a fast sinc function gridding algorithm for fourier inversion in computer tomography the fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the cartesian grid required for fast numerical fourier inversion although the fourier inversion method is recognized as being computationally faster than the back projection method for parallel ray projection data the artifacts resulting from inaccurate interpolation have generally limited application of the method this paper presents a computationally efficient gridding algorithm which can be used with direct fourier transformation to achieve arbitrarily small artifact levels the method has potential for application to other measurement geometries such as fan beam projections and diffraction tomography and nmr imaging

a stochastic roundoff error analysis for the fast fourier transform we study the accuracy of the output of the fast fourier transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication we compare the results with the corresponding ones for the direct algorithm for the discrete fourier transform and we give indications of the relative performances when different rounding schemes are used we also present the results of numerical experiments run to test the theoretical bounds and discuss their significance

perfect control of the robust servomechanism problem the problem of finding a robust servomechanism controller which achieves exact asymptotic tracking regulation for a modeled class of reference and disturbance signals arbitrarily good approximate error regulation agaer for reference disturbance signals which lie outside of this design class and arbitrarily good transient response is considered it is shown that a necessary condition that must be satisfied to accomplish this requirement is that the initial conditions of the servocompensator should be zero an explicit algorithm for designing such controllers is then given a simple controller called the high gain servomechanism controller is also presented which has the property that it gives agaer and arbitrarily good transient response when the robust exact asymptotic tracking regulation requirement is relaxed this controller shows that dynamics in a controller is not essential for achieving good error regulation in the presence of unknown unmeasurable disturbances it is also shown that using the controllers proposed exact error regulation occurs for a larger class of design signals called the extended disturbance reference input signals even if the conditions for perfect control are not satisfied some examples are included to illustrate the type of results obtained

rapid computation of the discrete fourier transform algorithms for the rapid computation of the forward and inverse discrete fourier transform for points which are nonequispaced or whose number is unrestricted are presented the computational procedure is based on approximation using a local taylor series expansion and the fast fourier transform fft the forward transform for nonequispaced points is computed as the solution of a linear system involving the inverse fourier transform this latter system is solved using the iterative method gmres with preconditioning numerical results are given to confirm the efficiency of the algorithms

weakly and strongly non minimum phase systems properties and limitations abstractin this paper a new classification of non minimum phase systems based on stabilisability of the internal dynamics is introduced according to this classification the non minimum phase systems are categorised as follows 1 weakly non minimum systems for which a finite value of minimum norm of the regulation error can be obtained and 2 strongly non minimum phase systems for which this minimum norm of the regulation becomes infinite due to the presence of both transient and steady state errors while for the weakly non minimum phase systems there is no steady state regulation error it is worth mentioning that based on this new classification a level of non minimum phase ness can be further defined for any given desired trajectory furthermore some numerical simulations are carried out to show consistency with theoretical analysis

a new method of solving noisy abel type equations abstract a new approach to solving noisy integral equations of the first kind is applied to the family of abel equations such equations play a role in stereology wicksell s unfolding problem medicine engineering and astronomy the method is based on an expansion in an arbitrary orthonormal basis coupled with exact inversion of the integral operator the inverse appears in the fourier coefficients of the expansion where it can be carried over to the usually well behaved basis elements in the form of the adjoint this method is an alternative to tikhonov regularization regularization of the inverse of the operator itself or a wavelet vaguelette singular value decomposition the method is particularly interesting in irregularity of the kernel the input or both because knowledge of the spectral properties of the operator is not required the method is also of interest in regular cases where these spectral properties are not sufficiently known or are hard to deal with for smooth input functions the simple basis of trigonometric functions yields input estimators whose mean integrated squared error converges at the optimal rate for the entire family of abel operators this can be shown when smooth wavelets are used for abel operators with index smaller than 1 2 and when the haar wavelet is used for operators with index larger than 1 2

a fast algorithm for the inversion of abels transform we present a new algorithm for the computation of the inverse abel transform a problem which emerges in many areas of physics and engineering we prove that the legendre coefficients of a given function coincide with the fourier coefficients of a suitable periodic function associated with its abel transform this allows us to compute the legendre coefficients of the inverse abel transform in an easy fast and accurate way by means of a single fast fourier transform the algorithm is thus appropriate also for the inversion of abel integrals given in terms of samples representing noisy measurements rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments

activation detection in fmri using a maximum energy ratio statistic obtained by adaptive spatial filtering an adaptive spatial filtering method is proposed that takes into account contextual information in fmri activation detection this filter replaces the time series of each voxel with a weighted average of time series of a small neighborhood around it the filter coefficients at each voxel are derived so as to maximize a test statistic designed to indicate the presence of activation this statistic is the ratio of the energy of the filtered time series in a signal subspace to the energy of the residuals it is shown that the filter coefficients and the maximum energy ratio can be found through a generalized eigenproblem this approach equates the filter coefficients to the elements of an eigenvector corresponding to the largest eigenvalue of a specific matrix while the largest eigenvalue itself becomes the maximum energy ratio that can be used as a statistic for detecting activation the distribution of this statistic under the null hypothesis is derived by a nonparametric permutation technique in the wavelet domain also in this paper we introduce a new set of basis vectors that define the signal subspace the space spanned by these basis vectors covers a wide range of possible hemodynamic response functions hrf and is applicable to both event related and block design fmri signal analysis this approach circumvents the need for a priori assumptions about the exact shape of the hrf resting state experimental fmri data were used to assess the specificity of the method showing that the actual false alarm rate of the proposed method is equal or less than its expected value analysis of simulated data and motor task fmri datasets from six volunteers using the method proposed here showed an improved sensitivity as compared to a conventional test with a similar statistic applied to spatially smoothed data

new stable numerical inversion of abel s integral equation abstract the 3 d image reconstruction from cone beam projections in computerized tomography leads naturally in the case of radial symmetry to the study of abel s type of integral equations a new method for the numerical solution of such equations when the experimental information is obtained through measured data on a discrete set of points is presented and rigorously analyzed

a sufficient condition for local controllability of nonlinear systems along closed orbits a computable sufficient condition for determining local controllability of a differential nonlinear control system along a reference closed orbit is presented comparisons to other results on local controllability along a reference trajectory are made

model independent method for fmri analysis this paper presents a fast method for delineation of activated areas of the brain from functional magnetic resonance imaging fmri time series data the steps of the work accomplished are as follows 1 it is shown that the detection performance evaluated by the area under the receiver operating characteristic curve is directly related to the signal to noise ratio snr of the composite image generated in the detection process 2 detection and segmentation of activated areas are formulated in a vector space framework in this formulation a linear transformation image combination method is shown to be desirable to maximize the snr of the activated areas subject to the constraint of removing inactive areas 3 an analytical solution for the problem is found 4 image pixel vectors and expected time series pattern signature for inactive pixels are used to calculate weighting vector and identify activated regions 5 signatures of the activated regions are used to segment different activities 6 segmented images by the proposed method are compared with those generated by the conventional methods correlation t statistic and z statistic detection performance and snrs of the images are compared the proposed approach outperforms the conventional methods of fmri analysis in addition it is model independent and does not require a priori knowledge of the fmri response to the paradigm since the method is linear and most of the work is done analytically numerical implementation and execution of the method are much faster than the conventional methods

a model selection rule for sinusoids in white gaussian noise the model selection problem for sinusoidal signals has often been addressed by employing the akaike 1974 information criterion aic and the minimum description length principle mdl the popularity of these criteria partly stems from the intrinsically simple means by which they can be implemented they can however produce misleading results if they are not carefully used the aic and mdl have a common form in that they comprise two terms a data term and a penalty term the data term quantifies the residuals of the model and the penalty term reflects the desideratum of parsimony while the data terms of the aic and mdl are identical the penalty terms are different in most of the literature the aic and mdl penalties are however both obtained by apportioning an equal weight to each additional unknown parameter be it phase amplitude or frequency by contrast we demonstrate that the penalties associated with the amplitude and phase parameters should be weighted differently than the penalty attached to the frequencies following the bayesian methodology we derive a model selection criterion for sinusoidal signals in gaussian noise which also contains the log likelihood and the penalty terms the simulation results disclose remarkable improvement in our selection rule over the commonly used mdl and aic

approximate solution of fractional vibration equation using jacobi polynomials an approximate method using jacobi polynomials is proposed for the approximate solutions of fractional vibration equation convergence and numerical stability of the proposed method are shown results are discussed for different cases of jacobi polynomials results are compared with existing analytical methods in this paper we present a method based on the jacobi polynomials for the approximate solution to fractional vibration equation fve of large membranes proposed method converts the fve into sylvester form of algebraic equations whose solution gives the approximate solution convergence analysis of the proposed method is given it is also shown that our approximate method is numerically stable numerical results are discussed for different values of wave velocities and fractional order involved in the fve these numerical results are shown through figures for particular cases of jacobi polynomials such as 1 legendre polynomial 2 chebyshev polynomial of second kind 3 chebyshev polynomial of third kind 4 chebyshev polynomial of fourth kind 5 gegenbauer polynomial the accuracy of the proposed method is proved by comparing results of our method and other exiting analytical methods comparison of results are presented in the form of tables for particular cases of fve and jacobi polynomials

robust gaussian and non gaussian matched subspace detection we address the problem of matched filter and subspace detection in the presence of arbitrary noise and interference or interfering signals that may lie in an arbitrary unknown subspace of the measurement space a minmax methodology developed to deal with this uncertainty can also be adapted to situations where partial information on the interference or other uncertainties is available this methodology leads to a hypothesis test with adequate levels of false alarm robustness and signal detection sensitivity the robust test is applicable to a large class of noise density functions in addition generalized likelihood ratio glr detectors are derived for the class of generalized gaussian noise the detectors are generalizations of the spl chi sup 2 t and f statistics used with gaussian noise which are themselves motivated in a new way by the robust test for matched filter detection these expressions are simpler and computationally efficient the robust test reduces to the conventional test when unlearned subspace interference is known to be absent the results demonstrate that when compared with the conventional detector the robust one trades off some detection performance in the absence of interference for the sake of robustness in its presence

nonlinear regression in parametric activation studies parametric study designs can reveal information about the relationship between a study parameter e g word presentation rate and regional cerebral blood flow rcbf in functional imaging the brain s responses in relation to study parameters might be nonlinear therefore the linear correlation coefficient as often used in the analysis of parametric studies might not be a proper characterization we present a noninteractive method which fits nonlinear functions of stimulus or task parameters to rcbf responses using second order polynomial expansions this technique is implemented in the context of the general linear model and statistical parametric mapping we also consider the usefulness of statistical inferences based on f fields about similarities and differences of these nonlinear responses in different groups this approach is illustrated with a 12 run h 2 o pet activation study using an auditory paradigm of increasing word presentation rates a patient who had recovered from severe aphasia and a normal control were studied we demonstrate the ability of this new technique to identify brain regions where rcbf is closely related to increasing word presentation rate in both subjects without constraining the nature of this relationship and where these nonlinear responses differ r1996academic press inc

approximate solution of abel integral equation this paper presents a new stable approximate inversion of abel integral equation by using the taylor expansion of the unknown function abel equation is approximately transformed to a system of linear equations for the unknown function together with its derivatives a desired solution can be determined by solving the resulting system according to cramer s rule this method gives a simple and closed form of approximate abel inversion which can be performed by symbolic computation the nth order approximation is exact for a polynomial of degree up to n abel integral equation is approximately expressed in terms of integrals of input data so the suggested approach is stable for experimental data with random noise an error analysis of this approach is given finally several numerical examples are given to illustrate the accuracy and stability of this method

likelihood based hypothesis tests for brain activation detection from mri data disturbed by colored noise a simulation study functional magnetic resonance imaging fmri data that are corrupted by temporally colored noise are generally preprocessed i e prewhitened or precolored prior to functional activation detection in this paper we propose likelihood based hypothesis tests that account for colored noise directly within the framework of functional activation detection three likelihood based tests are proposed the generalized likelihood ratio glr test the wald test and the rao test the fmri time series is modeled as a linear regression model where one regressor describes the task related hemodynamic response one regressor accounts for a constant baseline and one regressor describes potential drift the temporal correlation structure of the noise is modeled as an autoregressive ar model the order of the ar model is determined from practical null data sets using akaike s information criterion with penalty factor 3 as order selection criterion the tests proposed are based on exact expressions for the likelihood function of the data using monte carlo simulation experiments the performance of the proposed tests is evaluated in terms of detection rate and false alarm rate properties and compared to the current general linear model glm test which estimates the coloring of the noise in a separate step results show that theoretical asymptotic distributions of the glm glr and wald test statistics cannot be reliably used for computing thresholds for activation detection from finite length time series furthermore it is shown that for a fixed false alarm rate the detection rate of the proposed glr test statistic is slightly but statistically significantly improved compared to that of the common glm based tests finally simulations results reveal that all tests considered show seriously inferior performance if the order of the ar model is not chosen sufficiently high to give an adequate description of the correlation structure of the noise whereas the effects of slightly overmodeling are observed to be less harmful

efficient algorithms to solve singular integral equations of abel type in the present paper we obtain the approximate solution of abel s integral equation by using the following powerful efficient but simple methods i he s homotopy perturbation method hpm ii modified homotopy perturbation method mhpm iii adomian decomposition method adm and iv modified adomian decomposition method madm the validity and applicability of these techniques are illustrated through various particular cases which demonstrate their efficiency and simplicity in solving these types of integral equations compared with the other existing methods

activation detection in functional mri using subspace modeling and maximum likelihood estimation a statistical method for detecting activated pixels in functional mri fmri data is presented in this method the fmri time series measured at each pixel is modeled as the sum of a response signal which arises due to the experimentally controlled activation baseline pattern a nuisance component representing effects of no interest and gaussian white noise for periodic activation baseline patterns the response signal is modeled by a truncated fourier series with a known fundamental frequency but unknown fourier coefficients the nuisance subspace is assumed to be unknown a maximum likelihood estimate is derived for the component of the nuisance subspace which is orthogonal to the response signal subspace an estimate for the order of the nuisance subspace is obtained from an information theoretic criterion a statistical test is derived and shown to be the uniformly most powerful ump test invariant to a group of transformations which are natural to the hypothesis testing problem the maximal invariant statistic used in this test has an f distribution the theoretical f distribution under the null hypothesis strongly concurred with the experimental frequency distribution obtained by performing null experiments in which the subjects did not perform any activation task applications of the theory to motor activation and visual stimulation fmri studies are presented

penalized partially linear models using sparse representations with an application to fmri time series in this paper we consider modeling the nonparametric component in partially linear models plms using linear sparse representations e g wavelet expansions two types of representations are investigated namely orthogonal bases complete and redundant overcomplete expansions for bases we introduce a regularized estimator of the nonparametric part the important contribution here is that the nonparametric part can be parsimoniously estimated by choosing an appropriate penalty function for which the hard and soft thresholding estimators are special cases this allows us to represent in an effective manner a broad class of signals including stationary and or nonstationary signals and avoids excessive bias in estimating the parametric component we also give a fast estimation algorithm the method is then generalized to handle the case of overcomplete representations a large scale simulation study is conducted to illustrate the finite sample properties of the estimator the estimator is finally applied to real neurophysiological functional magnetic resonance imaging mri data sets that are suspected to contain both smooth and transient drift features

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

application of bayesian inference to fmri data analysis the methods of bayesian statistics are applied to the analysis of fmri data three specific models are examined the first is the familiar linear model with white gaussian noise in this section the jeffreys rule for noninformative prior distributions is stated and it is shown how the posterior distribution may be used to infer activation in individual pixels next linear time invariant lti systems are introduced as an example of statistical models with nonlinear parameters it is shown that the bayesian approach can lead to quite complex bimodal distributions of the parameters when the specific case of a delta function response with a spatially varying delay is analyzed finally a linear model with auto regressive noise is discussed as an alternative to that with uncorrelated white gaussian noise the analysis isolates those pixels that have significant temporal correlation under the model it is shown that the number of pixels that have a significantly large auto regression parameter is dependent on the terms used to account for confounding effects

contextual clustering for analysis of functional mri data presents a contextual clustering procedure for statistical parametric maps spm calculated from time varying three dimensional images the algorithm can be used for the detection of neural activations from functional magnetic resonance images fmri an important characteristic of spm is that the intensity distribution of background nonactive area is known whereas the distributions of activation areas are not the developed contextual clustering algorithm divides an spm into background and activation areas so that the probability of detecting false activations by chance is controlled i e hypothesis testing is performed unlike the much used voxel by voxel testing neighborhood information is utilized an important difference this is achieved by using a markov random field prior and iterated conditional modes icm algorithm however unlike in the conventional use of icm algorithm the classification is based only on the distribution of background the results from the authors simulations and human fmri experiments using visual stimulation demonstrate that a better sensitivity is achieved with a given specificity in comparison to the voxel by voxel thresholding technique the algorithm is computationally efficient and can be used to detect and delineate objects from a noisy background in other applications

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

an approximate method for abel inversion using chebyshev polynomials many problems in physics like reconstruction of the radially distributed emissivity from the line of sight projected intensity the 3 d image reconstruction from cone beam projections in computerized tomography etc lead naturally in the case of radial symmetry to the study of abel s type integral equation in this paper a new stable algorithm based on shifted chebyshev polynomial approximation is presented and analyzed first chebyshev operational matrix of integration p is constructed and then it is used to reduce the integral equation to a system of algebraic equation which can be solved easily the method is quite accurate and stable even though the approximations are performed using polynomials of degree up to 5 some test examples from the plasma diagnostics are illustrated to demonstrate the effectiveness and stability of the proposed method

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

performance limitations in reference tracking and path following for nonlinear systems we investigate limits of performance in reference tracking and path following and highlight an essential difference between them for a class of nonlinear systems we show that in reference tracking the smallest achievable l 2 norm of the tracking error is equal to the least amount of control energy needed to stabilize the zero dynamics of the error system we then show that this fundamental performance limitation does not exist when the control objective is to force the output to follow a geometric path without a timing law assigned to it this is true even when an additional desired speed assignment is required to be satisfied asymptotically or in finite time

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

a new abel inversion by means of the integrals of an input function with noise abel s integral equations arise in many areas of natural science and engineering particularly in plasma diagnostics this paper proposes a new and effective approximation of the inversion of abel transform this algorithm can be simply implemented by symbolic computation and moreover an nth order approximation reduces to the exact solution when it is a polynomial in r 2 of degree less than or equal to n approximate abel inversion is expressed in terms of integrals of input measurement data so the suggested approach is stable for experimental data with random noise an error analysis of the approximation of abel inversion is given finally several test examples used frequently in plasma diagnostics are given to illustrate the effectiveness and stability of this method

numerical solution of abel s integral equation by using legendre wavelets abstract a numerical method for solving abel s integral equation as singular volterra integral equations is presented the method is based upon legendre wavelet approximations the properties of legendre wavelet are first presented these properties are then utilized to reduce the singular volterra integral equations to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique