a review of geometric transformations for nonrigid body registration this paper provides a comprehensive and quantitative review of spatial transformations models for nonrigid image registration it explains the theoretical foundation of the models and classifies them according to this basis this results in two categories physically based models described by partial differential equations of continuum mechanics e g linear elasticity and fluid flow and basis function expansions derived from interpolation and approximation theory e g radial basis functions b splines and wavelets recent work on constraining the transformation so that it preserves the topology or is diffeomorphic is also described the final section reviews some recent evaluation studies the paper concludes by explaining under what conditions a particular transformation model is appropriate

oscillation criteria of first order linear difference equations with delay argument abstract this paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument u k p k u k 0 k n where u k u k 1 u k p n r n n and lim k k examples illustrating the results are given it is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument k

unwarping of unidirectionally distorted epi images echo planar imaging epi is a fast nuclear magnetic resonance imaging mri method unfortunately local magnetic field inhomogeneities induced mainly by the subject s presence cause significant geometrical distortion predominantly along the phase encoding direction which must be undone to allow for meaningful further processing so far this aspect has been too often neglected in this paper the authors suggest a new approach using an algorithm specifically developed for the automatic registration of distorted epi images with corresponding anatomically correct mri images they model the deformation field with splines which gives us a great deal of flexibility while comprising the affine transform as a special case the registration criterion is least squares interestingly the complexity of its evaluation does not depend on the resolution of the control grid the spline model gives the authors good accuracy thanks to its high approximation order the short support of splines leads to a fast algorithm a multiresolution approach yields robustness and additional speedup the algorithm was tested on real as well as synthetic data and the results were compared with a manual method a wavelet based sobolev type random deformation generator was developed for testing purposes a blind test indicates that the proposed automatic method is faster more reliable and more precise than the manual one

diffusion tensor analysis with invariant gradients and rotation tangents guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation tensor shape and orientation both have three degrees of freedom spanned by invariant gradients and rotation tangents respectively as an initial demonstration of the framework we create a tunable measure of tensor difference that can selectively respond to shape and orientation second to analyze the spatial gradient in a tensor volume a third order tensor our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented finally we apply the framework to decompose the fourth order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance including a direct approximation for the variance of tensor invariants such as fractional anisotropy

multiresolution feature extraction and selection for texture segmentation an approach is described for unsupervised segmentation of textured images local texture properties are extracted using local linear transforms that have been optimized for maximal texture discrimination local statistics texture energy measures are estimated at the output of an equivalent filter bank by means of a nonlinear transformation absolute value followed by an iterative gaussian smoothing algorithm this procedure generates a multiresolution sequence of feature planes with a half octave scale progression a feature reduction technique is then applied to the data and is determined by simultaneously diagonalizing scatter matrices evaluated at two different spatial resolutions this approach provides a good approximation of r a fisher s 1950 multiple linear discriminants and has the advantage of requiring no a priori knowledge this feature reduction methods appears to be an improvement on the commonly used karhunen loeve transform and allows efficient texture segmentation based on simple thresholding

oscillations of delay difference equations with oscillating coefficients abstract in this paper we establish sufficient conditions for the oscillation of all solutions of the delay difference equation y n 1 y n p n y n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer an open problem in ladas in international conference on differential equations stability and control dekker new york 1990 is solved

two dimensional spline interpolation for image reconstruction a scheme of two dimensional 2 d spline interpolation for image reconstruction is presented it is aimed to be applied in image data compression an impulse response derived from 2 d quadratic spline functions is theoretically analysed then it is approximated to be a finite duration one fir for practical implementation with this scheme compression rate of greater than 12 is achieved for x ray left ventricular cineangiogram image data compression

support and approximation properties of hermite splines abstract in this paper we formally investigate two mathematical aspects of hermite splines that are relevant to practical applications we first demonstrate that hermite splines are maximally localized in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties then we precisely quantify the approximation power of hermite splines for the reconstruction of functions and their derivatives it is known that the hermite and b spline approximation schemes have the same approximation order more precisely their approximation error vanishes as o t 4 when the step size t goes to zero in this work we show that they actually have the same asymptotic approximation error constants too therefore they have identical asymptotic approximation properties hermite splines combine optimal localization and excellent approximation power while retaining interpolation properties and closed form expression in contrast to existing similar functions these findings shed a new light on the convenience of hermite splines in the context of computer graphics and geometrical design

an oscillation criterion for a delay difference equation by using the method of yu zhang and qian in j s yu b g zhang x z qian oscillation of delay difference equations with oscillating coefficients j math anal appl 177 1993 432 444 we prove another oscillation criterion for all solutions of the following linear delay difference equation x n 1 x n p nx n k 0 n 0 1 2 where p n is a sequence of real numbers and k is a positive integer

estimating intensity variance due to noise in registered images applications to diffusion tensor mri image registration techniques which require image interpolation are widely used in neuroimaging research we show that signal variance in interpolated images differs significantly from the signal variance of the original images in native space we describe a simple approach to compute the signal variance in registered images based on the signal variance and covariance of the original images the spatial transformations computed by the registration procedure and the interpolation or approximation kernel chosen the method is general and could handle various sources of signal variability such as thermal noise and physiological noise provided that their effects can be assessed in the original images our approach is applied to diffusion tensor dt mri data assuming only thermal noise as the source of variability in the data we show that incorrect noise variance estimates in registered diffusion weighted images can affect dt parameters as well as indices of goodness of fit such as chi square maps in addition to dt mri we believe that this methodology would be useful any time parameter extraction methods are applied to registered or interpolated data such as in relaxometry and functional mri studies

least squares image resizing using finite differences we present an optimal spline based algorithm for the enlargement or reduction of digital images with arbitrary noninteger scaling factors this projection based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are b splines of any degree n a noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a for a given choice of basis functions the results of our method are consistently better than those of the standard interpolation procedure the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal to noise ratio the method can be generalized to include other classes of piecewise polynomial functions expressed as linear combinations of b splines and their derivatives

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

recursive regularization filters design properties and applications least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed for each problem there is a family of discrete regularization filters r filters which allow an efficient determination of the solutions these operators are stable symmetric lowpass filters with an adjustable scale factor two decomposition theorems for the z transform of such systems are presented one facilitates the determination of their impulse response while the other allows an efficient implementation through successive causal and anticausal recursive filtering a case of special interest is the design of r filters for the first and second order difference operators these results are extended for two dimensional signals and for illustration purposes are applied to the problem of edge detection this leads to a very efficient implementation 8 multiplies 10 adds per pixel of the optimal canny edge detector based on the use of a separable second order r filter

fast migration inversion with multivalued ray fields part 1 method validation test and application in 2d to marmousi we address the problem of building an efficient algorithm for ray based migration inversion in complex media characterized by multivalued traveltime since the multivalued ray maps can be computed by wavefront construction the key point lies in the interpolation of these multivalued coarse maps for this goal we propose two new strategies the first one is based on an interpolation of the rayfield after identifying the ray branches on the elementary cells of the coarse grid the second is based on an extrapolation of the arrivals from the coarse grid points both strategies are evaluated in two dimensions with the marmousi data set demonstrating the cpu efficiency and accuracy of both approaches since they both consider multiarrivals locally the associated overhead cost with respect to a single arrival implementation is quite low twice the computing time of the marmousi test case the application also reveals that the new interpolation strategy is more efficient in terms of cpu time compared to the extrapolation strategy but at the expense of a more complex numerical implementation and less accurate results

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

soft computing approach to feature extraction based on both wavelet theory and fuzzy theory a soft computing system scs is proposed for feature extraction of signals the proposed scs approach possesses the advantages of soft decision making on wavelet coefficients for feature extraction adaptive selectivity of mapping factors to coarse to fine resolution and compact form of feature representation with the scs feature extractor fuzzy sets are used to provide a robust representation for signal information and wavelet transform is used to decompose a signal into detail and approximation signals at a given resolution the detail and approximation signals are inputted to the proposed scs to extract signal features at that resolution level the sensitivity in feature extraction of the proposed approach can be adapted by tuning the fuzzy sets for the detail and approximation signals at different resolutions the signal can be examined and suitable features can be extracted examples of both one dimensional signals and two dimensional fingerprint images are used to illustrate the proposed soft computing approach for feature extraction and pattern recognition the results show that the extracted features are sensitive enough to distinguish the similar signal from the different ones and are robust enough to tolerate noise corruption

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

polynomial alias higher degree fuzzy transform of complex valued functions abstract in this article we propose a general approach to the computation of components of the direct higher degree fuzzy transform apart from the orthogonal bases of the subspaces of polynomials of weighted hilbert spaces with respect to a generalized uniform fuzzy partition which are used in all papers on fuzzy transform of higher degree we admit also the non orthogonal bases an advantage of using non orthogonal bases consists in the possibility of replacing orthogonal polynomials derivation of which by the gram schmidt orthogonalization process can be questionable difficult or imprecise by suitable non orthogonal polynomials of much simpler form we present a simple matrix calculus and show how it can be used to introduce the components of the direct higher degree fuzzy transform with the help of the monomial basis we prove a convergence theorem and an approximation theorem for the higher degree fuzzy transform the results are illustrated by examples including a comparison with standard methods

some oscillation criteria for difference equations abstract consider the difference equation x n 1 x n p n x n k 0 where p n is a sequence of nonnegative real numbers and k is a positive integer it is proved that all solutions oscillate if there exists some positive integer l such that lim sup n i 0 k pn i i 0 k j 1 k pn i j m 0 l 1 i 0 k j 0 m 1 pn jk 1 1 or lim sup n i 1 k pn i i 1 k j 1 k pn i j m 0 l 1 i 0 k j 0 m 1 pn jk 1 1

a further result on the oscillation of delay difference equations in this paper we are concerned with the delay difference equations of the form yn 1 yn pnyn k 0 n 0 1 2 where pn 0 and k is a positive integer we prove by using a new technique that n 0 i nn kpi ln i nn k pi 1 sign i nn k pi i n 1n k pi ln i n 1n k pi 1 sign i n 1n k pi guarantees that all solutions of equation oscillate which improves many previous well known results in particular our theorems also fit the case where n 1i n kpi kk 1 k 1 k 1 in addition we present a nonoscillation sufficient condition for equation

causal cubic splines formulations interpolation properties and implementations the paper presents two formulations of causal cubic splines with equidistant knots both are based on a causal direct b spline filter with parallel or cascade implementation in either implementation the causal part of the impulse response is realized with an efficient infinite impulse response iir structure while only the anticausal part is approximated with a finite order finite impulse response fir filter resulting cubic coefficients are computed from the causal b spline coefficients by using a third order output fir filter with either single input multiple output simo or multiple input multiple output mimo structure depending on the chosen formulation of the cubic spline the paper demonstrates and proves that the properties of the resulting causal splines are quite different whether they are based on a more popular b spline formulation or a bit neglected tridiagonal matrix formulation it is shown that the proposed low complexity but accurate causal interpolators can be realized for many practical applications with the delay of only a few samples

a necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments abstract in this article we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions in particular we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non negative examples and tables are given to illustrate our results

oscillation of delay difference equation we introduce some new techniques to analyze generalized characteristic equations to obtain several infinite sum sharp conditions for oscillation of the following delay difference equation xn 1 xn i 1mpi n xn ki 0 n 0 1 2 where pi n 0 and ki are positive integers

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

polynomial spline approximation of clarke s model we investigate polynomial spline approximation of stationary random processes on a uniform grid applied to clarke s model of time variations of path amplitudes in multipath fading channels with doppler scattering the integral mean square error mse for optimal and interpolation splines is presented as a series of spectral moments the optimal splines outperform the interpolation splines however as the sampling factor increases the optimal and interpolation splines of even order tend to provide the same accuracy to build such splines the process to be approximated needs to be known for all time which is impractical local splines on the other hand may be used where the process is known only over a finite interval we first consider local splines with quasioptimal spline coefficients then we derive optimal spline coefficients and investigate the error for different sets of samples used for calculating the spline coefficients in practice approximation with a low processing delay is of interest we investigate local spline extrapolation with a zero processing delay the results of our investigation show that local spline approximation is attractive for implementation from viewpoints of both low processing delay and small approximation error the error can be very close to the minimum error provided by optimal splines thus local splines can be effectively used for channel estimation in multipath fast fading channels

addendum b spline interpolation in medical image processing analyzes b spline interpolation techniques of degree 2 4 and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging lehmann et al 1999 it is shown that high degree b spline interpolation has superior fourier properties smallest interpolation error and reasonable computing times therefore high degree b splines are preferable interpolators for numerous applications in medical image processing particularly if high precision is required if no aliasing occurs this result neither depends on the geometric transform applied for the tests nor the actual content of images

new oscillation criteria for delay difference equations abstract this paper considers the delay difference equation x n 1 x n p n x n k 0 n 0 1 2 where pn is a sequence of nonnegative real numbers and k is a positive integer some new oscillation criteria for this equation are obtained our theorems improve all known results in the literature

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

new results for oscillation of delay difference equations in this paper we consider the delay difference equation xn 1 xn pnxn k 0 n 0 1 2 where pn is a sequence of nonnegative real numbers and k is a positive integer some new results for the oscillation of this equation are obtained our theorems improve all known results in the literature

a comparative study of open default theories the paper examines the definitions of open default theories known from the literature first it is shown that none of them is satisfactory either for formal or for intuitive reasons next a new approach is considered it is free from the obvious deficiencies of the known definitions but possesses their positive properties

oscillation results for difference equations with several oscillating coefficients this paper presents a new sufficient condition for the oscillation of all solutions of difference equations with several deviating arguments and oscillating coefficients corresponding difference equations of both retarded and advanced type are studied examples illustrating the results are also given

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

oscillations of delay difference equations in a critical state consider the first order delay difference equation xn i 1m pi n xn ki 0 n 0 1 2 in the case where lim infn pi n pi 0 ki 0 i 1 2 a necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding limiting equation xn i 1m pixn ki 0 n 0 1 2 admits a nonoscillatory solution it is to be pointed out that there is no result in this critical state for a delay difference equation with more than one delay argument

fast computation of the difference of low pass transform this paper defines the difference of low pass dolp transform and describes a fast algorithm for its computation the dolp is a reversible transform which converts an image into a set of bandpass images a dolp transform is shown to require o n2 multiplies and produce o n log n samples from an n sample image when gaussian low pass filters are used the result is a set of images which have been convolved with difference of gaussian dog filters from an exponential set of sizes a fast computation technique based on resampling is described and shown to reduce the dolp transform complexity to o n log n multiplies and o n storage locations a second technique cascaded convolution with expansion is then defined and also shown to reduce the computational cost to o n log n multiplies combining these two techniques yields an algorithm for a dolp transform that requires o n storage cells and requires o n multiplies

fast b spline transforms for continuous image representation and interpolation efficient algorithms for the continuous representation of a discrete signal in terms of b splines direct b spline transform and for interpolative signal reconstruction indirect b spline transform with an expansion factor m are described expressions for the z transforms of the sampled b spline functions are determined and a convolution property of these kernels is established it is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering fast computational algorithms based on the recursive implementations of these filters are proposed a b spline interpolator can also be characterized in terms of its transfer function and its global impulse response cardinal spline of order n the case of the cubic spline is treated in greater detail the present approach is compared with previous methods that are reexamined from a critical point of view it is concluded that b spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner

multiresolution approximation using shifted splines we consider the construction of least squares pyramids using shifted polynomial spline basis functions we derive the pre and post filters as a function of the degree n and the shift parameter spl delta we show that the underlying projection operator is entirely specified by two transfer functions acting on the even and odd signal samples respectively we introduce a measure of shift invariance and show that the most favorable configuration is obtained when the knots of the splines are centered with respect to the grid points i e spl delta 1 2 when n is odd and spl delta 0 when n is even the worst case corresponds to the standard multiresolution setting where the spline spaces are nested

a continuous time linear system identification method for slowly sampled data both direct and indirect methods exist for identifying continuous time linear systems a direct method estimates continuous time input and output signals from their samples and then use them to obtain a continuous time model whereas an indirect method estimates a discrete time model first both methods rely on fast sampling to ensure good accuracy in this paper we propose a more direct method where a continuous time linear model is directly fitted to the available samples this method produces an exact model asymptotically modulo some possible aliasing ambiguity even when the sampling rate is relatively slow we also state conditions under which the aliasing ambiguity can be resolved and we provide experiments showing that the proposed method is a valid option when a slow sampling frequency must be used but a large number of samples is available

a note on the stable model semantics for logic programs abstract the stable model semantics for logic programs is extended from ground literals onto open literals by augmenting the program language with an infinite set of new constants this in turn leads to a natural translation of logic programs into open default theories

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

computation of wavelet coefficients from average samples there exist efficient methods to compute the wavelet coefficients of a function f t from its point samples f t email protected email protected n however in many applications the available samples are average samples of the type f t email protected u t dt where the averaging function u t reflects the characteristic of the acquisition device in this work methods to compute the coefficients in a biorthogonal wavelet system from average samples are studied error estimations are obtained and using them the optimal values for the parameters in the proposed approximation rules are calculated the obtained error estimations can also be applied to the rules that compute the coefficients from point samples and thus these estimations can be used to compare and to choose between the different methods proposed in the literature the methods proposed here also allow us to compute the biorthogonal wavelet coefficients from the coefficients in another biorthogonal wavelet system

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

fast fully 3 d image reconstruction in pet using planograms we present a method of performing fast and accurate three dimensional 3 d backprojection using only fourier transform operations for line integral data acquired by planar detector arrays in positron emission tomography this approach is a 3 d extension of the two dimensional 2 d linogram technique of edholm by using a special choice of parameters to index a line of response lor for a pair of planar detectors rather than the conventional parameters used to index a lor for a circular tomograph all the lors passing through a point in the field of view fov lie on a 2 d plane in the four dimensional 4 d data space thus backprojection of all the lors passing through a point in the fov corresponds to integration of a 2 d plane through the 4 d planogram the key step is that the integration along a set of parallel 2 d planes through the planogram that is backprojection of a plane of points can be replaced by a 2 d section through the origin of the 4 d fourier transform of the data backprojection can be performed as a sequence of fourier transform operations for faster implementation in addition we derive the central section theorem for planogram format data and also derive a reconstruction filter for both backprojection filtering and filtered backprojection reconstruction algorithms with software based fourier transform calculations we provide preliminary comparisons of planogram backprojection to standard 3 d backprojection and demonstrate a reduction in computation time by a factor of approximately 15

quasi interpolation by means of filter banks we consider the problem of approximating a regular function f t from its samples f nt taken in a uniform grid quasi interpolation schemes approximate f t with a dilated version of a linear combination of shifted versions of a kernel t specifically fapprox t t af n t t n in a way that the polynomials of degree at most l 1 are recovered exactly these approximation schemes give order l i e the error is o tl where t is the sampling period recently quasi interpolation schemes using a discrete prefiltering of the samples f nt to obtain the coefficients af n have been proposed they provide tight approximation with a low computational cost in this work we generalize considering rational filter banks to prefilter the samples instead of a simple filter this generalization provides a greater flexibility in the design of the approximation scheme the upsampling and downsampling ratio r of the rational filter bank plays a significant role when r 1 the scheme has similar characteristics to those related to a simple filter approximation schemes corresponding to smaller ratios give less approximation quality but in return they have less computational cost and involve less storage load in the system

polynomial spline signal approximations filter design and asymptotic equivalence with shannon s sampling theorem the least squares polynomial spline approximation of a signal g t in l sub 2 r is obtained by projecting g t on s sup n r the space of polynomial splines of order n it is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g t with a b spline of order n more specifically the coefficients of the b spline interpolation of order 2n 1 of the sampled filtered sequence are identical to the coefficients of the least squares approximation of g t of order n it is shown that this approximation can be obtained from a succession of three basic operations prefiltering sampling and postfiltering which confirms the parallel with the classical sampling reconstruction procedure for bandlimited signals the frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift invariant basis functions of s sup n r the standard expansion in terms of b spline coefficients a representation in terms of sampled signal values and a representation using orthogonal basis functions

a minimum squared error framework for generalized sampling we treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a hilbert space our formulation is general and includes as special cases reconstruction from finitely many samples as well as uniform sampling of continuous time signals which are not necessarily bandlimited to obtain a good approximation of the signal in the reconstruction space from its samples we suggest two design strategies that attempt to minimize the squared norm error between the signal and its reconstruction the approaches we propose differ in their assumptions on the input signal if the signal is known to lie in an appropriately chosen subspace then we propose a method that achieves the minimal squared error on the other hand when the signal is not restricted we show that the minimal norm reconstruction cannot generally be obtained instead we suggest minimizing the worst case squared error between the reconstructed signal and the best possible but usually unattainable approximation of the signal within the reconstruction space we demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

sufficient conditions for the oscillation of linear delay difference equations with oscillating coefficients abstract in this paper sufficient conditions for the oscillation of all solutions of the delay difference equation xn 1 xn pnxn k 0 n 0 1 2 are established where the coefficient pn itself may be allowed to be oscillatory we also give an example to demonstrate the advantage of our results

a fast efficient computation of cubic spline interpolation in image codec it is shown that cubic spline interpolation csi can be performed by a fast and efficient computation for the encoding and decoding processes of image coding it requires substantially fewer additions and multiplications than the original csi algorithm furthermore a new type of overlap save scheme is utilized to solve the boundary condition problems that occur between two neighboring subimages in the actual image it is also shown that a very efficient nine point winograd discrete fourier transform wdft can be used to replace the fast fourier transform fft needed to implement the csi scheme in the modified jpeg encoder finally the proposed fast new csi algorithm with a compression ratio of 9 1 is used along with the jpeg standard to speed up the modified jpeg encoder decoder and still obtain a better quality of reconstructed image for higher compression ratios

interpolation artifacts in sub pixel image registration we consider the problem of registering aligning two images to sub pixel accuracy by optimization of objective functions constructed from the images intensity values we show that some widely used interpolation methods can introduce multiple local optima in the energy of the interpolated image which if not counter balanced by other terms can cause local optima in registration objective functions including the sum of squared differences cross correlation and mutual information we discuss different solutions to address the problem based on high degree b spline interpolation low pass filtering the images and stochastic integration numerical examples using synthetic and real signals and images are shown