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

linear and parabolic tau p transforms revisited new derivations for the conventional linear and parabolic p transforms in the classic continuous function domain provide useful insight into the discrete p transformations for the filtering of unwanted waves such as multiples the derivation of the p transform should define the inverse transform first and then compute the forward transform the forward transform usually requires a p direction deconvolution to improve the resolution in that direction it aids the wave filtering by improving the separation of events in the p domain the p direction deconvolution is required for both the linear and curvilinear p transformations for aperture limited data it essentially compensates for the finite length of the array for the parabolic p transform the deconvolution is required even if the input data have an infinite aperture for sampled data the derived p transform formulas are identical to the drt equations obtained by other researchers numerical examples are presented to demonstrate event

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

joint space frequency segmentation using balanced wavelet packet trees for least cost image representation we examine the question of how to choose a space varying filterbank tree representation that minimizes some additive cost function for an image the idea is that for a particular cost function e g energy compaction or quantization distortion some tree structures perform better than others while the wavelet tree represents a good choice for many signals it is generally outperformed by the best tree from the library of wavelet packet frequency selective trees the double tree library of bases performs better still by allowing different wavelet packet trees over all binary spatial segments of the image we build on this foundation and present efficient new pruning algorithms for both one and two dimensional 1 d and 2 d trees that will find the best basis from a library that is many times larger than the library of the single tree or double tree algorithms the augmentation of the library of bases overcomes the constrained nature of the spatial variation in the double tree bases and is a significant enhancement in practice use of these algorithms to select the least cost expansion for images with a rate distortion cost function gives a very effective signal adaptive compression scheme this scheme is universal in the sense that without assuming a model for the signal or making use of training data it performs very well over a large class of signal types in experiments it achieves compression rates that are competitive with the best training based schemes

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

adaptive transforms for image coding using spatially varying wavelet packets we introduce a novel adaptive image representation using spatially varying wavelet packets wps our adaptive representation uses the fast double tree algorithm introduced previously herley et al 1993 to optimize an operational rate distortion r d cost function as is appropriate for the lossy image compression framework this involves jointly determining which filter bank tree wp frequency decomposition to use and when to change the filter bank tree spatial segmentation for optimality the spatial and frequency segmentations must be done jointly not sequentially due to computational complexity constraints we consider quadtree spatial segmentations and binary wp frequency decompositions corresponding to two channel filter banks for application to image coding we present results verifying the usefulness and versatility of this adaptive representation for image coding using both a first order entropy rate measure based coder as well as a powerful space frequency quantization based spq based wavelet coder introduced by xiong et al 1993

aliasing in the tau p transform and the removal of spatially aliased coherent noise the tau p transform is a discretized radon transform the choice of discretization parameters is a very important part of performing the transform insufficient sampling in the tau direction leads to aliasing problems equivalent to those encountered in any one dimensional time series a simple graphical method illustrates that too coarse sampling in the p direction results in reconstructions containing data duplicated incorrectly at different spatial positions the spacing of these duplications is dependent on the temporal frequency of the data insufficient spatial sampling of the original seismic data causes events to plot at multiple p values in tau p domain again dependent on temporal frequency therefore to velocity filter spatially aliased noise efficiently multiple p values must be filtered the use of appropriate filters in the p f domain can substantially improve the noise rejection capabilities of velocity filters on spatially aliased noise while having little effect on desired reflection sig

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

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

upper bounds for fundamental solutions to non local diffusion equations with divergence free drift we consider some non local diffusion equations in the presence of a divergence free drift term where the diffusion operators are related to certain dirichlet forms of jump type we derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term our class of the velocity includes functions with the scale critical regularity and some growing functions at spatial infinity

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

new image compression techniques using multiwavelets and multiwavelet packets advances in wavelet transforms and quantization methods have produced algorithms capable of surpassing the existing image compression standards like the joint photographic experts group jpeg algorithm for best performance in image compression wavelet transforms require filters that combine a number of desirable properties such as orthogonality and symmetry however the design possibilities for wavelets are limited because they cannot simultaneously possess all of the desirable properties the relatively new field of multiwavelets shows promise in obviating some of the limitations of wavelets multiwavelets offer more design options and are able to combine several desirable transform features the few previously published results of multiwavelet based image compression have mostly fallen short of the performance enjoyed by the current wavelet algorithms this paper presents new multiwavelet transform and quantization methods and introduces multiwavelet packets extensive experimental results demonstrate that our techniques exhibit performance equal to or in several cases superior to the current wavelet filters

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

flexible tree structured signal expansions using time varying wavelet packets addresses the problem of finding the best time varying filter bank tree structured representation for a signal the tree is allowed to vary at regular intervals and the spacing of these changes can be arbitrarily short the question of how to choose tree structured representations of signals based on filter banks is considered wavelets and their adaptive version known as wavelet packets represent one approach that is popular wavelet packets are subband trees where the tree is chosen to match the characteristics of the signal variations where the tree varies over time have been proposed as the double tree and the time frequency tree algorithms time variation adds a further level of adaptivity in all of the approaches proposed so far the tree must be either fixed for the whole duration of the signal or fixed for its dyadic subintervals the solution that we propose as it allows more flexible variation is an advance on the wavelet packet algorithm the double tree algorithm and the recently proposed time frequency tree algorithm our solution is based on casting it in a dynamic programming dp setting focusing on compression applications we use a lagrangian cost of distortion spl lambda spl times rate as the objective function and explain our algorithm in detail pointing out its relation to existing approaches to the problem we demonstrate that the new algorithm indeed searches a larger library of representations than previously possible and that overcoming the constraint of dyadic time segmentations gives a significant improvement in practice

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

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

recovery of the acoustic impedance from reflection seismograms this paper examines the problem of recovering the acoustic impedance from a band limited normal incidence reflection seismogram the convolutional model for the seismogram is adopted at the outset and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible in the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace that is the band limited version of the reflectivity function into the standard recursion formula for the acoustic impedance the formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance moreover the averaging function is identical to that used in deconvolving the initial seismogram the advantage of these averages is that they are unique their disadvantage is that low frequency information which is crucial to making a g

spatial antialias filtering in the slowness frequency domain a rigorous explicit spatial antialias filter is designed and applied to spatially coarsely sampled seismic data by removing all energy above the first nyquist wavenumber and aliased energy that is folded back across the nyquist in the horizontal slowness frequency domain the spatial filtering in the slowness frequency domain is explicit free from any event linearity assumption and does not require any interpolation the spatially aliased energy is dispersive and present at small and large slownesses comparison of the output data after antialias spatial filtering with output data after conventional antialias frequency filtering shows that the filter removes the spatially aliased frequencies selectively at each slowness antialias low pass frequency filtering under or overcorrects for spatial aliasing at all slownesses a seismic gather can be spatially dealiased only at the expense of wavelet spectral changes dealiasing and preservation of amplitude variations with offset are not simultaneously possible

3 d prestack full wavefield inversion a full wavefield inversion algorithm for direct imaging of a 3 d compressional wave velocity distribution is based on the full 3 d scalar wave equation and operates on common source data recorded by areal arrays for each source the method involves reverse time extrapolation of the residual wavefield application of the image condition by crosscorrelation with the source wavefield at each time step produces a 3 d image whose amplitude at each point is proportional to the required velocity update at that point convergence to local minima is mitigated against by gradually increasing the wavenumber bandwidth in the estimated 3 d velocity distribution as iterations proceed starting from the smallest wavenumber the algorithm is illustrated by successful application to synthetic data for a multilayered monocline and for a multilayered structure with the geometry of the standard french model the latter demonstrates good performance with noisy unequally spaced data with significant elevation statics

wavelet radon domain dealiasing and interpolation of seismic data spatial aliasing is unavoidable in some seismic data and has serious effects on the performance of multichannel data processing and migration antialias filtering produces distortion of the signal through the removal of high frequency information in contrast dealiasing produces an unaliased estimate of the signal at all frequencies present in the original time series a new dealiasing algorithm is developed by exploiting the properties of seismic wavefields in the wavelet radon transform domain specifically the overlap of information between wavelet scales at the same frequency the effectiveness of the wavelet radon dealiasing algorithm is demonstrated through the processing of both synthetic and field seismic data

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

wavelet packet image coding using space frequency quantization we extend our previous work on space frequency quantization sfq for image coding from wavelet transforms to the more general wavelet packet transforms the resulting wavelet packet coder offers a universal transform coding framework within the constraints of filterbank structures by allowing joint transform and quantizer design without assuming a priori statistics of the input image in other words the new coder adaptively chooses the representation to suit the image and the quantization to suit the representation experimental results show that for some image classes our new coder gives excellent coding performance

fast adaptive wavelet packet image compression wavelets are ill suited to represent oscillatory patterns rapid variations of intensity can only be described by the small scale wavelet coefficients which are often quantized to zero even at high bit rates our goal is to provide a fast numerical implementation of the best wavelet packet algorithm in order to demonstrate that an advantage can be gained by constructing a basis adapted to a target image emphasis is placed on developing algorithms that are computationally efficient we developed a new fast two dimensional 2 d convolution decimation algorithm with factorized nonseparable 2 d filters the algorithm is four times faster than a standard convolution decimation an extensive evaluation of the algorithm was performed on a large class of textured images because of its ability to reproduce textures so well the wavelet packet coder significantly out performs one of the best wavelet coder on images such as barbara and fingerprints both visually and in term of psnr

seismic trace interpolation in the f x domain interpolation of seismic traces is an effective means of improving migration when the data set exhibits spatial aliasing a major difficulty of standard interpolation methods is that they depend on the degree of reliability with which the various geological events can be separated in this respect a multichannel interpolation method is described which requires neither a priori knowledge of the directions of lateral coherence of the events nor estimation of these directions the method is based on the fact that linear events present in a section made of equally spaced traces may be interpolated exactly regardless of the original spatial interval without any attempt to determine their true dips the predictability of linear events in the f x domain allows the missing traces to be expressed as the output of a linear system the input of which consists of the recorded traces the interpolation operator is obtained by solving a set of linear equations whose coefficients depend only on the spectrum of the spatial prediction filter defined by the recorded traces synthetic examples show that this method is insensitive to random noise and that it correctly handles curvatures and lateral amplitude variations assessment of the method with a real data set shows that the interpolation yields an improved migrated section

wave field transformations of vertical seismic profiles vertical seismic profile vsp data may be partitioned in a variety of ways by application of wave field transformations these transformations provide insights into the nature of the data and aid in the design of processing operations transformations are implemented in a reversible sequence that takes the observed vsp data from the depth time z t domain through the slowness time intercept p domain by a slant stack to the slowness frequency p domain by a 1 d fourier transform over to the wavenumber frequency k domain by resampling using the fourier central slice theorem and finally back to the z t domain by an inverse 2 d fourier transform multidimensional wave field transformations combined with k p and p filtering can be applied to wave field resampling interpolation and extrapolation separation of p waves and s waves separation of upgoing and downgoing waves and wave field decomposition for isolation identification and analysis of arrivals

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

adaptive wavelet packet basis selection for zerotree image coding image coding methods based on adaptive wavelet transforms and those employing zerotree quantization have been shown to be successful we present a general zerotree structure for an arbitrary wavelet packet geometry in an image coding framework a fast basis selection algorithm is developed it uses a markov chain based cost estimate of encoding the image using this structure as a result our adaptive wavelet zerotree image coder has a relatively low computational complexity performs comparably to state of the art image coders and is capable of progressively encoding images

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

a progressive transmission image coder using linear phase uniform filterbanks as block transforms this paper presents a novel image coding scheme using m channel linear phase perfect reconstruction filterbanks lpprfbs in the embedded zerotree wavelet ezw framework introduced by shapiro 1993 the innovation here is to replace the ezws dyadic wavelet transform by m channel uniform band maximally decimated lpprfbs which offer finer frequency spectrum partitioning and higher energy compaction the transform stage can now be implemented as a block transform which supports parallel processing and facilitates region of interest coding decoding for hardware implementation the transform boasts efficient lattice structures which employ a minimal number of delay elements and are robust under the quantization of lattice coefficients the resulting compression algorithm also retains all the attractive properties of the ezw coder and its variations such as progressive image transmission embedded quantization exact bit rate control and idempotency despite its simplicity our new coder outperforms some of the best image coders published previously in the literature for almost all test images especially natural hard to code ones at almost all bit rates

progressive image coding using trellis coded quantization in this work we present coding techniques that enable progressive transmission when trellis coded quantization tcq is applied to wavelet coefficients a method for approximately inverting tcq in the absence of least significant bits is developed results are presented using different rate allocation strategies and different entropy coders the proposed wavelet tcq coder yields excellent coding efficiency while supporting progressive modes analogous to those available in jpeg

seismic trace interpolation in the fourier transform domain a data adaptive interpolation method is designed and applied in the fourier transform domain f k or f kx ky for spatially aliased data the method makes use of fast fourier transforms and their cyclic properties thereby offering a significant cost advantage over other techniques that interpolate aliased data the algorithm designs and applies interpolation operators in the f k or f kx ky domain to fill zero traces inserted in the data in the t x or t x y domain at locations where interpolated traces are needed the interpolation operator is designed by manipulating the lower frequency components of the stretched transforms of the original data this operator is derived assuming that it is the same operator that fills periodically zeroed traces of the original data but at the lower frequencies and corresponds to the f k or f kx ky domain version of the well known f x or f x y domain trace interpolators the method is applicable to 2d and 3d data recorded sparsely in a horizontal plane the most comm

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

comparisons of interpolation methods the purpose of this article is to illustrate the characteristics of various interpolation methods seismic data need to be interpolated when the spatial sampling of acquired data is coarser than the spatial sampling required for a multichannel process this multichannel process can be as simple as stacking traces to reduce coherent noise or as complex as prestack depth migration interpolation requires some assumptions about the character of the seismic data one assumption in the methods considered for this article is that seismic events are linear in practice the data are not perfectly linear and we will show later the results of violating the assumption of linearity

pitfalls of using conventional and discrete radon transforms on poorly sampled data the least square discrete radon transform drt is currently one of the most popular methods used in the suppression of multiples and other coherent noise events on irregularly sampled data gathers used for prestack true amplitude analysis unfortunately in the absence of a priori information this technique suffers from the same aliasing problems as fourier and conventional p methods although the drt is able to reconstruct the original image more accurately than conventional p transforms a harmful by product is an increase in the amplitude of aliased events in the transform domain in particular the drt will boost the amplitude of the aliases of true events that fall outside the p analysis window to help reconstruct the input data these amplified aliases degrade signal periodicity in the p domain if muted they can destroy subtle amplitude changes necessary for amplitude variation with offset avo analysis at the very least one should carefully evaluate the choice of analysis wind