded approach for stable and fast contour evolution contour evolution algorithms have been extensively used for boundary detection and tracking in computer vision the standard solution based on partial differential equations and level sets requires the use of numerical methods of integration that are costly computationally and may have stability issues we present a morphological approach to contour evolution based on a new curvature morphological operator valid for surfaces of any dimension we approximate the numerical solution of the curve evolution pde by the successive application of a set of morphological operators defined on a binary level set and with equivalent infinitesimal behavior these operators are very fast do not suffer numerical stability issues and do not degrade the level set function so there is no need to reinitialize it moreover their implementation is much easier since they do not require the use of sophisticated numerical algorithms we validate the approach providing a morphological implementation of the geodesic active contours the active contours without borders and turbopixels in the experiments conducted the morphological implementations converge to solutions equivalent to those achieved by traditional numerical solutions but with significant gains in simplicity speed and stability

cyclic schemes for pde based image analysis we investigate a class of efficient numerical algorithms for many partial differential equations pdes in image analysis they are applicable to parabolic or elliptic pdes that have bounded coefficients and lead to space discretisations with symmetric matrices our schemes are easy to implement and well suited for parallel implementations on gpus since they are based on the explicit diffusion scheme in the parabolic case and the jacobi method in the elliptic case by supplementing these methods with cyclically varying time step sizes or relaxation parameters we achieve efficiency gains of several orders of magnitude we call the resulting algorithms fast explicit diffusion fed and fast jacobi fj methods to achieve a good compromise between efficiency and accuracy we show that one should use parameter cycles that result from factorisations of box filters for these cycles we establish stability results in the euclidean norm our schemes perform favourably in a number of applications including isotropic nonlinear diffusion filters with widely varying diffusivities as well as anisotropic diffusion methods for image filtering inpainting and regularisation in computer vision moreover they are equally suited for higher dimensional problems as well as higher order pdes and they can also be interpreted as efficient first order methods for smooth optimisation problems

adaptive control of linear time invariant systems via a wavelet network and applications to control lorenz chaos abstract compactly supported orthogonal wavelets have certain properties that are useful for controller design in this paper we explore the mechanism of a wavelet controller by integrating the controller with linear time invariant systems lti a necessary condition for effective control is that the compact support of the wavelet network covers the state space where the state trajectories stay closed form bounds on the design parameters of the wavelet controller are derived which guarantee asymptotic stability of wavelet controlled lti systems the same wavelet controller is then applied to the lorenz equations the control objective is to stabilize the lorenz system well into its normally chaotic region at one of its equilibria

wavelet based system identification for nonlinear control compactly supported orthogonal wavelets have certain properties that are useful for system identification and learning control drawing on the rich theory of wavelets we propose a system identification scheme based on orthogonal wavelets better accuracy of estimation can be obtained by adding more terms to the wavelet based identifier and these terms do not alter the coefficients of the existing terms these terms can be selectively added depending upon the region of interest for example we may require more terms in regions where the identified functions vary rapidly we illustrate the concepts by applying the procedure to determine a speed controller for an electric vehicle

predicting chaotic time series with wavelet networks a new technique wavelet network is introduced to predict chaotic time series by using this technique firstly we make accurate short term predictions of the time series from chaotic attractors secondly we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time finally we predict chaotic attractors by making long term predictions based on remarkably few data points where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors

a semi lagrangian scheme for the curve shortening flow in codimension 2 we consider the model problem where a curve in r 3 moves according to the mean curvature flow the curve shortening flow we construct a semi lagrangian scheme based on the feynman kac representation formula of the solutions of the related level set geometric equation the first step is to obtain an approximation of the associated codimension 1 problem formulated by ambrosio and soner where the squared distance from the initial curve is used as initial condition since the e sublevel of this evolution contains the curve the next step is to extract the curve itself by following an optimal trajectory inside each e sublevel we show that this procedure is robust and accurate as long as the fattening phenomenon does not occur moreover it can still single out the physically meaningful solution when it occurs

optimal rates of convergence of empirical bayes tests for the continuous one parameter exponential family one parameter exponential family is studied previous results on this problem construct empirical bayes tests via kernel density estimates they also obtain upper bounds for the unconditional regret at some prior distribution in this paper we discuss the general question of how difficult the above empirical bayes problem is and why empirical bayes rules based on kernel density estimates are useful asymptotic minimax type lower bounds are obtained for the unconditional regret and empirical bayes rules based on kernel density estimates are shown to possess a certain optimal asymptotic minimax property 1 introduction we investigate the following component decision problem let 0 g and consider testing ho 0 0 based on an observation x with x given 0 being distributed according to the exponential family

a convolution thresholding approximation of generalized curvature flows we construct a convolution thresholding approximation scheme for the geometric surface evolution in the case when the velocity of the surface at each point is a given function of the mean curvature conditions for the monotonicity of the scheme are found and the convergence of the approximations to the corresponding viscosity solution is proved we also discuss some aspects of the numerical implementation of such schemes and present several numerical results

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

nonparametric regression with errors in variables the effect of errors in variables in nonparametric regression estimation is examined to account for errors in covariates deconvolution is involved in the construction of a new class of kernel estimators it is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution in fact there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth it is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate which is rich enough for many practical applications furthermore to achieve optimality we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators

adaptive system identification based on generalized wavelet decomposition abstract in this paper a new adaptive structure based on wavelet packets filter bank wpfb is proposed the system identification of a finite impulse response fir is investigated by utilizing the presented adaptive scheme when the input signal is lowpass limited the recursive least squares rls estimation of the adaptive filter weights becomes ill conditioned in order to attain the stabilized convergence of the adaptive filters a regularization parameter is introduced for each wavelet packet to minimize the mean squares error mse of the adaptive filter weights the analytical expression of the optimal regularization parameter and the optimal initial condition of the rls algorithm is given the effectiveness of the proposed approach is demonstrated through simulation examples

solving the 2 d heat equations using wavelet galerkin method with variable time step a wavelet galerkin method to solve nonhomogenous 2 d heat equations in finite rectangular domains is presented integrals involving nonhomogenity terms with scaling function are evaluated based on available connection coefficients variable time step technique is used to delay the error blow up hence obtaining better results for solutions that reach steady state

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem