a sparse data fast fourier transform sdfft a multilevel algorithm that efficiently fourier transforms sparse spatial data to sparse spectral data with controllable error is presented the algorithm termed sparse data fast fourier transform sdfft is particularly useful for signal processing applications where only part of the k space is to be computed regardless of whether it is a regular region like an angular section of the ewald sphere or it consists of completely arbitrary points scattered in the k space in addition like the various nonuniform fast fourier transforms the o nlogn algorithm can deal with a sparse nonuniform spatial domain in this paper the parabolic reflector antenna problem is studied as an example to demonstrate its use in the computation of far field patterns due to arbitrary aperture antennas and antenna arrays the algorithm is also promising for various applications such as backprojection tomography diffraction tomography and synthetic aperture radar imaging

direct fourier reconstruction in fan beam tomography we consider the problem of reconstructing tomographic imagery from fan beam projections using the direct fourier method dfm previous dfm reconstructions from parallel beam projections produced images of quality comparable to that of filtered convolution back projection moreover the number of operations using dfm in the parallel beam case is proportional to n2 log n versus n3 for back projection 3 the fan beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required we derive bounds on the detector spacing in fan beam ct that enable direct fourier reconstruction and describe the full algorithm necessary for processing the fan beam data the feasibility of the method is demonstrated with an example a key result of this paper is that high quality imagery can be reconstructed from fan beam data using the dfm in 0 n2 log n operations

method of moments enhancement technique for the analysis of sierpinski pre fractal antennas the numerical analysis of highly iterated sierpinski microstrip patch antennas by the method of moments mom involves many tiny subdomain basis functions resulting in a very large number of unknowns the sierpinski pre fractal can be defined by an iterated function system ifs as a consequence the geometry has a multilevel structure with many equal subdomains this property together with a multilevel matrix decomposition algorithm mlmda implementation in which the mlmda blocks are equal to the ifs generating shape is used to reduce the computational cost of the frequency analysis of a sierpinski based structure

a high speed reconstruction from projections using direct fourier method with optimized parameters an experimental analysis some issues of the direct fourier method dfm implementation are discussed a hybrid spline linear interpolation for the dfm is proposed the results of comprehensive simulation research are presented the following reconstruction problems and parameters are emphasized interpolation increasing the radial density of the polar raster filtering the 2 d inverse fourier transformation dimension and considering the cases of noiseless and noisy input data for the a priori prescribed resolution of the reconstructed image values of reconstruction parameters have been determined which are optimal with regard to reconstruction quality and computation cost the computational requirements of the dfm algorithm which correspond to distinct interpolation schemes are compared to one another for ct and mr tomography respectively the estimations obtained are compared to computational characteristics of the convolution backprojection method

efficient computation of 1 d periodic layered mixed potentials for the analysis of leaky wave antennas with vertical elements an efficient mixed potential integral equation formulation is proposed for the analysis of one dimensional 1 d periodic leaky wave antennas lwas based on planar stratified configurations with inclusions of arbitrarily oriented metallic or dielectric perturbations both the transverse and vertical components of the mixed potential green s functions due to a 1 d phased array of dipoles in a layered medium are accelerated using suitable homogeneous medium asymptotic extractions from the standard spectral series of floquet harmonics a novel acceleration procedure is applied for the computation of the vertical potentials whose extracted terms can be expressed as potentials from a 1 d phased array of half line sources in a homogeneous medium their numerical calculation requires a suitable modification of the ewald method thus resulting in new modified spectral and spatial series having gaussian convergence even in the case of complex modes and improper harmonics numerical comparisons for the 1 d periodic potentials both in the case of bounded and unbounded e g leaky harmonics validate the efficiency and accuracy of the proposed acceleration technique the method is illustrated and verified by determining the dispersion behavior of both bound and leaky modes for several lwa test cases

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

multigrid tomographic inversion with variable resolution data and image spaces a multigrid inversion approach that uses variable resolutions of both the data space and the image space is proposed since the computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements the proposed algorithm further reduces the computation relative to conventional multigrid approaches which change only the image space resolution at coarse scales the advantage is particularly important for data rich applications where data resolutions may differ for different scales applications of the approach to bayesian reconstruction algorithms in transmission and emission tomography with a generalized gaussian markov random field image prior are presented both with a poisson noise model and with a quadratic data term simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed grid iterative coordinate descent method and a multigrid method with fixed data resolution

a comparison of acceleration procedures for the two dimensional periodic green s function alternate acceleration procedures are evaluated for the two dimensional free space periodic green s function two of these techniques yield exponential convergence rates and are preferable if high accuracies are required numerical aspects of these procedures are described

an efficient approach for the computation of 2 d green s functions with 1 d and 2 d periodicities in homogeneous media this paper presents an algorithm for the acceleration of the series involved in the computation of 2 d homogeneous green s functions with 1 d and 2 d periodicities the algorithm is based on an original implementation of the spectral kummer poisson s method and it can be applied to the efficient computation of a wide class of infinite series in the algorithm the number of asymptotic terms retained in kummer s transformation is externally controlled so that any of the series that has to be accelerated is split into one series with exponential convergence and another series with algebraic convergence of arbitrarily large order numerical simulations have shown that there is an ldquooptimumrdquo number of asymptotic terms retained in kummer s transformation for which the cpu time needed in the summation of the series is minimized the cpu times required by ewald s method for the evaluation of 2 d green s functions with 1 d and 2 d periodicities have been compared with those required by the present algorithm and the algorithm has been found to be between 1 2 and 3 times faster than ewald s method when working in ldquooptimumrdquo operation conditions

on the spatial bandwidth of scattered fields it is shown that the scattered fields are almost space bandlimited functions the effective bandwidth w is introduced and evaluated for a very general scattering system as well as the error made using functions bandlimited to w w for representing the scattered field the effective bandwidth is very simply related to the maximum dimension of the scattering system the error drops to negligible values for modest increases of w compared to w in the case of large scatterers important consequences of the above general results are finally stressed

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

an efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures an efficient multilevel fast multipole algorithm mlfma formalism to model radiation and scattering by from large planar microwave structures is presented the technique relies on an electric field integral equation efie formulation and a series expansion for the green dyadic based on the use of perfectly matched layers pml in this way a new pml mlfma is developed to efficiently evaluate matrix vector multiplications arising in the iterative solution of the scattering problem the computational complexity of the new algorithm scales down to o n for electrically large structures the theory is validated by means of several illustrative numerical examples

discretization of continuous spectra based on perfectly matched layers as a tool of analysis in physics wavefields are often expanded in a set of eigensolutions obtained from a sturm liouville problem for singular sturm liouville problems subject to radiation boundary conditions i e problems defined on an infinite domain this set of eigensolutions has continuous parts in this paper we will show that it is possible to approximate this continuous set of eigensolutions by a discrete set of eigensolutions of the same sturm liouville operator but subject to dirichlet boundary conditions in complex space the idea of dirichlet boundary conditions in complex space stems from the perfectly matched layer pml absorbing boundary condition the pml was introduced in 1994 j p berenger j comput phys 114 1994 pp 185 200 as an absorbing termination of a finite difference time domain grid these complex space dirichlet boundary conditions have been used recently to close open electromagnetic waveguide structures in the present paper we aim at developing a mathemati

an efficient computation scheme for the free space green s function of a two dimensional semiinfinite phased array a simple scheme is developed to compute the green s function of a periodic semiinfinite array in free space it is based on the spectral representation of the fields radiated by an infinite linear array of dipoles results related to successive linear arrays are added in the space domain this summation can be accelerated tremendously by using an elementary extrapolation technique the resulting formulation converges everywhere in the plane containing the array and the number of terms required to achieve a given precision increases slowly away from this plane

efficient computation of the 3d green s function for the helmholtz operator for a linear array of point sources using the ewald method the ewald method is applied to accelerate the evaluation of the green s function gf of an infinite equispaced linear array of point sources with linear phasing only a few terms are needed to evaluate ewald sums which are cast in terms of error functions and exponential integrals to high accuracy it is shown analytically that the choice of the standard optimal ewald splitting parameter e0 causes overflow errors at high frequencies period large compared to the wavelength and convergence rates are analyzed a recipe for selecting the ewald splitting parameter is provided

inner functions on the bidisk and associated hilbert spaces abstract matrix valued inner functions on the bidisk have a number of natural subspaces of the hardy space on the torus associated to them we study their relationship to agler decompositions regularity up to the boundary and restriction maps into one variable spaces we give a complete description of the important spaces associated to matrix rational inner functions the dimension of some of these spaces can be computed in a straightforward way and this ends up having an application to the study of three variable rational inner functions examples are included to highlight the differences between the scalar and matrix cases

rapidly convergent representations for 2d and 3d green s functions for a linear periodic array of dipole sources hybrid spectral spatial representations are introduced to rapidly calculate periodic scalar and dyadic green s functions of the helmholtz equation for 2d and 3d configurations with a 1d linear periodicity the presented schemes work seamlessly for any observation location near the array and for any practical array periodicities including electrically small and large periodicities the representations are based on the expansion of the periodic green s functions in terms of the continuous spectral integrals over the transverse to the array spectral parameters to achieve high convergence and numerical efficiency the introduced integral representations are cast in a hybrid form in terms of i a small number of contributions due to sources located around the unit cell of interest ii a small number of symmetric combinations of the floquet modes and iii an integral evaluated along the steepest descent path sdp the sdp integral is regularized by extracting the singular behavior near the saddle point of the integrand and integrating the extracted components in closed form efficient quadrature rules are established to evaluate this integral using a small number of quadrature nodes with arbitrary small error for a wide range of structure parameters strengths of the introduced approach are demonstrated via extensive numerical examples

series representation of green dyadics for layered media using pmls the green dyadics for closed layered media i e layered media bounded by a perfectly conducting plate at the bottom and top of the structure can be expanded in a discrete surface wave series for open layered media with semi infinite layers at the top and or bottom of the structure the discrete series needs to be complemented by a branch cut integral of space waves in this paper we present a technique to circumvent this branch cut integral by truncating the semi infinite layers with a perfectly matched layer pml that is backed by a perfect electric conductor pec it is demonstrated that in this way it is possible to obtain an accurate series or closed form representation for the green dyadic of the open layered medium the series allows a very efficient calculation and storage of the green dyadic if it is needed for multiple observation and or excitation points very close to the source the series loses efficiency it is shown that the determination of the surface waves in the pml truncated layered medium has the same complexity as the determination of the surface waves in a pec truncated layered medium without a pml

a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

a fast full wave analysis of scattering and radiation from large finite arrays of microstrip antennas a fast full wave analysis technique that can be used to analyze the scattering and radiation from large finite arrays of microstrip antennas is presented the technique discretizes the mixed potential integral equation mpie in the spatial domain by means of a full wave discrete complex image method the del operators on the green s functions are transferred from the singular kernel to the expansion and testing functions the resultant system of equations is solved using the biconjugate gradient bcg method in which the matrix vector product is evaluated efficiently using the fast fourier transform fft this results in an efficient and accurate computation of the scattering and radiation from finite arrays of microstrip antennas several numerical results are presented demonstrating the accuracy efficiency and capability of this technique

an algorithm for the numerical evaluation of the associated legendre functions that runs in time independent of degree and order abstract we describe a method for the numerical evaluation of normalized versions of the associated legendre functions p and q of degrees 0 1 000 000 and orders for arguments in the interval 1 1 our algorithm which runs in time independent of and is based on the fact that while the associated legendre functions themselves are extremely expensive to represent via polynomial expansions the logarithms of certain solutions of the differential equation defining them are not we exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated legendre differential equation and representing them via piecewise trivariate chebyshev expansions these precomputed expansions which allow for the rapid evaluation of the associated legendre functions over a large swath of parameter domain mentioned above are supplemented with asymptotic and series expansions in order to cover it entirely the results of numerical experiments demonstrating the efficacy of our approach are presented and our code for evaluating the associated legendre functions is publicly available

accelerating x ray data collection using pyramid beam ray casting geometries image reconstruction from its projections is a necessity in many applications such as medical ct security inspection and others this paper extends the 2 d fan beam method in to 3 d the algorithm called pyramid beam pb is based upon the parallel reconstruction algorithm in it allows fast capturing of the scanned data and in 3 d the reconstructions are based upon the discrete x ray transform the pb geometries are reordered to fit parallel projection geometry the underlying idea is to use the algorithm in by porting the proposed pb geometries to fit the algorithm in the complexity of the algorithm is comparable with the 3 d fft the results show excellent reconstruction qualities while being simple for practical use

julia theory for slice regular functions the theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and schur analysis however much less is known about their boundary behaviors in this paper we initiate the study of julia theory for slice regular functions more precisely we establish the quaternionic versions of the julia lemma the julia caratheodory theorem the boundary schwarz lemma the hopf lemma and the burns krantz rigidity theorem for slice regular self mappings of the open unit ball b h and of the right half space h we provide an explicit example to show that in quaternionic hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real in contrast to the complex version our result implies that the commonly believed version of the hopf lemma turns out to be totally wrong this new quaternionic version of the hopf lemma also improves osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis since the result involves the lie brackets which reflect the non commutative feature of quaternions

error control in the perfectly matched layer based multilevel fast multipole algorithm the development of efficient algorithms to analyze complex electromagnetic structures is of topical interest application of these algorithms in commercial solvers requires rigorous error controllability in this paper we focus on the perfectly matched layer based multilevel fast multipole algorithm pml mlfma a dedicated technique constructed to efficiently analyze large planar structures more specifically the crux of the algorithm viz the pertinent layered medium green functions is under investigation therefore particular attention is paid to the plane wave decomposition for 2 d homogeneous space green functions in very lossy media as needed in the pml mlfma the result of the investigations is twofold first upper bounds expressing the required number of samples in the plane wave decomposition as a function of a preset accuracy are rigorously derived these formulas can be used in 2 d homogeneous lossy media mlfmas second a more heuristic approach to control the error of the pml mlfma s green functions is presented the theory is verified by means of several numerical experiments

line detection in images through regularized hough transform the problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing traditionally the hough transform a special case of the radon transform has been widely used to solve this problem for binary images in this paper we pose the problem of detecting straight lines in gray scale images as an inverse problem our formulation is based on use of the inverse radon operator which relates the parameters determining the location and orientation of the lines in the image to the noisy input image the advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the hough based line detector through the incorporation of prior information in the form of regularization we discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications finally we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines this interpretation relates the problem of hough based line finding to the body of work on adaptive signal representation

derivatives of rational inner functions geometry of singularities and integrability at the boundary we analyze the singularities of rational inner functions rifs on the unit bidisk and study both when these functions belong to dirichlet type spaces and when their partial derivatives belong to hardy spaces we characterize derivative hp membership purely in terms of contact order a measure of the rate at which the zero set of an rif approaches the distinguished boundary of the bidisk we also show that derivatives of rifs with singularities fail to be in hp for p 32 and that higher non tangential regularity of an rif paradoxically reduces the hp integrability of its derivative we derive inclusion results for dirichlet type spaces from derivative inclusion for hp using agler decompositions and local dirichlet integrals we further prove that a restricted class of rifs fails to belong to the unweighted dirichlet space

fast calculation of multiple line integrals a line integral is defined as the integral of two dimensional data along a one dimensional straight line of given length and orientation line integrals are used in various forms of edge and line detectors in images and in the computation of the radon transform we present a recursive algorithm which enables approximation of discretized line integrals at all lengths orientations and locations to within a prescribed error bound in at most o n log n log log n operations where n is the number of data points furthermore for most applications in particular where even small amounts of noise are present in the data all of these integrals can be computed to the desired accuracy in about 24n log n operations

fast inversion of the radon transform using log polar coordinates and partial back projections in this paper a novel filtered back projection algorithm for inversion of a discretized radon transform is presented it makes use of invariance properties possessed by both the radon transform and its dual by switching to log polar coordinates both operators can be expressed in a displacement invariant manner explicit expressions for the corresponding transfer functions are calculated furthermore by dividing the back projection into several partial back projections inversion can be performed by means of finite convolutions and hence implemented by an fft algorithm in this way a fast and accurate reconstruction method is obtained

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

frequency domain green s function for a planar periodic semi infinite phased array i truncated floquet wave formulation this two part sequence deals with the derivation and physical interpretation of a uniform high frequency solution for the field radiated at finite distance by a planar semi infinite phased array of parallel elementary electric dipoles the field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent floquet waves fws together with corresponding diffracted rays which arise from scattering of the fw at the edge of the array the formal aspects of the solution are treated in the present paper they involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent different species of spectral poles define the various species of propagating and evanescent fw their interception by the steepest descent path sdp determines the variety of shadow boundaries for the edge truncated fw the uniform asymptotic reduction of the sdp integrals performed by the van der waerden 1951 procedure and yielding a variety of edge diffracted fields completes the formal treatment

a fast digital radon transform an efficient means for evaluating the hough transform a fast digital radon transform based on recursively defined digital straight lines is described which has the sequential complexity of n 2 log n additions for an n x n image this transform can be used to evaluate the hough transform to detect straight lines in a digital image whilst a parallel implementation of the hough transform algorithm is difficult because of global memory access requirements the fast digital radon transform is vectorizable and therefore well suited for parallel computation the structure of the fast algorithm is shown to be quite similar to the fft algorithm for decimation in frequency it is demonstrated that even for sequential computation the fast radon transform is an attractive alternative to the classical hough transform algorithm

a framework for discrete integral transformations ii the 2d discrete radon transform although naturally at the heart of many fundamental physical computations and potentially useful in many important image processing tasks the radon transform lacks a coherent discrete definition for two dimensional 2d discrete images which is algebraically exact invertible and rapidly computable we define a notion of 2d discrete radon transforms for 2d discrete images which is based on summation along lines of absolute slope less than 1 values at nongrid locations are defined using trigonometric interpolation on a zero padded grid our definition is shown to be geometrically faithful the summation avoids wrap around effects our proposal uses a special collection of lines in mathbb r 2 for which the transform is rapidly computable and invertible we describe a fast algorithm using o n log n operations where n n 2 is the number of pixels in the image the fast algorithm exploits a discrete projection slice theorem which associates the discrete radon transform with the pseudopolar fourier transform a averbuch et al siam j sci comput 30 2008 pp 764 784 our definition for discrete images converges to a natural continuous counterpart with increasing refinement

fast tomographic reconstruction via rotation based hierarchical backprojection we introduce a family of fast algorithms for tomographic backprojection in the parallel beam geometry the algorithms reduce the computational cost of backprojecting p projections onto an n times n pixel image from the conventional o n 2 p to o n 2 log p the new algorithms aggregate the projections in a hierarchical structure with images in the hierarchy formed by the rotation and addition of other images made up of fewer projections while these algorithms are related to existing fast algorithms this work places them within the signal processing framework providing a systematic means to optimize and adjust the trade off between computational cost and accuracy rotations are performed separably in order that higher order interpolators may be used with low computational cost the same ideas are applied to create a tomographic projection algorithm which computes projections of an n times n pixel image onto p view angles at a cost of o n 2 log p

a fast multilevel domain decomposition algorithm for radar imaging a novel algorithm for radar imaging is presented the method comprises two steps first a decomposition of the radar data domain into subdomains and computation of pertinent low resolution images second interpolation phase correction and aggregation of the low resolution images into the final high resolution one a multilevel algorithm is formulated via a recursive application of the domain decomposition and image aggregation steps the computational cost of the proposed algorithm is comparable to that of the fast fourier transform fft based techniques while it appears to be considerably more flexible than the latter

nevanlinna representations in several variables abstract we generalize to several variables the classical theorem of nevanlinna that characterizes the cauchy transforms of positive measures on the real line we show that for the loewner class a large class of analytic functions that have non negative imaginary part on the upper polyhalf plane there are representation formulae in terms of densely defined self adjoint operators on a hilbert space we find four different representation formulae and we show that every function in the loewner class has one of the four representations corresponding precisely to four different growth conditions at infinity

shear based fast hierarchical backprojection for parallel beam tomography we introduce a family of fast algorithms for 2 d parallel beam tomographic backprojection they aggregate the projections in a hierarchical structure involving the shearing and addition of sparsely sampled images the algorithms achieve a computational cost of o n2 logp when backprojecting an ntimesn pixel image from p projections the algorithms provide a systematic means guided by a fourier domain interpretation to adjust and optimize the tradeoff between computational cost and accuracy in an example with n 512 and p 1458 the algorithms provide high accuracy with more than an order of magnitude reduction in operation counts

realization and interpolation for schur agler class functions on domains with matrix polynomial defining function in cn abstract we consider a bitangential interpolation problem for operator valued functions defined on a general class of domains in c n including as particular cases cartan domains of types i iii which satisfy a type of von neumann inequality associated with the domain we show that any such function has a realization in terms of a unitary colligation and the defining polynomial for the domain we show how the solution of various classes of bitangential interpolation problems for this class of functions corresponds to a unitary extension of a particular partially defined isometry uniquely specified by the interpolation data criteria for existence of solutions are given 1 in terms of positivity of a certain kernel completely determined by the data or more generally 2 by the existence of a positive kernel solution of a certain generalized stein equation completely determined by the data

comments and corrections on the use of polar sampling theorems in ct in two recent papers an exact polar interpolation formula was used to reconstruct computer tomography ct imagery by a procedure known as direct fourier transform inversion the resulting imagery compared favorably to ct imagery reconstructed by filtered convolution back projection strictly speaking however the interpolation formula was inappropriately used since it is valid only for an odd number of azimuthal samples while in ct one uses an even number of samples when the number of samples n is large say 200 as in ct the error is not noticeable and the appropriateness of the formula has no practical significance however when n is small a large error can result we derive in this paper exact azimuthal interpolation formulas for n even and arbitrary n

a fast multipole method for layered media based on the application of perfectly matched layers the 2 d case an efficient fast multipole method fmm formalism to model scattering from two dimensional 2 d microstrip structures is presented the technique relies on a mixed potential integral equation mpie formulation and a series expression for the green functions based on the use of perfectly matched layers pml in this way a new fmm algorithm is developed to evaluate matrix vector multiplications arising in the iterative solution of the scattering problem novel iteration schemes have been implemented and a computational complexity of order o n is achieved the theory is validated by means of several illustrative numerical examples this paper aims at elucidating the pml fmm mpie concept and can be seen as a first step toward a pml based multilevel fast multipole algorithm mlfma for 3 d microstrip structures embedded in layered media

a caratheodory theorem for the bidisk via hilbert space methods if is an analytic function bounded by 1 on the bidisk mathbb d 2 and tau in partial mathbb d 2 is a point at which has an angular gradient nabla varphi tau then nabla varphi lambda to nabla varphi tau as nontangentially in mathbb d 2 this is an analog for the bidisk of a classical theorem of caratheodory for the disk for as above if tau in partial mathbb d 2 is such that the lim inf of 1 varphi lambda 1 lambda as is finite then the directional derivative d exists for all appropriate directions delta in mathbb c 2 moreover one can associate with and an analytic function h in the pick class such that the value of the directional derivative can be expressed in terms of h

integral equation solution for analyzing scattering from one dimensional periodic coated strips a set of integral equations based on the surface surface formulation are developed for analyzing electromagnetic scattering by one dimensional periodic structures to compare the accuracy efficiency and robustness of the formulation the electric field integral equation efie magnetic field integral equation mfie and combined field integral equation cfie are developed for analyzing the same structure for different excitations due to the periodicity of the structure the integral equations are formulated in the spectral domain using the fourier transform of the integrodifferential operators the generalized biconjugate gradient fast fourier transform method with subdomain basis functions is used to solve the matrix equation

authors reply to comments on direct fourier reconstruction in computer tomography in their comments fan and sanz discuss an interpolation formula used by us in our papers on direct fourier transform tomography 1 2 they raise questions relating to 1 the relation between our work and others and 2 the optimality of the interpolation procedure in direct fourier reconstruction in ct we show that 1 most of their points are irrelevant and 2 other points follow from an error which was already corrected by the authors in 3 prior to the publication of the comments

o n sup 2 log sub 2 n filtered backprojection reconstruction algorithm for tomography we present a new fast reconstruction algorithm for parallel beam tomography the new algorithm is an accelerated version of the standard filtered backprojection fbp reconstruction and uses a hierarchical decomposition of the backprojection operation to reduce the computational cost from o n sup 3 to o n sup 2 log sub 2 n we discuss the choice of the various parameters that affect the behavior of the algorithm and present numerical studies that demonstrate the cost versus distortion tradeoff comparisons with fourier reconstruction algorithms and a multilevel inversion algorithm by brandt et al both of which also have o n sup 2 log sub 2 n cost suggest that the proposed hierarchical scheme has a superior cost versus distortion performance it offers rms reconstruction errors comparable to the fbp with considerable speedup for an example with a 512 spl times 512 pixel image and 1024 views the speedup achieved with a particular implementation is over 40 fold with reconstructions visually indistinguishable from the fbp

scattering and radiation from by 1 d periodic metallizations residing in layered media an efficient technique is proposed to compute the scattering or radiation from by 1 d periodic structures residing in a layered background medium the technique is based on a mixed potential integral equation mpie combined with the method of moments mom solving for the unknown current density flowing within a unit cell of the periodic structure the formalism requires the knowledge of the pertinent layered medium green s functions with 1 d periodicity here these green s functions are derived in closed form by invoking the perfectly matched layer pml paradigm the stationary phase method is applied to determine the far field of the infinite periodic structure leading to a series of floquet modes in addition approximating this series leads to an efficient technique to estimate the scattering or radiation from by large but finite periodic structures with a 1 d periodic character the theory is illustrated and validated by means of various examples stemming from scattering and radiation applications from by antenna arrays residing on microstrip substrates the efficiency of the method is also demonstrated

properties of beurling type submodules via agler decompositions abstract in this paper we study operator theoretic properties of the compressed shift operators s z 1 and s z 2 on complements of submodules of the hardy space over the bidisk h 2 d 2 specifically we study beurling type submodules namely submodules of the form h 2 d 2 for inner using properties of agler decompositions of to deduce properties of s z 1 and s z 2 on model spaces h 2 d 2 h 2 d 2 results include characterizations in terms of of when a commutator s z j s z j has rank n and when subspaces associated to agler decompositions are reducing for s z 1 and s z 2 we include several open questions

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

operator monotone functions and lowner functions of several variables we prove generalizations of lowner s results on matrix monotone functions to several variables we give a characterization of when a function of d variables is locally monotone on d tuples of commuting self adjoint n by n matrices we prove a generalization to several variables of nevanlinna s theorem describing analytic functions that map the upper half plane to itself and satisfy a growth condition we use this to characterize all rational functions of two variables that are operator monotone

comparative analysis of acceleration techniques for 2 d and 3 d green s functions in periodic structures along one and two directions the problem of accelerating the calculation of the periodic green s function is addressed here for both 3 d and 2 d free space configurations in the 3 d case periodicity is considered both along one axis and along two generally skew axes a comprehensive review of the existing methods is first presented and some extensions are developed the possibility of treating the case of complex phase shifts between unit cells necessary for the study of complex modes in periodic structures is also investigated comparisons among the various acceleration methods are performed thus providing fundamental information on their actual efficiency in typical problems

a multilevel formulation of the method of moments em scattering the modeling of electromagnetic scattering and radiation problems using the method of moments mom is limited to resonant frequencies because of the extensive computational requirements of solving large matrix equations in this study a multilevel formulation of mom which allows substantial computational savings and thus extends the application of mom to higher frequencies is presented using a hierarchy of discretization levels the multilevel technique extracts different modal components of the solution by focusing on a specific portion of the spectrum of the solution at a given level the fundamental features of this process for the mom solution of the electric field integral equation efie are developed and implemented this multilevel mom allows the rapid evaluation of the current distributions on a variety of 2 d scatterers with thousands of unknowns in fewer than ten cycles and in fractions of the normal cpu times the method is stable fast suitable for multiple excitations and adaptable as a solve module for almost any mom code

on the degrees of freedom of scattered fields starting from the observation that fields differing less than a prescribed error cannot be resolved as distinct entities the degrees of freedom of the scattered field are introduced and then computed the degrees of freedom are shown to be practically equal to the nyquist number appropriate to the effective spatial bandwidth of the scattered field and to the extension of the observation domain accordingly a finite number of elements of information can be used to determine the scattered field to a prescribed approximation error it is also shown that the field representation can be made in terms of field values and simple sampling functions provided that a marginal increase in the approximation error is tolerated the results not only completely justify the use of sampling interpolation for representing scattering fields but also demonstrate that such representation is practically an optimal one an algorithm for the reconstruction of scattered fields given the maximum allowed error is then produced

fast antenna diagnosis algorithm using oblate spheroidal nonuniform grids a fast algorithm for the diagnosis of large antennas is developed the algorithm is based on the rayleigh sommerfeld rs backpropagation method accelerated by a multilevel nonuniform grid algorithm comparison between the field distributions reconstructed from measurements and the desired ones can be used for the localization of antenna defects and geometrical distortions the optimal performance is obtained by using oblate spheroidal grids which are tailored to both the measurement and antenna surfaces further acceleration can be achieved for particular cases via optimal sampling on the antenna surface the algorithm s performance is demonstrated for the example of a parabolic reflector with a number of surface distortions and compared to that achieved with spherical grids the accuracy and error controllability are studied in comparison to the direct rs integral computation

