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

classical solutions of singular monge ampere equations in a ball abstract our concern is on existence uniqueness and regularity of convex negative radially symmetric classical solutions to det d 2 u x u in b u 0 on b where d 2 u is the hessian of u b r n n 1 is the unit ball with boundary b b 0 0 is continuous and x t x t where x is the euclidean norm of x the main interest is in the case is singular at x 1 and or u 0 although several nonsingular cases are covered by the main result our approach to show existence exploits fixed point arguments and the shooting method uniqueness and regularity are achieved through suitable estimates

separability based multiscale basis selection and feature extraction for signal and image classification algorithms for multiscale basis selection and feature extraction for pattern classification problems are presented the basis selection algorithm is based on class separability measures rather than energy or entropy at each level the accumulated tree structured class separabilities obtained from the tree which includes a parent node and the one which includes its children are compared the decomposition of the node or subband is performed creating the children if it provides larger combined separability the suggested feature extraction algorithm focuses on dimensionality reduction of a multiscale feature space subject to maximum preservation of information useful for classification at each level of decomposition an optimal linear transform that preserves class separabilities and results in a reduced dimensional feature space is obtained classification and feature extraction is then performed at each scale and resulting soft decisions obtained for each area are integrated across scales the suggested algorithms have been tested for classification and segmentation of one dimensional 1 d radar signals and two dimensional 2 d texture and document images the same idea can be used for other tree structured local basis e g local trigonometric basis functions and even for nonorthogonal redundant and composite basis dictionaries

asymptotic behavior of large solution to elliptic equation of bieberbach rademacher type with convection terms we analyze the asymptotic behavior of solutions to nonlinear elliptic equation du u q b x f u in w subject to the singular boundary condition u x as dist x email protected 0 where w is a smooth bounded domain in r n email protected email protected rv r r 0 for some email protected c 2 a lim u l u and l nrv 1 our approach employs karamata regular variation theory combined with the method of lower and supper solution

boundary estimates of solutions for a class of monge ampere equations abstract on a smooth strictly convex bounded domain in r 2 we establish some boundary estimates of the solution to the complete hyperbolic affine sphere equation on the unit ball in r n we also give some sharp estimates of the solutions of a class of monge ampere equations as an application we prove the relative hyperbolic hypersurface constructed by solving two linked monge ampere equations has constant gauss kronecker curvature and has bounded principal curvatures

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

boundary behavior of solutions of monge ampere equations with singular righthand sides abstract in this paper by karamata regular variation theory and constructing super and subsolutions we obtain the asymptotic boundary behavior of convex solutions of monge ampere equations with singular righthand sides

boundary blow up solutions for a class of elliptic equations on a bounded domain this paper deals with existence of solutions of the quasilinear elliptic equation du f x u u in w satisfying the boundary blow up condition u x x email protected where email protected r n is a bounded domain with smooth boundary email protected our main result applies to the existence of blow up solutions of a class of equations which appear in stochastic control theory namely du a x g u email protected u email protected x in w where a x g u j x are suitable functions and l s are nonnegative constants our approach employs an approximation procedure combined with both a non monotone iteration variant of the method of lower and upper solutions and fixed point arguments

boundary blow up in nonlinear elliptic equations of bieberbach rademacher type the research of the first author was carried out at victoria university melbourne with the support of the australian government through detya the second author has been supported by grant 2 cex06 11 18 2006

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

singular elliptic problems existence non existence and boundary behavior abstract we deal with the existence uniqueness and boundary behavior of positive solutions of the semilinear elliptic equation u a x g u b x f u in under dirichlet boundary conditions where r n is a bounded domain with smooth boundary a b g f are continuous non negative real valued functions the main feature here is that either g or f or both of them are singular at 0 in the sense that g t f t t 0 and 0 are parameters our results require no symmetry from either a or b and no monotonicity on f or g penalty arguments as well as variational principles are exploited

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

extraction of features using m band wavelet packet frame and their neuro fuzzy evaluation for multitexture segmentation in this paper we propose a scheme for segmentation of multitexture images the methodology involves extraction of texture features using an overcomplete wavelet decomposition scheme called discrete m band wavelet packet frame dmbwpf this is followed by the selection of important features using a neuro fuzzy algorithm under unsupervised learning a computationally efficient search procedure is developed for finding the optimal basis based on some maximum criterion of textural measures derived from the statistical parameters for each of the subbands the superior discriminating capability of the extracted features for segmentation of various texture images over those obtained by several existing methods is established

boundary blow up for differential equations with indefinite weight we obtain results of existence and multiplicity of solutions for the second order equation x q t g x 0 with x t defined for all tis an element of 0 1 and such that x t infinity as t 0 and t 1 we assume g having superlinear growth at infinity and q t possibly changing sign on 0 1 c 2002 elsevier science usa all rights reserved

boundary behaviors for liouville s equation in planar singular domains abstract we study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary the conformal structure plays an essential role

abelian ideals of borel subalgebras and affine weyl groups abstract this paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a borel subalgebra our main result is a nice uniform formula for the dimension of maximal ideals in these posets we also obtain results on ad nilpotent ideals which complete the analysis started in j algebra 225 2000 130 258 2002 112

the maximum cardinality of minimal inversion complete sets in finite reflection groups abstract we compute for reflection groups of type a b d f 4 h 3 and for dihedral groups a statistic counting the maximal cardinality of a set of elements in the group whose generalized inversions yield the full set of inversions and which are minimal with respect to this property we also provide lower bounds for the e types that we conjecture to be the exact value of our statistic

boundary blow up elliptic problems with nonlinear gradient terms by karamata regular variation theory and perturbation method we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in rn q 0 g c1 0 g 0 0 g is regularly varying at infinity with index with 0 and b is nonnegative nontrivial in which may be vanishing on the boundary

existence and uniqueness of nontrivial radial solutions for k hessian equations abstract we prove existence and uniqueness of nontrivial radial solutions to the k hessian problem s k d 2 u f u in u 0 on where s k d 2 u is the k hessian operator of u is the open unit ball in r n f is a continuous function and may have k sublinear growth at with possible k superlinear growth at 0 and is a large parameter

boundary behavior of large solutions to the monge ampere equations with weights abstract this paper is concerned with the monge ampere equation det d 2 u x b x f u x x where is a strictly convex bounded smooth domain in r n with n 2 and b c which is positive in but may be vanishing on the boundary we find a new structure condition on f which plays a crucial role in the boundary behavior of strictly convex large solutions our results are obtained in a more general setting than those in cirstea and trombetti 2008 12 where f is regularly varying at infinity with index p n

on singular boundary value problems for the monge ampere operator abstract we consider different types of singular boundary value problems for the monge ampere operator the approach is based on existing regularity theory and a subsolution supersolution method nonexistence and uniqueness results are also given

existence and estimates of solutions to a singular dirichlet problem for the monge ampere equation given a strictly convex smooth and bounded domain in rn we establish the existence of a negative convex solution in c c with zero boundary value to the singular monge ampere equation det d2u p x g u an associated dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem estimates of solutions will also be given and regularity of solutions will be deduced from the estimates

texture classification and segmentation using wavelet packet frame and gaussian mixture model in this paper we propose a scheme for texture classification and segmentation the methodology involves an extraction of texture features using the wavelet packet frame decomposition this is followed by a gaussian mixture based classifier which assigns each pixel to the class each subnet of the classifier is modeled by a gaussian mixture model and each texture image is assigned to the class to which pixels of the image most belong this scheme shows high recognition accuracy in the classification of brodatz texture images it can also be expanded to an unsupervised texture segmentation using a kullback leibler divergence between two gaussian mixtures the proposed method was successfully applied to brodatz mosaic image segmentation and fabric defect detection

existence of boundary blow up solutions for a class of quasilinear elliptic systems with critical case abstract we consider the quasilinear elliptic system div u p 2 u u m 1 v n 1 div v q 2 v u m 2 v n 2 in where m 1 p 1 n 2 q 1 m 2 n 1 0 and r n is a smooth bounded domain subject to three different types of dirichlet boundary conditions u v or u v or u v on where 0 under several hypotheses on the parameters m 1 n 1 m 2 n 2 which is a critical case we show that the existence of positive solutions

singular boundary value problems for the monge ampere equation abstract given a strictly convex smooth and bounded domain in r n we consider solving the monge ampere equation det d 2 u f x u for solutions in c c with zero boundary value where the nonlinearity f x t could be singular at t 0 we will show that under some fairly general assumptions on f the above dirichlet problem admits a negative convex solution in uniqueness of such solutions is then established for a wide class of nonlinearities f x t as a consequence of a comparison principle

