dct domain watermarking techniques for still images detector performance analysis and a new structure a spread spectrum like discrete cosine transform dct domain watermarking technique for copyright protection of still digital images is analyzed the dct is applied in blocks of 8 spl times 8 pixels as in the jpeg algorithm the watermark can encode information to track illegal misuses for flexibility purposes the original image is not necessary during the ownership verification process so it must be modeled by noise two tests are involved in the ownership verification stage watermark decoding in which the message carried by the watermark is extracted and watermark detection which decides whether a given image contains a watermark generated with a certain key we apply generalized gaussian distributions to statistically model the dct coefficients of the original image and show how the resulting detector structures lead to considerable improvements in performance with respect to the correlation receiver which has been widely considered in the literature and makes use of the gaussian noise assumption as a result of our work analytical expressions for performance measures such as the probability of errors in watermark decoding and the probabilities of false alarms and of detection in watermark detection are derived and contrasted with experimental results

locally optimal detection of image watermarks in the wavelet domain using bessel k form distribution a uniformly most powerful watermark detector which applies the bessel k form bkf probability density function to model the noise distribution was proposed by bian and liang in this paper we derive a locally optimum lo detector using the same noise model since the literature lacks thorough discussion on the performance of the bkf lo nonlinearities the performance of the proposed detector is discussed in detail first we prove that the test statistic of the proposed detector is asymptotically gaussian and evaluate the actual performance of the proposed detector using the receiver operating characteristic roc then the large sample performance of the proposed detector is evaluated using asymptotic relative efficiency are and maximum are the experimental results show that the proposed detector has a good performance with or without attacks in terms of its roc curves particularly when the watermark is weak therefore the proposed method is suitable for wavelet domain watermark detection particularly when the watermark is weak

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

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

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

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

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

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

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

robust optimum detection of transform domain multiplicative watermarks digital watermarking is an emerging technique to protect data security and intellectual property right identification or verification of watermarking patterns can be achieved by detecting watermarks in received signals however one of the biggest challenges in watermarking detection is that the strengths of the watermark signals will change after being distorted by an attacker in a watermarking channel meanwhile the embedding strengths may be adapted to the original signals which are unknown at the receiver end further the original signals are often highly non gaussian although some work has been done on optimum detection of watermarks the uncertainty of watermark signal strengths and real statistical behavior of multimedia contents have not been taken into account simultaneously much more study is needed to enhance the performance of watermarking systems since multiplicative watermarks are robust and well suited for copyright protection this paper presents our investigation on robust optimum detection of multiplicative watermarks for sub band transformed domains such as the discrete cosine transform dct discrete wavelet transform dwt and pyramid transform a class of generalized correlators is constructed based on the generalized gaussian distributions thresholding methods to achieve a given false alarm rate and the performance analyses are provided the square root detector is designed and demonstrated to have near optimal performance for a large set of natural images and can be employed as a universally optimal detector or decoder for images and video the locally most powerful detection method is then extended to dft domain multiplicative watermarking with the magnitudes of coefficients modeled by the weibull distributions another class of detectors is built based on this statistical modeling the robust optimum detection of multiplicative watermarks can be applied to copyright notification enforcement and broadcast monitoring we have applied the robust optimum watermarking detection to combined audio and video watermarking

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

asymptotically optimal detection for additive watermarking in the dct and dwt domains most of the watermarking schemes that have been proposed until now employ a correlation detector matched filter the current paper proposes a new detector scheme that can be applied in the case of additive watermarking in the dct discrete cosine transform or dwt discrete wavelet transform domain certain properties of the probability density function of the coefficients in these domains are exploited thus an asymptotically optimal detector is constructed based on well known results of the detection theory experimental results prove the superiority of the proposed detector over the correlation detector

a globally convergent and consistent method for estimating the shape parameter of a generalized gaussian distribution we propose a novel methodology for estimating the shape parameter of a generalized gaussian distribution ggd this new method is based on a simple estimating equation derived from a convex shape equation the estimating equation is completely independent of gamma and polygamma functions thus no lookup table interpolation or additional subroutine to evaluate these functions are required for real time implementations of the proposed method which is in contrast to all existing methods furthermore we establish that the shape equation has a unique global root on the positive real line and the newton raphson root finding algorithm converges to the unique global root from any starting point in a semi infinite interval thetamin more importantly we show that the sample based shape estimating equation has a unique global root with probability tending to one and the root is consistent for the true shape parameter finally we prove via fixed point arguments that with probability tending to one the newton raphson algorithm converges to the unique global root of the sample shape estimating equation from any starting point in thetamin some numerical experiments are also provided to demonstrate the global convergence and the excellent finite sample performance of the proposed method

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

image representation by sign information image representation by sign data in its most general context is considered for the case when f is a two dimensional signal the authors study conditions under which f is determined by its sign or in other words by its quantization to 1 bit of information this study is carried out in two different directions first theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image an interesting paradigm arising in theoretical computer vision is then posed are the zero crossing of f convolved with a laplacian of a gaussian at a single channel enough for unambiguously representing f second the problem of the completeness of the representation when the position of the zero crossings is known only approximately is addressed it is shown that when sign f is sampled significant ambiguities are introduced in the representation experimental results are presented which were obtained from an iterative algorithm devised to reconstruct real images from multiscale sign information

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

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

parametric generalized gaussian density estimation the primary objective of this paper is to compare the large sample as well as the small sample properties of different methods for estimating the parameters of a three parameter generalized gaussian distribution three estimators namely the moment method mm the maximum likelihood ml and the moment newton step mns estimators are considered the applicability of general asymptotic optimality results of the efficient ml and mns estimation techniques is studied in the generalized gaussian context the asymptotic normal distributions of the estimators are obtained the asymptotic relative superiority of the ml estimator or its variant the mns estimator over the moment method is studied in terms of asymptotic relative efficiency based on this study it is concluded that deviations from normality in the underlying distribution of the data necessitate the use of the efficient ml or mns methods in the small sample case a detailed comparative study of the estimators is made possible by extensive monte carlo simulations from this study it is concluded that the maximum likelihood method is found to be significantly superior for heavy tailed distributions in a region of the parameter space corresponding to the vicinity of the gaussian distribution the moment method compares well with the other methods further the mns estimator is shown to perform best for light tailed distributions the simulation results are shown to lend support to analytically derived asymptotic results for each of the methods

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

the phase retrieval problem the phase retrieval problem arises in applications of electromagnetic theory in which wave phase is apparently lost or impractical to measure and only intensity data are available the mathematics of the problem provides unusual insights into the nature of electromagnetic fields the theory is reviewed and illustrated the basic issue of the phase retrieval problem stated for a one dimensional field is that although a unique fourier transform relation exists between the field f x in the fraunhofer plane and the field u x in the object plane the infinite fold phase ambiguity which appears as the result of the possibilities of conjugating the zeros of f z z x jy implies that additional information or processing of the object wave must be available to obtain the phase among the possible solutions which are described are reference beam addition apodization and the use of multiple intensity distributions permitting the use of iterative computational procedures in some applications

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

lightweight detection of additive watermarking in the dwt domain this article aims at lightweight blind detection of additive spread spectrum watermarks in the dwt domain we focus on two host signal noise models and two types of hypothesis tests for watermark detection as a crucial point of our work we take a closer look at the computational requirements of watermark detectors this involves the computation of the detection response parameter estimation and threshold selection we show that by switching to approximate host signal parameter estimates or even fixed parameter settings we achieve a remarkable improvement in runtime performance without sacrificing detection performance our experimental results on a large number of images confirm the assumption that there is not necessarily a tradeoff between computation time and detection performance

wavelet based texture retrieval using generalized gaussian density and kullback leibler distance we present a statistical view of the texture retrieval problem by combining the two related tasks namely feature extraction fe and similarity measurement sm into a joint modeling and classification scheme we show that using a consistent estimator of texture model parameters for the fe step followed by computing the kullback leibler distance kld between estimated models for the sm step is asymptotically optimal in term of retrieval error probability the statistical scheme leads to a new wavelet based texture retrieval method that is based on the accurate modeling of the marginal distribution of wavelet coefficients using generalized gaussian density ggd and on the existence a closed form for the kld between ggds the proposed method provides greater accuracy and flexibility in capturing texture information while its simplified form has a close resemblance with the existing methods which uses energy distribution in the frequency domain to identify textures experimental results on a database of 640 texture images indicate that the new method significantly improves retrieval rates e g from 65 to 77 compared with traditional approaches while it retains comparable levels of computational complexity

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

blind optimum detector for robust image watermarking in nonsubsampled shearlet domain in this paper we propose a blind robust image watermarking approach in the nonsubsampled shearlet domain using bessel k form bkf modeling nonsubsampled shearlet transform nsst is an effective multi scale and multi direction analysis method it not only can exactly compute the shearlet coefficients based on a multiresolution analysis but also can provide nearly optimal approximation for a piecewise smooth function in order to achieve more robustness and imperceptibility watermark information is embedded into the most significant nsst directional subband with the highest energy by simply modifying the nsst coefficient amplitude according to the watermark data by modeling the nsst coefficients with bkf probability density function the distribution of watermarked noisy coefficients is analytically calculated the tradeoff between the imperceptibility and robustness of watermark data is solved in a novel fashion at the receiver based on the maximum likelihood ml decision rule a blind statistical watermark detector is proposed experimental results on test images demonstrate that the proposed approach can provide better imperceptibility and robustness against various attacks such as additive white gaussian noise salt pepper noise median filtering jpeg compression rotation and scaling in comparison with the recently proposed techniques

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

blind intensity estimation from shot noise data the estimation of the intensity function of an inhomogeneous poisson process is considered when the observable data consists of sampled shot noise that results from passing the poisson process through an unknown linear time invariant system the proposed method consists of first estimating a histogram of the underlying point process the estimated histogram is used to construct a kernel estimate of the intensity function an estimate of the unknown impulse response of the linear time invariant system is constructed via a regularized backsubstitution of a discrete time convolution with the estimated histogram

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

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

on the modeling of dct and subband image data for compression image subband and discrete cosine transform coefficients are modeled for efficient quantization and noiseless coding quantizers and codes are selected based on laplacian fixed generalized gaussian and adaptive generalized gaussian models the quantizers and codes based on the adaptive generalized gaussian models are always superior in mean squared error distortion performance but generally by no more than 0 08 bit pixel compared with the much simpler laplacian model based quantizers and noiseless codes this provides strong motivation for the selection of pyramid codes for transform and subband image coding

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

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

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

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

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

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

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

asymptotically optimal detection in incompletely characterized non gaussian noise the problem of detecting a signal known except for amplitude in non gaussian noise is addressed the noise samples are assumed to be independent and identically distributed with a probability density function known except for a few parameters using a generalized likelihood ratio test it is proven that for a symmetric noise probability density function the detection performance is asymptotically equivalent to that obtained for a detector designed with a priori knowledge of the noise parameters a computationally more efficient but equivalent test is proposed and a computer simulation performed to illustrate the theory is described

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

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

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

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed