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

generalized transition waves and their properties in this paper we generalize the usual notions of waves fronts and propagation speeds in a very general setting these new notions which cover all usual situations involve uniform limits with respect to the geodesic distance to a family of hypersurfaces that are parametrized by time we prove the existence of new such waves for some time dependent reaction diffusion equations as well as general intrinsic properties some monotonicity properties and some uniqueness results for almost planar fronts the classification results which are obtained under some appropriate assumptions show the robustness of our general definitions 2012 wiley periodicals inc

symmetry and bifurcations of momentum mappings the zero set of a momentum mapping is shown to have a singularity at each point with symmetry the zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function the proof uses the kuranishi theory of deformations among the applications it is shown that the set of all solutions of the yang mills equations on a lorentz manifold has a singularity at any solution with symmetry in the sense of a pure gauge symmetry similarly the set of solutions of einstein s equations has a singularity at any solution that has spacelike killing fields provided the spacetime has a compact cauchy surface

segmentation of bright targets using wavelets and adaptive thresholding a general systematic method for the detection and segmentation of bright targets is developed we use the term bright target to mean a connected cohesive object which has an average intensity distribution above that of the rest of the image we develop an analytic model for the segmentation of targets which uses a novel multiresolution analysis in concert with a bayes classifier to identify the possible target areas a method is developed which adaptively chooses thresholds to segment targets from background by using a multiscale analysis of the image probability density function pdf a performance analysis based on a gaussian distribution model is used to show that the obtained adaptive threshold is often close to the bayes threshold the method has proven robust even when the image distribution is unknown examples are presented to demonstrate the efficiency of the technique on a variety of targets

automatic target segmentation by locally adaptive image thresholding a locally adaptive thresholding algorithm concerning the extraction of targets from a given field of background is proposed conventional histogram based or global type methods are deficient in detecting small targets of possibly low contrast as well the present research is notable for solving the mentioned problems by introducing 1 shape connectivity measure based on co occurrence statistics for threshold evaluation and 2 no target identification procedure for modeling a local processing paradigm in this manner thresholds are determined adaptively even in the presence of space varying noise or clutter experiments show that the results are reliable and even outperform those that manual operations can achieve for global thresholding

a multi dimensional bistable nonlinear diffusion equation in a periodic medium in this work we study the existence of wave solutions for a scalar reaction diffusion equation of bistable type posed in a multi dimensional periodic medium roughly speaking our result states that bistability ensures the existence of waves for both balanced and unbalanced reaction term here the term wave is used to describe either pulsating travelling wave or standing transition solution as a special case we study a two dimensional heterogeneous allen cahn equation in both cases of slowly varying medium and rapidly oscillating medium we prove that bistability occurs in these two situations and we conclude to the existence of waves connecting u 0 and u 1 moreover in a rapidly oscillating medium we derive a sufficient condition that guarantees the existence of pulsating travelling waves with positive speed in each direction

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

a new class of biorthogonal wavelet systems for image transform coding we construct general biorthogonal coifman wavelet systems a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair a time domain design method is employed and closed form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived the resulting filter coefficients are all dyadic fractions which is an attractive feature in the realization of multiplication free discrete wavelet transform even ordered systems in this family are symmetric which correspond to linear phase dual filters in particular three filterbanks fbs in this family are systematically verified to have competitive compression potential to the 9 7 tap biorthogonal wavelet fb by cohen et al 1992 which is currently the most widely used one in the field of wavelet transform coding in addition the proposed fb s have much smaller computational complexity in terms of floating point operations required in transformation and therefore indicate a better tradeoff between compression performance and computational complexity

filter design and comparison for two fast cwt algorithms the continuous wavelet transform cwt is a powerful technique for signal analysis direct cwt computation by fft requires o n log sub 2 n operations per scale where n is the data length the a trous algorithm and the shensa 1992 algorithm are two fast methods to compute cwt recursively that require only o n operations per scale both of them can be described by the multiresolution analysis mra structure but with different mra filters this paper proposes methods to design the mra filters of the two algorithms to improve their accuracy on cwt computation we begin with the formulation of the cwt computation error using the mra structure the mra filters of the two algorithms are then designed to minimize the error in either algorithm both the lowpass and bandpass mra filters can be optimized the a trous algorithm has closed form solutions for the two filters the shensa algorithm on the other hand has an analytic solution for the bandpass filter only finding the optimum lowpass filter requires a multidimensional numerical search simulation studies show that by using the proposed optimum filters the shensa algorithm in general outperforms the a trous algorithm

existence and convergence to a propagating terrace in one dimensional reaction diffusion equations we consider one dimensional reaction diffusion equations for a large class of spatially periodic nonlinearities including multistable ones and study the asymptotic behavior of solutions with heaviside type initial data our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front but by a layer of several fronts which we call a terrace existence and convergence to such a terrace is proven by using an intersection number argument without much relying on standard linear analysis hence on top of the peculiar phenomenon of propagation that our work highlights several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

implementation of split radix fft algorithms for complex real and real symmetric data a new algorithm is presented for the fast computation of the discrete fourier transform this algorithm belongs to that class of recently proposed 2n fft s which present the same arithmetic complexity the lowest among any previously published one moreover this algorithm has the advantage of being performed in place by repetitive use of a butterfly type structure without any data reordering inside the algorithm furthermore it can easily be applied to real and real symmetric data with reduced arithmetic complexity by removing all redundancy in the algorithm

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

pulsating solutions for multidimensional bistable and multistable equations we devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction diffusion equations in arbitrary dimension in both bistable and more general multistable frameworks in the multistable case the notion of a single front is not sufficient to understand the dynamics of solutions and we instead observe the appearance of a so called propagating terrace this roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered surprisingly for a given equation the shape of this terrace i e the involved intermediate states or even the cardinality of the family of fronts may depend on the direction of propagation

differential graded lie algebras and singularities of level sets of momentum mappings the germ of an analytic varietyx at a pointx x is said to bequadratic if it is bi analytically isomorphic to the germ of a cone defined by a system of homogeneous quadratic equations at the origin arms marsden and moncrief show in 2 that under certain conditions the analytic germ of a level set of a momentum mapping is quadratic we discuss related ideas in a more algebraic context by associating to an affine hamiltonian action a differential graded lie algebra which in the presence of an invariant positive complex structure is formal in the sence of 5

derived galois deformation rings abstract we define a derived version of mazur s galois deformation ring it is a pro simplicial ring r classifying deformations of a fixed galois representation to simplicial coefficient rings its zeroth homotopy group 0 r recovers mazur s deformation ring we give evidence that these rings r occur in the wild for suitable galois representations the langlands program predicts that 0 r should act on the homology of an arithmetic group we explain how the taylor wiles method can be used to upgrade such an action to a graded action of r on the homology

running fir and iir filtering using multirate filter banks the computational complexity of running fir finite impulse response and iir infinite impulse response filtering using multirate filter banks is considered no restrictions are put on signal filter or block lengths it is shown how to map long running convolutions into smaller ones by using filter banks based on aperiodic convolution algorithms and short time fourier transforms with the proposed approach good tradeoffs among computational complexity system architecture and input output delay are achieved

a discrete time multiresolution theory multiresolution analysis and synthesis for discrete time signals is described concepts of scale and resolution are first reviewed in discrete time the resulting framework allows one to treat the discrete wavelet transform octave band perfect reconstruction filter banks and pyramid transforms from a unified standpoint this approach is very close to previous work on multiresolution decomposition of functions of a continuous variable and the connection between these two approaches is made it is shown that they share many mathematical properties such as biorthogonality orthonormality and regularity however the discrete time formalism is well suited to practical tasks in digital signal processing and does not require the use of functional spaces as an intermediate step

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

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

local estimation of the uniform error threshold the theory behind selection of the uniform error threshold which equalizes the probability of misclassification in an image containing two classes is presented it is shown how this threshold can be estimated using local operations some examples and possible extensions are considered

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