boundary behavior of k convex solutions for singular k hessian equations abstract we discuss the existence and boundary behavior of k convex solution to the singular k hessian problem s k d 2 u x b x f u x x u x 0 x where s k d 2 u k 1 2 n is the k hessian operator r n n 2 is a smooth bounded strictly convex domain here the weight function b x is not necessarily bounded on another interest is that f u as u 0 our approach mainly relies on karamata s regular variation theory and the construction of suitable sub and super solutions

design of efficient m band coders with linear phase and perfect reconstruction properties this paper presents a new design technique for obtaining m band orthogonal coders where m 2 sup i the structures obtained using the proposed technique have the perfect reconstruction property furthermore all filters that constitute the subband coder are linear phase fir type filters in contrast with conventional design techniques that attempt to find a unitary alias component matrix in the frequency domain we carry out the design in the time domain based on time domain orthonormality constraints that the filters must satisfy the m band design problem is reduced to the problem of finding a suitable lowpass filter h sub 0 n once a suitable lowpass filter is found the remaining m 1 filters of the coder are obtained through the use of shuffling operators on the lowpass filter this approach leads to a set of filters that use the same numerical coefficient values in different shift positions allowing very efficient numerical implementation of the subband coder in addition by imposing further constraints on the lowpass branch impulse response h sub 0 n we are able to construct continuous bases of m channel wavelets with good regularity properties design examples are presented for four eight and 16 band coders along with examples of continuous wavelet bases that they generate

existence of entire explosive positive solutions of quasi linear elliptic equations in this paper our main purpose is to establish the existence of entire explosive positive radial solutions for quasi linear elliptic equations the main results of the paper extend previously known results both for the super linear and sub linear cases

boundary blow up solutions to the k hessian equation with a weakly superlinear nonlinearity abstract by constructing new sub and super solutions we are concerned with determining values of for which there exist k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 and is a smooth bounded strictly convex domain in r n n 2 we suppose that the nonlinearity behaves like u k ln u as u which is more complex and difficult to deal with than the nonlinearity grows like u p with p k or faster at infinity further several new results of nonexistence global estimates and estimates near the boundary for the solutions are also given

texture classification by wavelet packet signatures this correspondence introduces a new approach to characterize textures at multiple scales the performance of wavelet packet spaces are measured in terms of sensitivity and selectivity for the classification of twenty five natural textures both energy and entropy metrics were computed for each wavelet packet and incorporated into distinct scale space representations where each wavelet packet channel reflected a specific scale and orientation sensitivity wavelet packet representations for twenty five natural textures were classified without error by a simple two layer network classifier an analyzing function of large regularity d sub 20 was shown to be slightly more efficient in representation and discrimination than a similar function with fewer vanishing moments d sub 6 in addition energy representations computed from the standard wavelet decomposition alone 17 features provided classification without error for the twenty five textures included in our study the reliability exhibited by texture signatures based on wavelet packets analysis suggest that the multiresolution properties of such transforms are beneficial for accomplishing segmentation classification and subtle discrimination of texture

existence of positive boundary blow up solutions for quasilinear elliptic equations via sub and supersolutions abstract in this paper the existence of positive boundary blow up weak solution for the quasilinear elliptic equation div u p 2 u m x f u in a smooth bounded domain r n or global space r n are obtained under new conditions our proof is based on the method of sub and supersolutions

existence of large solutions for a class of quasilinear elliptic equations abstract we consider the quasilinear elliptic equation div u p 2 u a x f u u 0 in where p 1 a x 0 a x 0 a x c and f is continuous and non decreasing on 0 satisfies f 0 0 f s 0 for s 0 and the keller osserman condition 1 f s 1 p d s f s 0 s f t d t we establish conditions on the function a that are necessary and sufficient for the existence of positive solutions bounded and unbounded of the given equation

content based image retrieval using visually significant point features this paper presents a new image retrieval scheme using visually significant point features the clusters of points around significant curvature regions high medium and weak type are extracted using a fuzzy set theoretic approach some invariant color features are computed from these points to evaluate the similarity between images a set of relevant and non redundant features is selected using the mutual information based minimum redundancy maximum relevance framework the relative importance of each feature is evaluated using a fuzzy entropy based measure which is computed from the sets of retrieved images marked relevant and irrelevant by the users the performance of the system is evaluated using different sets of examples from a general purpose image database the robustness of the system is also shown when the images undergo different transformations

large solutions to the monge ampere equations with nonlinear gradient terms existence and boundary behavior abstract in this paper we obtain conditions about the existence and boundary behavior of strictly convex solutions to the monge ampere equations with boundary blow up det d 2 u x b x f u x u q x u and det d 2 u x b x f u x 1 u q x u where is a strictly convex bounded smooth domain in r n with n 2 q 0 n or q 0 n b c which is positive in but may vanish or blow up on the boundary f c 0 f 0 0 and f is strictly increasing on 0 or f c r f s 0 s r and f is strictly increasing on r

boundary blow up solutions to the k hessian equation with singular weights abstract in this paper we study the k convex solutions to the boundary blow up k hessian problem s k d 2 u h x u p for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator and is a smooth bounded strictly convex domain in r n n 2 we show the existence nonexistence uniqueness results global estimates and estimates near the boundary for the solutions our approach is largely based on the construction of suitable sub and super solutions

large and entire large solutions for a class of nonlinear problems abstract in this paper our main purpose is to establish the existence of entire large positive solutions to the quasilinear equation p u x h u we give also necessary conditions for the existence and nonexistence of such solutions we distinguish the cases when is radial and when it is nonradial

uniqueness of boundary blowup solutions to k curvature equation abstract we consider the boundary blowup problem for the k curvature equation i e h k u f u g d u in an n dimensional domain with the boundary condition u x as dist x 0 we prove the uniqueness result under some hypotheses

minimal inversion complete sets and maximal abelian ideals abstract let g be a simple lie algebra b a fixed borel subalgebra and w the weyl group of g in this article we study a relationship between the maximal abelian ideals of b and the minimal inversion complete sets of w the latter have been recently defined by malvenuto et al 2015 10

existence and boundary behavior of solutions of hessian equations with singular right hand sides abstract in this paper we establish the existence of viscosity solutions of hessian equations with singular right hand sides and obtain the asymptotic boundary behavior of solutions the asymptotic results generalize those for poisson equations and monge ampere equations and are more precise than obtained from hopf lemma

existence of large solutions for quasilinear elliptic problems with a gradient term abstract in this paper we prove the existence of the positive large solutions for the equation div u m 2 u x u m 1 x u x in the whole space r n n 3 where 0 0 is a continuous function and m 1 with r n 0 0 is required to satisfy some hypotheses detailed blow more precisely we will give a sufficient and necessary condition for the existence of an entire large radial solution and a necessary condition for the existence of the entire large positive radial solution

frame representations for texture segmentation we introduce a novel method of feature extraction for texture segmentation that relies on multichannel wavelet frames and 2 d envelope detection we describe and compare two algorithms for envelope detection based on 1 the hilbert transform and 2 zero crossings we present criteria for filter selection and discuss quantitatively their effect on feature extraction the performance of our method is demonstrated experimentally on samples of both natural and synthetic textures

existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems abstract we show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div u m 2 u p x g v div v n 2 v q x f u on r n where f and g are positive and non decreasing functions on 0 satisfying the keller osserman condition

optimal global and boundary asymptotic behavior of large solutions to the monge ampere equation abstract this paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the monge ampere equation det d 2 u b x f u x where is a strict convex and bounded smooth domain in r n with n 2 f c 1 0 or f c 1 r which is increasing in 0 or r and satisfies the keller osserman type condition b c is positive in but may vanish or blow up on the boundary properly we find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions moreover we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values in addition when f does not satisfy the keller osserman type condition and is a ball we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem

filtering for texture classification a comparative study in this paper we review most major filtering approaches to texture feature extraction and perform a comparative study filtering approaches included are laws masks 1980 ring wedge filters dyadic gabor filter banks wavelet transforms wavelet packets and wavelet frames quadrature mirror filters discrete cosine transform eigenfilters optimized gabor filters linear predictors and optimized finite impulse response filters the features are computed as the local energy of the filter responses the effect of the filtering is highlighted keeping the local energy function and the classification algorithm identical for most approaches for reference comparisons with two classical nonfiltering approaches co occurrence statistical and autoregressive model based features are given we present a ranking of the tested approaches based on extensive experiments

existence of explosive positive solutions of quasilinear elliptic equations abstract in this paper our main purpose is to consider the quasilinear equation div u p 2 u m x f u on a domain r n n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and m is a nonnegative continuous function with the property that any zero of m is contained in a bounded domain in such that m is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists for un boundary including r n we show that a similar result holds where u x as x within and u x as x

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

the existence and asymptotic behavior of boundary blow up solutions to the k hessian equation abstract in this paper we consider the existence and asymptotic behavior of k convex solution to the boundary blow up k hessian problem s k d 2 u h x f u for x u x as dist x 0 where k 1 2 n s k d 2 u is the k hessian operator is a smooth bounded strictly convex domain in r n n 2 h c is positive in but is not necessarily bounded on and f is a smooth positive function that satisfies the so called keller osserman condition further results are obtained for the special case that is a ball our approach to show the existence and asymptotic behavior exploits the method of sub and super solutions and karamata regular variation theory

the asymptotic behaviour of solutions with blow up at the boundary for semilinear elliptic problems abstract by constructing the comparison functions and the perturbed method it is showed that any solution u c 2 to the semilinear elliptic problems u k x g u x u satisfies lim d x 0 u x z d x 2 2 2 c 0 2 1 where is a bounded domain with smooth boundary in r n lim d x 0 k x d x c 0 2 c 0 0 2 2 g c 1 0 g 0 and g s s is increasing on 0 there exists 0 such that lim s g s g s 0 z s d t 2 g t s g t 0 t g s d s

nonlinear elliptic equations with singular boundary conditions abstract we consider the model problem formula where is a bounded region in r n with smooth boundary q 0 2 and k u u p p 1 or e u we show that there exists a classical solution

existence and boundary asymptotic behavior of large solutions of hessian equations abstract in this paper we establish the existence of large solutions of hessian equations and obtain a new boundary asymptotic behavior of solutions

on a k hessian equation with a weakly superlinear nonlinearity and singular weights abstract consider the existence nonexistence and global estimates of k convex solutions to the boundary blow up k hessian problem s k d 2 u x h x u x k ln u x 0 for x u x as dist x 0 here k 1 2 n s k d 2 u is the k hessian operator 0 is a smooth bounded strictly convex domain in r n n 2 and h x is a positive weight function which is singular near the boundary we first give the existence and nonexistence results of k convex solution to the above boundary blow up problem on a larger range of h and then we show that there is a k convex solution provided that h x grows fast near and u k ln u grows slow at it turns out that this case is more difficult to handle than the case in which h x grows slow near and u k ln u grows fast at this needs some new ingredients in the arguments

abelian ideals in a borel subalgebra of a complex simple lie algebra let mathfrak g be a complex simple lie algebra and mathfrak b a fixed borel subalgebra of mathfrak g we describe the abelian ideals in mathfrak b in a uniform way that is independent of the classification of complex simple lie algebras as an application we derive a formula for the maximal dimension of a commutative lie subalgebra of mathfrak g

existence non existence and asymptotic behavior of blow up entire solutions of semilinear elliptic equations abstract we are mainly concerned with existence non existence and the behavior at infinity of non negative blow up entire solutions of the equation u x f u in r n no monotonicity condition is assumed upon f and in fact we obtain solutions with a prescribed behavior both at infinity and at the origin the method used to get existence is based upon lower and upper solutions techniques while for non existence we explore radial symmetry estimates on an associated integral equation and the keller osserman condition

uniqueness of large positive solutions for a class of radially symmetric cooperative systems abstract this paper shows the uniqueness of the large solution of a class of radially symmetric sublinear elliptic systems of cooperative type this result is based on the strong maximum principle rather than on the blow up rates of the large solutions as it is common in most the available results for single equations

large solution of a semilinear elliptic problem we show that large positive solutions exist for the following email protected u q p x f u p in w r n n 3 in which the domain w is either bounded or equal to r n the nonnegative function p is continuous and may vanish on large parts of w if w r n then p must satisfy a decay email protected 0 email protected r dr furthermore we show that the given conditions on p are nearly optimal for equation p

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