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This paper is concerned with the question of reconstructing a vector in a finite-dimensional complex Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We present new invertibility results as well as an iterative algorithm that finds the least-square solution, which is robust in the presence of noise. We analyze its numerical performance by comparing it to the Cramer-Rao lower bound., Author(s): Radu Balan[sup.1] Author Affiliations: (1) Department of Mathematics and Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, 20742, College Park, MD, USA Introduction This paper [...]

A formulation of the problem of combining data from several sources is discussed in terms of random effects models. The unknown measurement precision is assumed not to be the same [...]

We present the design and testing of an algorithm for iterative refinement of the solution of linear equations where the residual is computed with extra precision. This algorithm was originally proposed in 1948 and analyzed in the 1960s as a means to compute very accurate solutions to all but the most ill-conditioned linear systems. However, two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard has essentially removed the first obstacle. To overcome the second obstacle, we show how the application of iterative refinement can be used to compute an error bound in any norm at small cost and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound. We report extensive test results on over 6.2 million matrices of dimensions 5, 10, 100, and 1000. As long as a normwise (componentwise) condition number computed by the algorithm is less than 1/max{10,[square root of (n)][[epsilon].sub.w], the computed normwise (componentwise) error bound is at most 2 max{10,[square root of (n)] * [[epsilon].sub.w]}, and indeed bounds the true error. Here, n is the matrix dimension and [[epsilon].sub.w] = [2.sup.-24] is the working precision. Residuals were computed in double precision (53 bits of precision). In other words, the algorithm always computed a tiny error at negligible extra cost for most linear systems. For worse conditioned problems (which we can detect using condition estimation), we obtained small correct error bounds in over 90% of cases. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra--Linear systems (direct and iterative methods), Error analysis General Terms: Algorithms, Reliability Additional Key Words and Phrases: Linear algebra, LAPACK, BLAS, floating-point arithmetic

A blind maximum likelihood equalization method is proposed for frequency selective fast fading Ricean channels. This method employs the expectation-maximization Viterbi algorithm (EMVA) developed in [1] for blind channel estimation and signal detection. Since the Viterbi algorithm (VA) is used to execute the E-phase of an expectation-maximization (EM) iteration, it requires that the observed sequence can be modelled as a finite-state hidden Markov process. We develop a hidden Markov model for frequency selective fast fading Ricean channels, so that the observed process can be viewed as the noisy output of a finite state machine (FSM), to which the VA is applicable. The EMVA is then employed to obtain a blind maximum likelihood estimate of the specular part of the channel and, for one special case, of a noise parameter measuring the total power of the additive and multiplicative channel noise components. Simulation results are presented which show that the EMVA achieves an accurate estimate of the channel specular part and has an error rate performance close to that of the maximum likelihood detector based on true parameters for the given FSM model. Index Terms--Blind equalization, expectation-maximization (EM) algorithm, Rayleigh fading channel, Ricean fading channel, Viterbi algorithm (VA).

This paper presents a dual-loop iteration algorithm for steady-state determination of current-programmed direct current to direct current (dc-dc) switching converters. The methodology starts with separating the gate signal from the feedback circuit to the power stage in order to convert a closed-loop configuration into an open-loop one. The first iteration loop applies the Newton's method together with a time-domain simulation technique to find the steady-state state variables when the transformed configuration is operating at a constant duty cycle. The second iteration loop is to find the steady-state duty cycle using the secant method. The algorithm includes all of the advantages of a previously developed time-domain simulation technique and solves the nonconvergence problem in the single-loop iteration method, due to the amplifying effect of poor guess values of the state variables on the duty cycle in the feedback path during the iteration. A current-programmed boost converter is illustrated. The performances of the single-loop and the proposed methods are compared. The theoretical results are verified with the experimental measurements. Index Terms - Circuit simulation, dc-dc power conversion, iterative methods, Jacobian matrices, power electronics, switched mode power supplies, switching circuits.

An iterative gradient algorithm is presented for determining the quantization levels in each of a number of independent sensors so arranged as to pick up a common signal field. The system is to satisfy the Neyman-Pearson criterion that the probability of detection be maximum for a preassigned false-alarm probability. In general a number of local maxima exist, and the proposed method enables efficient search for these by starting from a variety of initial trial values.

A new maintenance model for continuous-time Markov assumes that inspection is needed to determine the degree of deterioration, inspection and replacement times are non-negligible, and replacement time and cost and operating costs become higher with deterioration. Using iterative algorithms, the optimal maintenance policies for each of the five maintenance strategies of failure replacement, age replacement, sequential inspection, periodic inspection and continuous monitoring are determined.

In order to generate expected cell frequencies for log-linear models, the researcher has to resort to an iterative fitting procedure. A lucid example involving a three layer contingency table was presented by Muller and his associates (1970). Unfortunately, they only provide an incomplete iteration process. Using the same example, the author of this paper completes the entire iteration process, which is essential for the testing of the hypothesis., Recent development in the research methodology of social sciences have had grea impacts upon the field of higher education. Multivariate analysis techniques permit the exploration of the relationship between two [...]

A method is developed to generate the 3-D binary representation for a tree-like object from three mutually orthogonal projections. This is done by first backprojecting the binarized images from three directions and then iteratively removing artifacts in the backprojection. Three different algorithms have been developed: the Lagrange multiplier algorithm (LMA), the conjugate gradient algorithm (CGA), and the minimumvoxel representation algorithm (MRA). The performance of these algorithms under noise-free conditions is evaluated using mathematically projected images of a 3-D tree structure. While all three algorithms are capable of producing a relatively accurate reconstruction, the MRA is superior not only because it requires the least amount of computation but also because it uses binary instead of gray-scale information in the input images. Reconstruction of 3-D coronary arterial structures using MRA is further verified with x-ray images of a human chest phantom and shows a satisfactory performance. The result of this study should be valuable for 3-D imaging of blood vessels. view reconstruction, orthogonal projections

An iterative method based on higher order statistics is presented for parameter estimation of multichannel moving average processes. The algorithm, which globally convergences within finite steps, solves the third-order cumulant-based identification equations that generate the moving average parameter matrices. Computer simulations are carried out to demonstrate the performance of the algorithm.

Gabor's research on nonorthogonal signal modeling in the time-frequency space puts forward an iterative approach for estimating the Gabor's expansion coefficients. An evaluation of the method shows that the algorithm's convergence depends on the window function chosen. The results are used to draw up conditions that facilitate convergence of the algorithm for every signal represented in the time-frequency space.

A general approach to the implementation of highly selective spatial filters for pattern recognition leads to a nonlinear optimization problem. Three optimization algorithms, and climbing (direct search), simulated annealing, and the genetic algorithm, were investigated for implementation on hybrid electro-optical systems. Experimental results indicate the substantial superiority of the genetic algorithm in terms of operating speed and performance quality., An analysis of the applicability of hill climbing, simulated annealing and genetic optimization algorithms to an electro-optical system reveals the superiority of the genetic procedure in combination with optical processing. An array of 100-by-100 variables used in an experimental optimization problem is easily solved indicating that the application of the algorithms in the system results in high efficiency and performance quality including automatic compensation of aberrations and distortions.

A comparison of the expectation maximization (EM) algorithm and the iterative conditional estimation (ICE) algorithm was conducted to show that in the probability density function family, the EM is a particular case of the ICE algorithm. ICE algorithms have a wider application since they cover probability distributions where the concept of likelihood no longer counts. Signal processing examples are illustrated to support the conclusion.

The application of signal processing techniques for real-time resolution of coherent and noncoherent signals in antenna arrays is studied. An algorithm for the array processing problem is developed. It employs a conjugate gradient approach to accelerate the convergence of an iterative method used in the algorithm for recursive caluculation of the array weights vector. Examples are also given to demonstrate the algorithm's performance.

Byline: Daniel Solow, Hans Halim An open question pertaining to the simplex algorithm is where phase 1 terminates on the feasible region. Here, computer simulations support the conjecture that phase 1 terminates at a feasible point geometrically close to the starting point. This observation leads to a coordinate transformation for bounded Linear Programs (LP) that requires fewer iterations in phase 2. This is confirmed by computational experiments on random LPs and Netlib problems and shows generally increasing benefits as the number of negative coefficients in the minimisation objective function increases. Though not winning on all Netlib problems, the coordinate transformation saves 2–11% of the CPU time.

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential. Key Words: Computer Software, Genetic Analysis

Power control has been shown to be an effective way to increase capacity in wireless systems. In previous work on power control, it has been assumed that power levels can be assigned from a continuous range. In practice, however, power levels are assigned from a discrete set. In this work, we consider the minimization of the total power transmitted over given discrete sets of available power levels subject to maintaining an acceptable signal quality for each mobile. We have developed distributed iterative algorithms for solving a more general version of this integer programming problem, which is of independent interest, and have shown that they find the optimal solution in a finite number of iterations which is polynomial in the number of power levels and the number of mobiles. Index Terms--Cellular networks, distributed algorithms, integer programming, power control.

This paper addresses the problem of spatio-temporal segmentation of video sequences. An initial intensity segmentation method (watershed segmentation) provides a number of initial segments which are subsequently labeled, with a known number of labels, according to motion information. The label field is modeled as a Markov Random Field where the statistical spatial and temporal interactions are expressed on the basis of the initial watershed segments. The labeling criterion is the maximization of the conditional a posteriori probability of the label field given the motion hypotheses, the estimate of the label field of the previous frame, and the image intensities. For the optimization, an iterative motion estimation-labeling algorithm is proposed and experimental results are presented. Index Terms--Markov Random Fields, motion-based segmentation, region labeling, watershed segmentation, motion estimation.

Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of "local" functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph. In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes--either exactly or approximately--various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms. Index Terms--Belief propagation, factor graphs, fast Fourier transform, forward/backward algorithm, graphical models, iterative decoding, Kalman filtering, marginalization, sum-product algorithm, Tanner graphs, Viterbi algorithm.

A generalized state realization of the Wiberg type is called normal if symbol variables have degree 1 and state variables have degree 2. A natural graphical model of such a realization has leaf edges representing symbols, ordinary edges representing states, and vertices representing local constraints. Such a graph can be decoded by any version of the sum-product algorithm. Any state realization of a code can be put into normal form without essential change in the corresponding graph or in its decoding complexity. Group or linear codes are generated by group or linear state realizations. On a cycle-free graph, there exists a we!i-defined minimal canonical realization, and the sum-product algorithm is exact. However, the Cut-Set Bound shows that graphs with cycles may have a superior performance-complexity tradeoff, although the sum-product algorithm is then inexact and iterative, and minimal realizations are not well-defined. Efficient cyclic and cycle-free realizations of Reed-Muller (RM) codes are given as examples. The dual of a normal group realization, appropriately defined, generates the dual group code. The dual realization has the same graph topology as the primal realization, replaces symbol and state variables by their character groups, and replaces primal local constraints by their duals. This fundamental result has many applications, including to dual state spaces, dual minimal trellises, duals to Tanner graphs, dual input/output (I/O) systems, and dual kernel and image representations. Finally, a group code may be decoded using the dual graph, with appropriate Fourier transforms of the inputs and outputs; this can simplify decoding of high-rate codes. Index Terms--Codes, dual codes, graphical models, group codes, state realizations, sum-product algorithm.

We propose a novel, to our knowledge, algorithm of iterative simulated quenching with temperature rescaling for designing diffractive optical elements, based on an analogy between simulated annealing and statistical thermodynamics. The temperature is iteratively rescaled at the end of each quenching process according to ensemble statistics to bring the system back from a frozen imperfect state with a local minimum of energy to a dynamic state in a Boltzmann heat bath in thermal equilibrium at the rescaled temperature. The new algorithm achieves much lower cost function and reconstruction error and higher diffraction efficiency than conventional simulated annealing with a fast exponential cooling schedule and is easy to program. The algorithm is used to design binary-phase generators of large irregular spot arrays. The diffractive phase elements have trapezoidal apertures of varying heights, which fit ideal arbitrary-shaped apertures better than do trapezoidal apertures of fixed heights. [C] 2000 Optical Society of America OCIS codes: 050.1950, 050.1960, 050.1970, 090.1760, 090.1970.

We present a class of convolutional codes defined by a low-density parity-check matrix and an iterative algorithm of the decoding of these codes. The performance of this decoding is close to the performance of turbo decoding. Our simulation shows that for the rate R = 1/2 binary codes, the performance is substantionally better than for ordinary convolutional codes with the same decoding complexity per information bit. As an example, we constructed convolutional codes with memory M = 1025, 2049, and 4097 showing that we are about 1 dB from the capacity limit at a bit-error rate (BER) of [10.sup.-5] and a decoding complexity of the same magnitude as a Viterbi decoder for codes having memory M = 10. Index Terms - Convolutional codes, iterative decoding, low-density parity-check codes.