hom lie superalgebras and hom lie admissible superalgebras the purpose of this paper is to study hom lie superalgebras that is a superspace with a bracket for which the superjacobi identity is twisted by a homomorphism this class is a particular case of graded quasi lie algebras introduced by larsson and silvestrov in this paper we characterize hom lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of hom lie superalgebras deforming the orthosymplectic lie superalgebra also we prove a z2 graded version of a hartwig larsson silvestrov theorem which leads us to a construction of a q deformed witt superalgebra

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

zero crossings of a wavelet transform the completeness stability and application to pattern recognition of a multiscale representation based on zero crossings is discussed an alternative projection algorithm is described that reconstructs a signal from a zero crossing representation which is stabilized by keeping the value of the wavelet transform integral between each pair of consecutive zero crossings the reconstruction algorithm has a fast convergence and each iteration requires o n log sup 2 n computation for a signal of n samples the zero crossings of a wavelet transform define a representation which is particularly well adapted for solving pattern recognition problems as an example the implementation and results of a coarse to fine stereo matching algorithm are described

general synthesis procedures for fir lossless transfer matrices for perfect reconstruction multirate filter bank applications a recently reported procedure see ibid vol assp 35 p 476 492 apr 1987 and proc ieee int conf acoust speech sig processing apr 1987 p 2169 2172 for the design of m channel perfect reconstruction quadrature mirror filter qmf bands uses the concept of lossless alias component matrices the synthesis of such qmf banks centers around the generation of a lossless m m finite impulse response fir transfer matrix e z recent results for such generation have been somewhat ad hoc i e not sufficiently general a general procedure is outlined for the generation of such transfer matrices the procedure is based on a cascaded lattice structure derived from a state space viewpoint the structure is such that it generates only m m lossless fir transfer matrices regardless of parameter values and conversely any m m lossless fir transfer matrix can be obtained by a suitable choice of parameters these parameters turn out to be angles theta sub k and the structure is such that the number of angles is minimal a design example is presented to demonstrate the main results

theory and design of signal adapted fir paraunitary filter banks we study the design of signal adapted fir paraunitary filter banks using energy compaction as the adaptation criterion we present some important properties that globally optimal solutions to this optimization problem satisfy in particular we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi infinite programming sip problem the remaining filters are related to the first through a matrix eigenvector decomposition we discuss uniqueness and sensitivity issues the sip problem is solved using a discretization method and a standard simplex algorithm we also show how regularity constraints may be incorporated into the design problem to obtain globally optimal in the energy compaction sense filter banks with specified regularity we also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be dct based such constraints may also be incorporated into our optimization algorithm therefore we are able to obtain globally optimal filter banks subject to regularity and or computational complexity constraints numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters as well as the effects of the various constraints the conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for fir filter banks

ternary hom nambu lie algebras induced by hom lie algebras the need to consider n ary algebraic structures generalizing lie and poisson algebras has become increasingly important in physics and it should therefore be of interest to study the mathematical concepts related to n ary algebras the purpose of this paper is to investigate ternary multiplications as deformations of n lie structures constructed from the binary multiplication of a hom lie algebra a linear twisting map and a trace function satisfying certain compatibility conditions we show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of hom nambu lie algebras obtained using this construction

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

the role of linear semi infinite programming in signal adapted qmf bank design we consider the problem of designing a perfect reconstruction fir quadrature mirror filter qmf bank h g adapted to input signal statistics with the coding gain as the adaptation criterion maximization of the coding gain has so far been viewed as a difficult nonlinear constrained optimization problem it is shown that the coding gain depends only on the product filter p z h z h z sup 1 and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints termed linear semi infinite programming sip problems the sought for original filter h z is obtained by deflation and spectral factorization of p z with the sip formulation every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms the natural regularity properties inherent in the sip formulation enhance the performance of these algorithms we present a comprehensive theoretical analysis of the sip problem and its dual characterize the optimal filters and analyze uniqueness and sensitivity issues all these properties are intimately related to those of the input signal and bring considerable insight into the nature of the adaptation process we present discretization and cutting plane algorithms and apply both methods to several examples

theory of rate distortion optimal constrained filterbanks application to iir and fir biorthogonal designs we design filterbanks that are best matched to input signal statistics in m channel subband coders using a rate distortion criterion previous research has shown that unconstrained length paraunitary filterbanks optimized under various energy compaction criteria are principal component filterbanks that satisfy two fundamental properties total decorrelation and spectral majorization in this paper we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints our results apply to a broad class of rate distortion criteria including the conventional coding gain criterion as a special case a consequence of these properties is that optimal perfect reconstruction pr filterbanks take the form of the cascade of principal component filterbanks and a bank of pre and post conditioning filters the proof uses variational techniques and is applicable to a variety of constrained design problems in the second part of this paper we apply the theory above to practical filterbank design problems we give analytical expressions for optimal iir biorthogonal filterbanks our analysis validates a conjecture by several researchers we then derive the asymptotic limit of optimal fir biorthogonal filterbanks as filter length tends to infinity the performance loss due to fir constraints is quantified theoretically and experimentally the optimal filters are quite different from traditional filters finally a sensitivity analysis is presented

optimal subband filter banks for multiple description coding multiple description coding refers to the encoding of an information source into multiple bit streams in a manner that each bit stream independently represents a coarse description of the source while multiple descriptions jointly convey a higher fidelity source representation thus multiple description coding allows enhanced resilience to loss of individual descriptions and can be viewed as a form of source based diversity coding this paper addresses the design of subband filter banks to optimize the rate distortion performance of multiple description subband coding the problem is formulated in a lagrangian optimization framework and is solved directly in the frequency domain to produce the optimal filter spectral responses by varying the lagrangian parameter our design obtains all points on the redundancy distortion curve for gaussian wide sense stationary input processes we consider the cases of both orthonormal and more general biorthogonal filter banks we also analyze the connections between these two solutions and quantify the gap in their optimal performance application of the proposed methods to the popular first order autoregressive ar 1 model yields very interesting and insightful results in the two extreme cases of maximum and minimum redundancy our solutions degenerate to previously known results finally comparisons with popularly deployed filter banks reveal large gaps from the theoretical performance bound

yetter drinfeld modules for hom bialgebras the aim of this paper is to define and study yetter drinfeld modules over hom bialgebras a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism yetter drinfeld modules over a hom bialgebra with bijective structure map provide solutions of the hom yang baxter equation the category yhhd of yetter drinfeld modules with bijective structure maps over a hom bialgebra h with bijective structure map can be organized in two different ways as a quasi braided pre tensor category if h is quasitriangular respectively coquasitriangular the first respectively second quasi braided pre tensor category yhhd contains as a quasi braided pre tensor subcategory the category of modules respectively comodules with bijective structure maps over h

on the optimal choice of a wavelet for signal representation two techniques for finding the discrete orthogonal wavelet of support less than or equal to some given integer that leads to the best approximation to a given finite support signal up to a desired scale are presented the techniques are based on optimizing certain cost functions the first technique consists of minimizing an upper bound that is derived on the l sub 2 norm of error in approximating the signal up to the desired scale it is shown that a solution to the problem of minimizing that bound does exist and it is explained how the constrained minimization over the parameters that define discrete finite support orthogonal wavelets can be turned into an unconstrained one the second technique is based on maximizing an approximation to the norm of the projection of the signal on the space spanned by translates and dilates of the analyzing discrete orthogonal wavelet up to the desired scale both techniques can be implemented much faster than the optimization of the l sub 2 norm of either the approximation to the given signal up to the desired scale or that of the error in that approximation

hom lie algebra structures on semi simple lie algebras abstract hom lie algebras can be considered as a deformation of lie algebras in this note we prove that the hom lie algebra structures on finite dimensional simple lie algebras are trivial we find when a finite dimensional semi simple lie algebra admits non trivial hom lie algebra structures and the isomorphic classes of non trivial hom lie algebras are determined

the hom yang baxter equation hom lie algebras and quasi triangular bialgebras we study a twisted version of the yang baxter equation called the hom yang baxter equation hybe which is motivated by hom lie algebras three classes of solutions of the hybe are constructed one from hom lie algebras and the others from drinfeld s dual quasi triangular bialgebras each solution of the hybe can be extended to operators that satisfy the braid relations assuming an invertibility condition these operators give a representation of the braid group

relative hom hopf modules and total integrals let h be a monoidal hom hopf algebra and a a right h hom comodule algebra we first investigate the criterion for the existence of a total integral of a in the setting of monoidal hom hopf algebras also we prove that there exists a total integral h a if and only if any representation of the pair h a is injective in a functorial way as a corepresentation of h which generalizes doi s result finally we define a total quantum integral h hom h a and prove the following affineness criterion if there exists a total quantum integral and the canonical map a ba a h a bb 1 a b 0 b 1 is surjective then the induction functor a b k b k ah is an equivalence of categories

on the optimality of ideal filters for pyramid and wavelet signal approximation the reconstructed lowpass component in a quadrature mirror filter qmf bank provides a coarser resolution approximation of the input signal since the outputs of the two qmf branches are orthogonal the transformation that provides the maximum energy compaction in the lowpass channel is also the one that results in the minimum approximation error this property is used as a common strategy for the optimization of qmf banks orthogonal wavelet transforms and least squares pyramids a general solution is derived for the qmf bank that provides the optimal decomposition of an arbitrary wide sense stationary process this approach is extended to the continuous case to obtain the minimum error approximation of a signal at a given sampling rate in particular it is shown that the sine wavelet transform is optimal for the representation at all scales of signals with non increasing spectral density

iterative greedy algorithm for solving the fir paraunitary approximation problem in this paper a method for approximating a multi input multi output mimo transfer function by a causal finite impulse response fir paraunitary pu system in a weighted least squares sense is presented using a complete parameterization of fir pu systems in terms of householder like building blocks an iterative algorithm is proposed that is greedy in the sense that the observed mean squared error at each iteration is guaranteed to not increase for certain design problems in which there is a phase type ambiguity in the desired response which is formally defined in the paper a phase feedback modification is proposed in which the phase of the fir approximant is fed back to the desired response with this modification in effect it is shown that the resulting iterative algorithm not only still remains greedy but also offers a better magnitude type fit to the desired response simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank pcfb like filter banks and the fir pu interpolation problem concerning the pcfb design problem it is shown that as the mcmillan degree of the fir pu approximant increases the resulting filter bank behaves more and more like the infinite order pcfb consistent with intuition in particular this pcfb like behavior is shown in terms of filter response shape multiresolution coding gain noise reduction with zeroth order wiener filtering in the subbands and power minimization for discrete multitone dmt type transmultiplexers

bifrequency and bispectrum maps a new look at multirate systems with stochastic inputs in multirate digital signal processing we often encounter decimators interpolators and complicated interconnections of these with lti filters we also encounter cyclo wide sense stationary cwss processes and linear periodically time varying ptv systems it is often necessary to understand the effects of multirate systems on the statistical properties of their input signals some of these issues have been addressed earlier for example it has been shown that a necessary and sufficient addition for the output of an l fold interpolation filter to be wide sense stationary wss for all wss inputs is that the filter was an alias free l support however several questions of this nature remain unanswered for example what is the necessary and sufficient condition on a pair or more generally a bank of interpolation filters so that their outputs are jointly wss jwss for all jointly wss inputs what is the condition if only the sum of their outputs is required to be wss when is the output of an lptv system for example a uniform filter bank wss for all wss inputs some of these questions may appear to be simple generalizations of the above mentioned result for a single interpolation filter however the frequency domain approaches that proved this result are quite difficult to generalize to answer these questions the purpose of this paper is to provide these answers using analysis based on bifrequency maps and bispectra these tools are two dimensional 2 d fourier transforms that characterize linear time varying ltv systems and nonstationary random processes respectively we show that the questions raised above are addressed elegantly and in a geometrically insightful way using these tools we also derive a bifrequency characterization lossless ltv systems this may potentially lead to an increased understanding of these systems

design of fir paraunitary filter banks for subband coding using a polynomial eigenvalue decomposition the problem of paraunitary pu filter bank design for subband coding has received considerable attention in recent years not least because of the energy preserving property of this class of filter banks in this paper we consider the design of signal adapted finite impulse response fir pu filter banks using polynomial matrix evd pevd techniques modifications are proposed to an iterative time domain pevd method known as the sequential best rotation sbr2 algorithm which enables its effective application to the problem of fir orthonormal filter bank design for efficient subband coding by choosing an optimization scheme that maximizes the coding gain at each stage of the algorithm it is shown that the resulting filter bank behaves more and more like the infinite order principle component filter bank pcfb the proposed method is compared to state of the art techniques namely the iterative greedy algorithm iga the approximate evd aevd standard sbr2 and a fast algorithm for fir compaction filter design called the window method wm we demonstrate that for the calculation of the subband coder the wm approach offers a low cost alternative at lower coding gains while at moderate to high complexity the proposed approach outperforms the benchmarkers in terms of run time complexity aevd performs well at low orders while the proposed algorithm offers a better coding gain than the benchmarkers at moderate to high filter order for a number of simulation scenarios

optimized nonorthogonal transforms for image compression the transform coding of images is analyzed from a common standpoint in order to generate a framework for the design of optimal transforms it is argued that all transform coders are alike in the way they manipulate the data structure formed by transform coefficients a general energy compaction measure is proposed to generate optimized transforms with desirable characteristics particularly suited to the simple transform coding operation of scalar quantization and entropy coding it is shown that the optimal linear decoder inverse transform must be an optimal linear estimator independent of the structure of the transform generating the coefficients a formulation that sequentially optimizes the transforms is presented and design equations and algorithms for its computation provided the properties of the resulting transform systems are investigated in particular it is shown that the resulting basis are nonorthogonal and complete producing energy compaction optimized decorrelated transform coefficients quantization issues related to nonorthogonal expansion coefficients are addressed with a simple efficient algorithm two implementations are discussed and image coding examples are given it is shown that the proposed design framework results in systems with superior energy compaction properties and excellent coding results

on the spectral factor ambiguity of fir energy compaction filter banks this paper focuses on the design of signal adapted finite impulse response fir paraunitary pu filter banks optimized for energy compaction ec the design of such filter banks has been shown in the literature to consist of the design of an optimal fir compaction filter followed by an appropriate karhunen loe spl grave ve transform klt despite this elegant construction ec optimal filter banks have been shown to perform worse than common nonadapted filter banks for coding gain contrary to intuition here it is shown that this phenomenon is most likely due to the nonuniqueness of the compaction filter in terms of its spectral factors this nonuniqueness results in a finite set of ec optimal filter banks by choosing the spectral factor yielding the largest coding gain it is shown that the resulting filter bank behaves more and more like the infinite order principal components filter bank pcfb in terms of numerous objectives such as coding gain multiresolution noise reduction with zeroth order wiener filters in the subbands and power minimization for discrete multitone dmt type nonredundant transmultiplexers

matched sampling systems relation to wavelets and implementation using prcc filter banks this paper deals with two types of sampling systems namely the interpolation and approximation sampling systems closed form expressions are derived for the frequency responses of the filters used in these systems that are matched to the input process in the mean squared sense closed form expressions are also derived for the mean squared error between the input and the reconstructed processes for these matched sampling systems using these expressions it is shown that the meyer scaling function and wavelet or functions derived from these arise naturally in the context of subsampled bandlimited processes to implement these systems the perfect reconstruction circular convolution prcc filter bank is proposed as a framework for the frequency sampled implementation of these systems examples of matched interpolation and approximation sampling systems are provided and their performance is compared with some standard interpolators to demonstrate their efficacy

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

quasi hom lie algebras central extensions and 2 cocycle like identities this paper introduces the notion of a quasi hom lie algebra or simply a qhl algebra which is a natural generalization of hom lie algebras introduced in a previous paper j t hartwig d larsson s d silvestrov deformations of lie algebras using sigma derivations math qa 0408064 quasi hom lie algebras include also as special cases color lie algebras and superalgebras and can be seen as deformations of these by maps twisting the jacobi identity and skew symmetry the natural realm for these quasi hom lie algebras is generalizations deformations of the witt algebra delta of derivations on the laurent polynomials c t t 1 we also develop a theory of central extensions for qhl algebras which can be used to deform and generalize the virasoro algebra by centrally extending the deformed witt type algebras constructed here in addition we give a number of other interesting examples of quasi hom lie algebras among them a deformation of the loop algebra less

on hom algebras with surjective twisting abstract a hom associative structure is a set a together with a binary operation and a self map such that an twisted version of associativity is fulfilled in this paper we assume that is surjective we show that in this case under surprisingly weak additional conditions on the multiplication the binary operation is a twisted version of an associative operation as an application an earlier result fregier and gohr 1 on weakly unital hom algebras is recovered with a different proof in the second section consequences for the deformation theory of hom algebras with surjective twisting map are discussed

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

principal component filter banks for optimal multiresolution analysis an important issue in multiresolution analysis is that of optimal basis selection an optimal p band perfect reconstruction filter bank prfb is derived in this paper which minimizes the approximation error in the mean square sense between the original signal and its low resolution version the resulting prfb decomposes the input signal into uncorrelated low resolution principal components with decreasing variance optimality issues are further analyzed in the special case of stationary and cyclostationary processes by exploiting the connection between discrete time filter banks and continuous wavelets an optimal multiresolution decomposition of l sub 2 r is obtained analogous results are also derived for deterministic signals some illustrative examples and simulations are presented

symmetries and the u condition in hom yetter drinfeld categories let h s be a monoidal hom hopf algebra and hydhh the hom yetter drinfeld category over h then in this paper we first find sufficient and necessary conditions for hydhh to be symmetric and pseudosymmetric respectively second we study the u condition in hydhh and show that the hom yetter drinfeld module h adjoint resp h m coadjoint satisfies the u condition if and only if s2 id finally we prove that hydhh over a triangular resp cotriangular hom hopf algebra contains a rich symmetric subcategory

constructing new braided t categories over monoidal hom hopf algebras let autmhh h denote the set of all automorphisms of a monoidal hopf algebra h with bijective antipode in the sense of caenepeel and goyvaerts monoidal hom hopf algebras commun algebra 39 2216 2240 2011 and let g be a crossed product group autmhh h autmhh h the main aim of this paper is to provide new examples of braided t category in the sense of turaev crossed group categories arabian j sci eng sect c 33 2c 483 503 2008 for this purpose we first introduce a class of new categories mhydhh a b of a b yetter drinfeld hom modules with a b autmhh h then we construct a category mhyd h mhydhh a b a b g and show that such category forms a new braided t category generalizing the main constructions by panaite and staic generalized anti yetter drinfel d modules as components of a braided t category isr j math 158 349 366 2007 finally we compute an explicit new example of such braided t categories

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

the role of gtd in optimizing perfect reconstruction filter banks filter bank optimization for specific input statistics has been of great interest in both theory and practice in many signal processing applications in this paper we propose gtd generalized triangular decomposition filter banks as a subband coder for optimizing the theoretical coding gain we focus on perfect reconstruction orthonormal gtd filter banks and biorthogonal gtd filter banks we show that in both cases there are two fundamental properties in the optimal solutions namely total decorrelation and spectrum equalization the optimal solutions can be obtained by performing the frequency dependent gtd on the cholesky factor of the input power spectrum density matrices we also show that in both theory and numerical simulations the optimal gtd subband coders have superior performance than optimal traditional subband coders in addition the uniform bit loading scheme can with no loss of optimality be used in the optimal biorthogonal gtd coders which solves the granularity problem in the conventional optimum bit loading formula we then extend the use of gtd filter banks to wireless communication systems where linear precoding and zero forcing decision feedback equalization is used in frequency selective channels we consider the quality of service qos problem of minimizing the transmitted power subject to the bit error rate and total bit rate constraints optimal systems with orthonormal precoder and unconstrained precoder are both derived and shown to be related to the frequency dependent gtd of the channel frequency response

algebraic aspects of the quantum yang baxter equation in this paper we examine a variety of algebraic contexts in which the quantum yang baxter equation arises and derive methods for generating new solutions from given ones the solutions we describe are encoded in objects which have a module and a comodule structure over a bialgebra our work here is based in part on the ideas of dr1 dr2 suppose that r m m m m is a solution to the quantum yang baxter equation where m is a finite dimensional vector space over a field k in dr2 it was shown that r can be derived from a left h module structure and a right h comodule structure on m for some bialgebra h over k also see yet maj1 maj2 frt2 the module and comodule structure satisfy a natural compatibility condition m together with these structures is called a left quantum yang baxter h module every left quantum yang baxter module gives rise to a solution to the quantum yang baxter equation in a natural way we study the category hyb of left quantum yang baxter h modules and show that many of the familiar constructions in the module category hm can be made in hyb through these constructions solutions to the quantum yang baxter equation can be combined in various ways to generate new ones we place particular emphasis on the tensor product of objects of hyb and ways in which the tensor algebra t m of an object m of hyb can be viewed as an object of hyb using the tensor algebra we are able to extend certain solutions when dim m 2 to the quantum plane we turn the tensor algebra t m of an object m of hyb into an object of hyb by giving it a module algebra and a comodule algebra structure module algebra and comodule algebra structures play a very important role in this paper in section 9 we develop a calculus for uq sl2 starting with a right k slq 2 comodule algebra structure on the quantum plane k x y q where k slq 2 is the coordinate ring of quantum slq 2 over k this calculus has been described in other ways m s sud the right comodule algebra structure on k x y q accounts for the left uq sl2 module algebra structure on k x y q described in m s we show how the basic irreducible representations of tak follow by the calculus a module algebra action or comodule algebra action is a natural vehicle for extending structure on a submodule or subcomodule respectively to the subalgebra it generates the calculus and the solutions to the quantum yang baxter equation we describe in this paper are determined in this manner on small structures two dimensional in this case we believe that many more results will follow from this circle of ideas in section 2 we discuss basic aspects of coalgebra and bialgebras used in the sequel a good general reference for this is sweedler s book sb on hopf algebras in section 3 we review the quantum yang baxter equation in connection with quantum yang baxter modules the a r construction of faddeev reshetikhin and takhtajan frt is considered in this light and examined in detail in when r arises from two commuting operators even though this particular r is rather simple the algebra structure of a r is surprisingly complex using the scratchpad computer algebra system we determine the algebra structure of a r in an interesting specific case showing that our particular example has a finite non commutative groeb

results on principal component filter banks colored noise suppression and existence issues we have made explicit the precise connection between the optimization of orthonormal filter banks fbs and the principal component property the principal component filter bank pcfb is optimal whenever the minimization objective is a concave function of the subband variances of the fb this explains pcfb optimality for compression progressive transmission and various hitherto unnoticed white noise suppression applications such as subband wiener filtering the present work examines the nature of the fb optimization problems for such schemes when pcfbs do not exist using the geometry of the optimization search spaces we explain exactly why these problems are usually analytically intractable we show the relation between compaction filter design i e variance maximization and optimum fbs a sequential maximization of subband variances produces a pcfb if one exists but is otherwise suboptimal for several concave objectives we then study pcfb optimality for colored noise suppression unlike the case when the noise is white here the minimization objective is a function of both the signal and the noise subband variances we show that for the transform coder class if a common signal and noise pcfb klt exists it is optimal for a large class of concave objectives common pcfbs for general fb classes have a considerably more restricted optimality as we show using the class of unconstrained orthonormal fbs for this class we also show how to find an optimum fb when the signal and noise spectra are both piecewise constant with all discontinuities at rational multiples of spl pi

wavelet synthesis by alternating projections the coefficients of the quadrature mirror filters involved in orthonormal wavelet or wavelet packets signal decompositions are often chosen in an ad hoc manner in order to adapt such a decomposition to the signal being analyzed it may be pertinent to maximize an energy concentration criterion which leads to a constrained optimization problem a simple geometric interpretation of this problem is given and an alternating projection method is proposed to solve it

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images optimal hierarchical coding is sought for progressive or scalable image transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise given arbitrary analysis filters and assuming adequate knowledge of the noise statistics optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal and subband image decompositions experimental results illustrate the implementation and performance of the optimal coders

minlab minimum noise structure for ladder based biorthogonal filter banks we introduce a minimum noise structure for ladder based biorthogonal minlab filter bank the minimum noise structure ensures that the quantization error has a unity noise gain even though the filter bank is biorthogonal the coder has a very low design and implementation cost the perfect reconstruction property is structurally preserved optimal bit allocation and coding gain formulas are derived we show that the coding gain of the optimal minlab coder is always greater than or equal to unity for both ar 1 process and ma 1 process the minlab coder with two taps has a higher coding gain than the optimal orthonormal coder with an infinite number of taps in addition to its superior decorrelation ability it has many other desired features that make it a potentially valuable and attractive alternative to the orthonormal coder especially for a high fidelity compression

signal adapted multiresolution transform for image coding the authors consider the problem of designing multiresolution transforms that are adapted to the given image signal in the sense that they maximize the coding gain at each resolution level a simple alternating optimization algorithm is derived for solving this problem in the framework of the lattice realization of para unitary quadrature mirror filters the resulting large coding scheme is discussed in some detail and its performance is compared with that of the discrete cosine transform jpeg technique and with that of some nonadapted multiresolution transforms

theory and design of optimum fir compaction filters the problem of optimum fir energy compaction filter design for a given number of channels m and a filter order n is considered the special cases where n m and n spl infin have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively we deal with the more difficult case of m n spl infin for the two channel case and for a restricted but important class of random processes we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle this also corresponds to the optimal two channel fir filter bank that maximizes the coding gain under the traditional quantization noise assumptions with a minor extension this can also be used to generate optimal wavelets for the arbitrary m channel case we provide a very efficient suboptimal design method called the window method the method involves two stages that are associated with the above two special cases as the order increases the suboptimality becomes negligible and the filter converges to the ideal optimal solution we compare the window method with a previously introduced technique based on linear programming

invariant factors as approximation numbers a characterization of the invariant factors of an integral matrix as approximation numbers is given this characterization similar to the one known for the singular values of complex matrices is used to prove some recent results on invariant factors

time varying system identification and model validation using wavelets parametric identification of time varying tv systems is possible if each tv coefficient can be expanded onto a finite set of basis sequences the problem then becomes time invariant with respect to the parameters of the expansion the authors address the question of selecting this set of basis sequences they advocate the use of a wavelet basis because of its flexibility in capturing the signal s characteristics at different scales and discuss how to choose the optimal wavelet basis for a given system trajectory they also develop statistical tests to keep only the basis sequences that significantly contribute to the description of the system s time variation by formulating the problem as a regressor selection problem they apply an p test and an aic based approach for multiresolution analysis of tv systems the resulting algorithm can estimate tv ar or armax models and determine their orders they apply this algorithm to both synthetic and real speech data and compare it with the kalman filtering tv parameter estimator

efficient design of orthonormal wavelet bases for signal representation the efficient representation of a signal as a linear combination of elementary atoms or building blocks is central to much signal processing theory and many applications wavelets provide a powerful flexible and efficiently implementable class of such atoms in this paper we develop an efficient method for selecting an orthonormal wavelet that is matched to a given signal in the sense that the squared error between the signal and some finite resolution wavelet representation of it is minimized since the squared error is not an explicit function of the design parameters some form of approximation of this objective is required if conventional optimization techniques are to be used previous approximations have resulted in nonconvex optimization problems which require delicate management of local minima in this paper we employ an approximation that results in a design problem that can be transformed into a convex optimization problem and efficiently solved constraints on the smoothness of the wavelet can be efficiently incorporated into the design we show that the error incurred in our approximation is bounded by a function that decays to zero as the number of vanishing moments of the wavelet grows in our examples we demonstrate that our method provides wavelet bases that yield substantially better performance than members of standard wavelet families and are competitive with those designed by more intricate nonconvex optimization methods

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

generalized singular values interlacing inequalities and monotonic norms abstract we complete the results of an earlier paper by sodupe where interlacing properties for generalized singular values of matrices and submatrices are studied we show that there is a close relationship between these interlacing properties and the orthant monotonicity of the norms involved in the definition of the generalized singular values we briefly discuss interlacing properties for other kinds of a s numbers

eigenstructure algorithms for multirate adaptive lossless fir filters this paper addresses the problem of adaptively optimizing a two channel lossless finite impulse response fir filter bank which finds application in subband coding and wavelet signal analysis instead of using a gradient decent procedure with its inherent problem of becoming trapped in local minima of a nonquadratic cost function two eigenstructure algorithms are proposed both algorithms feature a priori bounds on the output variance at any convergent point which based on simulations lead to solutions that lie acceptably close to a global minimum point of an output variance objective function moreover a sufficient condition for such stationary points based on fixed point theory is shown it is shown that the convergence rate of both algorithms increases as the separation of eigenvalues of the input covariance matrix increases simulations for synthetic and real data support the conclusions

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

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

multirate filter banks and transform coding gain an analytical derivation of quantization error in filter banks using additive and signal dependent noise models is presented the coding gain over pcm is derived for both noise models interesting properties of coding gain in optimized filter banks with an ar 1 input signal model asymptotic performance of multirate systems and the analytical constraints required to achieve such performance are discussed connections to the recent developments of optimal subband systems are presented

recursive biorthogonal interpolating wavelets and signal adapted interpolating filter banks in this paper by combining the ideas of the recursive wavelets with second generation wavelets a family of recursive biorthogonal interpolating wavelets rbiws is developed the rbiws have simple shape parameter vectors on each level which allows a multichannel decomposition algorithm and provides a flexible structure for designing signal adapted interpolating filter banks in the single level case an efficient approach to design an optimum two channel biorthogonal interpolating filter bank is proposed which maximizes the coding gain under the traditional quantization noise assumption furthermore in the multilevel case using level wise optimization of the shape parameter vectors signal adapted tree structured recursive biorthogonal interpolating filter banks rbifbs are designed which are efficient in computation and can remarkedly improve the coding gain finally numerical results demonstrate the effectiveness of the proposed methods

prediction based lower triangular transform a new nonunitary transform called the prediction based lower triangular transform plt is introduced for signal compression the new transform has the same decorrelation property as the kahurnen loeve transform klt but its implementational cost is less than one half of klt compared with the klt the design cost of an m spl times m plt is much lower and is only of the order of o m sup 2 moreover the plt can be factorized into simple building blocks using two different factorizations we introduce two minimum noise structures that have roughly the same complexity as the direct implementation of plt these minimum noise structures have the following properties 1 its noise gain is unity even though the transform is nonunitary 2 perfect reconstruction is structurally guaranteed 3 it can be used for both lossy lossless compression we show that the coding gain of plt implemented using the minimum noise structure is the same as that of klt furthermore universal transform coders using plt are derived for ar 1 process the m spl times m plt has a closed form and needs only m 1 multiplications and additions

interlacing inequalities for generalized singular values abstract a generalization of the concept of singular value of a complex matrix introduced by j f maitre and n huu vinh is considered we give the relation between the singular values of two matrices and the singular values of their sum and we find cauchy interlacing type inequalities for some of these singular values

the category of yetter drinfel d hom modules and the quantum hom yang baxter equation in this paper we introduce the category of yetter drinfel d hom modules which is a braided monoidal category and show that the category of yetter drinfel d hom modules is a full monoidal subcategory of the left center of left hom module category also we study the equivalent relationship between the category of yetter drinfel d hom modules and the category of hom modules over the drinfel d double finally the faddeev reshetikhin takhtajan frt type theorem for the quantum hom yang baxter equation is investigated

consistent estimation of the cyclic autocorrelation the cyclic autocorrelation is often used to describe nonstationary random processes the authors investigate the conditions under which the cyclic autocorrelation can be estimated consistently in mean square for discrete time gaussian processes they extend and generalize results of hurd 1989 and refine results of boyles and gardner 1983 they derive necessary and sufficient conditions for consistency in mean square of an estimator which are in the form of a single sum of autocorrelation coefficients in the form of a double sum of autocorrelation coefficients in the bifrequency domain and in terms of the average spectrum they also discuss the rate of convergence for this estimator

algebraic theory of optimal filterbanks we introduce an optimality theory for finite impulse response fir filterbanks using a general algebraic point of view we consider an admissible set spl lscr of fir filterbanks and use scalability as the main notion based on which performance of the elements in spl lscr are compared we show that quantification of scalability leads naturally to a partial ordering on the set spl lscr an optimal solution is therefore represented by the greatest element in spl lscr it turns out that a greatest element does not necessarily exist in spl lscr hence one has to settle with one of the maximal elements that exist in spl lscr we provide a systematic way of finding a maximal element by embedding the partial ordering at hand in a total ordering this is done by using a special class of order preserving functions known as schur convex there is however a price to pay for achieving a total ordering there are infinitely many possible choices for schur convex functions and the optimal solution specified in spl lscr depends on this subjective choice an interesting aspect of the presented algebraic theory is that the connection between several concepts namely principal component filterbanks pcfbs filterbanks with maximum coding gain and filterbanks with good scalability is clearly revealed we show that these are simply associated with different extremal elements of the partial ordering induced on spl lscr by scalability

optimal filter banks for multiple description coding analysis and synthesis multiple description md coding is a source coding technique for information transmission over unreliable networks in md coding the coder generates several different descriptions of the same signal and the decoder can produce a useful reconstruction of the source with any received subset of these descriptions in this paper we study the problem of md coding of stationary gaussian sources with memory first we compute an approximate md rate distortion region for these sources which we prove to be asymptotically tight at high rates this region generalizes the md rate distortion region of el gamal and cover 1982 and ozarow 1980 for memoryless gaussian sources then we develop an algorithm for the design of optimal two channel biorthogonal filter banks for md coding of gaussian sources we show that optimal filters are obtained by allocating the redundancy over frequency with a reverse water filling strategy finally we present experimental results which show the effectiveness of our filter banks in the low complexity low rate regime

two applications of wavelet transforms in magnetic resonance imaging the authors describe two different applications of wavelet transforms in magnetic resonance imaging mri in each case the use of wavelets offers tangible benefits over the traditional fourier transform based imaging these stem from the simultaneous time and frequency localization properties of the wavelets the first application utilizes the localization properties of wavelets to acquire single t sub 2 weighted images in times which are short relative to current methods an interesting combinatorial scheduling problem arises from the multiscale structure inherent in the formulation of this procedure its solution is sketched and applied to the imaging process the second example is a technique for potentially instant imaging image acquisition is modeled as an inverse wavelet transform of the spin density in the region in question signal acquisition takes place in as few as one or two echoes after the excitation by rf pulses designed to excite a moving band of spins in a slice these sweeping rf pulses excite strips of spins which rephase at different times during acquisition they replace phase encoding gradients no fast pulsed gradients are required

m band biorthogonal interpolating wavelets via lifting scheme recently the lifting scheme was generalized to the multidimensional and multiband cases and was used to design m band interpolating scaling filters and their duals based on this idea we develop a new lifting pattern namely the progressive lifting pattern this pattern allows us to pairwise generate m band interpolating filterbanks and wavelets by the order from lowpass to highpass filters a complete lifting procedure is divided into m 1 simple steps in each step a pair of filters the l th filter and its dual are generated in this way an m band biorthogonal interpolating filterbank wavelet is determined by m m 1 lifting filters the first 2 m 1 lifting filters completely characterize the two scaling filters as well as the vanishing moments of bandpass and highpass filters the residual m 1 m 2 lifting filters are used to pairwise optimize the bandpass and highpass filters in terms of the criterion of stopband energy minimization the obtained m band biorthogonal interpolating filterbanks and wavelets provide excellent frequency characteristics in particular low stopband sidelobes furthermore the pattern is also utilized to design signal adapted interpolating filterbanks and their rational coefficient counterparts in terms of subband coding gain the obtained filterbanks achieve large subband coding gains the rational coefficient filterbanks preserve the biorthogonality and allow wavelet transforms from integers to integers and a unifying lossy lossless coding framework at the cost of a slight degradation in subband coding gain

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