wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

adaptive 2 d wavelet transform based on the lifting scheme with preserved vanishing moments in this paper we propose novel adaptive wavelet filter bank structures based on the lifting scheme the filter banks are nonseparable based on quincunx sampling with their properties being pixel wise adapted according to the local image features despite being adaptive the filter banks retain a desirable number of primal and dual vanishing moments the adaptation is introduced in the predict stage of the filter bank with an adaptation region chosen independently for each pixel based on the intersection of confidence intervals ici rule the image denoising results are presented for both synthetic and real world images it is shown that the obtained wavelet decompositions perform well especially for synthetic images that contain periodic patterns for which the proposed method outperforms the state of the art in image denoising

nonlinear wavelet transforms for image coding via lifting we investigate central issues such as invertibility stability synchronization and frequency characteristics for nonlinear wavelet transforms built using the lifting framework the nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework we also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels preliminary compression results for model and real world images demonstrate the promise of our techniques

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

on the new control functions for linear discrete delay systems new control functions are derived for linear difference equations with delay which can be used to construct a nontrivial solution of a boundary value problem with zero boundary conditions later a special case of invariant linear subspace is considered and corresponding control functions are constructed results for weakly nonlinear problems are also discussed

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

rotated orthogonal transform rot for motion compensation residual coding discrete cosine transform dct is the orthogonal transform that is most commonly used in image and video compression the motion compensation residual mc residual is also compressed with the dct in most video codecs however the mc residual has different characteristics from a nature image in this paper we develop a new orthogonal transform rotated orthogonal transform rot that can perform better on the mc residual than the dct for coding purposes we derive the proposed rot based on orthogonal constrained l1 norm minimization problem for its sparse property using the dct matrix as the starting point a better orthogonal transform matrix is derived in addition by exploring inter frame dependency and local motion activity transmission of substantial side information is avoided the experiment results confirm that with small computation overhead the rot is adaptive to change of local spatial characteristic of mc residual frame and provides higher compression efficiency for the mc residual than dct especially for high and complex motion videos

directionlets using in phase lifting for image representation directionlets allow a construction of perfect reconstruction and critically sampled multidirectional anisotropic basis yet retaining the separable filtering of standard wavelet transform however due to the spatially varying filtering and downsampling direction it is forced to apply spatial segmentation and process each segment independently because of this independent processing of the image segments directionlets suffer from the following two major limitations when applied to say image coding first failure to exploit the correlation across block boundaries degrades the coding performance and also induces blocking artifacts thus making it mandatory to use de blocking filter at low bit rates second spatial scalability i e minimum segment size or the number of levels of the transform is limited due to independent processing of segments we show that with simple modifications in the block boundaries we can overcome these limitations by what we call in phase lifting implementation of directionlets in the context of directionlets using in phase lifting we identify different possible groups of downsampling matrices that would allow the construction of a multilevel transform without forcing independent processing of segments both with and without any modifications in the segment boundary experimental results in image coding show objective and subjective improvements when compared with the directionlets applied independently on each image segment as an application using both the in phase lifting implementation of directionlets and the adaptive directional lifting we have constructed an adaptive directional wavelet transform which has shown improved image coding performance over these adaptive directional wavelet transforms

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

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

relative and trajectory controllability of linear discrete systems with constant coefficients and a single delay this paper studies the qualitative behavior of solutions of linear discrete systems with constant coefficients and a constant time delay under the influence of a control function in particular it deals with a nontrivial task of transferring an arbitrary solution to several prescribed positions trajectory controllability being more general and different from the problem of relative controllability frequently solved in the literature a kalman type controllability criterion is proved and in addition previous results are discussed and illustrative examples are given

a note on relative controllability of higher order linear delayed discrete systems this article is concerned with relative controllability of higher order linear discrete delayed systems with a single delay by a formula that uses special matrix functions discrete delayed matrix sine and cosine to compute the solution of the initial problem a criterion is found for relative controllability in terms of special matrix functions a control function is constructed to solve the problem of relative controllability it is shown that the transformation of a given system into a higher dimensional system without delays leads to a nonequivalent problem of relative controllability the results are illustrated by an example

controllability of linear discrete systems with constant coefficients and pure delay the purpose of this contribution is to develop a controllability method for linear discrete systems with constant coefficients and with pure delay to do this a representation of solutions with the aid of a discrete matrix delayed exponential is used such an approach leads to new conditions of controllability except for a criterion of relative controllability a control function is constructed as well

