explicit solutions of grassmannian σ models classical solutions of the two dimensional grassmannian models of respectively euclidean and minkowskian type are compared with each other some explicit solutions of both types are constructed and some of their graphs are presented

efficient arbitrary sampling rate conversion with recursive calculation of coefficients in this paper we present a novel algorithm for sampling rate conversion by an arbitrary factor theoretically sampling rate conversion of a discrete time dt sequence can be performed by converting the sequence to a series of continuous time ct impulses this series of impulses is filtered with a ct lowpass filter and the output is then sampled at the desired rate if the ct filter is chosen to have a rational transfer function then this system can be simulated using a dt algorithm for which both computation and memory requirements are low the dt implementation is comprised of a parallel structure where each branch consists of a time varying filter with one or two taps followed by a fixed recursive filter operating at the output sampling rate the coefficients of the time varying filters are calculated recursively this eliminates the need to store a large table of coefficients as is commonly done

a smooth signal generator based on quadratic b spline functions a hardware system of quadratic spline interpolation is proposed its properties are that switching noise with digital to analog d a converter chips does not appear and that the output is smooth without any analog lowpass filters that disorganize the phase characteristics

van kampen s embedding obstruction for discrete groups we give a lower bound to the dimension of a contractible mani fold on which a given group can act properly discontinuously in particular we show that the n fold product of nonabelian free groups cannot act prop erly discontinuously on r 2n 1

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

interpolation in digital modems ii implementation and performance for pt i see ibid vol 41 no 3 p 502 208 1993 properties of a specific class of interpolators that are based upon polynomials are discussed several implementations are described one of which is particularly convenient in practical hardware simulations demonstrate that simple interpolators give excellent performance in many cases two point linear interpolation is adequate if better performance is needed classical four point third order polynomials could be used better yet a novel four point interpolating filter with piecewise parabolic impulse response can have performance superior to that of the standard cubic interpolator and still be implemented much more simply the nco based control method presented in part i is shown to be equivalent to a conventional phase locked loop and its operation is verified by simulation

h infty optimal fractional delay filters fractional delay filters are digital filters to delay discrete time signals by a fraction of the sampling period since the delay is fractional the intersample behavior of the original analog signal becomes crucial in contrast to the conventional designs based on the shannon sampling theorem with the band limiting hypothesis the present paper proposes a new approach based on the modern sampled data h optimization that aims at restoring the intersample behavior beyond the nyquist frequency by using the lifting transform or continuous time blocking the design problem is equivalently reduced to a discrete time h optimization which can be effectively solved by numerical computation softwares moreover a closed form solution is obtained under an assumption on the original analog signals design examples are given to illustrate the advantage of the proposed method

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

digital two rate iir and hybrid iir fir filters for sampling rate conversion the paper discusses structures design methods and efficiency of digital two rate filters for sampling rate alteration by integer or rational ratios four canoic filter structures are demonstrated the direct form the parallel form the ladder lattice forms and a structure based on canonic all pass sections in addition a structure exploiting numerator coefficient symmetry is presented an extended class of filters hybrid iir fir where the passband equalization is performed by a combination of poles and off the unitcircle zeros is introduced we also describe a design method which iteratively utilizes the remez algorithm this simple extension of the mcclellan et al program 24 facilitates the design of virtually any kind of filter including hybrid filters and or two rate filters several design examples are presented to visualize the different aspects of two rate filtering

analysis of coefficient quantization errors in state space digital filters in this paper we examine the sensitivity of state space digital filters to coefficient quantization errors a cost function using an l p norm criteria is used to measure the deviation in frequency response of the filter the cost function is used to indicate the connection between roundoff noise and filter sensitivity to coefficient quantization errors then based on this cost function it is shown that low roundoff noise state space digital filters filters in balanced coordinates and filters based on polynomials orthogonal on the unit circle have low sensitivity to coefficient quantization errors furthermore the sensitivity of the filters is shown to depend on the second order modes thereby exhibiting certain robustness and immunity to filter bandwidth

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

a new class of explicit interpolatory splines and related measurement estimation this paper proposes a new and rich class of cardinal splines called generalized mk splines gmk splines which manifest several beneficial properties especially the interpolation property and the ability to obtain a construction with a desired frequency response the gmk spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted b splines of that degree in a unified framework the gmk spline bases form riesz bases of their generating spaces and the gmk spline frequency responses meet strang fix conditions furthermore we provide search methods for minimal support gmk splines and local optimal gmk spline frequency responses under different given passband frequencies several related estimates are theoretically assessed including energy magnitude maximum and approximation error of gmk spline interpolated signals in signal processing applications the gmk splines behave as interpolating fir filters with simple implementation and some of them attain lower error than other traditional interpolation system responses such as cubic cardinal b spline and linear spline when approximating to an ideal lowpass filter experimental results show that the gmk spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction

description of surfaces associated with grassmannian sigma models on minkowski space we construct and investigate smooth orientable surfaces in su n algebras the structural equations of surfaces associated with grassmannian sigma models on minkowski space are studied using moving frames adapted to the surfaces the first and second fundamental forms of these surfaces as well as the relations between them as expressed in the gauss weingarten and gauss codazzi ricci equations are found the scalar curvature and the mean curvature vector expressed in terms of a solution of grassmanian sigma model are obtained

paley wiener characterization of kernels for finite rate of innovation sampling exact reconstruction of finite rate of innovation signals can be achieved by employing customized sampling kernels that satisfy certain frequency domain properties we impose compact support in time as an additional constraint considering frequency domain reconstruction we derive conditions for admissible sampling kernels and corresponding sampling rates our constructive kernel design methodology is based on the paley wiener theorem for compactly supported functions the new kernels satisfy generalized strang fix conditions and have specific polynomial modulated exponential reproducing properties unlike exponential splines which have a support that is directly proportional to the number of exponentials they can generate the proposed kernels have a support that is independent of that number to analyze noise robustness we consider a special member of the class that has a sum of modulated splines sms form in the time domain and optimize its parameters to minimize the noise variance the sum of sincs sos kernel reported in the literature is an instance of this construction in noise robustness analysis sms kernels show improvement in mean squared error mse compared with the state of the art alternatives in continuous time noise the improvement in mse is about 2 db for low signal to noise ratio snr and 7 db for high snr in the case of discrete white gaussian noise the mse is lower by as much as 25 db by using a higher order sms kernel compared with the sos kernel for snrs in the range of 10 15 db

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

gauge equivalence of sigma models with non compact grassmannian manifolds the gauge equivalence ge of sigma models associated with non compact grassmannian manifolds is investigated with emphasis on the necessary restrictions for the choice of gauge elements in such cases the importance of ge in solving a non linear system with the help of inverse scattering data of its gauge related counterpart is demonstrated the gauge relations between generalised landau lifshitz ll and non linear schrodinger nls type equations and also between non linear sigma models and generalised sine sinh gordon equations for non compact su p q s u u v u s t manifolds are established using h gauge invariance of ll the ge is extended to some higher order specific non linear systems the gauge connection among various ll and nls equations are schematically represented along with the recovery of earlier results important new results some with significant non compact structures are discovered

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

classical nonlinear σ models on grassmann manifolds of compact or noncompact type grassmannian models are reexamined in the light of a new geometrical result namely the cartan immersion of a riemannian symmetric space g h into its isometry group g is not always a diffeomorphism onto the set m introduced by eichenherr and forger nucl phys b 164 528 1980 this is the case in particular for grassmann manifolds of all types their set m is shown to be disconnected and its structure is completely analyzed as a consequence a new constraint must be introduced in the backlund transformation bt method proposed by harnad saint aubin and shnider commun math phys 92 329 1984 it is shown however to always be satisfied for grassmann manifolds of compact type i e g compact but the problem remains open in most other cases on the other hand the bt method is extended to the euclidean regime

explicit forms of backlund transformations for the grassmannian and cp n 1 sigma models a fairly wide class of classical solutions of the euclidean two dimensional grassmannian and cpn 1 sigma models has been constructed explicitly and elementarily by the present author starting from these classical solutions we derive the explicit forms of the backlund transformations following harnad et al properties of the backlund transformations for the grassmannian models are discussed in particular a simple interpretation of the backlund transformations for the cpn 1 model is obtained generalization of these results to the noncompact grassmannian sigma models is straightforward

a proof of the hilbert smith conjecture for actions by lipschitz maps the classical hilbert 5th problem 14 asks whether every nite dimensional locally euclidean topological group is necessarily a lie group it was solved in the a rmative by von neumann 23 for compact groups in 1933 and by gleason 11 and by montgomery and zippin 20 for locally compact groups in 1952 a more general version of the hilbert 5th problem called the hilbert smith conjecture asserts that among all locally compact groups only lie groups g can act e ectively on nite dimensional manifolds m i e each g g e moves at least one point of m 28 it follows from the work of newman 24 and smith 29 that this conjecture is equivalent to its special case when the acting group g is the group of p adic integers ap in 1946 bochner and montgomery 3 proved the hilbert smith conjecture for groups g acting e ectively on a manifold m by di eomorphisms a simpler geometrical proof was obtained by skopenkov and the authors 25 using the idea of smooth homogeneity a compact subset k m of a smooth manifold m is said to be smoothly ambiently homogeneous i e for each x y k there exists a di eomorphism h m k x m k y it was shown that this property implies that k is a smooth submanifold of m therefore g k is a lie group the proof reveals a close relationship between homogeneity and taming theory for compact subsets of rn which are pinched by tangent balls the latter problem was investigated in the past by various authors 6 10 12 16 17 see also a very interesting paper by hahn 13 an interesting approach to the hilbert smith conjecture is via wild cantor sets in rn with strong homogeneity properties note that the antoine necklace

digital methods for conversion between arbitrary sampling frequencies the paper presents filtering methods for interfacing time discrete systems with different sampling frequencies the methods are applicable for sampling rate conversion between any two sampling frequencies the conversion ratio may even be irrational or slowly time varying interpolation by irrational factors requires digital filters with nonperiodically varying coefficients this is dealt with in two ways 1 all possible coefficient values are precalculated this is in a sense possible because of the finite resolution needed or 2 the coefficients can be updated in real time using either fir or iir filters the first solution requires a huge coefficient memory the second scheme on the other hand is computationally intensive while discussing both of these solutions more practical intermediate schemes incorporating both fir and iir type filters are suggested the suggested practical implementations are either based on analog reconstruction filters where the derived digital filter coefficients are functions of the distances between current input and output samples or digital interpolators combined with simple analog interpolation schemes for finding the desired values in between the uniform output samples from the digital interpolator

on the design of fir digital differentiators which are maximally linear at the frequency pi p p in positive integers digital differentiators dds which are maximally linear at the spot frequency omega pi p p in positive integers are proposed for operation over a narrow band of frequencies the suggested dds besides giving zero phase error over the entire band of frequencies pi

infinite dimensional lie algebras acting on the solution space of various sigma models infinite dimensional lie algebras of infinitesimal transformations acting on the solution space of various two dimensional models are investigated the main tools are i takasaki s interpretation commun math phys 94 35 1984 of the solutions of the associated linear system in terms of points in an infinite dimensional grassmann manifold and ii mikhailov s reduction procedure physica d 3 73 1981 for linear systems takasaki s approach leads for the models with values in a lie group g to a set of transformations that has the structure of the loop algebra g r t t 1 where g is the lie algebra of g this algebra has already been encountered by dolan phys rev lett 47 1371 1981 and by wu nucl phys b 211 160 1983 among others the models with a wess zumino term are also considered the algebraic structure is found to be the same finally mikhailov s procedure is used to study the models with values in a riemannian symmetric space rss g h which is not a lie group th

a class of linear phase fir filters for decimation interpolation and narrow band filtering a new class of linear phase finite impulse response fir digital filters for decimation and interpolation is discussed the transfer function of these filters is of the form h z a zd b z where d is the decimation or interpolation ratio and a zd is realized at the lower sampling rate the polynomial a zd is adjusted so that the overall magnitude response presents an equiripple behavior in the pass band whereas b z provides an equiripple stopband response an efficient iterative procedure for the design of the filters of this type is presented and the optimal selection of the orders of a zd and b z is considered several examples show how the new decimators and interpolators require significantly fewer multiplications per second than equivalent conventional linear phase fir and elliptic designs the performance of these linear phase filters is about the same as that of the recursive filters of martinez and parks with only powers of zdin the denominator some characteristics of the new filters such as zero positions coefficient sensitivity and output noise variance due to roundoff errors are discussed and compared with conventional fir designs furthermore it is shown that by cascading multistage new decimators and interpolators we obtain efficient narrow band filters requiring considerably fewer multiplications per output sample than equivalent elliptic designs

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

warped distance for space variant linear image interpolation the problem of image interpolation using linear techniques is dealt with in this paper conventional space invariant methods are revisited and changed into space variant ones by introducing the concept of the warped distance among the pixels of an image a better perceptual rendition of the image details is obtained in this way this effect is proved both via the evaluation of the response to an idealized sigmoidal edge model and with experiments on real world images the computational costs of the proposed approach are very small when compared to those of state of the art nonlinear interpolation operators

proper actions of lattices on contractible manifolds every lattice in a connected semi simple lie group g acts properly discontinuously by isometries on the contractible manifold g k k a maximal compact subgroup of g we prove that if acts on a contractible manifold w and if either 1 the action is properly discontinuous or 2 w is equipped with a complete riemannian metric the action is by isometries and with unbounded orbits g is simple with finite center and rank 1 then dimw dimg k

least squares design of higher order nonrecursive differentiators a method is described that can be used to design non recursive linear phase higher order differentiators that can perform differentiation over any frequency range the method is based on formulating the absolute mean square error between the amplitude responses of the practical and ideal differentiator as a quadratic function the coefficients of the differentiators are obtained by solving a set of linear equations this method leads to a lower mean square error and is computationally more efficient than both the eigenfilter method and the method based on the remez exchange algorithm design of differentiators based on minimization of the relative mean square error is also carried out finally the method is extended to the design of frequency selective higher order differentiators

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

solutions of euclidean σ models on noncompact grassmann manifolds a method for constructing explicit solutions of classical two dimensional euclidean models on grassmann manifolds of noncompact type and their supersymmetric extensions is presented it generalizes the holomorphic method developed by din zakrzewski and sasaki in the compact case by taking properly into account the geometry of indefinite metric spaces the solutions obtained are local in the sense that they may become singular on certain curves in the plane and the signature of the solution may change across such a curve

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