fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

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

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

shift variance and nonstationarity of linear periodically shift variant systems and applications to generalized sampling reconstruction processes we present a systematic analysis on shift variance of linear t periodically shift variant t lpsv systems with continuous time input and output we first determine how far a t lpsv system h is away from being shift invariant in terms of two measures namely the distance and the angle between h and the subspace of linear shift invariant systems we then consider the level and the index of shift variance svi of h via commutator of h and the shift operator and examine the amount of shift variance of h under particular input all these quantities are characterized in terms of the so called shift variant part and the shift variance kernel of h moreover we study nonstationarity of wide sense cyclostationary random processes possibly induced by t lpsv systems we define and calculate a measure of nonstationarity nst of the output as the svi of the autocorrelation operator the nst can be used to characterize performance loss when the output is treated with a wss approach the expected shift variance of h subject to particular wss random process is also calculated we apply the above results to generalized sampling reconstruction processes gsrps in addition we consider the approximation error of the gsrp and relate it to the shift variance measures three sampling schemes namely the orthogonal the consistent and the minimax regret sampling are examined in detail we show that for both deterministic and wss random inputs the minimax regret gsrp always introduces less shift variance than the consistent counterparts numerical results for the gsrps of b splines of various orders are provided

multiresolution registration of remote sensing imagery by optimization of mutual information using a stochastic gradient image registration is the process by which we determine a transformation that provides the most accurate match between two images the search for the matching transformation can be automated with the use of a suitable metric but it can be very time consuming and tedious we introduce a registration algorithm that combines a simple yet powerful search strategy based on a stochastic gradient with two similarity measures correlation and mutual information together with a wavelet based multiresolution pyramid we limit our study to pairs of images which are misaligned by rotation and or translation and present two main results first we demonstrate that in our application mutual information may be better suited for sub pixel registration as it produces consistently sharper optimum peaks than correlation then we show that the stochastic gradient search combined with either measure produces accurate results when applied to synthetic data as well as to multitemporal or multisensor collections of satellite data mutual information is generally found to optimize with one third the number of iterations required by correlation results also show that a multiresolution implementation of the algorithm yields significant improvements in terms of both speed and robustness over a single resolution implementation

non universal critical behaviour in mean field theory of inhomogeneous systems abstract we consider the layered magnetic systems with inhomogeneous inter layer couplings and study their critical properties within the framework of mean field theory we consider two kinds of distribution of the couplings i quasi periodic according to the fibonacci sequence and ii smoothly inhomogeneous in which the couplings deviate from the bulk couplings by an amount of al 2 where l measures the distance from a free surface according to analytical and accurate numerical results the critical behaviour of both the problems is non universal and the corresponding critical exponents are coupling dependent

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

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

nonuniversal behavior for aperiodic interactions within a mean field approximation we study the spin 1 2 ising model on a bethe lattice in the mean field limit with the interaction constants following one of two deterministic aperiodic sequences the fibonacci or period doubling one new algorithms of sequence generation were implemented which were fundamental in obtaining long sequences and therefore precise results we calculate the exact critical temperature for both sequences as well as the critical exponents ensuremath beta ensuremath gamma and ensuremath delta for the fibonacci sequence the exponents are classical while for the period doubling one they depend on the ratio between the two exchange constants the usual relations between critical exponents are satisfied within error bars for the period doubling sequence therefore we show that mean field like procedures may lead to nonclassical critical exponents

aperiodic ising quantum chains some years ago luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic ising systems in this article we show how luck s criterion can be derived within an exact renormalization scheme for ising quantum chains with coupling constants modulated according to substitution rules luck s conjectures for this case are confirmed and refined among other outcomes we give an exact formula for the correlation length critical exponent for arbitrary two letter substitution sequences with marginal fluctuations of the coupling constants

a family of cyclic quartic fields arising from modular curves we study a family of cyclic quartic fields arising from the covering of modular curves x1 16 xo 16 an integral basis and a fundamental system of units are found it is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field recently 0 lecacheux 9 10 and h darmon 4 showed how to use coverings of modular curves to obtain cyclic extensions of q in particular they were able to give a geometric construction of a family of cyclic quintic fields discovered by e lehmer 11 the covering x1 n x0 n for n 2 has degree q n 2 and group z nz x i 1 for the quintic case they took n 25 which gave a cyclic covering of degree 10 then took the subcovering of degree 5 an important ingredient in the construction was the fact that xo 25 has genus 0 this also occurs for n 1 10 12 13 16 18 these all give trivial or quadratic coverings except for n 7 9 13 16 18 the values n 7 9 18 yield cubic extensions and can be shown to yield the family of polynomials x3 ax2 a 3 x 1 namely the simplest cubic fields 17 however it should be remarked that every cyclic cubic extension of q comes from a polynomial of this form if a is allowed to be rational similarly we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form x2 ax 1 with a rational the case n 13 is treated by lecacheux 9 it might be suspected that the sextic fields she obtains are the same as the simplest sextics constructed by m n gras 6 however these latter fields were found by taking the fixed field of an element of order 6 in pgl2 q aut q x therefore they come from a covering of curves of genus 0 but x1 13 has genus 2 alternatively these sextic fields must be different because the discriminants of the quadratic and cubic subfields are different in the present paper we study the case n 16 as above it might be hoped that this case would give a geometric construction of the quartic fields studied received march 14 1990 revised october 22 1990 1980 mathematics subject classification 1985 revision primary 1 i r 16 research supported in part by n s f 1991 american mathematical society 0025 5718 91 1 00 25 per page

comparison of interpolating methods for image resampling when resampling an image to a new set of coordinates for example when rotating an image there is often a noticeable loss in image quality to preserve image quality the interpolating function used for the resampling should be an ideal low pass filter to determine which limited extent convolving functions would provide the best interpolation five functions were compared a nearest neighbor b linear c cubic b spline d high resolution cubic spline with edge enhancement a 1 and e high resolution cubic spline a 0 5 the functions which extend over four picture elements c d e were shown to have a better frequency response than those which extend over one a or two b pixels the nearest neighbor function shifted the image up to one half a pixel linear and cubic b spline interpolation tended to smooth the image the best response was obtained with the high resolution cubic spline functions the location of the resampled points with respect to the initial coordinate system has a dramatic effect on the response of the sampled interpolating function the data are exactly reproduced when the points are aligned and the response has the most smoothing when the resampled points are equidistant from the original coordinate points thus at the expense of some increase in computing time image quality can be improved by resampled using the high resolution cubic spline function as compared to the nearest neighbor linear or cubic b spline functions

segment adaptive gradient angle interpolation we introduce a new edge directed interpolator based on locally defined straight line approximations of image isophotes spatial derivatives of image intensity are used to describe the principal behavior of pixel intersecting isophotes in terms of their slopes the slopes are determined by inverting a tridiagonal matrix and are forced to vary linearly from pixel to pixel within segments image resizing is performed by interpolating along the approximated isophotes the proposed method can accommodate arbitrary scaling factors provides state of the art results in terms of psnr as well as other quantitative visual quality metrics and has the advantage of reduced computational complexity that is directly proportional to the number of pixels

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

isotropic polyharmonic b splines scaling functions and wavelets in this paper we use polyharmonic b splines to build multidimensional wavelet bases these functions are nonseparable multidimensional basis functions that are localized versions of radial basis functions we show that rabut s elementary polyharmonic b splines do not converge to a gaussian as the order parameter increases as opposed to their separable b spline counterparts therefore we introduce a more isotropic localization operator that guarantees this convergence resulting into the isotropic polyharmonic b splines next we focus on the two dimensional quincunx subsampling scheme this configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach however up until now the design of appropriate filters for the quincunx scheme has mainly been done using the mcclellan transform in our approach we start from the scaling functions which are the polyharmonic b splines and as such explicitly known and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi orthogonal wavelet transform e g orthonormal b spline and dual the filters are automatically specified by the scaling relations satisfied by these functions we prove that the isotropic polyharmonic b spline wavelet converges to a combination of four gabor atoms which are well separated in the frequency domain we also show that these wavelets are nearly isotropic and that they behave as an iterated laplacian operator at low frequencies we describe an efficient fast fourier transform based implementation of the discrete wavelet transform based on polyharmonic b splines