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

curved wavelet transform for image coding the conventional two dimensional wavelet transform used in existing image coders is usually performed through one dimensional 1 d filtering in the vertical and horizontal directions which cannot efficiently represent edges and lines in images the curved wavelet transform presented in this paper is carried out by applying 1 d filters along curves rather than being restricted to vertical and horizontal straight lines the curves are determined based on image content and are usually parallel to edges and lines in the image to be coded the pixels along these curves can be well represented by a small number of wavelet coefficients the curved wavelet transform is used to construct a new image coder the code stream syntax of the new coder is the same as that of jpeg2000 except that a new marker segment is added to the tile headers results of image coding and subjective quality assessment show that the new image coder performs better than or as well as jpeg2000 it is particularly efficient for images that contain sharp edges and can provide a psnr gain of up to 1 67 db for natural images compared with jpeg2000

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

adaptive directional lifting based wavelet transform for image coding we present a novel 2 d wavelet transform scheme of adaptive directional lifting adl in image coding instead of alternately applying horizontal and vertical lifting as in present practice adl performs lifting based prediction in local windows in the direction of high pixel correlation hence it adapts far better to the image orientation features in local windows the adl transform is achieved by existing 1 d wavelets and is seamlessly integrated into the global wavelet transform the predicting and updating signals of adl can be derived even at the fractional pixel precision level to achieve high directional resolution while still maintaining perfect reconstruction to enhance the adl performance a rate distortion optimized directional segmentation scheme is also proposed to form and code a hierarchical image partition adapting to local features experimental results show that the proposed adl based image coding technique outperforms jpeg 2000 in both psnr and visual quality with the improvement up to 2 0 db on images with rich orientation features

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

stable invariant manifolds for parabolic dynamics abstract we consider nonautonomous equations v a t v in a banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates e c t for some function t this corresponds to the existence of a generalized exponential dichotomy which is known to be robust when t t this behavior can be described as a type of parabolic dynamics we consider the general case of nonuniform exponential dichotomies for which the lyapunov stability is not uniform we show that for any sufficiently small perturbation f of a generalized exponential dichotomy there is a stable invariant manifold for the perturbed equation v a t v f t v we also consider the case of exponential contractions which allow a simpler treatment and we show that they persist under sufficiently small nonlinear perturbations

three dimensional subband coding with motion compensation three dimensional 3 d frequency coding is an alternative approach to hybrid coding concepts used in today s standards the first part of this paper presents a study on concepts for temporal axis frequency decomposition along the motion trajectory in video sequences it is shown that if a two band split is used it is possible to overcome the problem of spatial inhomogeneity in the motion vector field mvf which occurs at the positions of uncovered and covered areas in these cases original pixel values from one frame are placed into the lowpass band signal while displaced frame difference values are embedded into the highpass band this technique is applicable with arbitrary mvf s examples with block matching and interpolative motion compensation are given derivations are first performed for the example of two tap quadrature mirror filters qmf s and then generalized to any linear phase qmf s with two band analysis and synthesis stages arranged as cascade structures higher resolution frequency decompositions are realizable in the second part of the paper encoding of the temporal axis subband signals is discussed a parallel filterbank scheme was used for spatial subband decomposition and adaptive lattice vector quantization was employed to approach the entropy rate of the 3 d subband samples coding results suggest that high motion video sequences can be encoded at significantly lower rates than those achievable with conventional hybrid coders main advantages are the high energy compaction capability and the nonrecursive decoder structure in the conclusion the scheme is interpreted more generally viewed as a motion compensated short time spectral analysis of video sequences which can adapt to the quickness of changes although a 3 d multiresolution representation of the picture information is produced a true multiresolution representation of motion information based on spatio temporal decimation and interpolation of the mvf is regarded as the still missing part

design of signal adapted multidimensional lifting scheme for lossy coding this paper proposes a new method for the design of lifting filters to compute a multidimensional nonseparable wavelet transform our approach is stated in the general case and is illustrated for the 2 d separable and for the quincunx images results are shown for the jpeg2000 database and for satellite images acquired on a quincunx sampling grid the design of efficient quincunx filters is a difficult challenge which has already been addressed for specific cases our approach enables the design of less expensive filters adapted to the signal statistics to enhance the compression efficiency in a more general case it is based on a two step lifting scheme and joins the lifting theory with wiener s optimization the prediction step is designed in order to minimize the variance of the signal and the update step is designed in order to minimize a reconstruction error application for lossy compression shows the performances of the method

direction adaptive discrete wavelet transform for image compression we propose a direction adaptive dwt da dwt that locally adapts the filtering directions to image content based on directional lifting with the adaptive transform energy compaction is improved for sharp image features a mathematical analysis based on an anisotropic statistical image model is presented to quantify the theoretical gain achieved by adapting the filtering directions the analysis indicates that the proposed da dwt is more effective than other lifting based approaches experimental results report a gain of up to 2 5 db in psnr over the conventional dwt for typical test images subjectively the reconstruction from the da dwt better represents the structure in the image and is visually more pleasing

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

efficient least absolute deviation adaptive wavelet filter bank compact representation of signals and images is a key for many applications compactness is often achieved through linear transforms with good energy concentration property we present an adaptive wavelet filter bank with fixed number of vanishing moments plus additional local adaptation proposed adaptation method is conducted at each sample according to the least absolute deviation lad criterion fixed vanishing moments provide for polynomial annihilation adaptation is aimed to achieve maximum sparseness for a wider class of signals such as sine waves lad criterion results in more accurate adaptation on sudden changes of signal statistics in this paper an efficient lad realization is proposed in spite of nonexistence of the closed form solution combining least squares and lad criterion we have achieved unbiased adaptation robust to noise due to its simplicity and acceptable computational speed the proposed scheme is a good candidate for the real world applications in this paper advantages of the proposed scheme are shown in signal denoising and reconstruction

adaptive lifting schemes with perfect reconstruction we propose a framework for constructing adaptive wavelet decompositions using the lifting scheme a major requirement is that perfect reconstruction is possible without any overhead cost we restrict ourselves to the update lifting stage it is assumed that the update filter utilizes local gradient information to adapt itself to the signal in the sense that smaller gradients evoke stronger update filters as a result sharp transitions in a signal will not be smoothed to the same extent as regions that are more homogeneous the approach taken in this paper differs from other adaptive schemes found in the literature in the sense that no bookkeeping is required in order to have perfect reconstruction

stable manifolds for nonautonomous equations without exponential dichotomy abstract we consider nonautonomous ordinary differential equations v a t v in banach spaces and under fairly general assumptions we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v a t v f t v which corresponds to the set of negative lyapunov exponents of the original linear equation the main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity i e a small deviation from the classical notion of uniform exponential dichotomy in fact we showed that essentially any linear equation v a t v admits a nonuniform exponential dichotomy and thus the above assumption only concerns the smallness of the nonuniformity of the dichotomy this smallness is a rather common phenomenon at least from the point of view of ergodic theory almost all linear variational equations obtained from a measure preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity we emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small only sufficiently small our approach is related to the notion of lyapunov regularity which goes back to lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations

a 2 d orientation adaptive prediction filter in lifting structures for image coding lifting style implementations of wavelets are widely used in image coders a two dimensional 2 d edge adaptive lifting structure which is similar to daubechies 5 3 wavelet is presented the 2 d prediction filter predicts the value of the next polyphase component according to an edge orientation estimator of the image consequently the prediction domain is allowed to rotate spl plusmn 45 spl deg in regions with diagonal gradient the gradient estimator is computationally inexpensive with additional costs of only six subtractions per lifting instruction and no multiplications are required

orientation adaptive subband coding of images in the subband coding of images directionality of image features has thus far been exploited very little the proposed subband coding scheme utilizes orientation of local image features to avoid the highly objectionable gibbs like phenomena observed at reconstructed image edges with conventional subband schemes at low bit rates at comparable bit rates the subjective image quality obtained by our orientation adaptive scheme is considerably enhanced over a conventional separable subband coding scheme as well as other separable approaches such as the jpeg compression standard

lossless image compression based on optimal prediction adaptive lifting and conditional arithmetic coding the optimal predictors of a lifting scheme in the general n dimensional case are obtained and applied for the lossless compression of still images using first quincunx sampling and then simple row column sampling in each case the efficiency of the linear predictors is enhanced nonlinearly directional postprocessing is used in the quincunx case and adaptive length postprocessing in the row column case both methods are seen to perform well the resulting nonlinear interpolation schemes achieve extremely efficient image decorrelation we further investigate context modeling and adaptive arithmetic coding of wavelet coefficients in a lossless compression framework special attention is given to the modeling contexts and the adaptation of the arithmetic coder to the actual data experimental evaluation shows that the best of the resulting coders produces better results than other known algorithms for multiresolution based lossless image coding