orthogonal bandelet bases for geometric images approximation this paper introduces orthogonal bandelet bases to approximate images having some geometrical regularity these bandelet bases are computed by applying parametrized alpert transform operators over an orthogonal wavelet basis these bandeletization operators depend upon a multiscale geometric ow that is adapted to the image at each wavelet scale this bandelet construction has a hierarchical structure over wavelet coef cients taking advantage of existing regularity among these coef cients it is proved that c images having singularities along calpha curves are approximated in a best orthogonal bandelet basis with an optimal asymptotic error decay fast algorithms and compression applications are described

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

quantization of filter bank frame expansions through moving horizon optimization this paper describes a novel approach to quantization in oversampled filter banks the new technique is based on moving horizon optimization does not rely on an additive white noise quantization model and allows stability to be explicitly enforced in the associated nonlinear feedback loop moreover the quantization structure proposed here includes sigmadelta and linear predictive subband quantizers as a special case and in general outperforms them

the finite element approximation of hamilton jacobi bellman equations abstract this paper deals with the finite element approximation of hamilton jacobi bellman equations we establish a convergence and a quasi optimal l error estimate involving a weakly coupled systems of quasi variational inequalities for the solution of which an interative scheme of monotone kind is introduced and analyzed

decision feedback equalization via separating hyperplanes the design of finite length decision feedback equalization dfe forward and feedback filters under the assumption of genie aided feedback and independent and equally likely transmitted symbols is considered it is shown that the problem of determining dfe filters that minimize the probability of symbol error at high signal to noise ratio snr is equivalent to finding the hyperplane that maximally separates two given finite groups of points in a finite dimensional euclidean space the latter task can be formulated as a quadratic program which is readily solved numerically it is also shown that the problem of finding finite length dfe filters that minimize the probability of symbol error at any snr subject to a certain separation condition is a convex optimization problem the case where the transmitted data is coded using a runlength limited code is also investigated examples show that this criterion yields a performance that is better than zero forcing dfe on severely distorted channels at high snr

block decision feedback equalization a natural generalization of the conventional decision feedback equalizer dfe based on block processing and maximum a posteriori decisions is presented this block dfe is indexed by two parameters depending on the block length p and the number of decisions q

delayed decision feedback sequence estimation the authors delayed decision feedback sequence estimation ddfse for uncoded pam signals is considered estimates on the performance of the algorithm are given and simulation results are provided for several examples a more general form of ddfse applicable to coded modulation systems is also presented as an example detection of trellis coded qpsk quadrature phase shift keyed signals over intersymbol interference channels is discussed

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

truncation length for viterbi decoding a bound is derived and analyzed for the bit error rate ber of a viterbi decoder with survivor truncation estimates of the snr signal to noise ratio loss on the awgn additive white gaussian noise channel due to truncation are obtained for convolutional codes larger truncation lengths are required than the smallest value that does not effectively decrease the code s free distance especially at low e sub b n sub 0

sparse geometric image representations with bandelets this paper introduces a new class of bases called bandelet bases which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow this geometric flow indicates directions in which the image gray levels have regular variations the image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm bandelet bases lead to optimal approximation rates for geometrically regular images for image compression and noise removal applications the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion comparisons are made with wavelet image compression and noise removal algorithms

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

the contourlet transform an efficient directional multiresolution image representation the limitations of commonly used separable extensions of one dimensional transforms such as the fourier and wavelet transforms in capturing the geometry of image edges are well known in this paper we pursue a true two dimensional transform that can capture the intrinsic geometrical structure that is key in visual information the main challenge in exploring geometry in images comes from the discrete nature of the data thus unlike other approaches such as curvelets that first develop a transform in the continuous domain and then discretize for sampled data our approach starts with a discrete domain construction and then studies its convergence to an expansion in the continuous domain specifically we construct a discrete domain multiresolution and multidirection expansion using nonseparable filter banks in much the same way that wavelets were derived from filter banks this construction results in a flexible multiresolution local and directional image expansion using contour segments and thus it is named the contourlet transform the discrete contourlet transform has a fast iterated filter bank algorithm that requires an order n operations for n pixel images furthermore we establish a precise link between the developed filter bank and the associated continuous domain contourlet expansion via a directional multiresolution analysis framework we show that with parabolic scaling and sufficient directional vanishing moments contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves finally we show some numerical experiments demonstrating the potential of contourlets in several image processing applications

moving horizon estimation for hybrid systems we propose a state smoothing algorithm for hybrid systems based on moving horizon estimation mhe by exploiting the equivalence between hybrid systems modeled in the mixed logic dynamical form and piecewise affine systems we provide sufficient conditions on the time horizon and the penalties on the state at the beginning of the estimation horizon to guarantee asymptotic convergence of the mhe scheme moreover we propose two practical algorithms for the computation of penalties that allow us to implement mhe by solving a mixed integer quadratic program

a new family of nonredundant transforms using hybrid wavelets and directional filter banks we propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks we convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks where we form a nonredundant transform family which exhibits both directional and nondirectional basis functions we demonstrate the potential of the proposed transforms using nonlinear approximation in addition we employ the proposed family in two key image processing applications image coding and denoising and show its efficiency for these applications

directionlets anisotropic multidirectional representation with separable filtering in spite of the success of the standard wavelet transform wt in image processing in recent years the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions one dimensional 1 d discontinuities in images edges and contours that are very important elements in visual perception intersect too many wavelet basis functions and lead to a nonsparse representation to efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions a more complex multidirectional m dir and anisotropic transform is required we present a new lattice based perfect reconstruction and critically sampled anisotropic m dir wt the transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two dimensional wt unlike in the case of some other directional transform constructions e g curvelets contourlets or edgelets the corresponding anisotropic basis functions directionlets have directional vanishing moments along any two directions with rational slopes furthermore we show that this novel transform provides an efficient tool for nonlinear approximation of images achieving the approximation power o n sup 1 55 which while slower than the optimal rate o n sup 2 is much better than o n sup 1 achieved with wavelets but at similar complexity

reduced complexity sequence estimation using state partitioning a reduced state sequence estimator rsse for linear dispersive channels is described the technique is based on partitioning the set of all channel states in a way that defines a trellis with fewer states the new technique generalizes trellises that were previously obtained by duel hallen and heegard 1989 and eyouboglu and qureshi 1988 while the approach followed by duel hallen results in powers of two state trellises the new technique generates trellises with an arbitrary number of states using the state partitioning approach one could also generalize the trellises obtained by eyouboglu therefore the approach described in this paper provides a better performance complexity tradeoff

l error estimate for a system of elliptic quasi variational inequalities with noncoercive operators this paper deals with the standard finite element approximation of a noncoercive system of quasi variational inequalities qvis arising in stochastic control problems we improve a result obtained in boulbrachene comput math appl 45 983 989 2003 and establish the optimal l convergence order making use of the concepts of subsolutions and discrete regularity

approximation and compression with sparse orthonormal transforms we propose a new transform design method that targets the generation of compression optimized transforms for next generation multimedia applications the fundamental idea behind transform compression is to exploit regularity within signals such that redundancy is minimized subject to a fidelity cost multimedia signals in particular images and video are well known to contain a diverse set of localized structures leading to many different types of regularity and to nonstationary signal statistics the proposed method designs sparse orthonormal transforms sots that automatically exploit regularity over different signal structures and provides an adaptation method that determines the best representation over localized regions unlike earlier work that is motivated by linear approximation constructs and model based designs that are limited to specific types of signal regularity our work uses general nonlinear approximation ideas and a data driven setup to significantly broaden its reach we show that our sot designs provide a safe and principled extension of the karhunen loeve transform klt by reducing to the klt on gaussian processes and by automatically exploiting non gaussian statistics to significantly improve over the klt on more general processes we provide an algebraic optimization framework that generates optimized designs for any desired transform structure multiresolution block lapped and so on with significantly better n term approximation performance for each structure we propose a new prototype codec and test over a database of images simulation results show consistent increase in compression and approximation performance compared with conventional methods

adaptive multiresolution analysis based on anisotropic triangulations a simple greedy refinement procedure for the generation of data adapted triangulations is proposed and studied given a function of two variables the algorithm produces a hierarchy of triangulations dj j 0 and piecewise polynomial approximations of on these triangulations the refinement procedure consists in bisecting a triangle in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between and its piecewise polynomial approximation after is bisected the hierarchical structure allows us to derive various approximation tools such as multiresolution analysis wavelet bases adaptive triangulations based either on greedy or optimal cart trees as well as a simple encoding of the corresponding triangulations we give a general proof of convergence in the lp norm of all these approximations numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio which is dictated by the local hessian of in case of c2 functions

polynomial smoothing of time series with additive step discontinuities this paper addresses the problem of estimating simultaneously a local polynomial signal and an approximately piecewise constant signal from a noisy additive mixture the approach developed in this paper synthesizes the total variation filter and least square polynomial signal smoothing into a unified problem formulation the method is based on formulating an l1 norm regularized inverse problem a computationally efficient algorithm based on variable splitting and the alternating direction method of multipliers admm is presented algorithms are derived for both unconstrained and constrained formulations the method is illustrated on experimental data involving the detection of nano particles with applications to real time virus detection using a whispering gallery mode detector

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

guaranteed robustness properties of multivariable nonlinear stochastic optimal regulators we study the robustness of optimal regulators for nonlinear deterministic and stochastic multiinput dynamical systems under the assumption that all state variables can be measured we show that under mild assumptions such nonlinear regulators have a guaranteed infinite gain margin moreover they have a guaranteed 50 percent gain reduction margin and a 60 degree phase margin in each feedback channel provided that the system is linear in the control and the penalty to the control is quadratic thus extending the well known properties of lq regulators to nonlinear optimal designs these results are also valid for infinite horizon average cost stochastic optimal control problems

bifurcation and optimal stochastic control abstract two questions concerning bifuraction theory and optimal stochastic control are considered first in a few examples we give the interpretation of a bifurcation in terms of optimal stochastic control next we introduce the analogue of the lowest eigenvalue for the nonlinear operator associated with the hamilton jacobi bellman equations of optimal stochastic control author

optimal switching with constraints and utility maximization of an indivisible market this work focuses on optimal switching with constraints our motivation stems from utility maximization of an indivisible market the dynamic programming approach is used the value function is characterized as the unique viscosity solution of a quasi variational inequality the unbounded domain introduces new challenges by studying the sample paths of the diffusion at the boundary a sufficient condition for the continuity of the value function is provided yielding the desired characterization not only are the results of this work applicable to the utility maximization problem but also they can be used for general optimal switching problems with finite regimes

rate distortion optimized tree structured compression algorithms for piecewise polynomial images this paper presents novel coding algorithms based on tree structured segmentation which achieve the correct asymptotic rate distortion r d behavior for a simple class of signals known as piecewise polynomials by using an r d based prune and join scheme for the one dimensional case our scheme is based on binary tree segmentation of the signal this scheme approximates the signal segments using polynomial models and utilizes an r d optimal bit allocation strategy among the different signal segments the scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying r d behavior d r spl sim c sub 0 2 sup c1r thus improving over classic wavelet schemes we also prove that the computational complexity of the scheme is of o nlogn we then show the extension of this scheme to the two dimensional case using a quadtree this quadtree coding scheme also achieves an exponentially decaying r d behavior for the polygonal image model composed of a white polygon shaped object against a uniform black background with low computational cost of o nlogn again the key is an r d optimized prune and join strategy finally we conclude with numerical results which show that the proposed quadtree coding scheme outperforms jpeg2000 by about 1 db for real images like cameraman at low rates of around 0 15 bpp

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

space frequency quantization for wavelet image coding a new class of image coding algorithms coupling standard scalar quantization of frequency coefficients with tree structured quantization related to spatial structures has attracted wide attention because its good performance appears to confirm the promised efficiencies of hierarchical representation this paper addresses the problem of how spatial quantization modes and standard scalar quantization can be applied in a jointly optimal fashion in an image coder we consider zerotree quantization zeroing out tree structured sets of wavelet coefficients and the simplest form of scalar quantization a single common uniform scalar quantizer applied to all nonzeroed coefficients and we formalize the problem of optimizing their joint application we develop an image coding algorithm for solving the resulting optimization problem despite the basic form of the two quantizers considered the resulting algorithm demonstrates coding performance that is competitive often outperforming the very best coding algorithms in the literature

on the importance of combining wavelet based nonlinear approximation with coding strategies this paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory contrasting linear and nonlinear approximation spaces we show that there are interesting classes of functions random processes which are much more compactly represented by wavelet based nonlinear approximation these classes include locally smooth signals that have singularities and provide a model for many signals encountered in practice in particular for images however we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate distortion sense based on this observation we construct compression techniques and formulate the family of functions stochastic processes for which they provide efficient descriptions in a rate distortion sense we show that this family invariably leads to besov spaces yielding a natural relationship among besov smoothness linear nonlinear approximation order and compression performance in a rate distortion sense the designed compression techniques show similarities to modern high performance transform codecs allowing us to establish relevant rate distortion estimates and identify performance limits

optimal control of stochastic integrals and hamilton jacobi bellman equations i we consider the solution of a stochastic integral control problem and we study its regularity in particular we characterize the optimal cost as the maximum solution of begin gathered forall

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

representation and compression of multidimensional piecewise functions using surflets we study the representation approximation and compression of functions in m dimensions that consist of constant or smooth regions separated by smooth m 1 dimensional discontinuities examples include images containing edges video sequences of moving objects and seismic data containing geological horizons for both function classes we derive the optimal asymptotic approximation and compression rates based on kolmogorov metric entropy for piecewise constant functions we develop a multiresolution predictive coder that achieves the optimal rate distortion performance for piecewise smooth functions our coder has near optimal rate distortion performance our coder for piecewise constant functions employs surflets a new multiscale geometric tiling consisting of m dimensional piecewise constant atoms containing polynomial discontinuities our coder for piecewise smooth functions uses surfprints which wed surflets to wavelets for piecewise smooth approximation both of these schemes achieve the optimal asymptotic approximation performance key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity we also extend our results to the corresponding discrete function spaces for sampled data we provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities for approximation of discrete data we propose a new scale adaptive dictionary that contains few elements at coarse and fine scales but many elements at medium scales simulation results on synthetic signals provide a comparison between surflet based coders and previously studied approximation schemes based on wedgelets and wavelets

on singular stochastic control problems for diffusion with jumps we consider some cases of control problems for diffusion processes with jumps when the payoff functional does not depend explicitly on the control we prove the continuity of the optimal cost with a quasi variational inequality interpreting the problems as the limit of an impulse control problem when the cost of impulse tends to zero moreover we show the existence of an optimal control for some particular situations

multistep detector for linear isi channels incorporating degrees of belief in past estimates this paper formulates the channel equalization problem in the framework of constrained maximum likelihood estimation this allows us to highlight key issues including the need to summarize past data and to apply a finite alphabet constraint over a sliding optimization window the approach adopted here leads to embellishments of the usual nonadaptive decision feedback equalizer and its multistep extensions it includes a provision for degrees of belief in past estimates which addresses the problem of error propagation

decision feedback equalization using multiple hyperplane partitioning for detecting isi corrupted m ary pam signals a decision feedback equalizer scheme is derived based on multiple hyperplane partitioning of signal space for detecting m ary pulse amplitude modulation symbols transmitted through a noisy intersymbol interference channel the proposed scheme is based on the fact that the optimal bayesian decision boundary separating two neighboring signal classes is asymptotically piecewise linear and consists of several hyperplanes when the signal to noise ratio tends to infinity an algorithm is developed to determine these hyperplanes which are then used to partition the observation signal space the resulting detector can closely approximate the optimal bayesian detector at an advantage of considerably reduced detector complexity

an ode comparison theorem with application in the optimal exit time control problem the optimal exit time control oetc problem tries to find the feedback control law with a reasonable cost that can keep the system state inside a certain subset of the state space called the safe set for the longest time under random perturbations the symmetry property of its solutions has been proved previously when the state dimension is higher than one in this note a comparison theorem is established that compares the solutions to two odes arising in the 1 d oetc problem the symmetry of solutions to the 1 d oetc problem is proved using this comparison theorem an example is presented to show how the symmetry result can help to solve the oetc problem analytically in certain cases

a new proof for the existence and uniqueness of the discrete evolutionary hjb equations in this paper we establish a new proof for the existence and uniqueness of the discrete solution of evolutionary hjb equations which can be approximated by a weakly coupled system of coercive elliptic quasi variational inequalities ovis for that some properties of the presented algorithm cf e g boulaaras and haiour 2013 boulbrachene and haiour 2001 using the theta scheme with respect to the t variable combined with a finite element spatial approximation are proved

regularity for obstacle problems in infinite dimensional hilbert spaces in this paper we study fully nonlinear obstacle type problems in hilbert spaces we introduce the notion of q elliptic equation and prove existence uniqueness and regularity of viscosity solutions of q elliptic obstacle problems in particular we show that solutions of concave problems with semiconvex obstacles are in the space

estimation of signal dependent noise level function in transform domain via a sparse recovery model this paper proposes a novel algorithm to estimate the noise level function nlf of signal dependent noise sdn from a single image based on the sparse representation of nlfs noise level samples are estimated from the high frequency discrete cosine transform dct coefficients of nonlocal grouped low variation image patches then an nlf recovery model based on the sparse representation of nlfs under a trained basis is constructed to recover nlf from the incomplete noise level samples confidence levels of the nlf samples are incorporated into the proposed model to promote reliable samples and weaken unreliable ones we investigate the behavior of the estimation performance with respect to the block size sampling rate and confidence weighting simulation results on synthetic noisy images show that our method outperforms existing state of the art schemes the proposed method is evaluated on real noisy images captured by three types of commodity imaging devices and shows consistently excellent sdn estimation performance the estimated nlfs are incorporated into two well known denoising schemes nonlocal means and bm3d and show significant improvements in denoising sdn polluted images

sur les equations de monge ampere in this paper we study the real monge arapere equations det d2u f x in 0 u convex in 0 u 0 on 0 and we introduce a new method for solving these equations which enables us to show the existence of regular solutions this method uses only p d e techniques and does not use any geometrical results furthermore it enables us to solve quasilinear monge ampere equations

moving horizon design of discrete coefficient fir filters we show how the discrete coefficient filter design problem can be solved with a moving horizon optimization approach the computation time of this procedure is determined by the optimization horizon and does not grow exponentially with the filter length spl sigma spl delta design methods are a special case of the proposed procedure

spatial sparsity induced prediction sip for images and video a simple way to reject structured interference we propose a prediction technique that is geared toward forming successful estimates of a signal based on a correlated anchor signal that is contaminated with complex interference the corruption in the anchor signal involves intensity modulations linear distortions structured interference clutter and noise just to name a few the proposed setup reflects nontrivial prediction scenarios involving images and video frames where statistically related data is rendered ineffective for traditional methods due to cross fades blends clutter brightness variations focus changes and other complex transitions rather than trying to solve a difficult estimation problem involving nonstationary signal statistics we obtain simple predictors in linear transform domain where the underlying signals are assumed to be sparse we show that these simple predictors achieve surprisingly good performance and seamlessly allow successful predictions even under complicated cases none of the interference parameters are estimated as our algorithm provides completely blind and automated operation we provide a general formulation that allows for nonlinearities in the prediction loop and we consider prediction optimal decompositions beyond an extensive set of results on prediction and registration the proposed method is also implemented to operate inside a state of the art compression codec and results show significant improvements on scenes that are difficult to encode using traditional prediction techniques

wavelet domain approximation and compression of piecewise smooth images the wavelet transform provides a sparse representation for smooth images enabling efficient approximation and compression using techniques such as zerotrees unfortunately this sparsity does not extend to piecewise smooth images where edge discontinuities separating smooth regions persist along smooth contours this lack of sparsity hampers the efficiency of wavelet based approximation and compression on the class of images containing smooth c sup 2 regions separated by edges along smooth c sup 2 contours for example the asymptotic rate distortion r d performance of zerotree based wavelet coding is limited to d r spl lsim 1 r well below the optimal rate of 1 r sup 2 in this paper we develop a geometric modeling framework for wavelets that addresses this shortcoming the framework can be interpreted either as 1 an extension to the zerotree model for wavelet coefficients that explicitly accounts for edge structure at fine scales or as 2 a new atomic representation that synthesizes images using a sparse combination of wavelets and wedgeprints anisotropic atoms that are adapted to edge singularities our approach enables a new type of quadtree pruning for piecewise smooth images using zerotrees in uniformly smooth regions and wedgeprints in regions containing geometry using this framework we develop a prototype image coder that has near optimal asymptotic r d performance d r spl lsim logr sup 2 r sup 2 for piecewise smooth c sup 2 c sup 2 images in addition we extend the algorithm to compress natural images exploring the practical problems that arise and attaining promising results in terms of mean square error and visual quality

how good is quantized model predictive control with horizon one model predictive control is increasingly being used in areas where decision variables are constrained to finite or countably infinite sets well known fields include power electronics signal processing and telecommunications typically the applications utilize high speed sampling and thus there is an incentive to reduce computational burden one way of achieving this is to use small optimization horizons this raises the question as to the optimality and performance of control laws with short horizons in this paper we give necessary and sufficient conditions for horizon one quantized model predictive control to be equivalent to the use of larger horizons we also explore situations where horizon one is near optimal

optimal correction problem of a multidimensional stochastic system abstract we consider a stochastic dynamic system which is governed by a multidimensional diffusion process with constant drift and diffusion coefficients the correction corresponds to an additive input which is under control there is no limit on the rate of input into the system the objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted it is proved that hamilton jacobi bellman s equation of the problem has a solution which corresponds to the optimal cost of the problem an existence of optimal policy is proved

nonlinear and nonseparable bidimensional multiscale representation based on cell average representation the aim of this paper is to construct a new nonlinear and nonseparable multiscale representation of piecewise continuous bidimensional functions this representation is based on the definition of a linear projection and a nonlinear prediction operator which locally adapts to the function to be represented this adaptivity of the prediction operator proves to be very interesting for image encoding in that it enables a considerable reduction in the number of significant coefficients compared with other representations applications of this new nonlinear multiscale representation to image compression and super resolution conclude this paper

two parameter optimal stopping and bi markov processes in this paper the optimal stopping problem is solved for particular two parameter processes here called bi markov processes a subsequent potential theory is developed with respect to a pair of one parameter semi groups we introduce a new notion of harmonicity for two variable functions and we interpret it in the framework of the theory of bi markov processes

asymptotic bayesian decision feedback equalizer using a set of hyperplanes we present a signal space partitioning technique for realizing the optimal bayesian decision feedback equalizer dfe it is known that when the signal to noise ratio snr tends to infinity the decision boundary of the bayesian dfe is asymptotically piecewise linear and consists of several hyperplanes the proposed technique determines these hyperplanes explicitly and uses them to partition the observation signal space the resulting equalizer is made up of a set of parallel linear discriminant functions and a boolean mapper unlike the existing signal space partitioning technique of kim and moon 1998 which involves complex combinatorial search and optimization in design our design procedure is simple and straightforward and guarantees to achieve the asymptotic bayesian dfe

multiscale image segmentation using wavelet domain hidden markov models we introduce a new image texture segmentation algorithm hmtseg based on wavelets and the hidden markov tree hmt model the hmt is a tree structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform since the hmt is particularly well suited to images containing singularities edges and ridges it provides a good classifier for distinguishing between textures utilizing the inherent tree structure of the wavelet hmt and its fast training and likelihood computation algorithms we perform texture classification at a range of different scales we then fuse these multiscale classifications using a bayesian probabilistic graph to obtain reliable final segmentations since hmtseg works on the wavelet transform of the image it can directly segment wavelet compressed images without the need for decompression into the space domain we demonstrate the performance of hmtseg with synthetic aerial photo and document image segmentations

the nonsubsampled contourlet transform theory design and applications in this paper we develop the nonsubsampled contourlet transform nsct and study its applications the construction proposed in this paper is based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks the result is a flexible multiscale multidirection and shift invariant image decomposition that can be efficiently implemented via the a trous algorithm at the core of the proposed scheme is the nonseparable two channel nonsubsampled filter bank nsfb we exploit the less stringent design condition of the nsfb to design filters that lead to a nsct with better frequency selectivity and regularity when compared to the contourlet transform we propose a design framework based on the mapping approach that allows for a fast implementation based on a lifting or ladder structure and only uses one dimensional filtering in some cases in addition our design ensures that the corresponding frame elements are regular symmetric and the frame is close to a tight one we assess the performance of the nsct in image denoising and enhancement applications in both applications the nsct compares favorably to other existing methods in the literature

optimal n term approximation by linear splines over anisotropic delaunay triangulations anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data in particular from image data in previous work we have proposed a locally adaptive method for efficient image approximation called adaptive thinning which relies on linear splines over anisotropic delaunay triangulations in this paper we prove asymptotically optimal n term approximation rates for linear splines over anisotropic delaunay triangulations where our analysis applies to relevant classes of target functions a piecewise linear horizon functions across holder smooth boundaries b functions of w p regularity where 2 p 1 c piecewise regular horizon functions of w 2 regularity where 1

coherent multiscale image processing using quaternion wavelets coherent multiscale image processing using quaternion wavelets by wai lam chan this thesis develops a quaternion wavelet transform qwt as a new multiscale analysis tool for geometric image features the qwt is a near shift invariant tight frame representation whose coefficients sport a magnitude and three phases two phases encode local image shifts while the third contains textural information the qwt can be computed using a dual tree filter bank with linear computational complexity we develop the qwt by applying an alternative theory of 2 d hilbert transforms and analytic signals to the 1 d complex wavelet transform to demonstrate the properties of the qwt s coherent magnitude phase representation we develop a efficient and accurate algorithm for estimating the local geometrical structure of images and a multiscale algorithm for estimating the disparity between a pair of images the latter algorithm has potentials for various applications in image registration and flow estimation it uses an interesting multiscale phase unwrapping procedure and features linear complexity and sub pixel accuracy

conditions for optimality of naive quantized finite horizon control this paper presents properties of a control law which quantizes the unconstrained solution to a unitary horizon quadratic programme this naive quantized control law underlies many popular algorithms such as converters and decision feedback equalities and is easily shown to be globally optimal for horizon one however the question arises as to whether it is also globally optimal for horizons greater than one i e whether it solves a finite horizon quadratic programme where decision variables are restricted to belonging to a quantized set by using dynamic programming we develop sufficient conditions for this to hold the present analysis is restricted to first order plants however this case already raises a number of highly non trivial issues the results can be applied to arbitrary horizons and quantized sets which may contain a finite or an infinite though countable number of elements

multidimensional signal space partitioning using a minimal set of hyperplanes for detecting isi corrupted symbols a signal space partitioning technique is presented for detecting symbols transmitted through intersymbol interference channels the decision boundary is piecewise linear and is made up of several hyperplanes the goal here is to minimize the number of hyperplanes for a given performance measure namely the minimum distance between any signal and the decision boundary unlike in voronoi partitioning individual hyperplanes are chosen to separate signal clusters rather than signal pairs the convex regions associated with individual signals which together form the overall decision region generally overlap or coincide among in class signals the technique leads to an asymptotically optimum detector when the target distance is set at half the minimum distance associated with the maximum likelihood sequence detector complexity and performance can be easily traded as the target distance is a flexible design parameter

space frequency quantization for image compression with directionlets the standard separable 2 d wavelet transform wt has recently achieved a great success in image processing because it provides a sparse representation of smooth images however it fails to efficiently capture 1 d discontinuities like edges or contours these features being elongated and characterized by geometrical regularity along different directions intersect and generate many large magnitude wavelet coefficients since contours are very important elements in the visual perception of images to provide a good visual quality of compressed images it is fundamental to preserve good reconstruction of these directional features in our previous work we proposed a construction of critically sampled perfect reconstruction transforms with directional vanishing moments imposed in the corresponding basis functions along different directions called directionlets in this paper we show how to design and implement a novel efficient space frequency quantization sfq compression algorithm using directionlets our new compression method outperforms the standard sfq in a rate distortion sense both in terms of mean square error and visual quality especially in the low rate compression regime we also show that our compression method does not increase the order of computational complexity as compared to the standard sfq algorithm

wavelet footprints theory algorithms and applications wavelet based algorithms have been successful in different signal processing tasks the wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients discontinuities often carry relevant signal information and therefore they represent a critical part to analyze we study the dependency across scales of the wavelet coefficients generated by discontinuities we start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual theorem 1 we then introduce the notion of footprints which are scale space vectors that model discontinuities in piecewise polynomial signals exactly we show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints this also leads to efficient approximation of piecewise smooth functions finally we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising compression and nonblind deconvolution in the case of compression we also prove that at high rates footprint based algorithms attain optimal performance theorem 3

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

rate distortion analysis of directional wavelets the inefficiency of separable wavelets in representing smooth edges has led to a great interest in the study of new 2 d transformations the most popular criterion for analyzing these transformations is the approximation power transformations with near optimal approximation power are useful in many applications such as denoising and enhancement however they are not necessarily good for compression therefore most of the nearly optimal transformations such as curvelets and contourlets have not found any application in image compression yet one of the most promising schemes for image compression is the elegant idea of directional wavelets diws while these algorithms outperform the state of the art image coders in practice our theoretical understanding of them is very limited in this paper we adopt the notion of rate distortion and calculate the performance of the diw on a class of edge like images our theoretical analysis shows that if the edges are not sharp the diw will compress them more efficiently than the separable wavelets it also demonstrates the inefficiency of the quadtree partitioning that is often used with the diw to solve this issue we propose a new partitioning scheme called megaquad partitioning our simulation results on real world images confirm the benefits of the proposed partitioning algorithm promised by our theoretical analysis

reduced state sequence estimation with set partitioning and decision feedback a reduced state sequence estimator for linear intersymbol interference channels is described the estimator uses a conventional viterbi algorithm with decision feedback to search a reduced state subset trellis that is constructed using set partitioning principles the complexity of maximum likelihood sequence estimation mlse due to the length of the channel memory and the size of the signal set is systematically reduced an error probability analysis shows that a good performance complexity tradeoff can be obtained in particular the results indicate that the required complexity to achieve the performance of mlse is independent of the size of the signal set for large enough signal sets simulation results are provided for two partial response systems a simple technique for quadrature partial response signaling qprs is described that eliminates the quasicatastrophic nature of the ml trellis

value functions and the dirichlet problem for isaacs equation in a smooth domain in this paper we investigate probabilistic solutions of the dirichlet problem for the elliptic isaacs equation in a smooth bounded domain in euclidean space