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

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

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

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

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

centralized and distributed semiparametric compression of piecewise smooth functions this paper introduces novel wavelet based semiparametric centralized and distributed compression methods for a class of 1 d piecewise smooth functions classical centralized compression schemes are based on a relatively complex nonlinear encoder and a simple linear decoder recently a new paradigm in compression called distributed source coding has emerged this setup involves multiple encoders where each one partially observes the source and a centralized decoder first we focus on the dual situation of the centralized compression with a simple encoder and a complex decoder we show that by incorporating parametric estimation into the decoding procedure it is possible to achieve the same rate distortion performance as that of a conventional wavelet based compression scheme second we consider the distributed compression scenario where each independent encoder partially observes the 1 d piecewise smooth function we propose a new wavelet based distributed compression scheme that uses parametric estimation to perform joint decoding our analysis shows that it is possible for the proposed scheme to achieve the same compression performance as that of a joint encoding scheme

hyperanalytic denoising a new threshold rule for the estimation of a deterministic image immersed in noise is proposed the full estimation procedure is based on a separable wavelet decomposition of the observed image and the estimation is improved by introducing the new threshold to estimate the decomposition coefficients the observed wavelet coefficients are thresholded using the magnitudes of wavelet transforms of a small number of replicates of the image the replicates are calculated by extending the image into a vector valued hyperanalytic signal more than one hyperanalytic signal may be chosen and either the hypercomplex or riesz transforms are used to calculate this object the deterministic and stochastic properties of the observed wavelet coefficients of the hyperanalytic signal at a fixed scale and position index are determined a universal threshold is calculated for the proposed procedure an expression for the risk of an individual coefficient is derived the risk is calculated explicitly when the universal threshold is used and is shown to be less than the risk of universal hard thresholding under certain conditions the proposed method is implemented and the derived theoretical risk reductions substantiated

spatially adaptive wavelet based multiscale image restoration in this paper we present a new spatially adaptive approach to the restoration of noisy blurred images which is particularly effective at producing sharp deconvolution while suppressing the noise in the flat regions of an image this is accomplished through a multiscale kalman smoothing filter applied to a prefiltered observed image in the discrete separable 2 d wavelet domain the prefiltering step involves constrained least squares filtering based on optimal choices for the regularization parameter this leads to a reduction in the support of the required state vectors of the multiscale restoration filter in the wavelet domain and improvement in the computational efficiency of the multiscale filter the proposed method has the benefit that the majority of the regularization or noise suppression of the restoration is accomplished by the efficient multiscale filtering of wavelet detail coefficients ordered on quadtrees not only does this lead to potential parallel implementation schemes but it permits adaptivity to the local edge information in the image in particular this method changes filter parameters depending on scale local signal to noise ratio snr and orientation because the wavelet detail coefficients are a manifestation of the multiscale edge information in an image this algorithm may be viewed as an edge adaptive multiscale restoration approach

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

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

box and ball system as a realization of ultradiscrete nonautonomous kp equation cellular automata which are realized by dynamics of several kinds of balls in an infinite array of boxes are investigated they show soliton patterns even in the case when each box has arbitrary capacity the analytical expression for the soliton patterns are obtained using ultradiscretization of the nonautonomous discrete kp equation

a simple method to build oversampled filter banks and tight frames this paper presents conditions under which the sampling lattice for a filter bank can be replaced without loss of perfect reconstruction this is the generalization of common knowledge that removing up downsampling will not lose perfect reconstruction the results provide a simple way of building over sampled filter banks if the original filter banks are orthogonal these oversampled banks construct tight frames of l2 z n when iterated as an example a quincunx lattice is used to replace the rectangular one of the standard wavelet transform this replacement leads to a tight frame that has a higher sampling in both time and frequency the frame transform is nearly shift invariant and has intermediate scales an application of the transform to image fusion is also presented

hierarchical segmentation based image coding using hybrid quad binary trees a novel segmentation based image approximation and coding technique is proposed a hybrid quad binary qb tree structure is utilized to efficiently model and code geometrical information within images compared to other tree based representation such as wedgelets the proposed qb tree based method is more efficient for a wide range of contour features such as junctions corners and ridges especially at low bit rates

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

ultra discrete permanent and the consistency of max plus linear equations abstract in this paper we investigate the consistency conditions for three classes of the max plus linear equations including the case corresponding to the convex polyhedra which was the subject of the previous paper the necessary and sufficient conditions for the existence of finite solutions are expressed in the form of single equations by using the ultra discrete permanent

enhanced b wavelets via mixed composite packets a modified b wavelet construction with enhanced filter characteristics is considered the design comprises a superposition of tessellated integer dilated sister wavelet functions we here propose a cascaded filter bank realization of this wavelet family together with some notable extensions we prove that modifications of low order members exist in the multiresolution subspace spanned by the half translates of the original wavelets and hence that the resulting modified wavelet coefficients can be computed as convolutions of the undecimated original wavelet coefficients finite impulse response filters are thus designed and incorporated into a b wavelet packet architecture such that the mainlobe to sidelobe ratio of the resulting wavelet filter characteristic is improved this is achieved by designing the filters so that zeros are introduced near to the maxima of the harmonics it is shown that the numbers of zeros can be balanced with the length of the corresponding filters by controlling the modification order several constructions are presented we prove that two such constructions satisfy the perfect reconstruction property for all orders the resulting modified wavelets preserve many of the properties of the original b wavelets such as differentiability number of vanishing moments symmetry anti symmetry finite support and the existence of a closed form expression

signal and image approximation using interval wavelet transform in signal approximation classical wavelet synthesis are known to produce gibbs like phenomenon around discontinuities when wavelet coefficients in the cone of influence of the discontinuities are quantized by analyzing a function in a piecewise manner filtering across discontinuities can be avoided using this principle the interval wavelet transform can generate sparser representations in the vicinity of discontinuities than classical wavelet transforms this work introduces two new constructions of interval wavelets and shows how they can be used for image compression and upscaling

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

bilinear equations and backlund transformation for a generalized ultradiscrete soliton solution ultradiscrete soliton equations and backlund transformation for a generalized soliton solution are presented the equations include the ultradiscrete korteweg de vries kdv equation or the ultradiscrete toda equation in a special case we also express the solution by the ultradiscrete permanent which is defined by ultradiscretizing the signature free determinant that is the permanent moreover we discuss a relation between backlund transformations for discrete and ultradiscrete kdv equations

wavelet based statistical signal processing using hidden markov models wavelet based statistical signal processing techniques such as denoising and detection typically model the wavelet coefficients as independent or jointly gaussian these models are unrealistic for many real world signals we develop a new framework for statistical signal processing based on wavelet domain hidden markov models hmms that concisely models the statistical dependencies and non gaussian statistics encountered in real world signals wavelet domain hmms are designed with the intrinsic properties of the wavelet transform in mind and provide powerful yet tractable probabilistic signal models efficient expectation maximization algorithms are developed for fitting the hmms to observational signal data the new framework is suitable for a wide range of applications including signal estimation detection classification prediction and even synthesis to demonstrate the utility of wavelet domain hmms we develop novel algorithms for signal denoising classification and detection

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

generalized l m and r statistics abstract a class of statisticss generalizing u statistics and l statistics and containing other varieties of statistics as well such as trimmed u statistics is studied using the differentiable statistical function approach differential approximations are obtained and the influence curves of these generalized l statistics are derived these results are employed to establish asymptotic normality for such statistics parallel generalizations of m and r statistics are noted strong convergence berry esseen rates and computational aspects are discussed

elimination of variables and the determinant of the max plus algebra the determinant of the max plus algebra is proposed by considering the consistency condition of a class of max plus equations which correspond to the convex polyhedra some examples of the determinant and the relationship between the ultradiscrete permanent are also shown

on bounds of shift variance in two channel multirate filter banks critically sampled multirate fir filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage we assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged we express these changes by a so called commutator we then derive a sharp upper bound for shift variance via the operator norm of the commutator which is independent of the input signal its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator leading to a matrix norm which depends on frequency this bound can be regarded as a worst case instance holding for all input signals for two channel fir filter banks with perfect reconstruction pr we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used moreover the framework allows to identify the signals for which the upper bound is almost reached as so called near maximizers of the frequency dependent matrix norm for unitary pr filter banks these near maximizers are shown to be narrow band signals to complement this worst case bound we derive an additional bound on shift variance for input signals with given amplitude spectra where we use wide band model spectra instead of narrow band signals like the operator norm this additional bound is based on the above frequency dependent matrix norm we provide results for various critically sampled two channel filter banks such as quadrature mirror filters pr conjugated quadrature filters wavelets and biorthogonal filters banks

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 backlund transformation equations for the ultradiscrete kp equation the backlund transformation for the ultradiscrete kp equation is proposed an algorithm to eliminate variables from the ultradiscrete linear equations is proposed the consistency condition for the backlund transformation equations is obtained via the algorithm