filtering and frequency interpretations of singular spectrum analysis new filtering and spectral interpretations of singular spectrum analysis ssa are provided it is shown that the variables reconstructed from diagonal averaging of reduced rank approximations to the trajectory matrix can be obtained from a noncausal convolution filter with zero phase characteristics the reconstructed variables are readily constructed using a two pass filtering algorithm that is well known in the signal processing literature when the number of rows in the trajectory matrix is much larger than number of columns many results reported in the signal processing literature can be used to derive the properties of the resulting filters and their spectra new features of the reconstructed series are revealed using these results two examples are used to illustrate the results derived in this paper

reconstruction in diffraction ultrasound tomography using nonuniform fft we show an iterative reconstruction framework for diffraction ultrasound tomography the use of broad band illumination allows significant reduction of the number of projections compared to straight ray tomography the proposed algorithm makes use of forward nonuniform fast fourier transform nufft for iterative fourier inversion incorporation of total variation regularization allows the reduction of noise and gibbs phenomena while preserving the edges the complexity of the nufft based reconstruction is comparable to the frequency domain interpolation gridding algorithm whereas the reconstruction accuracy in sense of the l sup 2 and the l sup spl infin norm is better

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

small periodic solutions generated by sublinear terms abstract the paper is concerned with hopf bifurcations in systems of autonomous ordinary differential equations with a parameter the principal distinction between usual theorems on hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter we present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium sublinear nonlinearities play the main role in the results obtained

tomographic image reconstruction using the theory of convex projections the method of convex projections is applied to reconstruct an image in computer tomography this appears to the first time that the method has been used to obtain geometry free reconstruction from ray sum data it is shown that a special set of convex projections duplicates the result of the algebraic reconstruction technique art the similarities and differences of these two methods are discussed it is pointed out that use of a priori information enhances the quality of the results especially when partial data have been used in which case art fails simulations and reconstruction of a ct image are also furnished to demonstrate the feasibility of this method

estimation of a signal waveform from noisy data using low rank approximation to a data matrix an analysis and improvement of a data adaptive signal estimation algorithm are presented perturbation analysis of a reduced rank data matrix is used to reveal its statistical properties the obtained information is used for calculating the performance of the toeplitz restoration algorithm of d tufts et al 1982 this analysis leads to improvements of the methods and the predicted improvements are demonstrated by simulation and comparison with the cramer rao bounds

active microwave imaging a model based approach a model based approach that improves the spatial resolution of a buried inhomogeneous object using an active microwave imaging system is discussed algorithms are developed for quasi real time processing using a microwave camera specifically constructed for civil engineering applications using reflection mode imaging the model is generated by numerically solving an integral equation representing the ideal scattered field of the object the model is used to develop an optimal spatial filter to process the measured field finally the inverse problem is solved using an imaging algorithm based on diffraction tomography and fourier transforms experimental results indicate a reasonable performance of the approach

local geometry of singular real analytic surfaces let v r n be a compact real analytic surface with isolated singularities and assume its smooth part v 0 is equipped with a riemannian metric that is induced from some analytic riemannian metric on r n we prove 1 each point of v has a neighborhood which is quasi isometric naturally and almost isometrically to a union of metric cones and horns glued at their tips 2 a full asymptotic expansion for any p v of the length of v n q dist q p r as r 0 0 3 a gauss bonnet theorem saying that each singular point contributes 1 l 2 where l is the coefficient of the linear term in the expansion of 2 4 the l 2 stokes theorem selfadjointness and discreteness of the laplace beltrami operator on vo an estimate on the heat kernel and a gauss bonnet theorem for the l 2 euler characteristic as a central tool we use resolution of singularities

image recovery in computer tomography from partial fan beam data by convex projections for the image recovery process the authors use the convex projections method also known as the method of projections onto convex sets pocs several incomplete data geometries including those associated with limited source travel and beam blocking internal opacities are considered to enable the recovery several prior knowledge constraints including one associated with the directivity of the image vector are used the overall recovery algorithm can be practically implemented by exploiting the toeplitz structure of key operators

continuity of closest rank p approximations to matrices in signal processing the singular value decomposition and rank characterization of matrices play prominent roles the mapping which associates with any complex m n matrix x its closest rank p approximation x p need not be continuous when the pth and the p 1 st singular values of x are equal this mapping maps in fact a matrix to a set of matrices furthermore an example is given to show that large errors in computing x p can be expected when p is sufficiently close to p 1 it is finally shown that this mapping is closed in the sense of zangwill the property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping e g see 1

a parameter estimation scheme for damped sinusoidal signals based on low rank hankel approximation most of the existing algorithms for parameter estimation of damped sinusoidal signals are based only on the low rank approximation of the prediction matrix and ignore the hankel property of the prediction matrix we propose a modified kumaresan tufts mkt algorithm exploiting both rank deficient and hankel properties of the prediction matrix computer simulation results demonstrate that compared with the original kumaresan tufts 1982 algorithm and the matrix pencil algorithm the mkt algorithm has a lower noise threshold and can estimate the parameters of signal with larger damping factors

rational invariant subspace approximations with applications subspace methods such as music minimum norm and esprit have gained considerable attention due to their superior performance in sinusoidal and direction of arrival doa estimation but they are also known to be of high computational cost in this paper new fast algorithms for approximating signal and noise subspaces and that do not require exact eigendecomposition are presented these algorithms approximate the required subspace using rational and power like methods applied to the direct data or the sample covariance matrix several esprit as well as music type methods are developed based on these approximations a substantial computational saving can be gained comparing with those associated with the eigendecomposition based methods these methods are demonstrated to have performance comparable to that of music yet will require fewer computations to obtain the signal subspace matrix

calibration free b0 correction of epi data using structured low rank matrix recovery we introduce a structured low rank algorithm for the calibration free compensation of field inhomogeneity artifacts in echo planar imaging epi mri data we acquire the data using two epi readouts that differ in echo time using time segmentation we reformulate the field inhomogeneity compensation problem as the recovery of an image time series from highly undersampled fourier measurements the temporal profile at each pixel is modeled as a single exponential which is exploited to fill in the missing entries we show that the exponential behavior at each pixel along with the spatial smoothness of the exponential parameters can be exploited to derive a 3 d annihilation relation in the fourier domain this relation translates to a low rank property on a structured multi fold toeplitz matrix whose entries correspond to the measured k space samples we introduce a fast two step algorithm for the completion of the toeplitz matrix from the available samples in the first step we estimate the null space vectors of the toeplitz matrix using only its fully sampled rows the null space is then used to estimate the signal subspace which facilitates the efficient recovery of the time series of images we finally demonstrate the proposed approach on spherical mr phantom data and human data and show that the artifacts are significantly reduced

accelerated dynamic mri exploiting sparsity and low rank structure k t slr we introduce a novel algorithm to reconstruct dynamic magnetic resonance imaging mri data from under sampled k t space data in contrast to classical model based cine mri schemes that rely on the sparsity or banded structure in fourier space we use the compact representation of the data in the karhunen louve transform klt domain to exploit the correlations in the dataset the use of the data dependent kl transform makes our approach ideally suited to a range of dynamic imaging problems even when the motion is not periodic in comparison to current klt based methods that rely on a two step approach to first estimate the basis functions and then use it for reconstruction we pose the problem as a spectrally regularized matrix recovery problem by simultaneously determining the temporal basis functions and its spatial weights from the entire measured data the proposed scheme is capable of providing high quality reconstructions at a range of accelerations in addition to using the compact representation in the klt domain we also exploit the sparsity of the data to further improve the recovery rate validations using numerical phantoms and in vivo cardiac perfusion mri data demonstrate the significant improvement in performance offered by the proposed scheme over existing methods

super resolution reconstruction in frequency domain optical coherence tomography using the finite rate of innovation principle the standard approach to signal reconstruction in frequency domain optical coherence tomography fdoct is to apply the inverse fourier transform to the measurements this technique offers limited resolution due to heisenberg s uncertainty principle we propose a new super resolution reconstruction method based on a parametric representation we consider multilayer specimens wherein each layer has a constant refractive index and show that the backscattered signal from such a specimen fits accurately in to the framework of finite rate of innovation fri signal model and is represented by a finite number of free parameters we deploy the high resolution prony method and show that high quality super resolved reconstruction is possible with fewer measurements about one fourth of the number required for the standard fourier technique to further improve robustness to noise in practical scenarios we take advantage of an iterated singular value decomposition algorithm cadzow denoiser we present results of monte carlo analyses and assess statistical efficiency of the reconstruction techniques by comparing their performance against the cramer rao bound reconstruction results on experimental data obtained from technical as well as biological specimens show a distinct improvement in resolution and signal to reconstruction noise offered by the proposed method in comparison with the standard approach

spectral extrapolation of spatially bounded images mri application a spectral extrapolation algorithm for spatially bounded images is presented an image is said to be spatially bounded when it is confined to a closed region and is surrounded by a background of zeros with prior knowledge of the spatial domain zeros the extrapolation algorithm extends the image s spectrum beyond a known interval of low frequency components the result which is referred to as the finite support solution has space variant resolution features near the edge of the support region are better resolved than those in the center the resolution of the finite support solution is discussed as a function of the number of known spatial zeros and known spectral components a regularized version of the finite support solution is included for handling the case where the known spectral components are noisy for both the noiseless and noisy cases the resolution of the finite support solution is measured in terms of its impulse response characteristics and compared to the resolution of the zerofilled and nyquist solutions the finite support solution is superior to the zerofilled solution for both the noisy and noiseless data cases when compared to the nyquist solution the finite support solution may be preferred in the noisy data case examples using medical image data are provided

signal enhancement a composite property mapping algorithm a signal enhancement algorithm is developed that seeks to recover a signal from noise contaminated distorted measurements made on that signal this object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing the measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties conditions for the algorithm s convergence are established in which the primary requirement is that a composite property mapping be closed this is a relatively unrestricted condition in comparison to that required of most existing signal enhancement algorithms

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

the method of parameter functionalization in the hopf bifurcation problem depending on a parameter for an extensive bibliography see l the greater part of these investigations exploit information about not only the linear terms of the right hand member of the system l the linear operator fl a 0 but also information about the terms of higher orders in the taylor expansion in x of the function f a x with the help of such information one can answer the questions on the number of arising self oscillations their stability their dependence on a parameter etc proofs of corresponding assertions use as a rule an analytic technique such as varied forms of the implicit function theorem the theory of invariant manifolds and the like at the same time in problems arising during the study of complicated physical technological etc processes often the only rather complete information available is that concerning the linear terms of the right hand member of the system 1 in such circumstances it is very difficult to apply an analytic technique for studying the problem of generation of self oscillations yet as things turn out 2 in some cases the very fact of generation of small self oscillations can be picked out of information about the linear terms of the right hand member of the system 1 of course under lack of information about the high order terms in the taylor expansion in x of f a x one can say next to nothing about properties of arising self oscillations this article contains a topological proof of the hopf theorem in this proof the method of parameter functionalization 3 introduced by krasnosel skii in another situation has much significance employment of topological considerations makes it possible to throw aside usual assumptions of differentiability of the right hand member of the system 1 this provides an opportunity to investigate systems l the right hand side of which contains for example hysteresis type or relay type nonlinearities the proof presented below goes through without changes in the case of functional differential equations with lagging arguments 4 we do not present here the exact formulations of such assertions because our purpose is the demonstration of the method in s 61 the method of parameter functionalization has been applied to the investigation of bifurcation of long periodic solutions of differential equations and mappings

computationally efficient maximum likelihood estimation of structured covariance matrices by invoking the extended invariance principle exip we present herein a computationally efficient method that provides asymptotic for large samples maximum likelihood aml estimation for structured covariance matrices and is referred to as the aml algorithm a closed form formula for estimating the hermitian toeplitz covariance matrices that makes aml computationally simpler than most existing hermitian toeplitz matrix estimation algorithms is derived although the aml covariance matrix estimator can be used in a variety of applications we focus on array processing our simulation study shows that aml enhances the performance of angle estimation algorithms such as music by making them very close to the corresponding cramer rao bound crb for uncorrelated signals numerical comparisons with several structured and unstructured covariance matrix estimators are also presented

estimating localized sources of diffusion fields using spatiotemporal sensor measurements we consider diffusion fields induced by a finite number of spatially localized sources and address the problem of estimating these sources using spatiotemporal samples of the field obtained with a sensor network within this framework we consider two different time evolutions the case where the sources are instantaneous as well as the case where the sources decay exponentially in time after activation we first derive novel exact inversion formulas for both source distributions through the use of green s second theorem and a family of sensing functions to compute generalized field samples these generalized samples can then be inverted using variations of existing algebraic methods such as prony s method next we develop a novel and robust reconstruction method for diffusion fields by properly extending these formulas to operate on the spatiotemporal samples of the field finally we present numerical results using both synthetic and real data to verify the algorithms proposed herein

partial discreteness a novel prior for magnetic resonance image reconstruction an important factor influencing the quality of magnetic resonance mr images is the reconstruction method that is employed and specifically the type of prior knowledge that is exploited during reconstruction in this work we introduce a new type of prior knowledge partial discreteness pd where a small number of regions in the image are assumed to be homogeneous and can be well represented by a constant magnitude in particular we mathematically formalize the partial discreteness property based on a gaussian mixture model gmm and derive a partial discreteness image representation that characterizes the salient features of partially discrete images a constant intensity in homogeneous areas and texture in heterogeneous areas the partial discreteness representation is then used to construct a novel prior dedicated to the reconstruction of partially discrete mr images the strength of the proposed prior is demonstrated on various simulated and real k space data based experiments with partially discrete images results demonstrate that the pd algorithm performs competitively with state of the art reconstruction methods being flexible and easy to implement

fri sampling on structured nonuniform grids application to super resolved optical imaging the annihilating filter forms the core of many state of the art techniques for finite rate of innovation fri signal reconstruction and spectral estimation in its standard manifestation the annihilating filter is setup in the discrete time domain with uniform sampling and is used to estimate the parameters of a signal that is expressible as a sum of exponentials in this paper we consider annihilation in the continuous domain using translation operators codenamed opera and show that fri signals can be reconstructed using measurements taken on a structured nonuniform sampling sns grid the parameters are estimable uniquely under certain conditions the transition from uniform sampling to sns comes with specific advantages the measurements could be made in bursts from high signal to noise ratio snr regions which directly enhances parameter estimation accuracy the sampling interval could be made larger than in the uniform case which leads to a reduction in the mean squared error mse and closely spaced exponential parameters could be resolved better we present theoretical developments related to the sns grid and demonstrate accurate parameter estimation monte carlo performance analysis in the presence of noise shows that the opera with the sns approach yields a significant decrease in mse up to 10 to 20 db for certain snr ranges compared with the uniform sampling scenario considering an application to frequency domain optical coherence tomography we show that the proposed methodology results in higher resolution and higher reconstruction accuracy than the standard techniques

exploitation of symmetries for image reconstruction in linearized variable density diffraction tomography the problem of reconstructing an object s weakly varying compressibility and density distributions in three dimensional 3d acoustic diffraction tomography is studied based on the fourier diffraction projection theorem for acoustic media it is demonstrated that the 3d fourier components of an object s compressibility and density distributions can be decoupled algebraically thereby providing a method for separately reconstructing the distributions this is facilitated by the identification and exploitation of tomographic symmetries and the rotational invariance of the imaging model the developed reconstruction methods are investigated by use of computer simulation studies the application of the proposed image reconstruction strategy to other tomography problems is discussed

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

synchronized double frequency oscillations in a class of weakly resonant systems subharmonic oscillations in the weak resonant hopf bifurcation in control systems is studied 11

efficient sinc function interpolation technique for center padded data an efficient zooming fft fast fourier transform algorithm that allows center padding sinc function interpolation of 2 d images is presented this algorithm avoids the phase shifts that would be introduced if the efficient skinner interpolation method is used output pruning is incorporated to allow efficient determination of a zoomed subimage time savings of more than 50 can be achieved example images illustrating the use of the algorithm in conjunction with zooming and arma autoregressive moving average modeling of data are given

propagation of singularities for the wave equation on conic manifolds for the wave equation associated to the laplacian on a compact manifold with boundary with a conic metric with respect to which the boundary is metrically a point the propagation of singularities through the boundary is analyzed under appropriate regularity assumptions the diffracted non direct wave produced by the boundary is shown to have sobolev regularity greater than the incoming wave

a majorize minimize framework for rician and non central chi mr images the statistics of many mr magnitude images are described by the non central chi ncc family of probability distributions which includes the rician distribution as a special case these distributions have complicated negative log likelihoods that are nontrivial to optimize this paper describes a novel majorize minimize framework for ncc data that allows penalized maximum likelihood estimates to be obtained by solving a series of much simpler regularized least squares surrogate problems the proposed framework is general and can be useful in a range of applications we illustrate the potential advantages of the framework with real and simulated data in two contexts 1 mr image denoising and 2 diffusion profile estimation in high angular resolution diffusion mri the proposed approach is shown to yield improved results compared to methods that model the noise statistics inaccurately and faster computation relative to commonly used nonlinear optimization techniques

application of a maximum likelihood estimator in an experimental study in ultrasonic diffraction tomography a recently proposed approach to the inverse problem of detecting the presence and estimating the location of a known object from data collected in a set of diffraction tomographic experiments is evaluated experimental data are used to validate of the filtered backpropagation algorithms used and their robustness to modeling errors and to severe limitations in the angular coverage of the tomographic data is demonstrated a potential application to medical imaging of soft tissue is illustrated

hopf bifurcations in resonance 2 1 abstract in this paper hopf bifurcations for equations of control theory are studied in strong resonance case 2 1 the harmonic linearization approach and topological methods allow to give the sharp analysis of the problem for the case of quadratic principal nonlinearity

spectral and scattering theory for symbolic potentials of order zero the spectral and scattering theory is investigated for a generalization to scattering metrics on two dimensional compact manifolds with boundary of the class of smooth potentials on r2 which are homogeneous of degree zero near infinity the most complete results require the additional assumption that the restriction of the potential to the circle s at infinity be morse generalized eigenfunctions associated to the essential spectrum at non critical energies are shown to originate both at minima and maxima although the latter are not germane to the l2 spectral theory asymptotic completeness is shown both in the traditional l2 sense and in the sense of tempered distributions this leads to a definition of the scattering matrix the structure of which will be described in a future publication

quantitative imaging using a time domain eigenfunction method an inverse scattering method that uses eigenfunctions of a scattering operator at a single frequency is extended to include the full range of frequencies present in the incident pulse waveform the resulting so called time domain eigenfunction method is shown to yield a modulated version of the scattering potential the potential is recovered by a demodulation process using cross correlation with a reference including an adaptive delay in the reference is shown to compensate partially for the linearization of the born approximation and to extend its valid range the k space window of the time domain solution is expressed in terms of the incident waveform and shown to be smoother than that of a single frequency solution the time domain method is examined using both calculated and measured data in the calculations an exact solution for scattering from one or multiple nonconcentric cylinders is used to obtain the scattered field in the measurements a novel ring transducer system was employed to obtain the incident and total fields the results of simulations and experiments show that the method is robust and accurate for the size of objects considered and that the point resolution approaches one half the wavelength at the pulse center frequency

sequences with positive semidefinite fourier transforms a sequence is said to be positive if its fourier transform exclusively takes on real nonnegative values as a function of frequency positive sequences play a prominent role in contemporary signal processing and system theory to illustrate this point it is well known that the factorization theorem is extensively used in studies related to wide sense stationary random signals and linear systems the ability to appropriately factorize a fourier transform is contingent on that transform being positive semidefinite this paper is partially tutorial in that some fundamental positive sequence properties found in dispersed sources are first reviewed this is followed by the development of several new properties these properties are in turn used to develop an efficient algorithm for finding that positive sequence which lies closest to a given nonpositive sequence in the least squares error sense interest in this approximation problem arises from the fact that although a given sequence may be theoretically positive practical considerations often result in its realization being nonpositive for instance unbiased autocorrelation lag estimates can lead to nonpositive spectral density function estimates

the information content of mr images the theoretical information content defined by c e shannon 1948 is proposed as an objective measure of mr magnetic resonance image quality this measure takes into account the contrast to noise ratio cnr scan resolution and field of view it is used to derive an optimum in the tradeoff problem between image resolution and cnr and as a criterion to assess the usefulness of high resolution 512 sup 2 mr images the result tells that for a given total acquisition time an optimum value of the resolution can be found this optimum is very broad to apply shannon s theory on information constant to mr images a model for the spatial spectral power density of these images is required such a model has been derived from experimental observations of ordinary mr images as well as from theoretical considerations

quasi bandlimited properties of radon transforms and their implications for increasing angular sampling densities the n dimensional n d radon transform which forms the mathematical basis for a broad variety of tomographic imaging applications can be viewed as an n d function in n d sinogram space accurate reconstruction of continuous or discrete tomographic images requires full knowledge of the radon transform in the corresponding n d sinogram space in practice however one can have only a finite set of discrete samples of the radon transform in the sinogram space one often derives the desired full knowledge of the radon transform from its discrete samples by invoking various interpolation algorithms according to the wittaker shannon sampling theorem a necessary condition for a full and unique recovery of the radon transform from its discrete samples is that the radon transform itself be bandlimited therefore it is necessary to analyze the bandlimited properties of the radon transform in this work the authors analyze explicitly the bandlimited properties of the radon transform and show that the radon transform is mathematically quasi bandlimited or essentially bandlimited in two quantitative senses and can essentially be treated as bandlimited in practice the quasi bandlimited properties can be used for increasing the angular sampling density of the radon transform

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

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

a new reconstruction approach for reflection mode diffraction tomography reflection mode diffraction tomography rm dt is an inversion scheme used to reconstruct the acoustical refractive index distribution of a scattering object in this work we reveal the existence of statistically complementary information inherent in the backscattered data and propose reconstruction algorithms that exploit this information for achieving a bias free reduction of image variance in rm dt images such a reduction of image variance can potentially enhance the detectability of subtle image features when the signal to noise ratio of the measured scattered data is low in rm dt the proposed reconstruction algorithms are mathematically identical but they propagate noise and numerical errors differently we investigate theoretically and validate numerically the noise properties of images reconstructed using one of the reconstruction algorithms for several different multifrequency sources and uncorrelated data noise

higher order nonlinear diffraction tomography inversion of the rytov series nonlinear tomographic reconstruction algorithms are developed for inversion of data measured in scattering experiments in which the complex phase of the wavefields is modeled by an arbitrarily large possibly infinite number of terms in the rytov series the algorithms attain the form of a volterra series of nonlinear operators with the usual filtered backpropagation algorithm of diffraction tomography as the leading linear term a computer simulation study is included to illustrate the performance of the algorithms for the case of scattering objects with cylindrical symmetry

a fast and accurate multilevel inversion of the radon transform a number of imaging technologies reconstruct an image function from its radon projection using the convolution backprojection method the convolution is an o n2 log n algorithm where the image consists of n x n pixels while the backprojection is an o n3 algorithm thus constituting the major computational burden of the convolution backprojection method an o n2 log n multilevel backprojection method is presented here when implemented with a fourier domain postprocessing technique also presented here the resulting image quality is similar or superior to the image quality of the classical backprojection technique

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

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

sampling of planar curves theory and fast algorithms we introduce a continuous domain framework for the recovery of a planar curve from a few samples we model the curve as the zero level set of a trigonometric polynomial we show that the exponential feature maps of the points on the curve lie on a low dimensional subspace we show that the null space vector of the feature matrix can be used to uniquely identify the curve given a sufficient number of samples the worst case theoretical guarantees show that the number of samples required for unique recovery depends on the bandwidth of the underlying trigonometric polynomial which is a measure of the complexity of the curve we introduce an iterative algorithm that relies on the low rank property of the feature maps to recover the curves when the samples are noisy or when the true bandwidth of the curve is unknown we also demonstrate the preliminary utility of the proposed curve representation in the context of image segmentation

asymmetric pulse modeling for fri sampling we consider sampling and reconstruction of finite rate of innovation fri signals such as a train of pulses where the pulses have varying degrees of asymmetry we address the problem of asymmetry modeling starting from a given symmetric prototype we show that among the class of unitary operators that are linear and invariant to translation and scale the fractional hilbert frh operator is unique for parametrically modeling pulse asymmetry the frh operator is obtained by a trigonometric interpolation between the standard hilbert and identity operators where the interpolation weights are determined by the degree of asymmetry the frh operators are also steerable which allows for estimation of the asymmetry factors in addition to the delays and amplitudes using the high resolution spectral estimation techniques that are used for solving standard fri problems we also develop the discrete counterpart using discrete frh operators and show that all the desirable properties carry over smoothly to the discrete setting as well we derive closed form expressions for the cramer rao bounds and hammersley chapman robbins bound on the variances of the estimators for continuous and discrete parameters respectively experimental results show that the proposed estimators have variances that meet the lower bounds we demonstrate an application of the proposed discrete frh methodology on real electrocardiogram ecg signals in the presence of noise specifically we show how the asymmetry of qrs complexes in various channels of an ecg signal could be modeled accurately

structured low rank approximation this paper concerns the construction of a structured low rank matrix that is nearest to a given matrix the notion of structured low rank approximation arises in various applications ranging from signal enhancement to protein folding to computer algebra where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system the task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix this paper addresses some of the theoretical and numerical issues involved in the problem two procedures for constructing the nearest structured low rank matrix are proposed the procedures are flexible enough that they can be applied to any lower rank any linear structure and any matrix norm in the measurement of nearness the techniques can also be easily implemented by utilizing available optimization packages the special case of symmetric toeplitz structure using the frobenius matrix norm is used to exemplify the ideas throughout the discussion the concept rather than the implementation details is the main emphasis of the paper

sampling curves with finite rate of innovation in this paper we extend the theory of sampling signals with finite rate of innovation fri to a specific class of two dimensional curves which are defined implicitly as the zeros of a mask function here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials and therefore has finite rate of innovation an associated edge image which is discontinuous on the predefined parametric curve is proved to satisfy a set of linear annihilation equations we show that it is possible to reconstruct the parameters of the curve i e to detect the exact edge positions in the continuous domain based on the annihilation equations robust reconstruction algorithms are also developed to cope with scenarios with model mismatch moreover the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing e g image up sampling we demonstrate one potential application of the annihilation algorithm with examples in edge preserving interpolation experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges and improving snrs

low rank modeling of local k space neighborhoods loraks for constrained mri recent theoretical results on low rank matrix reconstruction have inspired significant interest in low rank modeling of mri images existing approaches have focused on higher dimensional scenarios with data available from multiple channels timepoints or image contrasts the present work demonstrates that single channel single contrast single timepoint k space data can also be mapped to low rank matrices when the image has limited spatial support or slowly varying phase based on this we develop a novel and flexible framework for constrained image reconstruction that uses low rank matrix modeling of local k space neighborhoods loraks a new regularization penalty and corresponding algorithm for promoting low rank are also introduced the potential of loraks is demonstrated with simulated and experimental data for a range of denoising and sparse sampling applications loraks is also compared against state of the art methods like homodyne reconstruction l1 norm minimization and total variation minimization and is demonstrated to have distinct features and advantages in addition while calibration based support and phase constraints are commonly used in existing methods the loraks framework enables calibrationless use of these constraints

one step image reconstruction from incomplete data in computer tomography the authors develop a projection operator that simultaneously projects onto the set of all functions satisfying raysum constraints in parallel beam ct the projector can be directly extended to the fan beam case through the process of rebinning the projector generates a solution that is closest to the initial estimate among all the functions that are consistent with the available raysum data it realizes the art algebraic reconstruction technique in one step the projector furnishes the one step projection reconstruction ospr for any arbitrary configurations of missing data because the projection is one step there can be a significant reduction in the number of online computations and memory requirements especially when the missing data exhibit some pattern within a view or between views

k space deep learning for accelerated mri the annihilating filter based low rank hankel matrix approach aloha is one of the state of the art compressed sensing approaches that directly interpolates the missing k space data using low rank hankel matrix completion the success of aloha is due to the concise signal representation in the k space domain thanks to the duality between structured low rankness in the k space domain and the image domain sparsity inspired by the recent mathematical discovery that links convolutional neural networks to hankel matrix decomposition using data driven framelet basis here we propose a fully data driven deep learning algorithm for k space interpolation our network can be also easily applied to non cartesian k space trajectories by simply adding an additional regridding layer extensive numerical experiments show that the proposed deep learning method consistently outperforms the existing image domain deep learning approaches

efficient denoising algorithms for large experimental datasets and their applications in fourier transform ion cyclotron resonance mass spectrometry modern scientific research produces datasets of increasing size and complexity that require dedicated numerical methods to be processed in many cases the analysis of spectroscopic data involves the denoising of raw data before any further processing current efficient denoising algorithms require the singular value decomposition of a matrix with a size that scales up as the square of the data length preventing their use on very large datasets taking advantage of recent progress on random projection and probabilistic algorithms we developed a simple and efficient method for the denoising of very large datasets based on the qr decomposition of a matrix randomly sampled from the data this approach allows a gain of nearly three orders of magnitude in processing time compared with classical singular value decomposition denoising this procedure called urqrd uncoiled random qr denoising strongly reduces the computer memory footprint and allows the denoising algorithm to be applied to virtually unlimited data size the efficiency of these numerical tools is demonstrated on experimental data from high resolution broadband fourier transform ion cyclotron resonance mass spectrometry which has applications in proteomics and metabolomics we show that robust denoising is achieved in 2d spectra whose interpretation is severely impaired by scintillation noise these denoising procedures can be adapted to many other data analysis domains where the size and or the processing time are crucial

reconstruction from projections under time frequency constraints low pass filtering computed tomography ct images to reduce noise may smooth or modify image features which are very important to the physician image features are often more easily identified and processed in the time frequency plane the authors use time frequency distributions for spatially varying filtering of noisy ct images constraining time frequency representation coefficients of the projection data or of the reconstructed image to be zero in certain regions of the time frequency plane the authors consider two different applications 1 filtering the projection data and then performing image reconstruction and 2 filtering the reconstructed image directly criteria minimized subject to constraints may be either a deterministic minimum weighted perturbation of the given projection data or a stochastic minimum mean square error in colored gaussian noise results show improvement over processing the image with a linear spatially invariant filter

consistency conditions and linear reconstruction methods in diffraction tomography because an image can be reconstructed from knowledge of its radon transform rt the task of reconstructing an image is tantamount to that of estimating its rt based upon the fourier diffraction projection fdp theorem from the statistical perspective of unbiased reduction of image variance the author previously proposed an infinite family of estimation methods for obtaining the rt from the scattered data in diffraction tomography dt in this work using the fdp theorem the authors define the diffraction radon transform drt which can be treated as the data function in dt subsequently using strategies similar to those that analyze the consistency conditions on the exponential radon transform in two dimensional 2 d single photon emission computed tomography with uniform attenuation the author studied the consistency condition on the drt and the author shows that there is a hierarchy of estimation methods that actually project the noisy data function onto its consistency space in different ways in terms of a weighted inner product of the consistency and inconsistency parts of a noisy data function the author further demonstrates that a subset of the family of estimation methods can be interpreted as orthogonal projections onto the consistency space of the drt in particular the statistically suboptimal estimation method in the family corresponds to an orthogonal projection associated with an ordinary inner product of the consistency and inconsistency parts of a noisy data function

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

recovery of damped exponentials using structured low rank matrix completion we introduce a structured low rank matrix completion algorithm to recover a series of images from their under sampled measurements where the signal along the parameter dimension at every pixel is described by a linear combination of exponentials we exploit the exponential behavior of the signal at every pixel along with the spatial smoothness of the exponential parameters to derive an annihilation relation in the fourier domain this relation translates to a low rank property on a structured matrix constructed from the fourier samples we enforce the low rank property of the structured matrix as a regularization prior to recover the images since the direct use of current low rank matrix recovery schemes to this problem is associated with high computational complexity and memory demand we adopt an iterative re weighted least squares algorithm which facilitates the exploitation of the convolutional structure of the matrix novel approximations involving 2 d fast fourier transforms are introduced to drastically reduce the memory demand and computational complexity which facilitates the extension of structured low rank methods to large scale 3 d problems we demonstrate our algorithm in the mr parameter mapping setting and show improvement over the state of the art methods

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

the double hopf bifurcation with 2 1 resonance many problems of physical interest involve the nonlinear interaction of two oscillators with different frequencies when these frequencies are incommensurate the interaction is straightforward involving the amplitude rather than the phase of each oscillator when the frequencies are in a ratio closely corresponding to a rational fraction of small denominator phase locking occurs and the dynamics is much richer in this paper we investigate fully the important case where the frequencies are in the ratio 2 1 the analysis allows for small departures from this ratio and treats not only the generic problem for small amplitudes but also an important degenerate case which appears to capture most of the dynamics at larger amplitudes

solution to the full vectorial 3 d inverse source problem by multilevel fast multipole method inspired hierarchical disaggregation a 3 d imaging approach based on a multilevel fast multipole method inspired hierarchical disaggregation of the broadband plane wave spectrum is introduced to solve the full vectorial electromagnetic inverse source problem based on the knowledge of the radiated fields in terms of complexity this algorithm is identical to fast fourier transform accelerated 3 d imaging techniques in terms of applicability the hierarchical disaggregation is advantageous for commonly encountered spherical k space data since the method can be directly combined with the near field fast multipole translations both far and near field data can be handled conveniently the investigation includes an analysis of the inherent point spread function and numerical results

reconstruction of binary shapes from blurred images via hankel structured low rank matrix recovery with the dominance of digital imaging systems we are often dealing with discrete domain samples of an analog image due to physical limitations all imaging devices apply a blurring kernel on the input image before taking samples to form the output pixels in this paper we focus on the reconstruction of binary shape images from few blurred samples this problem has applications in medical imaging shape processing and image segmentation our method relies on representing the analog shape image in a discrete grid much finer than the sampling grid we formulate the problem as the recovery of a rank r matrix that is formed by a hankel structure on the pixels we further propose efficient admm based algorithms to recover the low rank matrix in both noiseless and noisy settings we also analytically investigate the number of required samples for successful recovery in the noiseless case for this purpose we study the problem in the random sampling framework and show that with mathcal o r log 4 n 1 n 2 random samples where the size of the image is assumed to be n 1 times n 2 we can guarantee the perfect reconstruction with high probability under mild conditions we further prove the robustness of the proposed recovery in the noisy setting by showing that the reconstruction error in the noisy case is bounded when the input noise is bounded simulation results confirm that our proposed method outperform the conventional total variation minimization in the noiseless settings

towards generalized fri sampling with an application to source resolution in radioastronomy it is a classic problem to estimate continuous time sparse signals like point sources in a direction of arrival problem or pulses in a time of flight measurement the earliest occurrence is the estimation of sinusoids in time series using prony s method this is at the root of a substantial line of work on high resolution spectral estimation the estimation of continuous time sparse signals from discrete time samples is the goal of the sampling theory for finite rate of innovation fri signals both spectral estimation and fri sampling usually assume uniform sampling but not all measurements are obtained uniformly as exemplified by a concrete radioastronomy problem we set out to solve thus we develop the theory and algorithm to reconstruct sparse signals typically sum of sinusoids from nonuniform samples we achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements these uniform samples are known to satisfy the annihilation equations a valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint thanks to this new approach we unify a variety of fri based methods we demonstrate the versatility and robustness of the proposed approach with five fri reconstruction problems namely dirac reconstructions with irregular time or fourier domain samples fri curve reconstructions dirac reconstructions on the sphere and point source reconstructions in radioastronomy the proposed algorithm improves substantially over state of the art methods and is able to reconstruct point sources accurately from irregularly sampled fourier measurements under severe noise conditions

application of autoregressive moving average parametric modeling in magnetic resonance image reconstruction the modeling of data is an alternative to conventional use of the fast fourier transform fft algorithm in the reconstruction of magnetic resonance mr images the application of the fft leads to artifacts and resolution loss in the image associated with the effective window on the experimentally truncated phase encoded mr data the transient error modeling method treats the mr data as a subset of the transient response of an infinite impulse filter h z b z ia z thus the data are approximated by a deterministic autoregressive moving average arma model the algorithm for calculating the filter coefficients is described it is demonstrated that using the filter coefficients to reconstruct the image removes the truncation artifacts and improves the resolution however determining the autoregressive ar portion of the arma filter by algorithms that minimize the forward and backward prediction errors e g burg leads to significant image degradation the moving average ma portion is determined by a computationally efficient method of solving a finite difference equation with initial values special features of the mr data are incorporated into the transient error model the sensitivity to noise and the choice of the best model order are discussed mr images formed using versions of the transient error reconstruction tere method and the conventional fft algorithm are compared using data from a phantom and a human subject finally the computational requirements of the algorithm are addressed

maximum likelihood estimation of object location in diffraction tomography the problem is formulated within the context of diffraction tomography where the complex phase of the diffracted wavefield is modeled using the rytov approximation and the measurements consist of noisy renditions of this complex phase at a single frequency the log likelihood function is computed for the case of additive zero mean gaussian white noise and shown to be expressible in the form of the filtered backpropagation algorithm of diffraction tomography in this form however the filter function is no longer the rho filter appropriate to least square reconstruction but is now the generalized projection propagation of the object centered at the origin onto the line s parallel to the measurement line s but passing through the origin this result allows the estimation problem to be solved via a diffraction tomographic imaging procedure where the noisy data is filtered and backpropagated in a first step and the point of maximum value of the resulting image is then the maximum likelihood ml estimate of the object s location the authors include a calculation of the cramer rao bound for the estimation error and a computer simulation study illustrating the estimation procedure

vector entropy optimization based neural network approach to image reconstruction from projections in this paper we propose a multiobjective decision making based neural network model and algorithm for image reconstruction from projections this model combines the hopfield s model and multiobjective decision making approach we develop a weighted sum optimization based neural network algorithm the dynamical process of the net is based on minimization of a weighted sum energy function and euler s iteration and apply this algorithm to image reconstruction from computer generated noisy projections and siemens somatson dr scanner data respectively reconstructions based on this method is shown to be superior to conventional iterative reconstruction algorithms such as the multiplicate algebraic reconstruction technique mart and convolution from the point of view of accuracy of reconstruction computer simulation using the multiobjective method shows a significant improvement in image quality and convergence behavior over the conventional algorithms

a comparative study of three reconstruction methods for a limited view computer tomography problem three novel methods are used to reconstruct a simulated image from a set of incomplete data spanning a 160 degrees angular range these methods are the squashing affine transformation of j a reeds and l a shepp 1987 the circular interpolation method derived from the theory of j j clark m r plamer and p d lawrence 1985 and the geometry free reconstruction using the theory of convex projections these methods are briefly explained and their reconstructions are compared for the case of limited angular views

fast subspace decomposition many recent signal processing techniques in various areas e g array signal processing system identification and spectrum estimation require a step of extracting the low dimensional subspace from a large space this step is usually called subspace decomposition which is conventionally accomplished by an eigendecomposition ed since an ed requires o m sup 3 flops for an m spl times m matrix it may represent a barrier to the real time implementation of these algorithms the authors present a fast algorithm for signal subspace decomposition which is termed fsd that exploits the special matrix structure low rank plus identity associated with signal subspace algorithms and requires only o m sup 2 d flops where d often spl lt m denotes the signal subspace dimension unlike most of alternatives the dimension of the signal subspace is not assumed known apriori and is estimated as part of the procedure moreover theoretical analysis has been conducted and its results show the strong consistency of the new detection scheme and the asymptotic equivalence between fsd and ed in estimating the signal subspace their new approach can also exploit other matrix structure common in signal processing areas e g toeplitz or hankel and thus achieve another order of computational reduction moreover it can be easily implemented in parallel to reduce further the computation time to as little as o md using o m simple processors