membrane bound turing patterns motivated by recent observations in biological cells we study turing patterns in bounded regions where the nonlinear chemical reactions occur on the boundary and where reagent transport occurs in the bulk within a generic model we formulate the stability problem and discuss the conditions for the occurrence of a turing instability by choosing other model parameters to be unequal we find that turing patterns exist even in the case of equal diffusion constants finally a recently introduced computation technique is utilized to follow the nascent pattern into the highly nonlinear regime

design of regular wavelets using a three step lifting scheme we propose structural multidimensional multichannel filter banks with desirable numbers of vanishing moments for the analysis and synthesis banks for a two channel filter bank we use a three step lifting scheme as opposed to the conventional two step lifting method in order to provide more symmetry between the analysis and synthesis filters we show that the resulting filters have more regularity lower frame bounds ratio and better frequency selectivity we also extend our design to a general multichannel framework three steps of lifting scheme provides us a degree of freedom which we can benefit from toward a more flexible design

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

an orientation selective orthogonal lapped transform a novel critically sampled orientation selective orthogonal lapped transform called the lapped hartley transform lht is derived in a first step overlapping basis functions are generated by modulating basis functions of a 2 d block hartley transform by a cosine wave to achieve invertibility and orthogonality an iterative filter is applied as prefilter in the analysis and as postfilter in the synthesis operation respectively alternatively filtering can be restricted to analysis or synthesis ending up with a biorthogonal transform lht pr lht po a statistical analysis based on a 4000 image data base shows that the lht and lht po have better redundancy removal properties than other block or lapped transforms finally image compression and noise removal examples are given showing the advantages of the lht especially in images containing oriented textures

a filter bank for the directional decomposition of images theory and design the authors introduce a directionally oriented 2 d filter bank with the property that the individual channels may be critically sampled without loss of information the passband regions of the component filters are wedge shaped and thus provide directional information it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed the authors discuss the theory for directional decomposition and reconstruction in addition implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered

the reaction diffusion problem with dynamical boundary condition abstract the aim of this paper is to prove the existence and uniqueness of classical solutions to a coupled system of three parabolic and one ordinary differential equations with the latter determined on the boundary this system describes the receptor toxin antibody interaction problem taking into account diffusion of reacting species

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

nonuniform directional filter banks with arbitrary frequency partitioning directional filter banks dfbs are highly desired in directional representation of images in this correspondence we propose a 2 d nonsubsampled nonuniform directional filter bank nudfb and its design method the proposed nudfb has nonuniform wedge shaped subbands and allows arbitrary frequency partitioning schemes it can extract directional information according to the directional distribution of images this attractive advantage cannot be achieved by the existing directional transforms the design method of the proposed nudfb is based upon the pseudopolar fourier transform by utilizing the geometry property of the pseudopolar grid we employ a 1 d nonsubsampled nonuniform filter bank to obtain a set of nonuniform wedge shaped subbands during the design process only 1 d operations are involved and thus the difficulty encountered in the design of 2 d fan filters is avoided to demonstrate the potential of the proposed nudfb an example on image directional decomposition is given

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

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

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

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

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

positive solutions of quasilinear parabolic systems with nonlinear boundary conditions abstract the aim of this paper is to investigate the existence uniqueness and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions including a system of quasilinear parabolic and ordinary differential equations also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady state solution the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations it is shown that the time dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a porous medium type of problem a heat transfer problem and a two component competition model in ecology these applications illustrate some very interesting distinctive behavior of the time dependent solutions between density independent and density dependent diffusions

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

design of linear phase recombination nonuniform filter banks this correspondence proposes a novel method for designing a class of recombination nonuniform filter banks rnfbs with the linear phase lp property in the structure of the proposed nonuniform filter bank fb certain channels of an m channel uniform fb are merged by synthesis filters of transmultiplexers tmuxs yielding nonuniform subbands by a detailed analysis of the frequency characteristics of the rnfbs we derive the existence and matching conditions that are necessary for constructing the proposed rnfbs having good frequency responses with those conditions being satisfied the design problem becomes that of a set of lp uniform fbs by using the parks mcclellan algorithm we are able to design the near perfect reconstruction lp rnfbs in a very simple and efficient way as demonstrated by two examples

liouville type theorems for fourth order elliptic equations in a half plane consider an elliptic equation wawoa2p0 0 in the half plane x y 00 0 with boundary conditions s sy 0 o if y 0 x 0 and bj p 0 if y 0 x 0 w 0 and for some e so iw 0 w 4 0 and ii w 0 both methods can be extended to other linear elliptic boundary value problems in a half plane 0 introduction a liouville type theorem for an elliptic operator p d of order 2m with constant coefficients asserts that if 0 1 p d oinr n and 0 2 c x o rn as r oo where r jxl then p x is a polynomial there are two simple and entirely different methods for proving such a result method 1 suppose p d is a positive operator in the sense that 0 3 j p d o r bdx co j lb12dx co 0 r n r n for every 0 which decays at oo say like 0 r m if p d is homogenous then from 0 2 we can deduce by applying elliptic estimates to a scaled solution that d p o r m if lal m n as r oo received by the editors april 6 1995 1991 mathematics subject classification primary 35j40

critically sampled wavelets with composite dilations wavelets with composite dilations provide a general framework for the construction of waveforms defined not only at various scales and locations as traditional wavelets but also at various orientations and with different scaling factors in each coordinate as a result they are useful to analyze the geometric information that often dominate multidimensional data much more efficiently than traditional wavelets the shearlet system for example is a particular well known realization of this framework which provides optimally sparse representations of images with edges in this paper we further investigate the constructions derived from this approach to develop critically sampled wavelets with composite dilations for the purpose of image coding not only do we show that many nonredundant directional constructions recently introduced in the literature can be derived within this setting but we also introduce new critically sampled discrete transforms that achieve much better nonlinear approximation rates than traditional discrete wavelet transforms and outperform the other critically sampled multiscale transforms recently proposed

directional multiscale modeling of images using the contourlet transform the contourlet transform is a new two dimensional extension of the wavelet transform using multiscale and directional filter banks the contourlet expansion is composed of basis images oriented at various directions in multiple scales with flexible aspect ratios given this rich set of basis images the contourlet transform effectively captures smooth contours that are the dominant feature in natural images we begin with a detailed study on the statistics of the contourlet coefficients of natural images using histograms to estimate the marginal and joint distributions and mutual information to measure the dependencies between coefficients this study reveals the highly non gaussian marginal statistics and strong interlocation interscale and interdirection dependencies of contourlet coefficients we also find that conditioned on the magnitudes of their generalized neighborhood coefficients contourlet coefficients can be approximately modeled as gaussian random variables based on these findings we model contourlet coefficients using a hidden markov tree hmt model with gaussian mixtures that can capture all interscale interdirection and interlocation dependencies we present experimental results using this model in image denoising and texture retrieval applications in denoising the contourlet hmt outperforms other wavelet methods in terms of visual quality especially around edges in texture retrieval it shows improvements in performance for various oriented textures

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

optimal design of nonuniform fir transmultiplexer using semi infinite programming this correspondence considers an optimum nonuniform finite impulse response fir transmultiplexer design problem subject to specifications in the frequency domain our objective is to minimize the sum of the ripple energy for all the individual filters subject to the specifications on amplitude and aliasing distortions and to the passband and stopband specifications for the individual filters this optimum nonuniform transmultiplexer design problem can be formulated as a quadratic semi infinite programming problem the dual parametrization algorithm is extended to this nonuniform transmultiplexer design problem if the lengths of the filters are sufficiently long and the set of decimation integers is compatible then a solution exists since the problem is formulated as a convex problem if a solution exists then the solution obtained is unique and the local solution is a global minimum

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

a framework for discrete integral transformations ii the 2d discrete radon transform although naturally at the heart of many fundamental physical computations and potentially useful in many important image processing tasks the radon transform lacks a coherent discrete definition for two dimensional 2d discrete images which is algebraically exact invertible and rapidly computable we define a notion of 2d discrete radon transforms for 2d discrete images which is based on summation along lines of absolute slope less than 1 values at nongrid locations are defined using trigonometric interpolation on a zero padded grid our definition is shown to be geometrically faithful the summation avoids wrap around effects our proposal uses a special collection of lines in mathbb r 2 for which the transform is rapidly computable and invertible we describe a fast algorithm using o n log n operations where n n 2 is the number of pixels in the image the fast algorithm exploits a discrete projection slice theorem which associates the discrete radon transform with the pseudopolar fourier transform a averbuch et al siam j sci comput 30 2008 pp 764 784 our definition for discrete images converges to a natural continuous counterpart with increasing refinement

efficient algorithm for solving semi infinite programming problems and their applications to nonuniform filter bank designs an efficient algorithm for solving semi infinite programming problems is proposed in this paper the index set is constructed by adding only one of the most violated points in a refined set of grid points by applying this algorithm for solving the optimum nonuniform symmetric antisymmetric linear phase finite impulse response fir filter bank design problems the time required to obtain a globally optimal solution is much reduced compared with that of the previous proposed algorithm

image coding using dual tree discrete wavelet transform in this paper we explore the application of 2 d dual tree discrete wavelet transform ddwt which is a directional and redundant transform for image coding three methods for sparsifying ddwt coefficients i e matching pursuit basis pursuit and noise shaping are compared we found that noise shaping achieves the best nonlinear approximation efficiency with the lowest computational complexity the interscale intersubband and intrasubband dependency among the ddwt coefficients are analyzed three subband coding methods i e spiht ebcot and tce are evaluated for coding ddwt coefficients experimental results show that tce has the best performance in spite of the redundancy of the transform our ddwt tce scheme outperforms jpeg2000 up to 0 70 db at low bit rates and is comparable to jpeg2000 at high bit rates the ddwt tce scheme also outperforms two other image coders that are based on directional filter banks to further improve coding efficiency we extend the ddwt to an anisotropic dual tree discrete wavelet packets addwp which incorporates adaptive and anisotropic decomposition into ddwt the addwp subbands are coded with tce coder experimental results show that addwp tce provides up to 1 47 db improvement over the ddwt tce scheme outperforming jpeg2000 up to 2 00 db reconstructed images of our coding schemes are visually more appealing compared with dwt based coding schemes thanks to the directionality of wavelets

bivariate shrinkage functions for wavelet based denoising exploiting interscale dependency most simple nonlinear thresholding rules for wavelet based denoising assume that the wavelet coefficients are independent however wavelet coefficients of natural images have significant dependencies we only consider the dependencies between the coefficients and their parents in detail for this purpose new non gaussian bivariate distributions are proposed and corresponding nonlinear threshold functions shrinkage functions are derived from the models using bayesian estimation theory the new shrinkage functions do not assume the independence of wavelet coefficients we show three image denoising examples in order to show the performance of these new bivariate shrinkage rules in the second example a simple subband dependent data driven image denoising system is described and compared with effective data driven techniques in the literature namely visushrink sureshrink bayesshrink and hidden markov models in the third example the same idea is applied to the dual tree complex wavelet coefficients

multidimensional mapping based complex wavelet transforms although the discrete wavelet transform dwt is a powerful tool for signal and image processing it has three serious disadvantages shift sensitivity poor directionality and lack of phase information to overcome these disadvantages we introduce multidimensional mapping based complex wavelet transforms that consist of a mapping onto a complex function space followed by a dwt of the complex mapping unlike other popular transforms that also mitigate dwt shortcomings the decoupled implementation of our transforms has two important advantages first the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy this allows us to create a directional nonredundant complex wavelet transform with potential benefits for image coding systems to the best of our knowledge no other complex wavelet transform is simultaneously directional and nonredundant the second advantage of our approach is the flexibility to use any dwt in the transform implementation as an example we exploit this flexibility to create the complex double density dwt a shift insensitive directional complex wavelet transform with a low redundancy of 3 sup m 1 2 sup m 1 in m dimensions no other transform achieves all these properties at a lower redundancy to the best of our knowledge by exploiting the advantages of our multidimensional mapping based complex wavelet transforms in seismic signal processing applications we have demonstrated state of the art results

improved structures of maximally decimated directional filter banks for spatial image analysis this paper introduces an improved structure for directional filter banks dfbs that preserves the visual information in the subband domain the new structure achieves this outcome while preserving both the efficient polyphase implementation and the exact reconstruction property the paper outlines a step by step framework in which to examine the dfb and within this framework discusses how through the insertion of post sampling matrices visual distortions can be removed in addition to the efficient tree structure attention is given to the form and design of efficient linear phase filters most notably linear phase iir prototype filters are presented together with the design details these filters can enable the dfb to have more than a three fold improvement in complexity reduction over quadrature mirror filters qmfs

positive solutions of quasilinear parabolic systems with dirichlet boundary condition abstract coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems including systems of parabolic and ordinary differential equations are investigated the aim of this paper is to show the existence uniqueness and asymptotic behavior of time dependent solutions also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system the elliptic operators in both systems are allowed to be degenerate in the sense that the density dependent diffusion coefficients d i u i may have the property d i 0 0 for some or all i 1 n and the boundary condition is u i 0 using the method of upper and lower solutions we show that a unique global classical time dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions and if the maximal and minimal solutions coincide then the steady state solution is unique and the time dependent solution converges to the unique solution applications of these results are given to three model problems including a scalar polynomial growth problem a coupled system of polynomial growth problem and a two component competition model in ecology

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

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

nonlinear diffusion problem with dynamical boundary value a nonlinear degenerate convection diffusion initial boundary value problem is studied in a bounded domain a dynamical boundary condition containing the time derivative of a solution is prescribed on the one part of the boundary this models a non perfect contact on the boundary the existence and uniqueness of a weak solution in corresponding function spaces is proved using the backward euler method for the time discretization error estimates for time discrete approximations are derived

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

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

multiresolution direction filterbanks theory design and applications this paper presents new developments of directional filterbanks dfbs the motivation for the paper is the existence of multiresolution and multidirection orthogonal transform for two dimensional 2 d discrete signals based on the frequency supports of the ideal transform a new uniformly maximally decimated dfb with six highpass directional subbands and two lowpass subands is introduced the uniform dfb udfb can be implemented by a binary tree structure of two channel filterbanks the filterbank employed in the tree is shown to be alias free decimation and permissible the udfb is then extended to a nonuniform case nudfb which is still maximally decimated by combining the two lowpass subbands the nudfb yields nonuniform frequency division which composes of one lowpass filter with a decimation factor of one fourth and six highpass directional filters with a decimation factor of one eighth the new dfbs offer alternative image decompositions which overcome the limited directional selectivity of the separable wavelets and the limited multiresolution of the conventional dfb the lowpass subband of the nudfb can be used to obtain a multiresolution representation by simply reiterating the same nudfb decomposition on the other hand the directional subbands can also be further refined by simply applying a two channel conventional dfb at each highpass component a simple design method yielding near orthogonal uniform and nonuniform multidimensional filterbanks is presented finally the performances of the newly proposed nudfb are compared with other conventional transforms in nonlinear approximation image denoising and texture classification to demonstrate its potential

a framework for discrete integral transformations i the pseudopolar fourier transform the fourier transform of a continuous function evaluated at frequencies expressed in polar coordinates is an important conceptual tool for understanding physical continuum phenomena an analogous tool suitable for computations on discrete grids could be very useful however no exact analogue exists in the discrete case in this paper we present the notion of pseudopolar grid pp grid and the pseudopolar fourier transform ppft which evaluates the discrete fourier transform at points of the pp grid the pp grid is a type of concentric squares grid in which the radial density of squares is twice as high as usual the pp grid consists of equally spaced samples along rays where different rays are equally spaced in slope rather than angle we develop a fast algorithm for the ppft with the same complexity order as the cartesian fast fourier transform the algorithm is stable invertible requires only one dimensional operations and uses no approximate interpolations we prove that the ppft is invertible and develop two algorithms for its inversion iterative and direct both with complexity o n 2 log n where n times n is the size of the reconstructed image the iterative algorithm applies conjugate gradients to the gram operator of the ppft since the transform is ill conditioned we introduce a preconditioner which significantly accelerates the convergence the direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps first it resamples the pp grid to a cartesian frequency grid and then recovers the image from the cartesian frequency grid

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

a new class of two channel biorthogonal filter banks and wavelet bases proposes a novel framework for a new class of two channel biorthogonal filter banks the framework covers two useful subclasses i causal stable iir filter banks ii linear phase fir filter banks there exists a very efficient structurally perfect reconstruction implementation for such a class filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity the properties of such a class are discussed in detail the design of the analysis synthesis systems reduces to the design of a single transfer function very simple design methods are given both for fir and iir cases zeros of arbitrary multiplicity at aliasing frequency can be easily imposed for the purpose of generating wavelets with regularity property in the iir case two new classes of iir maximally flat filters different from butterworth filters are introduced the filter coefficients are given in closed form the wavelet bases corresponding to the biorthogonal systems are generated the authors also provide a novel mapping of the proposed 1 d framework into 2 d the mapping preserves the following i perfect reconstruction ii stability in the iir case iii linear phase in the fir case iv zeros at aliasing frequency v frequency characteristic of the filters

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

space time frequency processing of acoustic wave fields theory algorithms and applications consider a nonparametric representation of acoustic wave fields that consists of observing the sound pressure along a straight line or a smooth contour l defined in space the observed data contains implicit information of the surrounding acoustic scene both in terms of spatial arrangement of the sources and their respective temporal evolution we show that such data can be effectively analyzed and processed in what we call the space time frequency representation space consisting of a gabor representation across the spatio temporal manifold defined by the spatial axis l and the temporal axis t in the presence of a source the spectral patterns generated at l have a characteristic triangular shape that changes according to certain parameters such as the source distance and direction the number of sources the concavity of l and the analysis window size yet in general the wave fronts can be expressed as a function of elementary directional components most notably plane waves and far field components furthermore we address the problem of processing the wave field in discrete space and time i e sampled along l and t where a gabor representation implies that the wave fronts are processed in a block wise fashion the key challenge is how to chose and customize a spatio temporal filter bank such that it exploits the physical properties of the wave field while satisfying strict requirements such as perfect reconstruction critical sampling and computational efficiency we discuss the architecture of such filter banks and demonstrate their applicability in the context of real applications such as spatial filtering deconvolution and wave field coding

translation invariant contourlet transform and its application to image denoising most subsampled filter banks lack the feature of translation invariance which is an important characteristic in denoising applications in this paper we study and develop new methods to convert a general multichannel multidimensional filter bank to a corresponding translation invariant ti framework in particular we propose a generalized algorithme agrave trous which is an extension of the algorithme agrave trous introduced for 1 d wavelet transforms using the proposed algorithm as well as incorporating modified versions of directional filter banks we construct the ti contourlet transform tict to reduce the high redundancy and complexity of the tict we also introduce semi translation invariant contourlet transform stict then we employ an adapted bivariate shrinkage scheme to the stict to achieve an efficient image denoising approach our experimental results demonstrate the benefits and potential of the proposed denoising approach complexity analysis and efficient realization of the proposed ti schemes are also presented

a class of multiresolution directional filter banks in this paper we introduced a class of directional filter banks dfbs having the previously proposed uniform dfb udfb as a special case except for the udfb each dfb in this class can be used to decompose an image yielding up to 12 directions while maintaining perfect reconstruction and maximal decimation a multiresolution representation can be obtained by repeating the same decomposition at the lowpass band the permissible property of the filter banks in cases of being implemented by a tree structure and by direct implementation is discussed the result shows that only one dfb in the class called the uniform quincunx dfb uqdfb satisfies the permissible property when being implemented directly without using the tree structure the nonuniform quincunx dfb nuqdfb is then constructed from the uqdfb by merging its two lowpass subbands an alternative structure for constructing the nuqdfb is presented the new structure while yielding the same frequency partitioning allows the dfb to be realized with complexity comparable to that of the separable wavelet filter bank the connection between the discrete filter bank and the continuous directional wavelet is also established numerical experiments on directional feature extractions image denoising and nonlinear approximation are presented at the end of the paper to demonstrate the potential of the nuqdfb

multiresolution image representation using combined 2 d and 1 d directional filter banks in this paper effective multiresolution image representations using a combination of 2 d filter bank fb and directional wavelet transform wt are presented the proposed methods yield simple implementation and low computation costs compared to previous 1 d and 2 d fb combinations or adaptive directional wt methods furthermore they are nonredundant transforms and realize quad tree like multiresolution representations in applications on nonlinear approximation image coding and denoising the proposed filter banks show visual quality improvements and have higher psnr than the conventional separable wt or the contourlet