sample rate conversion using b spline interpolation for ofdm based software defined radios this paper proposes arbitrary ratio sample rate conversion src architectures and a simpler b spline interpolation algorithm for orthogonal frequency division multiplexing ofdm based software defined radios sdrs with multiband and multi channel capabilities different from conventional standalone digital front end designs for sdrs the proposed src architectures combine the b spline interpolation with ofdm modulation and equalization for ofdm transmitter and receiver respectively with this combined design the passband droop introduced by the b spline interpolation can be more efficiently compensated using frequency domain pre distortion instead of conventional time domain pre filtering and hence an overall system complexity reduction is achieved a novel multi period b spline interpolation and re sampling structure is then constructed and an interpolation algorithm with lower implementation complexity than that of the conventional farrow structure is further developed the src performance is also analysed by deriving the signal to peak distortion ratio formulas which can be used as design tools for determining the required orders of b splines in the ofdm transmitter and receiver respectively finally src examples used in a high speed multiband multi channel microwave backhaul system are given and compared with conventional polyphase filterbank interpolation to demonstrate the practicality and performance of the proposed src architectures and interpolation algorithm

optimized compact support interpolation kernels in this paper we investigate the problem of designing compact support interpolation kernels for a given class of signals by using calculus of variations we simplify the optimization problem from an nonlinear infinite dimensional problem to a linear finite dimensional case and then find the optimum compact support function that best approximates a given filter in the least square sense l2 norm the benefit of compact support interpolants is the low computational complexity in the interpolation process while the optimum compact support interpolant guarantees the highest achievable signal to noise ratio snr our simulation results confirm the superior performance of the proposed kernel compared to other conventional compact support interpolants such as cubic spline

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

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

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

optimization of mutual information for multiresolution image registration we propose a new method for the intermodal registration of images using a criterion known as mutual information our main contribution is an optimizer that we specifically designed for this criterion we show that this new optimizer is well adapted to a multiresolution approach because it typically converges in fewer criterion evaluations than other optimizers we have built a multiresolution image pyramid along with an interpolation process an optimizer and the criterion itself around the unifying concept of spline processing this ensures coherence in the way we model data and yields good performance we have tested our approach in a variety of experimental conditions and report excellent results we claim an accuracy of about a hundredth of a pixel under ideal conditions we are also robust since the accuracy is still about a tenth of a pixel under very noisy conditions in addition a blind evaluation of our results compares very favorably to the work of several other researchers

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

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

image interpolation by adaptive 2 d autoregressive modeling and soft decision estimation the challenge of image interpolation is to preserve spatial details we propose a soft decision interpolation technique that estimates missing pixels in groups rather than one at a time the new technique learns and adapts to varying scene structures using a 2 d piecewise autoregressive model the model parameters are estimated in a moving window in the input low resolution image the pixel structure dictated by the learnt model is enforced by the soft decision estimation process onto a block of pixels including both observed and estimated the result is equivalent to that of a high order adaptive nonseparable 2 d interpolation filter this new image interpolation approach preserves spatial coherence of interpolated images better than the existing methods and it produces the best results so far over a wide range of scenes in both psnr measure and subjective visual quality edges and textures are well preserved and common interpolation artifacts blurring ringing jaggies zippering etc are greatly reduced

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

fast point based 3 d alignment of live cells typical time intervals between acquisitions of three dimensional 3 d images of the same cell in live cell imaging are in the orders of minutes in the meantime the live cell can move in a water basin on the stage this movement can hamper the studies of intranuclear processes we propose a fast point based image registration method for the suppression of the movement of a cell as a whole in the image data first centroids of certain intracellular objects are computed for each image in a time lapse series then a matching between the centroids which have the maximal number of pairs is sought between consecutive point sets by a 3 d extension of a two dimensional fast point pattern matching method which is invariant to rotation translation local distortion and extra missing points the proposed 3 d extension assumes rotations only around the z axis to retain the complexity of the original method the final step involves computing the optimal fully 3 d transformation between images from corresponding points in the least squares manner the robustness of the method was evaluated on generated data the results of the simulations show that the method is very precise and its correctness can be estimated this article also presents two practical application examples namely the registration of images of hp1 domains and the registration of images of telomeres more than 97 of time consecutive images were successfully registered the results show that the method is very well suited to live cell imaging

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

multiscale curvature based shape representation using b spline wavelets this paper presents a new multiscale curvature based shape representation technique with application to curve data compression using b spline wavelets the evolution of the curve is implemented in the b spline scale space which enjoys a number of advantages over the classical gaussian scale space for instance the availability of fast algorithms the b spline wavelet transforms are used to efficiently estimate the multiscale curvature functions based on the curvature scale space image we introduce a coarse to fine matching algorithm which automatically detects the dominant points and uses them as knots for curve interpolation

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

variational b spline level set a linear filtering approach for fast deformable model evolution in the field of image segmentation most level set based active contour approaches take advantage of a discrete representation of the associated implicit function we present in this paper a different formulation where the implicit function is modeled as a continuous parametric function expressed on a b spline basis starting from the active contour energy functional we show that this formulation allows us to compute the solution as a restriction of the variational problem on the space spanned by the b splines as a consequence the minimization of the functional is directly obtained in terms of the b spline coefficients we also show that each step of this minimization may be expressed through a convolution operation because the b spline functions are separable this convolution may in turn be performed as a sequence of simple 1 d convolutions which yields an efficient algorithm as a further consequence each step of the level set evolution may be interpreted as a filtering operation with a b spline kernel such filtering induces an intrinsic smoothing in the algorithm which can be controlled explicitly via the degree and the scale of the chosen b spline kernel we illustrate the behavior of this approach on simulated as well as experimental images from various fields

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

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

approximation error of shifted signals in spline spaces spline signal spaces offer several advantages for the representation of signals compared with the more traditional signal spaces of bandlimited signals among them are the finite support of b splines simple manipulations like differentiation and integration etc a major disadvantage however is that spline signal spaces are not closed under signal shifts in order to assess the approximation error introduced by shifting a spline signal the approximation error norm and its average are evaluated furthermore an upper bound on the expected normalized approximation error is derived using reid s inequality

simple algorithms and architectures for b spline interpolation it is proved that the toeplitz binary value matrix inversion associated with mth order b spline interpolation can be implemented using only 2 m 1 additions pipelined architectures are developed for real time b spline interpolation based on simple running average filters it is shown that an ideal interpolating function which is approximated by a truncated sinc function with m half cycles can be implemented using b splines with m 2 multiplies with insignificant loss of performance the coefficients at the knots of the truncated sinc function can be approximated using coefficients which are powers of two the resulting implementation requires only m 4m 6 additions it is believed that the truncated sinc function approximated by zero order b spline functions actually achieves the best visual performance

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

adaptive morphological filters based on a multiple orientation vector field dependent on image local features this paper addresses the formulation of adaptive morphological filters based on spatially variant structuring elements the adaptivity of these filters is achieved by modifying the shape and orientation of the structuring elements according to a multiple orientation vector field this vector field is provided by means of a bank of directional openings which can take into account the possible multiple orientations of the contours in the image after reviewing and formalizing the definition of the spatially variant dilation erosion opening and closing the proposed structuring elements are described these spatially variant structuring elements are based on ellipses which vary over the image domain adapting locally their orientation according to the multiple orientation vector field and their shape the eccentricity of the ellipses according to the distance to relevant contours of the objects the proposed adaptive morphological filters are used on gray level images and are compared with spatially invariant filters with spatially variant filters based on a single orientation vector field and with adaptive morphological bilateral filters results show that the morphological filters based on a multiple orientation vector field are more adept at enhancing and preserving structures which contains more than one orientation

preperiodic points for quadratic polynomials with small cycles over quadratic fields given a number field k and a polynomial f z in k z one can naturally construct a finite directed graph g f k whose vertices are the k rational preperiodic points of f with an edge alpha rightarrow beta if and only if f alpha beta the dynamical uniform boundedness conjecture of morton and silverman suggests that for fixed integers n ge 1 and d ge 2 there are only finitely many isomorphism classes of directed graphs g f k as one ranges over all number fields k of degree n and polynomials f z in k z of degree d in the case n d 1 2 poonen has given a complete classification of all directed graphs which may be realized as g f mathbb q for some quadratic polynomial f z in mathbb q z under the assumption that f does not admit rational points of large period the purpose of the present article is to continue the work begun by the author faber and krumm on the case n d 2 2 by combining the results of the previous article with a number of new results we arrive at a partial result toward a theorem like poonen s with a similar assumption on points of large period but over all quadratic extensions of mathbb q

non additive interval valued f transform this article proposes a new interval valued fuzzy transform its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built the main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward this interpretation relates to a traditional sampling reconstruction framework where little is known about the sampling and or reconstructing kernels numerous properties of the proposed approach are proved that could be useful for function analysis and comparison in the experimental section we illustrate some properties of the proposed transform while highlighting interesting features of the obtained framework

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

a tensor b spline approach for solving the diffusion pde with application to optical diffusion tomography optical diffusion tomography odt is a modern non invasive medical imaging modality which requires mathematical modelling of near infrared light propagation in tissue solving the odt forward problem equation accurately and efficiently is crucial typically the forward problem is represented by a diffusion pde and is solved using the finite element method fem on a mesh which is often unstructured tensor b spline signal processing has the attractive features of excellent interpolation and approximation properties multiscale properties fast algorithms and does not require meshing this paper introduces tensor b spline methodology with arbitrary spline degree tailored to solve the odt forward problem in an accurate and efficient manner we show that our tensor b spline formulation induces efficient and highly parallelizable computational algorithms exploitation of b spline properties for integration over irregular domains proved valuable the tensor b spline solver was tested on standard problems and on synthetic medical data and compared to fem including state of the art odt forward solvers results show that 1 a significantly higher accuracy can be achieved with the same number of nodes 2 fewer nodes are required to achieve a prespecified accuracy 3 the algorithm converges in significantly fewer iterations to a given error these findings support the value of tensor b spline methodology for high performance odt implementations this may translate into advances in odt imaging for biomedical research and clinical application

ising model on an icosahedral quasilattice the ising model on the three dimensional icosahedral quasilattice is studied by use of a monte carlo simulation the authors treat both cases where the spins are located on the vertices of the lattice and the centres of the lattice they investigate the critical phenomena on the basis of finite size scaling it is shown that the critical exponents are universal among regular lattices and quasilattices the critical temperatures of both models are found to be higher than that of the simple cubic lattice though the average coordination numbers of the three lattices are all six

efficient implementation of all digital interpolation b splines are commonly used for continuous representation of discrete time signals this kind of representation proves to be very useful in applications such as image interpolation rotation and edge detection in all these applications the first step is to compute the b spline coefficients of the signal and this involves the use of an iir noncausal filter called the direct b spline filter the signal reconstruction is achieved using the indirect b spline filter which in many applications operates at a higher rate we introduce a simplified implementation of the signal reconstruction part that will significantly reduce the overall complexity we also show that the direct b spline filter can safely be replaced with a short fir filter without compromising the performance of the traditional method numerous examples show both visually and numerically that the differences between this method and the traditional one are indeed very small finally we report the performance of these newly proposed methods in other image processing applications such as edge detection and least squares approximation

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

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

fast implementation of scale space by interpolatory subdivision scheme while the scale space approach has been widely used in computer vision there has been a great interest in fast implementation of scale space filtering we introduce an interpolatory subdivision scheme iss for this purpose in order to extract the geometric features in a scale space representation discrete derivative approximations are usually needed hence a general procedure is also introduced to derive exact formulae for numerical differentiation with respect to this iss then from iss an algorithm is derived for fast approximation of scale space filtering moreover the relationship between the iss and the whittaker shannon sampling theorem and the commonly used spline technique is discussed as an example of the application of iss technique we present some examples on fast implementation of spl lambda spl tau spaces as introduced by gokmen and jain 1997 which encompasses various famous edge detection filters it is shown that the iss technique demonstrates high performance in fast implementation of the scale space filtering and feature extraction

l 2 sampled signal reconstruction with causality constraints part i setup and solutions this paper studies the problem of reconstructing an analog signal from its sampled measurements in which the sampler acquisition device is given and the reconstructor interpolator hold is the design parameter we formulate this problem as an l2 wiener kalman filtering like optimization problem and place the main emphasis on a systematic incorporation of causality constraints into the design procedure specifically the optimization problem is solved under the constraint that the interpolation kernel is l causal for a given i n i e that its impulse response is zero in the time interval ih where h is the sampling period we present a closed form state space solution of the problem whose computational complexity does not depend on i and which can be efficiently calculated and implemented

a fast and accurate fourier algorithm for iterative parallel beam tomography we use a series expansion approach and an operator framework to derive a new fast and accurate fourier algorithm for iterative tomographic reconstruction this algorithm is applicable for parallel ray projections collected at a finite number of arbitrary view angles and radially sampled at a rate high enough that aliasing errors are small the conjugate gradient cg algorithm is used to minimize a regularized spectrally weighted least squares criterion and we prove that the main step in each iteration is equivalent to a 2 d discrete convolution which can be cheaply and exactly implemented via the fast fourier transform fft the proposed algorithm requires o n sup 2 logn floating point operations per iteration to reconstruct an n spl times n image from p view angles as compared to o n sup 2 p floating point operations per iteration for iterative convolution backprojection algorithms or general algebraic algorithms that are based on a matrix formulation of the tomography problem numerical examples using simulated data demonstrate the effectiveness of the algorithm for sparse and limited angle tomography under realistic sampling scenarios although the proposed algorithm cannot explicitly account for noise with nonstationary statistics additional simulations demonstrate that for low to moderate levels of nonstationary noise the quality of reconstruction is almost unaffected by assuming that the noise is stationary

b spline design of maximally flat and prolate spheroidal type fir filters a digital fir filter is described that offers excellent passband and stopband characteristics for general applications design formulae include parameters that adjust the magnitude response from one having characteristics like the maximally flat designs of hermann 1971 and kaiser 1975 1979 to one having characteristics like the minimum sidelobe energy approximations of kaiser and saramaki 1989 the impulse response coefficients are more straightforward to obtain than these filter designs while offering preferable response characteristics in many instances unlike fir filters designed by window or frequency sampling methods the filter coefficients are determined from the inverse fourier transform in closed form once b splines have been used to replace sharp transition edges of the magnitude response although the filters are developed in the frequency domain a convergence window is identified in the convolution series and compared with windows of popular fir filters by means of example adjustment of the transitional parameter is shown to produce a filter response that rivals the stopband attenuation and transition width of prolate spheroidal designs the design technique is extended to create additional transitional filters from prototype window functions such as the transitional hann window filter the filters are particularly suitable for precision filtering and reconstruction of sampled physiologic and acoustic signals common to the health sciences but will also be useful in other applications requiring low passband and stopband errors

phase vector incompressible registration algorithm for motion estimation from tagged magnetic resonance images tagged magnetic resonance imaging has been used for decades to observe and quantify motion and strain of deforming tissue it is challenging to obtain 3 d motion estimates due to a tradeoff between image slice density and acquisition time typically interpolation methods are used either to combine 2 d motion extracted from sparse slice acquisitions into 3 d motion or to construct a dense volume from sparse acquisitions before image registration methods are applied this paper proposes a new phase based 3 d motion estimation technique that first computes harmonic phase volumes from interpolated tagged slices and then matches them using an image registration framework the approach uses several concepts from diffeomorphic image registration with a key novelty that defines a symmetric similarity metric on harmonic phase volumes from multiple orientations the material property of harmonic phase solves the aperture problem of optical flow and intensity based methods and is robust to tag fading a harmonic magnitude volume is used in enforcing incompressibility in the tissue regions the estimated motion fields are dense incompressible diffeomorphic and inverse consistent at a 3 d voxel level the method was evaluated using simulated phantoms human brain data in mild head accelerations human tongue data during speech and an open cardiac data set the method shows comparable accuracy to three existing methods while demonstrating low computation time and robustness to tag fading and noise

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

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

rapid computation of the continuous wavelet transform by oblique projections we introduce a fast simple method for computing the real continuous wavelet transform cwt the approach has the following attractive features it achieves o n complexity per scale the filter coefficients can be analytically obtained by a simple integration and the algorithm is faster than a least squares approach with negligible loss in accuracy our method is to use p wavelets per octave and to approximate them with their oblique projection onto a space defined by a compactly supported scaling function the wavelet templates are expanded to larger sizes octaves using the two scale relation and zero padded filtering error bounds are presented to justify the use of an oblique projection over an orthogonal one all the filters are fir with the exception of one filter which is implemented using a fast recursive algorithm

choquet integrals as projection operators for quantified tomographic reconstruction in this paper we propose to investigate and analyze a new method for performing quantified projection and back projection in emission tomography this method is based on using non summative kernels capacities and asymmetric choquet integral to obtain imprecise projected values i e intervals instead of usual reconstructed pixel values validation studies using numerical and physical single photon computed emission tomography spect phantoms were used to demonstrate links between the length of these reconstructed intervals and the stochastic noise level in reconstructed slices

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

aperiodic ising model on the bethe lattice exact results we consider the ising model on the bethe lattice with aperiodic modulation of the couplings which has been studied numerically in phys rev e 77 041113 2008 here we present a relevance irrelevance criterion and solve the critical behavior exactly for marginal aperiodic sequences we present analytical formulas for the continuously varying critical exponents and discuss a relationship with the surface critical behavior of the aperiodic quantum ising chain

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

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

modified b splines for the sampling of bandlimited functions it is shown that there is an optimal finite linear combination of b splines denominated modified b splines such that a pertinent low frequency condition called m flatness is satisfied a profound relationship of the modified b splines with the beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling

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

evaluation of optimization methods for nonrigid medical image registration using mutual information and b splines a popular technique for nonrigid registration of medical images is based on the maximization of their mutual information in combination with a deformation field parameterized by cubic b splines the coordinate mapping that relates the two images is found using an iterative optimization procedure this work compares the performance of eight optimization methods gradient descent with two different step size selection algorithms quasi newton nonlinear conjugate gradient kiefer wolfowitz simultaneous perturbation robbins monro and evolution strategy special attention is paid to computation time reduction by using fewer voxels to calculate the cost function and its derivatives the optimization methods are tested on manually deformed ct images of the heart on follow up ct chest scans and on mr scans of the prostate acquired using a bffe tl and t2 protocol registration accuracy is assessed by computing the overlap of segmented edges precision and convergence properties are studied by comparing deformation fields the results show that the robbins monro method is the best choice in most applications with this approach the computation time per iteration can be lowered approximately 500 times without affecting the rate of convergence by using a small subset of the image randomly selected in every iteration to compute the derivative of the mutual information from the other methods the quasi newton and the nonlinear conjugate gradient method achieve a slightly higher precision at the price of larger computation times

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

reversible fast and high quality grid conversions a new grid conversion method is proposed to resample between two 2 d periodic lattices with the same sampling density the main feature of our approach is the symmetric reversibility which means that when using the same algorithm for the converse operation then the initial data is recovered exactly to that purpose we decompose the lattice conversion process into at most three successive shear operations the translations along the shear directions are implemented by 1 d fractional delay operators which revert to simple 1 d convolutions with appropriate filters that yield the property of symmetric reversibility we show that the method is fast and provides high quality resampled images applications of our approach can be found in various settings such as grid conversion between the hexagonal and the cartesian lattice or fast implementation of affine transformations such as rotations

a new family of spline based biorthogonal wavelet transforms and their application to image compression in this paper we design a new family of biorthogonal wavelet transforms and describe their applications to still image compression the wavelet transforms are constructed from various types of interpolatory and quasiinterpolatory splines the transforms use finite impulse response and infinite impulse response filters that are implemented in a fast lifting mode

on causal estimation from bandlimited stationary sequences this paper considers the problem of estimating a stationary sequence y from the observation of a stationary correlated sequence x by means of a causal linear filter thereby it is assumed that the spectral density x of x vanishes on a subset of the unit circle of positive lebesgue measure such that the classical derivation of the estimation filter based on the spectral factorization of x can not be applied the paper derives the transfer function of such an estimation filter discusses its stability behavior and applies the result to the causal reconstruction of deterministic signals from its samples

adaptive fuzzy segmentation of magnetic resonance images an algorithm is presented for the fuzzy segmentation of two dimensional 2 d and three dimensional 3 d multispectral magnetic resonance mr images that have been corrupted by intensity inhomogeneities also known as shading artifacts the algorithm is an extension of the 2 d adaptive fuzzy c means algorithm 2 d afcm presented in previous work by the authors this algorithm models the intensity inhomogeneities as a gain field that causes image intensities to smoothly and slowly vary through the image space it iteratively adapts to the intensity inhomogeneities and is completely automated in this paper the authors fully generalize 2 d afcm to three dimensional 3 d multispectral images because of the potential size of 3 d image data they also describe a new faster multigrid based algorithm for its implementation they show using simulated mr data that 3 d afcm yields lower error rates than both the standard fuzzy c means fcm algorithm and two other competing methods when segmenting corrupted images its efficacy is further demonstrated using real 3 d scalar and multispectral mr brain images

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

percolation in two dimensional quasicrystals by monte carlo simulations the authors examine whether or not two dimensional quasi periodic lattices belong to the same universality class as periodic lattices and they study bond and site percolation in penrose tiling and its dual lattice by making use of monte carlo simulations for the sake of comparison they also investigate percolation in periodic systems such as a square kagome and dice lattice for all these lattices they evaluate several critical exponents such as alpha related to the total number of clusters beta related to the percolation strength gamma related to the mean cluster size nu related to the correlation length tau related to cluster sizes and the fractal dimension d their results indicate that universality holds in two dimensional lattices with or without periodicity where the coordination number could be either single valued or multi valued

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