the invertibility of rotation invariant radon transforms let r denote the radon transform on rn that integrates a function over hyperplanes in given smooth positive measures depending on the hyperplane we characterize the measures for which r is rotation invariant we prove rotation invariant transforms are all one to one and hence invertible on the domain of square integrable functions of compact support l02 rn we prove the hole theorem if f l02rn and r f 0 for hyperplanes not intersecting a ball centered at the origin then f is zero outside of that ball using the theory of fourier integral operators we extend these results to the domain of distributions of compact support on rn our results prove invertibility for a mathematical model of positron emission tomography and imply a hole theorem for the constantly attenuated radon transform as well as invertibility for other radon transforms

irregular phased array tiling by means of analytic schemata driven optimization the design of subarrayed planar phased arrays characterized by an irregular organization of domino shaped tiles is addressed starting from optimal tiling theorems drawn from mathematical theory an enumerative approach able to retrieve the optimal clustering providing the maximum aperture coverage and the best radiation performance is proposed to deal with the synthesis of low medium size rectangular arrays based on the same optimal theorems and still exploiting the algorithmic procedures at the basis of the enumerative approach an innovative schemata based optimization method is introduced for designing large arrays as well a set of representative results concerned with ideal as well as real radiating elements is reported and discussed to highlight the features and the potentialities of the proposed analytically based design framework

range characterization of radon transforms on quaternionic projective spaces the characterization of the ranges of radon transforms is one of the most important subjects in integral geometry in fact for the radon transforms on euclidean spaces this subject has been studied from the several points of view since john s result 18 in which john showed that the range of the x ray radon transform on the 3 dimensional euclidean space r 3 is characterized as a kernel of some second order ultrahyperbolic differential operator see e g 1 2 4 5 10 13 14 15 in particular gonzalez 4 gave a simple characterization of the range for the radon transform on r moreover recently gonzalez obtained remarkable results on this subject 6

filtered backprojection and the exponential radon transform on etudie des conditions sur la mesure qui assurent la validite de 7b r t 7b rf k f e ou e 7b r t k pour une transformee de radon generalisee 7b r et une grande classe de filtres k correspondant a des fonctions delta approchees e

identification of linear stochastic systems via second and fourth order cumulant matching the identification problem for time invariant single input single output linear stochastic systems driven by non gaussian white noise is considered the system is not restricted to be minimum phase and it is allowed to contain all pass components a least squares criterion that involves matching the second and the fourth order cumulant functions of the noisy observations is proposed knowledge of the probability distribution of the driving noise is not required an order determination criterion that is a modification of the akaike information criterion is also proposed strong consistency of the proposed estimator is proved under certain sufficient conditions simulation results are presented to illustrate the method

a functional from geometry with applications to discrepancy estimates and the radon transform estimates of discrepancy or irregularities of distribution are obtained for measures without atoms two estimators are used the half space or separation discrepancy ds and a geometric functional 1a a representation formula for the generalized energy integral pa is developed norm inequalities for the radon transform are obtained as an application of the continuous discrepancy results integral geometric notions play a prominent role

computerized tomography with unknown sources we give a reconstruction method for the attenuation coefficient in a 2d cross section of a 3d object from tomographic data which does not require knowledge of the positions or the intensity of the sources and the sources may even be inside the object such a situation arises in emission computerized tomography the method is based on consistency conditions in the range of the relevant integral transforms the paper contains a detailed description of the algorithm and numerical results for computer generated data

the limiting behavior of lms a realization oriented analysis is given of the gradient noise misadjustment and lag misadjustment performance of the lms least mean square algorithm new formulas are given for both of these components of excess mean square error it is shown that the traditional formula for lag misadjustment needs to be modified by adding further terms due to gradient noise and noise variance to perform the analysis it is necessary to study the convergence with probability one of the noise free fixed parameter lms algorithm convergence is found under simple conditions that improve on those previously obtained

deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes nongaussian linear processes are considered it is shown that the phase of the transfer function can be estimated under broad conditions this is not true of gaussian linear processes and in this sense gaussian linear processes are atypical the asymptotic behavior of a phase estimate is determined the phase estimates make use of bispectral estimates these ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase a number of computational illustrations are given

a generalized least absolute deviation method for parameter estimation of autoregressive signals this paper proposes a generalized least absolute deviation glad method for parameter estimation of autoregressive ar signals under non gaussian noise environments the proposed glad method can improve the accuracy of the estimation of the conventional least absolute deviation lad method by minimizing a new cost function with parameter variables and noise error variables compared with second and high order statistical methods the proposed glad method can obtain robustly an optimal ar parameter estimation without requiring the measurement noise to be gaussian moreover the proposed glad method can be implemented by a cooperative neural network nn which is shown to converge globally to the optimal ar parameter estimation within a finite time simulation results show that the proposed glad method can obtain more accurate estimates than several well known estimation methods in the presence of different noise distributions

reconstruction from uniformly attenuated spect projection data using the dbh method an algorithm was developed for the 2 d reconstruction of truncated and nontruncated uniformly attenuated data acquired from single photon emission computed tomography spect the algorithm is able to reconstruct data from half scan 180deg and short scan 180deg fan angle acquisitions for parallel and fan beam geometries respectively as well as data from full scan 360deg acquisitions the algorithm is a derivative backprojection and hilbert transform dbh method which involves the backprojection of differentiated projection data followed by an inversion of the finite weighted hilbert transform the kernel of the inverse weighted hilbert transform is solved numerically using matrix inversion numerical simulations confirm that the dbh method provides accurate reconstructions from half scan and short scan data even when there is truncation however as the attenuation increases finer data sampling is required

strongly consistent identification algorithms and noise insensitive mse criteria windowed cumulant projections of nongaussian linear processes yield autocorrelation estimators which are immune to additive gaussian noise of unknown covariance by establishing strong consistency of these estimators strongly consistent and noise insensitive recursive algorithms are developed for parameter estimation these computationally attractive schemes are shown to be optimal with respect to a modified mean square error mse criterion which implicitly exploits the high signal to noise ratio domain of cumulant statistics the novel mse objective function is expressed in terms of the noisy process but it is shown to be a scalar multiple of the standard mse criterion as if the latter was computed in the absence of noise simulations illustrate the performance of the proposed algorithms and compare them with the conventional algorithms

a unifying maximum likelihood view of cumulant and polyspectral measures for non gaussian signal classification and estimation classification and estimation of non gaussian signals observed in additive gaussian noise of unknown covariance are addressed using cumulants or polyspectra by integrating ideas from pattern recognition and model identification asymptotically optimum maximum likelihood classifiers and arma autoregressive moving average parameter estimators are derived without knowledge of the data distribution identifiability of noncausal and nonminimum phase arma models is established using a finite number of cumulant or polyspectral lags of any order greater than two a unifying view of cumulant and polyspectral discriminant measures utilizes these lags and provides a common framework for development and performance analysis of novel and existing estimation and classification algorithms tentative order determination and model validation tests for non gaussian arma processes are described briefly illustrative simulations are also presented

graph connection laplacian methods can be made robust to noise recently several data analytic techniques based on graph connection laplacian gcl ideas have appeared in the literature at this point the properties of these methods are starting to be understood in the setting where the data is observed without noise we study the impact of additive noise on these methods and show that they are remarkably robust as a by product of our analysis we propose modifications of the standard algorithms that increase their robustness to noise we illustrate our results in numerical simulations

integral geometry on grassmann manifolds and calculus of invariant differential operators abstract in this paper we mainly deal with two problems in integral geometry the range characterizations and construction of inversion formulas for radon transforms on higher rank grassmann manifolds the results will be described explicitly in terms of invariant differential operators we will also study the harmonic analysis on grassmann manifolds using the method of integral geometry in particular we will give eigenvalue formulas and radial part formulas for invariant differential operators

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

regularization approach for inverting the exponential radon transforms ert in this paper we study the problem of inverting the exponential radon transforms with non exact data a regularization approach is introduced to find approximate solutions of the problem

a new algorithm for the design of linear prediction error filters using cumulant based mse criteria proposes a new algorithm for the design of minimum phase linear prediction error lpe filters using two new cumulant higher order statistics based mse criteria when the given stationary random signal x n is nongaussian and contaminated by gaussian noise it is shown that the designed lpe filters based on the proposed criteria are identical to the conventional correlation second order statistics based lpe filter as if x n were noise free measurements as correlation based lpe filters coefficients of the designed cumulant based lpe filters can be obtained by solving a set of symmetric toeplitz linear equations using the well known computationally efficient levinson durbin recursion moreover the proposed two criteria are applicable for any cumulant order m spl ges 3 and one of the proposed criteria for m 3 reduces to delopoulos and giannakis 1992 third order cumulant based mse criterion some simulation results are then provided to support the analytical results

the identification problem for the attenuated x ray transform we study the problem of recovery both the attenuation a and the source f in the attenuated x ray transform in the plane we study the linearization as well it turns out that there is a natural hamiltonian flow that determines which singularities we can recover if the perturbation delta a is supported in a compact set that is non trapping for that flow then the problem is well posed otherwise it may not be and least in the case of radial a f it is not we present uniqueness and non uniqueness results for both the linearized and the non linear problem as well as a h older stability estimate

asymptotically optimal estimation of ma and arma parameters of non gaussian processes from high order moments a description is given of an asymptotically minimum variance algorithm for estimating the ma moving average and arma autoregressive moving average parameters of non gaussian processes from sample high order moments the algorithm uses the statistical properties covariances and cross covariances of the sample moments explicitly a simpler alternative algorithm that requires only linear operations is also presented the latter algorithm is asymptotically minimum variance in the class of weighted least squares algorithms

adaptive deconvolution and identification of nonminimum phase fir systems based on cumulants a novel adaptive deconvolution and system identification scheme is introduced for a linear non minimum phase finite impulse response fir system driven by non gaussian white noise the adaptive scheme is based on approximating the fir system by noncausal autoregressive ar models and using higher order cumulants of the system output as such it is a blind equalization deconvolution scheme the set of updated ar parameters is obtained by using a gradient type algorithm and by using higher order cumulants instead of time samples of the output signal it is demonstrated by means of extensive simulations that the adaptive scheme works well for both stationary and nonstationary cases as expected it outperforms the autocorrelation based gradient method for nonminimum phase system identification and deconvolution performance comparisons to existing methods are given using as figures of merit the probability of errors in the restored input sequence computational complexity and convergence rate

representation theoretic patterns in three dimensional cryo electron microscopy i the intrinsic reconstitution algorithm in this paper we reveal the formal algebraic structure underlying the intrinsic reconstitution algorithm introduced by singer and shkolnisky in 9 for determining three dimensional macromolecular structures from images obtained by an electron microscope inspecting this algebraic structure we obtain a conceptual explanation for the admissibility correctness of the algorithm and a proof of its numerical stability in addition we explain how the various numerical observations reported in that work follow from basic representation theoretic principles

design of broadband planar arrays based on the optimization of aperiodic tilings antenna arrays based on aperiodic tilings have been shown to exhibit low sidelobe levels and modest bandwidths over which grating lobes are suppressed in addition compared to conventional periodic arrays these arrays are naturally thinned i e mean inter element spacing is greater than the generation of these arrays involves placing array elements at the locations of the vertices of an aperiodic tiling to obtain a realizable design the entire array is then scaled and truncated to achieve a desired minimum element spacing and aperture size this paper demonstrates that it is possible to greatly extend the bandwidth of these arrays by incorporating a simple perturbation scheme into the basic array generation process the implementation of this perturbation scheme is straightforward and it lends itself well to being combined with an optimization technique such as the genetic algorithm it is successfully used to generate arrays that have large bandwidths peak sidelobe level db with no grating lobes of up to a minimum element spacing of moreover the flexibility of this technique will be further demonstrated by introducing a slight variation of the basic scheme that is capable of generating arrays with extremely wide bandwidths an example will be presented for an array design that has a bandwidth corresponding to a minimum element spacing of up to

consistent identification of stochastic linear systems with noisy input output data abstract a novel criterion is introduced for parametric errors in variables identification of stochastic linear systems excited by non gaussian inputs the new criterion is at least theoretically insensitive to a class of input output disturbances because it implicitly involves higher than second order cumulant statistics in addition it is shown to be equivalent to the conventional mean squared error mse as if the latter was computed in the ideal case of noise free input output data the sampled version of the criterion converges to the novel mse and guarantees strongly consistent parameter estimators the asymptotic behavior of the resulting parameter estimators is analyzed and guidelines for minimum variance experiments are discussed briefly informative enough input signals and persistent of excitation conditions are specified computatonally attractive recursive least squares variants are also developed for on line implementation of arma modeling and their potential is illustrated by applying them to time delay estimation in low snr environment the performance of the proposed algorithms and comparisons with conventional methods are corroborated using simulated data

null spaces and ranges for the classical and spherical radon transforms abstract let r be the classical radon transform that integrates a function over hyperplanes in r n and let sm be the transform that integrates a function over spheres containing the origin in r n we prove continuity results for both transforms and explicitly give the null space of r for a class of square integrable functions on the exterior of a ball in r n as well as the null space of sm for square integrable functions on a ball we show sm l 2 r n l 2 r n is one one and we characterize the range of sm on classes of smooth functions and square integrable functions by certain moment conditions if g x is a schwartz function on r n that is zero to infinite order at x 0 we prove moment conditions sufficient for g to be in the range of sm c r n we apply our results on sm to existence and uniqueness theorems for solutions to a characteristic initial value problem for the darboux partial differential equation

detection in correlated impulsive noise using fourth order cumulants we consider detection and estimation in correlated impulsive noise the non gaussian impulsive noise is modeled as the sum of two linear processes a nominal part and an impulsive part this model admits correlated impulsive bursts lasting many data samples identifiability of the noise model is established using fourth and second order cumulants under this model the correlated time series can be whitened and an appropriate memoryless nonlinearity applied to attenuate the impulsive events a detection statistic is then formed from the output of the nonlinearity in the threshold detection case the use of cumulants allows identification of the noise in the presence of the signal to be detected obviating the need for noise only training records simulation results with a sample size of 512 show small loss in detector performance versus an ideal detector with no impulsive part present

estimation and detection in non gaussian noise using higher order statistics one of the primary applications of higher order statistics has been for detection and estimation of nongaussian signals in gaussian noise of unknown covariance this is motivated by the fact that higher order cumulants of gaussian processes vanish we study the opposite problem namely detection and estimation in nongaussian noise we estimate cumulants of nongaussian processes in the presence of unknown deterministic and or gaussian signals which allows either parametric or nonparametric estimation of the covariance of the nongaussian noise our approach is to augment existing second order detection methods using cumulants we propose solutions for detection of deterministic signals based on matched filters and the generalized likelihood ratio test which incorporate cumulants where the resulting solutions are valid under either detection hypotheses this allows for single record detection and obviates the need for noise only training records the problem of estimating signal strength in the presence of nongaussian noise of unknown covariance is also considered and a cumulant based solution is proposed which uses a single data record examples are used throughout to illustrate our proposed methods

determination of an unknown coefficient in a nonlinear heat equation abstract the problem of determining an unknown term k u in the equation k u u t k u u x x is considered in this paper applying tikhonov s regularization approach we develop a procedure to find an approximate stable solution to the unknown coefficient from the overspecified data

a variational method using fractional order hilbert spaces for tomographic reconstruction of blurred and noised binary images we provide in this article a refined functional analysis of the radon operator restricted to axisymmetric functions and show that it enjoys strong regularity properties in fractional order hilbert spaces this study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects for which we have available one single blurred and noised snapshot we propose a variational approach to handle this problem consisting in solving a minimization problem settled in adapted fractional order hilbert spaces we show the existence of solutions and then derive first order necessary conditions for optimality in the form of optimality systems

vector diffusion maps and the connection laplacian we introduce vector diffusion maps vdm a new mathematical framework for organizing and analyzing massive high dimensional data sets images and shapes vdmis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods such as lle isomap and laplacian eigenmaps while existing methods are either directly or indirectly related to the heat kernel for functions over the data vdm is based on the heat kernel for vector fields vdm provides tools for organizing complex data sets embedding them in a low dimensional space and interpolating and regressing vector fields over the data in particular it equips the data with a metric which we refer to as the vector diffusion distance in the manifold learning setup where the data set is distributed on a low dimensional manifold embedded in we prove the relation between vdm and the connection laplacian operator for vector fields over the manifold 2012 wiley periodicals inc

the lattice structure of the set of domino tilings of a polygon abstract fix a polygon p with vertical and horizontal sides we first recall how each tiling of p with dominoes i e rectangles 2 1 can be encoded by a height function such an encoding induces a lattice structure on the set tp of the tilings of p we give some applications of this structure and we especially describe the order of meet irreducible elements of tp

adaptive iir algorithms based on high order statistics recursive estimation algorithms are derived for moving average and autoregressive moving average processes these algorithms can be used for adaptive infinite impulse response iir filtering of non gaussian processes the behavior of these algorithms is illustrated by extensive numerical examples

design of subarrayed linear and planar array antennas with sll control based on an excitation matching approach in this paper the synthesis of subarrayed antennas with subarrays having different weights and dimensions is considered the problem is formulated as the definition of the element clustering into spatially contiguous subarrays and the corresponding weights to obtain a pattern with the lowest i e minimum achievable sidelobe level sll the problem is addressed with an unconstrained approach based on an excitation matching method where the synthesis of subarray weights and sizes is simultaneously carried out successively a hybrid version of such an approach is proposed to directly enforce into the optimization procedure the sll constraints a set of representative results concerned with both sum and difference patterns are shown furthermore some comparisons with state of the art hybrid evolutionary based techniques are reported as well

on the range of the radon d plane transform and its dual we present direct group theoretic proofs of the range theorem for the radon d plane transform on r the original proof by richter involves extensive use of local coordinate calculations on g d n the grassmann manifold of affine a planes in r we show that moment conditions are not sufficient to describe this range when d n in contrast to the compactly supported case finally we show that the dual rf plane transform maps g g d n surjectively onto f r

on estimating noncausal nonminimum phase arma models of non gaussian processes the authors address the problem of estimating the parameters of non gaussian arma autoregressive moving average processes using only the cumulants of the noisy observation the measurement noise is allowed to be colored gaussian or independent and identically non gaussian distributed the arma model is not restricted to be causal or minimum phase and may even contain all pass factors the unique parameter estimates of both the ma and ar parts are obtained by linear equations the structure of the proposed algorithm facilitates asymptotic performance evaluation of the parameter estimators and model order selection using cumulant statistics the method is computationally simple and can be viewed as the least squares solution to a quadratic model fitting of a sampled cumulant sequence identifiability issues are addressed simulations are presented to illustrate the proposed algorithm

distances between measures from 1 dimensional projections as implied by continuity of the inverse radon transform distances between measures on ir d are determined from distances between their 1 dimensional projections the method employed involves considering the 1 dimensional projections to be the radon transform of the measures crucial to the main theorem is a continuity result for the inverse radon transform focus is restricted to the prohorov dual bounded lipschitz and d k metrics which metrize weak convergence of probability measures these metrics are related to each other and to the sobolev norms the d k results extend from measures to generalized functions

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

range descriptions for the spherical mean radon transform abstract the transform considered in the paper averages a function supported in a ball in r n over all spheres centered at the boundary of the ball this radon type transform arises in several contemporary applications e g in thermoacoustic tomography and sonar and radar imaging range descriptions for such transforms are important in all these areas for instance when dealing with incomplete data error correction and other issues four different types of complete range descriptions are provided some of which also suggest inversion procedures necessity of three of these appropriately formulated conditions holds also in general domains while the complete discussion of the case of general domains would require another publication

fast adaptive algorithms for ar parameters estimation using higher order statistics time varying statistics in linear filtering and linear estimation problems necessitate the use of adaptive or time varying filters in the solution with the rapid availability of vast and inexpensive computation power models which are non gaussian even nonstationary are being investigated at increasing intensity statistical tools used in such investigations usually involve higher order statistics hos the classical instrumental variable iv principle has been widely used to develop adaptive algorithms for the estimation of arma processes despite the great number of iv methods developed in the literature the cumulant based procedures for pure autoregressive ar processes are almost nonexistent except lattice versions of iv algorithms this paper deals with the derivation and the properties of fast transversal algorithms hence by establishing a relationship between classical iv methods and cumulant based ar estimation problems new fast adaptive algorithms fast transversal recursive instrumental variable ftriv and generalized least mean squares glms are proposed for the estimation of ar processes the algorithms are seen to have better performance in terms of convergence speed and misadjustment even in low snr the extra computational complexity is negligible the performance of the algorithms as well as some illustrative tracking comparisons with the existing adaptive ones in the literature are verified via simulations the conditions of convergence are investigated for the glms