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

discrete radon transform has an exact fast inverse and generalizes to operations other than sums along lines gotz druckmuller and independently brady have defined a discrete radon transform drt that sums an image9s pixel values along a set of aptly chosen discrete lines complete in slope and intercept the transform is fast o n 2 log n for an n n image it uses only addition not multiplication or interpolation and it admits a fast exact algorithm for the adjoint operation namely backprojection this paper shows that the transform additionally has a fast exact although iterative inverse the inverse reproduces to machine accuracy the pixel by pixel values of the original image from its drt without artifacts or a finite point spread function fourier or fast fourier transform methods are not used the inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise also introduced are generalizations of the drt that combine pixel values along lines by operations other than addition for example there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal to noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude

a non commutative julia inequality we prove a julia inequality for bounded non commutative functions on polynomial polyhedra we use this to deduce a julia inequality for holomorphicfunctions on classical domains in mathbb c d we look at differentiability at a boundary point for functions that have a certain regularity there

regularization of mixed potential layered media green s functions for efficient interpolation procedures in planar periodic structures the problem of improving the computational efficiency in the numerical analysis of planar periodic structures is investigated here using the mixed potential integral equation mpie approach a new regularization of the periodic green s functions pgfs that are involved in the analysis of multilayered structures is introduced based on the effective medium concept this regularization involves extracting the singularities of the pgfs up to second order terms the resulting regularized pgf is very smooth and amenable to interpolation thus optimized interpolation procedures for the pgfs can be applied resulting in a considerable reduction of computation time without any significant effect on the accuracy another benefit of the regularization is that it significantly enhances the convergence of the series for both the vector and scalar potential pgfs the theoretical formulation is fully validated with various numerical results for both two dimensional 2 d and one dimensional 1 d layered media periodic structures

a high performance upgrade of the perfectly matched layer multilevel fast multipole algorithm for large planar microwave structures an improvement of the perfectly matched layer multilevel fast multipole algorithm pml mlfma for simulating large planar microwave structures is presented by exploiting the low rank property of the pml mlfma multimodal aggregation and disaggregation matrices considerable reductions in memory usage and computation time are obtained the method has been extensively validated demonstrating complete error controllability when simulating large planar microwave structures reductions in memory requirements and cpu time of more than 60 and 40 have been achieved

accelerated computation of the free space green s function gradient of infinite phased arrays of dipoles in this communication we provide an efficient algorithm for the evaluation of the semi infinite array green s function sagf for a semi infinite planar periodic phased array of dipoles in free space for observation points not too far from the array plane the algorithm uses a hybrid spectral spatial representation of the green s function accelerated with the levin t method that we show to be faster than the shanks method for observation points sufficiently far away from the array plane we show that the sagf is efficiently evaluated by using asymptotic field expressions asymptotics is also used to explain the loss of accuracy of the levin t accelerator in certain regions and a correction procedure is proposed to overcome this problem

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

integrability and regularity of rational functions motivated by recent work in the mathematics and engineering literature we study integrability and non tangential regularity on the two torus for rational functions that are holomorphic on the bidisk one way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable a concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree the dimension count is accomplished by constructing a natural pair of commuting contractions on a finite dimensional hilbert space and studying their joint generalized eigenspaces non tangential regularity of rational functions on the polydisk is also studied one result states that rational inner functions on the polydisk have non tangential limits at every point of the n torus an algebraic characterization of higher non tangential regularity is given we also make some connections with the earlier material and prove that rational functions on the bidisk which are square integrable on the two torus are non tangentially bounded at every point several examples are provided date september 20 2015 2010 mathematics subject classification primary 26c secondary 47a57 46c05 32a40 30c15

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

quasi periodic green s functions of the helmholtz and laplace equations a classical problem of free space green s function g0 representations of the helmholtz equation is studied in various quasi periodic cases i e when an underlying periodicity is imposed in less dimensions than is the dimension of an embedding space exponentially convergent series for the free space quasi periodic g0 and for the expansion coefficients dl of g0 in the basis of regular cylindrical in two dimensions and spherical in three dimension 3d waves or lattice sums are reviewed and new results for the case of a one dimensional 1d periodicity in 3d are derived from a mathematical point of view a derivation of exponentially convergent representations for schlomilch series of cylindrical and spherical hankel functions of any integer order is accomplished exponentially convergent series for g0 and lattice sums dl hold for any value of the bloch momentum and allow g0 to be efficiently evaluated also in the periodicity plane the quasi periodic green s functions of the laplace equation are obtained from the corresponding representations of g0 of the helmholtz equation by taking the limit of the wave vector magnitude going to zero the derivation of relevant results in the case of a 1d periodicity in 3d highlights the common part which is universally applicable to any of remaining quasi periodic cases the results obtained can be useful for the numerical solution of boundary integral equations for potential flows in fluid mechanics remote sensing of periodic surfaces periodic gratings and infinite arrays of resonators coupled to a waveguide in many contexts of simulating systems of charged particles in molecular dynamics for the description of quasi periodic arrays of point interactions in quantum mechanics and in various ab initio first principle multiple scattering theories for the analysis of diffraction of classical and quantum waves

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

comments on direct fourier reconstruction in computer tomography in this correspondence several theoretical aspects of direct fourier reconstruction in cat as presented in 1 3 are revisited specifically the following points are discussed 1 a lack of appropriate references to analogous previous work in the theory of the proposed interpolation technique and its application in physics and engineering areas 1 2 some extensions and remarks on well known trigonometric interpolation formulas 1 3 and 3 some ambiguities related to the optimality of the reconstruction technique and the exactness of the interpolation procedure 1

a julia wolff caratheodory theorem for infinitesimal generators in the unit ball we prove a julia wolff carathedory theorem on angular derivatives of infinitesimal generators of one parameter semigroups of holomorphic self maps of the unit ball b n c n starting from results recently obtained by bracci and shoikhet

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

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

line detection as an inverse problem application to lung ultrasound imaging this paper presents a novel method for line restoration in speckle images we address this as a sparse estimation problem using both convex and non convex optimization techniques based on the radon transform and sparsity regularization this breaks into subproblems which are solved using the alternating direction method of multipliers thereby achieving line detection and deconvolution simultaneously we include an additional deblurring step in the radon domain via a total variation blind deconvolution to enhance line visualization and to improve line recognition we evaluate our approach on a real clinical application the identification of b lines in lung ultrasound images thus an automatic b line identification method is proposed using a simple local maxima technique in the radon transform domain associated with known clinical definitions of line artefacts using all initially detected lines as a starting point our approach then differentiates between b lines and other lines of no clinical significance including z lines and a lines we evaluated our techniques using as ground truth lines identified visually by clinical experts the proposed approach achieves the best b line detection performance as measured by the f score when a non convex ell text p regularization is employed for both line detection and deconvolution the f scores as well as the receiver operating characteristic roc curves show that the proposed approach outperforms the state of the art methods with improvements in b line detection performance of 54 40 and 33 for text f 0 5 text f 1 and text f 2 respectively and of 24 based on roc curve evaluations

complex multipole beam approach to electromagnetic scattering problems a novel approach to reducing the matrix size associated with the method of moments mom solution of the problem of electromagnetic scattering from arbitrary shaped closed bodies is presented the key step in this approach is to represent the scattered field in terms of a series of beams produced by multipole sources located in a complex space on the scatterer boundary the fields generated by these multipole sources resemble the gabor basis functions by utilizing the properties of the gabor series guidelines for selecting the orders as well as locations of the multipole sources are developed it is shown that the present approach not only reduces the number of unknowns but also generates a generalized impedance matrix with a banded structure and a low condition number the accuracy of the proposed method is verified by comparing the numerical results with those derived by using the method of moments

time domain weyl s identity and the causality trick based formulation of the time domain periodic green s function time domain floquet modes for analyzing scattering radiation from periodically arranged planar scatterers sources have been presented recently in a series of papers by felsen and capolino in these papers they have investigated phenomenological details of transient scattering radiation by such structures typically time domain floquet modes are derived either by fourier transforming their frequency domain counterparts or using spatial synthesis using poisson summation this paper proposes an alternative means to develop similar expressions and is motivated by the possibility of easily extending the method to the analysis of periodic structures in a layered medium as will be shown the time domain weyl s identity can be used to construct the periodic green s function using both homogeneous and inhomogeneous plane waves we will also derive expressions using the causality trick this technique involves using time symmetric non causal signals however it yields much simpler expressions that involve only homogeneous time domain plane waves and the results are identical to those obtained using the time domain weyl s identity for the time region of interest finally we will use the whittaker formulation which relies on real signals to derive time domain floquet modes this formulation is applicable for simultaneous excitation only

canonical agler decompositions and transfer function realizations a seminal result of agler proves that the natural de branges rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift invariant pieces agler s decomposition is non constructive a problem remedied by work of ball sadosky vinnikov which uses scattering systems to produce agler decompositions through concrete hilbert space geometry this method while constructive so far has not revealed the rich structure shown to be present for special classes of functions inner and rational inner functions in this paper we show that most of the important structure present in these special cases extends to general bounded analytic functions we give characterizations of all agler decompositions we prove the existence of coisometric transfer function realizations with natural state spaces and we characterize when schur functions on the bidisk possess analytic extensions past the boundary in terms of associated hilbert spaces

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

ct reconstruction from parallel and fan beam projections by a 2 d discrete radon transform the discrete radon transform drt was defined by abervuch et al as an analog of the continuous radon transform for discrete data both the drt and its inverse are computable in o n2logn operations for images of size n n in this paper we demonstrate the applicability of the inverse drt for the reconstruction of a 2 d object from its continuous projections the drt and its inverse are shown to model accurately the continuum as the number of samples increases numerical results for the reconstruction from parallel projections are presented we also show that the inverse drt can be used for reconstruction from fan beam projections with equispaced detectors

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions