on the finite escape phenomena for matrix riccati equations we show a necessary and sufficient condition for the solution of riccati differential equations to escape to the infinity in finite time this implies that we obtain a necessary and sufficient condition for the existence of the solution of all time t geq 0 this condition is given as a rank condition of the matrix which we refer to as the escape criterion matrix this matrix consists of the equation matrices and an equilibrium point of riccati differential equation

image representation by one bit fourier phase theory sampling and coherent image model the problem of recovering an image from its fourier transform phase quantized to one bit or equivalently finding the locations of the zero crossings of the real part of the fourier transform is addressed theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image it is then shown that sampling in the frequency domain presents a major obstacle to obtaining good reconstruction results the one bit fourier phase reconstruction problem is then considered when the original image is coherent i e the image phase is random and highly uncorrelated examples are given which demonstrate that the information loss produced by frequency sampling is not as severe as that in the nonclassical problem motivated by digital phase only holograms a known random diffuser is used as the image phase and a well known iterative reconstruction procedure is extended to incorporate the knowledge of the image phase at each stage of the iteration this reconstruction method produces good image quality by using a few iterations unlike its noncoherent counterpart

representation of images through group delay functions in this correspondence we propose a new representation for images through the use of group delay functions we provide algorithms for the computation of the group delay functions and for the recovery of images from them we give several examples to show that most of the perceptually significant information of an image is retained in this representation we derive the minimum phase equivalent images from the fourier transform ft magnitude as well as from the phase through their respective group delay functions we compare these images to those obtained through iterative reconstruction from the ft magnitude and phase respectively

signal representation in the combined phase spatial space reconstruction and criteria for uniqueness motivated by the major roles of the spectral phase in signal structure and by previous results on the importance of localized representation in images we develop conditions for unique representation of a signal by a combination of its spectral fourier phase and spatial samples methods for signal reconstruction from various combinations of spatial and phase information are introduced it is concluded that most of the previous results in the area of phase only representation and reconstruction are special cases of the general approach presented here

computationally attractive reconstruction of bandlimited images from irregular samples an efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed the novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two dimensional 2 d trigonometric polynomials where the matrix is of block toeplitz type with toeplitz blocks this system is solved iteratively by the conjugate gradient cg method we show that the use of so called adaptive weights in the establishment of the block toeplitz matrix can be seen as efficient preconditioning the superiority of the new method over conventional approaches is demonstrated by numerical experiments

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

mathematical considerations for the problem of fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent function from the magnitude of its fourier transform this so called phase retrieval problem will first be posed under its different underlying models we will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued sequence x from the magnitude of the output of a linear distortion hx j j 1 cdots n a number of important results will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval number of feasible solutions stability of the essential

direct phase retrieval a direct noniterative approach to retrieving a multidimensional complex image i e its phase can vary from pixel to pixel from the magnitude of its fourier transform is developed the uniqueness of the reconstruction is shown to be a direct consequence of the existence of zero surfaces or sheets in the multidimensional z transforms of the image the analytic properties of these zero sheets enable one to distinguish the zero sheet belonging to the image from that belonging to the complex conjugate of its reflection in the coordinate origin it is thereby possible to recover the unique most compact image form of the original image two dimensional examples are presented

methods for reconstruction of 2 d sequences from fourier transform magnitude the gerchberg saxton gs algorithm and its generalizations have been the main tools for phase retrieval unfortunately it has been observed that the reconstruction using these algorithms does not always converge to the correct result even if the desired solution satisfies the uniqueness condition in this paper we propose a new deautocorrelation algorithm and a few auxiliary techniques we recommend that a combination of the iterative fourier transform ift algorithm with our new algorithm and techniques can improve the probability of success of phase retrieval a pragmatic procedure is illustrated different reconstruction examples that are difficult to reconstructed using the single ift algorithm are used to show the robustness and effectiveness of the new combination of algorithms if the given fourier modulus data contain no noise it is sometimes possible to get a perfect reconstruction even when the signal to noise ratio snr of the fourier modulus data is only 10 db a meaningful result remains reachable for our examples a concept concerning the intrinsic ambiguity of phase retrieval is suggested we emphasize the necessity of verification of the solution since the available phase retrieval algorithms are incompetent for distinguishing between an intrinsically ambiguous solution and the true solution

optimal reconstruction of images from localized phase the importance of localized phase in signal representation is investigated the convergence rate of the pocs algorithm projection onto convex sets used for image reconstruction from spectral phase is defined and analyzed and the characteristics of images optimally reconstructed from phase only information are presented it is concluded that images of geometric form are most efficiently reconstructed from their spectral phase whereas images of symmetric form have the poorest convergence characteristics the transition between the two extremes is shown to be continuous the results provide a new approach and analysis of the previously reported advantages of the localized phase representation over the global approach and suggest possible compression schemes

fast reconstruction of linearly distorted signals a product expansion of the inverse operator is used to derive a class of iterative algorithms that have a pth order rate of convergence compared to the linear algorithm the number of iterations required to obtain a given reconstruction is reduced from p sup n to n the primary difference between these algorithms and the linear algorithm is that they update both the estimate of x and the operator d rather than the estimate alone several methods that may be used to incorporate linear and nonlinear constraints are also proposed finally some examples of linear inverse problems are presented and the differences in rates of convergence for the linear and the pth order algorithms are compared

signal reconstruction from signed fourier transform magnitude in this paper we show that a one dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed fourier transform magnitude magnitude and 1 bit of phase information in addition we develop a numerical algorithm to reconstruct a one dimensional or multidimensional sequence from its fourier transform magnitude reconstruction examples obtained using this algorithm are also provided

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

psf estimation via gradient domain correlation this paper proposes an efficient method to estimate the point spread function psf of a blurred image using image gradients spatial correlation a patch based image degradation model is proposed for estimating the sample covariance matrix of the gradient domain natural image based on the fact that the gradients of clean natural images are approximately uncorrelated to each other we estimated the autocorrelation function of the psf from the covariance matrix of gradient domain blurred image using the proposed patch based image degradation model the psf is computed using a phase retrieval technique to remove the ambiguity introduced by the absence of the phase experimental results show that the proposed method significantly reduces the computational burden in psf estimation compared with existing methods while giving comparable blurring kernel

an algorithm to find two dimensional signals with specified zero crossings we present an iterative method for finding two dimensional band limited functions with predetermined zero crossings this problem arises when we want to generate a prescribed binary image at the output of a diffraction limited imaging system with high contract recording an important application of this image generation problem is in the design of masks for microlithography after describing the algorithm a proof of its convergence based on the properties of nonexpansive mappings is given the relevance of this algorithm to the image construction problem is then discussed and we conclude by illustrating two patterns generated by our proposed method

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

on global convergence of gradient descent algorithms for generalized phase retrieval problem abstract in this paper we study the generalized phase retrieval problem to recover a signal x c n from the measurements y r a r x 2 r 1 2 m the problem can be reformulated as a least squares minimization problem although the cost function is nonconvex the global convergence of gradient descent algorithms from a random initialization is studied when m is large enough we improve the known result of the local convergence from a spectral initialization when the signal x is real valued we prove that the cost function is local convex near the solution x to accelerate the gradient descent we review and apply several efficient line search methods with exact line search stepsize we also perform a comparative numerical study of the line search methods and the alternative projection method numerical simulations demonstrate the superior ability of lbfgs algorithm than other algorithms

reconstruction of nonperiodic two dimensional signals from zero crossings in this correspondence we present new results on the reconstruction of two dimensional signals from zero crossing or threshold crossing information specifically we develop new theoretical results which state conditions under which two dimensional bandlimited signals are uniquely specified to within a scale factor with this information unlike previous results in this area our new results do not constrain the signals to be periodic or bandpass

significance of group delay functions in signal reconstruction from spectral magnitude or phase in this paper we discuss the problem of signal reconstruction from spectral magnitude or phase using group delay functions we define two separate group delay functions for a signal one is derived from the magnitude and the other from the phase of the fourier transform of the signal the group delay functions offer insight into the problem of signal reconstruction and suggest methods for reconstructing signals from partial information such as spectral magnitude or phase we examine the problem of signal reconstruction from spectral magnitude or phase on the basis of these two group delay functions and derive the conditions for signal reconstruction based on existing iterative and noniterative algorithms for signal reconstruction we propose new algorithms for some special classes of signals the algorithms are illustrated with several examples our study shows that the relative importance of spectral magnitude and phase depends on the nature of signals speech signals are used to illustrate the importance of spectral magnitude and picture signals are used to illustrate the importance of phase in signal reconstruction problems using the group delay functions we explain the convergence behavior of the existing iterative algorithms for signal reconstruction

a new direct algorithm for image reconstruction from fourier transform magnitude in this paper we present a new direct solution to the problem of reconstructing a two dimensional discrete signal of finite support from knowledge of only its fourier transform magnitude and support using the autocorrelation function of the unknown signal zeros are calculated of a polynomial whose coefficients correspond to the unknown image or the image reversed from these zeros a set of linear equations is developed whose solution yields either the original image or the image rotated by 180

novel criteria of uniqueness for signal reconstruction from phase an approach for ascertaining whether a signal is uniquely determined by its fourier transform phase is proposed it is shown that uniqueness corresponds to the nonsingularity of a matrix which can be formed from the finite length real sequence the criterion of uniqueness for reconstructing a one dimensional finite length sequence from its phase is discussed the criterion of uniqueness for reconstructing a multidimensional finite length sequence is presented

a note on iterative fourier transform phase reconstruction from magnitude in this correspondence a well known iterative procedure for fourier transform phase reconstruction from magnitude usually referred to as the gerchberg saxton fienup procedure is revisited the convergence of this algorithm has raised some controversy in the recent literature in this correspondence we first point out a potential source of error in the numerical implementation of the algorithm then we present a conjecture which would explain why this algorithm sometimes fails

on two dimensional hilbert integral equations generalized minimum phase signals and phase retrieval one dimensional 1 d causal signals admit hilbert integral relations between the real and imaginary parts of their fourier spectra for 1 d minimum phase signals the log magnitude and phase spectra also admit such hilbert relations in this paper we extend these results to 2 d signals we first establish the hilbert integral equations for 2 d first quadrant signals for continuous domain 2 d signals we present the partial hilbert transform relations between the real and imaginary parts of the spectrum for the discrete domain counterpart we show that the partial hilbert transform does not suffice which motivates us to introduce the composite hilbert transform considering the problem of phase retrieval we show that in both 1 d 2 d and continuous discrete domains there exists a generalized class of signals which subsumes the well known class of 1 d minimum phase signals with a rational transfer function for which hilbert integral relations exist between the log magnitude and phase spectra we introduce them under the nomenclature of generalized minimum phase signals for this class of signals phase retrieval is possible by using the hilbert transform without the requirement of a rational transfer function furthermore we show that a generalized minimum phase signal admits a stable convolutional inverse which also belongs to the same class simulation results demonstrating accurate reconstruction of 1 d and 2 d generalized minimum phase signals from their magnitude spectra are presented to support the theoretical developments

understanding blind deconvolution algorithms blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown recent algorithms have afforded dramatic progress yet many aspects of the problem remain challenging and hard to understand the goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally we explain the previously reported failure of the naive map approach by demonstrating that it mostly favors no blur explanations we show that using reasonable image priors a naive simulations map estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior on the other hand we show that since the kernel size is often smaller than the image size a map estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur the plethora of recent deconvolution techniques makes an experimental evaluation on ground truth data important as a first step toward this experimental evaluation we have collected blur data with ground truth and compared recent algorithms under equal settings additionally our data demonstrate that the shift invariant blur assumption made by most algorithms is often violated

uber pro p fundamentalgruppen markierter arithmetischer kurven let k be a global field p an odd prime number dierent from char k and s t disjoint finite sets of primes of k let g t k p g k t p k be the galois group of the maximal p extension of k which is unramified outside s and completely split at t we prove the existence of a finite set of primes s0 which can be chosen disjoint from any given setm of dirichlet density zero such that the cohomology of g t s0 k p coincides with the etale cohomology of the associated marked arithmetic curve in particular cd g t s0 k p 2 furthermore we can choose s0 in such a way that k t s s0 p realizes the maximal p extension kp p of the local field kp for all p2 s s0 the cup product h 1 g t s0 k p fp h 1 g t s0 k p fp h 2 g t s0 k p fp is surjective and the decomposition groups of the primes in s establish a free product inside g t s0 k p this generalizes previous work of the author where similar results were shown in the case t under the restrictive assumption p cl k and p 2 k

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

higher massey products in the cohomology of mild pro p groups abstract translating results due to j labute into group cohomological language a schmidt proved that a finitely presented pro p group g is mild and hence of cohomological dimension cd g 2 if h 1 g f p u v as f p vector space and the cup product h 1 g f p h 1 g f p h 2 g f p maps u v surjectively onto h 2 g f p and is identically zero on v v this has led to important results in the study of p extensions of number fields with restricted ramification in particular in the case of tame ramification in this paper we extend labute s theory of mild pro p groups with respect to weighted zassenhaus filtrations and prove a generalization of the above result for higher massey products which allows to construct mild pro p groups with defining relations of arbitrary degree we apply these results to one relator pro p groups and obtain some new evidence of an open question due to serre

signal reconstruction from noisy partial information of its transform the usual assumption that the available partial information is noiseless is replaced by a more realistic statistical model which compensates for the presence of noise the signal reconstruction is thus viewed as a parameter estimation problem for which the em iterative algorithm of a p dempster n m laird and d b rubin j roy stat soc b vol 39 p 1 37 1977 is especially suitable the posterior probability of the signal increases from iteration to iteration till the signal converges to a stationary point of the posterior distribution each iteration involves one transformation followed by an inverse transformation usually discrete fourier transformation dft and inverse dft algorithms for reconstruction of both one and two dimensional signals from their spectral magnitude spectral phase or modified short time fourier transform are typical examples of the proposed scheme

phase retrieval using a window function the problem considered is that of reconstructing a signal from the fourier intensity of the signal and the fourier intensities of two windowed signals one by a window w n and the other by its complementary window 1 w n several conditions are developed under which a signal can be uniquely specified to within several trivial ambiguities such as sign translation and time reversal from the given conditions a possible reconstruction algorithm derived from the gerchberg saxton algorithm is presented

bayesian image reconstruction from partial image and aliased spectral intensity data an image reconstruction problem motivated by x ray fiber diffraction analysis is considered the experimental data are sums of the squares of the amplitudes of particular sets of fourier coefficients of the electron density and a part of the electron density is known the image reconstruction problem is to estimate the unknown part of the electron density the image a bayesian approach is taken in which a prior model for the image is based on the fact that it consists of atoms i e the unknown electron density consists of separated sharp peaks currently used heuristic methods are shown to correspond to certain maximum a posteriori estimates of the fourier coefficients an analytical solution for the bayesian minimum mean square error estimate is derived simulations show that the minimum mean square error estimate gives good results even when there is considerable data loss and out performs the maximum a posteriori estimates

on stability of reconstruction from fourier transform modulus the lower bound on the condition number of reconstruction of two classes of sequences from their fourier transform magnitude ftm is derived the lower bound for one class is shown to be 1 and for the other class to be 1 2 n sup q 2 where q is the dimensionality of the sequence and n is the number of nonzero elements in each dimension stability of reconstruction from the ftm and the space domain phase is discussed it is found that randomizing the space domain phase improves reconstruction robustness experimental results are presented to verify theoretical predictions

partitioning algorithms for 1 d and 2 d discrete phase retrieval problems with disconnected support the discrete phase retrieval problem with disconnected support is to reconstruct a discrete time signal whose support is the union of disjoint intervals from the magnitude of its discrete fourier transform we use the good thomas 1958 fft mapping to transform the one dimensional 1 d version of this problem into a two dimensional 2 d discrete phase retrieval problem we then solve the latter problem by partitioning it into a set of 1 d phase retrieval problems the discrete and modulated radon transforms are used to formulate two coupled 1 d problems the solution to which specifies solutions to the other 1 d problems this effectively partitions the original disconnected support 1 d problem into smaller 1 d problems which may be solved in parallel small amounts of noise in the data can be rejected by using more than two coupled 1 d problems the 2 d phase retrieval problem with disconnected support is also considered numerical examples are provided including comparison with the hybrid i o algorithm

one and two dimensional minimum and nonminimum phase retrieval by solving linear systems of equations the discrete phase retrieval problem is to reconstruct a discrete time signal whose support is known and compact from the magnitude of its discrete fourier transform we formulate the problem as a linear system of equations our methods do not require polynomial rooting tracking zero curves of algebraic functions or any sort of iteration like previous methods our solutions obviate the stagnation problems associated with iterative algorithms and our solutions are computationally simpler and more stable than alternative noniterative algorithms furthermore our methods can explicitly accommodate noisy fourier magnitude information through the use of total least squares type techniques we assume either of the following two types of a priori knowledge of the signal 1 a band of known values which may be zeros or 2 some known values of a subminimum phase signal whose zeros lie inside a disk of radius greater than unity we illustrate our methods with nonminimum phase one dimensional 1 d and two dimensional 2 d signals

analytical construction of a negative semidefinite solution of algebraic riccati equations and its application a sufficient condition for the existence of a symmetric negative semidefinite solution of algebraic matrix riccati equations is given and at the same time the negative semidefinite solution is constructed by an analytical method a few properties of the negative semidefinite solution are also investigated it is shown that the negative semidefinite solution having those properties plays a more important role than a positive semidefinite solution in the existence problem of the solution of the corresponding riccati differential difference equations the results are applied to this existence problem

on the duality of time and frequency domain signal reconstruction from partial information certain results on one dimensional signal representation and reconstruction from partial information in the frequency domain are related to and derived using relevant results obtained in the time or positional information domain in particular we adopt logan s approach and apply his results concerning zero crossing representation of certain bandpass signals we also apply voelcker s results on demodulation of analytic ssb signals by an am detector exploiting the duality of the fourier stieltjes transform and its inverse we rederive and extend to continuous one dimensional signals some results concerning the representation of fourier transformed discrete time finite sequences by partial information such as one bit or complete fourier phase fourier magnitude or signed magnitude the duality and interrelationship of characteristics and constraints attributed to time and frequency signal representation create further possibilities of drawing results and interpretations from one to its counter domain in particular we indicate possible extensions of certain results to problems of n dimensional signals and to issues of stochastic signals we also highlight basic issues concerning the structure of signals and their reconstruction from partial information

the kummerian property and maximal pro p galois groups abstract for a prime number p we give a new restriction on pro p groups g which are realizable as the maximal pro p galois group g f p for a field f containing a root of unity of order p this restriction arises from kummer theory and the structure of the maximal p radical extension of f we study it in the abstract context of pro p groups g with a continuous homomorphism g 1 p z p and characterize it cohomologically and in terms of 1 cocycles on g this is used to produce new examples of pro p groups which do not occur as maximal pro p galois groups of fields as above

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

solving phaselift by low rank riemannian optimization methods for complex semidefinite constraints a framework phaselift was recently proposed to solve the phase retrieval problem in this framework the problem is solved by optimizing a cost function over the set of complex hermitian positive semidefinite matrices this approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite hermitian matrices where the desired minimizers are known to have low rank this paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds it is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally a rank based optimality condition for stationary points is given and optimization algorithms based on state of the art riemannian optimization and dynamically reducing rank are proposed empirical evaluations are performed using the phaselift problem the new approach is shown to be an effective method of phase retrieval with

circular sets of prime numbers and p extensions of the rationals zusammenfassung sei p eine ungerade primzahl und s eine endliche menge von primzahlen kongruent 1 modulo p wir zeigen da die gruppe gs q p von kohomologischer dimension 2 ist wenn das verschlingungsdiagramm zu s und p einer gewissen technischen bedingung genaugt und wir zeigen da gs q p in diesen faallen eine dualitaatsgruppe ist au erdem untersuchen wir das zerlegungsverhalten von primstellen in der erweiterung qs p q und wir setzen die kohomologie von gs q p zur etalkohomolgie des schemas spec z is in beziehung schlie lich berechnen wir den dualisierenden modul abstract let p be an odd prime number and let s be a nite set of prime numbers congruent to 1 modulo p we prove that the group gs q p has cohomological dimension 2 if the linking diagram attached to s and p satis es a certain technical condition and we show that gs q p is a duality group in these cases furthermore we investigate the decomposition behaviour of primes in the extension qs p q and we relate the cohomology of gs q p to the etale cohomology of the scheme spec z i s finally we calculate the dualizing module

multidimensional signal representation by zero crossings and algebraic study multidimensional signal representation by zero crossings arises in many problems in physics and engineering its applications span several areas such as signal modulation computer vision and data compression in this paper mathematical conditions under which signals are uniquely specified in terms of their real zero crossings are presented these results apply to multidimensional signals i e the dimension n geqq 2 modeled by polynomials this model is appropriate for the z transform of discrete signals and can also be used for certain two dimensional images the central theorems shown in this paper establish conditions under which a multivariate polynomial p is uniquely determined from its real zero crossings furthermore the regularity of a real zero of pwill be shown equivalent to the existence of zero crossings this property is interesting since it shows a deep relationship between the theory of polynomials and the representations used in practical applications finally this paper generalizes

on the approximate factorization of 2 d polynomials a method is presented for the approximate factorization of unfactorable bivariate polynomials an efficient iterative algorithm is developed to minimize an appropriate criterion the basic ideas can be easily extended to the case of polynomials in more than two independent variables an example is presented to illustrate the utility of the method

analytical formulae for reconstruction of certain discrete signals from phase level and line crossings we provide simple and explicit formulae for reconstructing any member of a class of discrete time signals from the frequencies at which its fourier phase crosses any specific level of constant phase or a linear phase line with integer slope provided that the number of crossings equals the length of the signal support unlike previous closed form solutions solution of an ill conditioned system of linear equations is not required the associated uniqueness results reduce in special cases to previous results for reconstruction from fourier transform real and imaginary part zero crossings

on the computational model of a kind of deconvolution problem it is known that discretization of a continuous deconvolution problem can alleviate the ill posedness of the problem the currently used circulant matrix model however does not play such a role moreover the approximation of deconvolution problems by circulant matrix model is rational only if the size of the kernel function is very small we propose an aperiodic model of deconvolution for discrete and finite deconvolution problems the new model is an exact one in the general case the new model can lead to a nonsingular system of equations that has a lower condition number than the circulant one and the related computations in the deconvolution can be done efficiently by means of the dft technique as in the ease for circulant matrices the rationality of the new model holds without regard to the size of the kernel and the image the use of the aperiodic model is illustrated by gradient based algorithms

polynomial system of equations and its applications to the study of the effect of noise on multidimensional fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent signal from the magnitude of its fourier transform we will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued signal x from the magnitude of the output of a linear distortion hx j j 1 n an important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval conditioning of the problem and stability of the essentially unique solution will be addressed

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

phase reconstruction from magnitude of band limited multidimensional signals abstract in this paper the problem of phase reconstruction from magnitude of multidimensional band limited functions is considered it is shown that any irreducible band limited function f z1 zn zi c i 1 n is uniquely determined from the magnitude of f x1 xn f x1 xn xi r i 1 n except for 1 linear shifts i 1z1 n2n i r i 1 n and 2 conjugation f z 1 z n

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

image representation by complex cell responses we present an analysis of the representation of images as the magnitudes of their transform with complex valued gabor wavelets such a representation is a model for complex cells in the early stage of visual processing and of high technical usefulness for image understanding because it makes the representation insensitive to small local shifts we show that if the images are band limited and of zero mean then reconstruction from the magnitudes is unique up to the sign for almost all images

the self duality of discrete short time fourier transform and its applications the self duality of short time fourier transform stft is an elegant property and is useful in shedding light on the construction of stft and its resolution capability in this paper the discrete version of self duality is studied and the property is interpreted in the context of resolution capabilities of time frequency distributions in addition two applications are provided as showcases of these insights obtained from the interpretation in the first application the problem of stft synthesis is considered and self duality serves as an important indication of whether the synthesis problem at hands is properly formulated in the second application a new kind of high resolution time frequency distribution is constructed based on the understandings obtained by contrasting two of the most popular time frequency analysis tools namely the stft and the wigner distribution

existence theorem for the difference riccati equations to obtain a necessary and sufficient condition for the existence of a solution of discrete riccati equations a new approach is proposed and an interesting relation between the equilibrium points of riccati equations and the existence of the solution is shown the results can be applied to nonsymmetric riccati equations however the results in this paper do not improve our resources for the optimal control problem or filtering problem though roughly speaking the known results on those problems are included

excitation retrieval of microwave linear arrays from phaseless far field data a methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed the approach combines tools from convex optimization to solve the phase retrieval problem and a simple measurement procedure to mitigate the non uniqueness of the solution numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed they show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem in addition the proposed approach that only calls for readily available routines is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases

on the uniqueness of the phase retrieval problem from far field amplitude only data the phase retrieval problem has gained considerable attention since the wave phase is lost or impractical to measure in several applications in electromagnetic engineering quite a few approaches have been proposed recently for the extraction of phase from amplitude only data however few of them investigate the uniqueness of the problem prior to the solution in this communication the problem of the uniqueness of the phase retrieval problem from far field amplitude only data is examined in detail it is shown that uniqueness is guaranteed if two experiments can be performed that differ by a quadratic term in the phase conversely experiments creating only linear phase differences in the data do not guarantee uniqueness this result is demonstrated by considering the problems of a scattering target illuminated in two experiments differing by a quadratic phase shift corresponding to two different values of defocusing of the incident wave

mild pro 2 groups and 2 extensions of q with restricted ramification abstract using the mixed lie algebras of lazard we extend the results of the first author on mild groups to the case p 2 in particular we show that for any finite set s 0 of odd rational primes we can find a finite set s of odd rational primes containing s 0 such that the galois group of the maximal 2 extension of q unramified outside s is mild we thus produce a projective system of such galois groups which converge to the maximal pro 2 quotient of the absolute galois group of q unramified at 2 and our results also allow results of alexander schmidt on pro p fundamental groups of marked arithmetic curves to be extended to the case p 2 over a global field which is either a function field of characteristic 2 or a totally imaginary number field

signal reconstruction from fourier transform sign information in this paper we present new results on the reconstruction of signals from only the sign of the real part of the fourier transform specifically we develop new theoretical results which state conditions under which two dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals furthermore we apply this result to the problem of reconstructing two dimensional signals from their zero crossings we also present two algorithms for reconstructing a signal from sign information in either the time or frequency domain

optical wavefront reconstruction theory and numerical methods optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in nasa s hubble space telescope hst nasa s success with reconstruction algorithms on the hst has led to an effort to develop software that can aid and in some cases replace complicated expensive and error prone hardware among the many applications is hst s replacement the next generation space telescope ngst indent this work details the theory of optical wavefront reconstruction reviews some numerical methods for this problem and presents a novel numerical technique that we call extended least squares we compare the performance of these numerical methods for potential inclusion in prototype ngst optical wavefront reconstruction software we begin with a tutorial on rayleigh sommerfeld diffraction theory