the discrete wavelet transform for a symmetric anti symmetric multiwavelet family on the interval the chui lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval this requires in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets four new functions of each type at each edge of the interval i e one more than required just to preserve the approximation order the criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions and a specific choice is examined in applications to both noiseless and noisy data these are accomplished through the development of interval basis extensions for both an earlier multiwavelet projection based prefilter postfilter combination and a multiwavelet vector thresholding algorithm

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

the discrete multiple wavelet transform and thresholding methods we propose thresholding for multiwavelets considering the coefficient vectors as a whole rather than thresholding individual elements a multivariate universal threshold is obtained using the spl chi sup 2 distribution simulations indicate that using the ghm multiwavelet with appropriate preprocessing our method outperforms univariate thresholding of both ghm and daubechies wavelet decompositions

the application of multiwavelet filterbanks to image processing multiwavelets are a new addition to the body of wavelet theory realizable as matrix valued filterbanks leading to wavelet bases multiwavelets offer simultaneous orthogonality symmetry and short support which is not possible with scalar two channel wavelet systems after reviewing this theory we examine the use of multiwavelets in a filterbank setting for discrete time signal and image processing multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank we describe two methods repeated row and approximation deapproximation for obtaining such a vector input stream from a one dimensional 1 d signal algorithms for symmetric extension of signals at boundaries are then developed and naturally integrated with approximation based preprocessing we describe an additional algorithm for multiwavelet processing of two dimensional 2 d signals two rows at a time and develop a new family of multiwavelets the constrained pairs that is well suited to this approach this suite of novel techniques is then applied to two basic signal processing problems denoising via wavelet shrinkage and data compression after developing the approach via model problems in one dimension we apply multiwavelet processing to images frequently obtaining performance superior to the comparable scalar wavelet transform

equivalent lagrangians multidimensional case we generalize a theorem known for one dimensional nonsingular equivalent lagrangians l and to the multidimensional case in particular we prove that the matrix which relates the left hand sides of the euler lagrange equations obtained from l and is such that the trace of all its integer powers are constants of the motion we construct several multidimensional examples in which the elements of are functions of position velocity and time and prove that in some cases equivalence prevails even if det 0

the equivariant inverse problem and the maxwell equations in affirmative the equivariant inverse problem for maxwell type euler lagrange expressions is solved this allows the proof of the uniqueness of the maxwell equations

optimization of functional mri for detection decoding and high resolution imaging of the response patterns of cortical columns the capacity of functional mri fmri to resolve cortical columns depends on several factors these include the spatial scale of the columnar pattern the point spread of the fmri response the voxel size and the signal to noise ratio snr considering thermal and physiological noise however it remains unknown how these factors combine and what is the voxel size that optimizes fmri of cortical columns here we combine current knowledge into a quantitative model of fmri of realistic patterns of cortical columns with different spatial scales and degrees of irregularity we compare different approaches for identifying patterns of cortical columns including univariate and multivariate based detection multi voxel pattern analysis mvpa based decoding and high resolution imaging and reconstruction of the pattern of cortical columns we present the dependence of the performance of each approach on the parameters of the imaged pattern as well as those of the data acquisition in addition we predict voxel sizes that optimize fmri of cortical columns under various scenarios we found that all measures associated with multivariate detection and decoding could be approximately calculated from a measure we termed multivariate contrast to noise ratio mv cnr which is a function of the contrast to noise ratio cnr and number of voxels furthermore mv cnr implied that the optimal voxel width for detection and decoding is independent of changes in response amplitude snr and imaged volume that are not caused by changes in voxel size for regular patterns optimal voxel widths for detection decoding and imaging reconstructing the pattern of cortical columns were approximately half the main cycle length of the organization optimal voxel widths for irregular patterns were less dependent on the main cycle length and differed between univariate detection multivariate detection and decoding and reconstruction we compared the effects of different factors of gradient echo fmri at 3 tesla t gradient echo fmri at 7t and spin echo fmri at 7t on the detection decoding and reconstruction measures considered and found that in all cases the width of the fmri point spread had the most significant effect in contrast different response amplitudes and noise characteristics played a relatively minor role we recommend specific voxel widths for optimal univariate detection for multivariate detection and decoding and for high resolution imaging of cortical columns under these three data acquisition scenarios our study supports the planning optimization and interpretation of high resolution fmri of cortical columns and the decoding of information conveyed by these columns

alternative lagrangians for spherically symmetric potentials a new geometrical account of the existence of alternative s equivalent lagrangians is given for a particle moving in three dimensional euclidean space under the influence of a central force the fact that the lagrangian for this familiar problem is not unique was pointed out by henneaux and shepley j math phys 23 2101 1982 the present analysis makes the reasons for the phenomenon more transparent than theirs

balanced multiwavelets theory and design this article deals with multiwavelets which are a generalization of wavelets in the context of time varying filter banks and with their applications to signal processing and especially compression by their inherent structure multiwavelets are fit for processing multichannel signals this is the main issue in which we are interested first we review material on multiwavelets and their links with multifilter banks and especially time varying filter banks then we have a close look at the problems encountered when using multiwavelets in applications and we propose new solutions for the design of multiwavelets filter banks by introducing the so called balanced multiwavelets

interpolating multiwavelet bases and the sampling theorem this paper considers the classical sampling theorem in multiresolution spaces with scaling functions as interpolants as discussed by xia and zhang 1993 for an orthogonal scaling function to support such a sampling theorem the scaling function must be cardinal interpolating they also showed that the only orthogonal scaling function that is both cardinal and of compact support is the haar function which is not continuous this paper addresses the same question but in the multiwavelet context where the situation is different this paper presents the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal the scaling functions thereby support a shannon like sampling theorem such wavelet bases are appealing because the initialization of the discrete wavelet transform prefiltering is the identity operator

quadrature prefilters for the discrete wavelet transform discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data prefilters based on lagrange interpolants are derived here for biorthogonal compact support wavelet systems providing exact subspace projection in cases of local polynomial smoothness the resulting divergence acceleration in a nonpolynomial test case is examined irregular sampling rates are also accommodated

multiwavelet bases with extra approximation properties this paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order because multiwavelet bases normally lack important properties that traditional wavelet bases of equal approximation order possess the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step prefiltering this paper examines the properties and design of orthogonal multiwavelet bases with approximation order 1 that possess those properties that are normally absent for these balanced bases so named by lebrun and vetterli see proc ieee int conf acoust speech signal process icassp munich germany vol 3 p 2473 76 1997 and ibid vol 46 p 1119 24 1998 prefiltering can be avoided by reorganizing the multiwavelet filter bank the development in this paper draws from results regarding the approximation order of m band wavelet bases the main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of z sup 2r 1 z sup 1 1 for traditional wavelets r 1 this specializes to the usual factor z 1 sup k

translation invariant denoising using multiwavelets translation invariant ti single wavelet denoising was developed by coifman and donoho 1994 and they show that ti is better than non ti single wavelet denoising on the other other hand strela et al 1994 have found that non ti multiwavelet denoising gives better results than non ti single wavelets we extend coifman and donoho s ti single wavelet denoising scheme to multiwavelets experimental results show that ti multiwavelet denoising is better than the single case for soft thresholding

balanced multiwavelet bases based on symmetric fir filters this paper describes a basic difference between multiwavelets and scalar wavelets that explains without using zero moment properties why certain complications arise in the implementation of discrete multiwavelet transforms assuming we wish to avoid the use of prefilters in implementing the discrete multiwavelet transform it is suggested that the behavior of the iterated filter bank associated with a multiwavelet basis of multiplicity r is more fully revealed by an expanded set of r sup 2 scaling functions spl phi sub i j this paper also introduces new k balanced orthogonal multiwavelet bases based on symmetric fir filters the nonlinear design equations arising in this work are solved using the grobner basis the minimal length k balanced multiwavelet bases based on even length symmetric fir filters are better behaved than those based on odd length symmetric fir filters as illustrated by special relations they satisfy and by examples constructed

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

adaptive multiwavelet initialization the multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals the multiple scaling functions extract the lowpass information of the input data set and the multiwavelets extract the bandpass information when the lowpass filters associated with the scaling functions are cascaded a multiresolution pyramid decomposition reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs however there is only one data set available to initialize this process this paper addresses the question of how to best modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application the best prefilters are determined by the minimization of the energy of preselected decomposition components the resulting decomposition is therefore signal adaptive and under appropriate conditions perfect reconstruction of the input data set can be achieved with a proper postfiltering the detailed discussions in the paper are focussed on the two wavelet and two scaling function case the general multiwavelet case is addressed at the end of the paper a compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors

equivalence and s equivalence of vector tensor lagrangians it will be proven that if a gauge invariant lagrangian density having the local form l l gij ai ai j is such that its euler lagrange equations ei l 0 have the same set of solutions as ei l0 0 where l0 g1 2fijfij then l and cl0 are equivalent for same constant c i e ei l ei cl0 from a previous result it follows that l cl0 d eg1 2 where d is a divergence and e is a constant

improved optimization for the robust and accurate linear registration and motion correction of brain images linear registration and motion correction are important components of structural and functional brain image analysis most modern methods optimize some intensity based cost function to determine the best registration to date little attention has been focused on the optimization method itself even though the success of most registration methods hinges on the quality of this optimization this paper examines the optimization process in detail and demonstrates that the commonly used multiresolution local optimization methods can and do get trapped in local minima to address this problem two approaches are taken 1 to apodize the cost function and 2 to employ a novel hybrid global local optimization method this new optimization method is specifically designed for registering whole brain images it substantially reduces the likelihood of producing misregistrations due to being trapped by local minima the increased robustness of the method compared to other commonly used methods is demonstrated by a consistency test in addition the accuracy of the registration is demonstrated by a series of experiments with motion correction these motion correction experiments also investigate how the results are affected by different cost functions and interpolation methods

on the inverse problem of the calculus of variations in field theory the inverse problem of the calculus of variations is investigated in the case of field theory uniqueness of the action principle is demonstrated for the vector laplace equation in a non decomposable riemannian space as well as for the harmonic map equation

multiwavelet neural network and its approximation properties a model of multiwavelet based neural networks is proposed its universal and l sup 2 approximation properties together with its consistency are proved and the convergence rates associated with these properties are estimated the structure of this network is similar to that of the wavelet network except that the orthonormal scaling functions are replaced by orthonormal multiscaling functions the theoretical analyses show that the multiwavelet network converges more rapidly than the wavelet network especially for smooth functions to make a comparison between both networks experiments are carried out with the lemarie meyer wavelet network the daubechies2 wavelet network and the ghm multiwavelet network and the results support the theoretical analysis well in addition the results also illustrate that at the jump discontinuities the approximation performance of the two networks are about the same

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

compactly supported wavelets and boundary conditions abstract wavelet bases subject to various homogeneous boundary conditions on an interval are constructed they are obtained from daubechies compactly supported wavelets which are restricted to the interval following an idea by y meyer and then suitably modified the sobolev spaces arising from homogeneous boundary problems are characterized in terms of wavelet coefficients in these bases

radial bias is not necessary for orientation decoding multivariate pattern analysis can be used to decode the orientation of a viewed grating from fmri signals in early visual areas although some studies have reported identifying multiple sources of the orientation information that make decoding possible a recent study argued that orientation decoding is only possible because of a single source a coarse scale retinotopically organized preference for radial orientations here we aim to resolve these discrepant findings we show that there were subtle but critical experimental design choices that led to the erroneous conclusion that a radial bias is the only source of orientation information in fmri signals in particular we show that the reliance on a fast temporal encoding paradigm for spatial mapping can be problematic as effects of space and time become conflated and lead to distorted estimates of a voxel s orientation or retinotopic preference when we implement minor changes to the temporal paradigm or to the visual stimulus itself by slowing the periodic rotation of the stimulus or by smoothing its contrast energy profile we find significant evidence of orientation information that does not originate from radial bias in an additional block paradigm experiment where space and time were not conflated we apply a formal model comparison approach and find that many voxels exhibit more complex tuning properties than predicted by radial bias alone or in combination with other known coarse scale biases our findings support the conclusion that radial bias is not necessary for orientation decoding in addition our study highlights potential limitations of using temporal phase encoded fmri designs for characterizing voxel tuning properties

projection based prefiltering for multiwavelet transforms we introduce a method for initializing the multiwavelet decomposition algorithm the initialization procedure is the orthogonal projection of the input signal into the space defined by the multiscaling function the approach will always have a solution places no restrictions on the input except that it be contained within l sub 2 and can be implemented in a fast algorithm we present the details of our approach and compare it with another proposed method of prefiltering

a new prefilter design for discrete multiwavelet transforms in conventional wavelet transforms prefiltering is not necessary due to the lowpass property of a scaling function this is no longer true for multiwavelet transforms a few research papers on the design of prefilters have appeared but the existing prefilters are usually not orthogonal which often causes problems in coding moreover the condition on the prefilters was imposed based on the first step discrete multiwavelet decomposition we propose a new prefilter design that combines the ideas of the conventional wavelet transforms and multiwavelet transforms the prefilters are orthogonal but nonmaximally decimated they are derived from a very natural calculation of multiwavelet transform coefficients in this new prefilter design multiple step discrete multiwavelet decomposition is taken into account our numerical examples by taking care of the redundant prefiltering indicate that the energy compaction ratio with the geronimo hardin massopust 1994 2 wavelet transform and our new prefiltering is better than the one with daubechies d sub 4 wavelet transform

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed