Abstract: The distorted Born iterative method (DBIM) is used to solve two-dimensional inverse scattering problems, thereby providing another general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations are performed using the DBIM and the method proposed previously by the authors (Int. J. Imaging Syst. Technol., vol.1, no.1, p.100-8, 1989) called the Born iterative method (BIM) for several cases in which the conditions for the first-order Born approximation are not satisfied. The results show that each method has its advantages; the DBIM shows faster convergence rate compared to the BIM, while the BIM is more robust to noise contamination compared to the DBIM. >

Abstract: A method is presented for the selection of a set of sensor locations from a larger candidate set for the purpose of on-orbit identification and correlation of Large Space Structures. The method ranks the candidate sensor locations according to their contribution to the linear independence of the target modal partitions. In an iterative maner, the locations which do not contribute significantly are removed. The final sensor configuration tends to maximize determinant of the corresponding Fisher Information Matrix.

Abstract: A procedure to calculate a highly quantized, blazed phase structure is presented. Characteristics that are concentrated on are a high diffraction efficiency and a large signal-to-noise ratio. The calculation techniques are based on iterative Fourier-transform algorithms. Stagnation problems are discussed, and methods to overcome them are described.

Abstract: Direct analytical methods are discussed for solving Poisson equations of the general form Delta u=f on a rectangular domain. Some embedding techniques that may be useful when boundary conditions (obtained from stereo and occluding boundary) are defined on arbitrary contours are described. The suggested algorithms are computationally efficient owing to the use of fast orthogonal transforms. Applications to shape from shading, lightness and optical flow problems are also discussed. A proof for the existence and convergence of the flow estimates is given. Experiments using synthetic images indicate that results comparable to those using multigrid can be obtained in a very small number of iterations. >

Abstract: A three-dimensional iterative reconstruction algorithm which incorporates models of the geometric point response in the projector-backprojector is presented for parallel, fan, and cone beam geometries. The algorithms have been tested on an IBM 3090-600S supercomputer. The iterative EM reconstruction algorithm is 50 times longer with geometric response and photon attenuation models than without modeling these physical effects. An improvement in image quality in the reconstruction of projection data collected from a single-photon-emission computed tomography (SPECT) imaging system has been observed. Significant improvements in image quality are obtained when the geometric point response and attenuation are appropriately compensated. It is observed that resolution is significantly improved with attenuation correction alone. Using phantom experiments, it is observed that the modeling of the spatial system response imposes a smoothing without loss of resolution. >

Abstract: A theory is presented for the computation of three-dimensional motion and structure from dynamic imagery, using only line correspondences. The traditional approach of corresponding microfeatures (interesting points-highlights, corners, high-curvature points, etc.) is reviewed and its shortcomings are discussed. Then, a theory is presented that describes a closed form solution to the motion and structure determination problem from line correspondences in three views. The theory is compared with previous ones that are based on nonlinear equations and iterative methods.

Abstract: The blur identification problem is formulated as a constrained maximum-likelihood problem. The constraints directly incorporate a priori known relations between the blur (and image model) coefficients, such as symmetry properties, into the identification procedure. The resulting nonlinear minimization problem is solved iteratively, yielding a very general identification algorithm. An example of blur identification using synthetic data is given. >

Abstract: The authors study a two-dimensional reconstruction algorithm due to D. C. Barber and B. H. Brown, applied to a linearized electrostatic inverse problem. First, the authors demonstrate how this algorithm fits within the framework of inverses of generalized Radon transforms studied by G. Beylkin. Second, an iterative improvement of the Barber–Brown algorithm is constructed based on the conjugate residual method. Several numerical results obtained with this iterative algorithm are presented.

Abstract: Discretization of steady-state eddy-current equations may lead to linear system Ax=b in which the complex matrix A is not Hermitian, but may be chosen symmetric. In the positive definite Hermitian case, an iterative algorithm for solving this system can be defined. The residual vectors can be made mutually orthogonal by means of a two-term recursion relation which leads to the well-known conjugate gradient (CG) method. The proposed method is illustrated by comparing it with other methods for some eddy current examples. >

Abstract: A new theory for the calculation of proper elements is presented This theory defines an explicit algorithm applicable to any chosen set of orbits and accounts for the effect of shallow resonances on secular frequencies The proper elements are computed with an iterative algorithm and the behavior of the iteration can be used to define a quality code

Abstract: An effort to unify three major numerical methods in electromagnetics is presented. Harrington's direct method of moments, the iterative methods, and the reaction integral equation method are shown to be generally equivalent and are unified as the generalised method of moments. It is shown that the reaction integral equation method is in general a moment method, and that the moment method, when defined in a symmetric space, generally satisfies the reaction theorem, and therefore reciprocity. A broad, though limited, equivalence between the moment and the iterative methods is also demonstrated. A numerical example is discussed to illustrate these and other points.

Abstract: T.W. Ridler and E.S. Calvard (ibid., vol.SMC-8, p.630-2, Aug. 1978) presented a method of picture thresholding that was further mathematically developed by H.J. Trussel (ibid., vol.SMC-9, p.311, 1979). The principle of this method is to evaluate the unique threshold T for any image with a bimodal histogram. The present paper reinterprets the procedure as an algorithm designed to optimize the conversion of a multiple gray-level picture to a bimodel picture while maintaining as closely as possible the average luminance of the picture. >

Abstract: Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.

Abstract: This article describes three approximation methods I used to solve the growth model (Model 1) studied by the National Bureau of Economic Research's nonlinear rational-expectations-modeling group project, the results of which were summarized by Taylor and Uhlig (1990). The methods involve computing exact solutions to models that approximate Model 1 in different ways. The first two methods approximate Model 1 about its nonstochastic steady state. The third method works with a version of the model in which the state space has been discretized. A value function iteration method is used to solve that model.

Abstract: Recent results on the asymptotically optimal design of sliding windows for virtual circuits in high speed, geographically dispersed data networks in a stationary environment are exploited here in the synthesis of algorithms for adapting windows in realistic, non-stationary environments. The algorithms proposed here require each virtual circuit's source to measure the round trip response times of its packets and to use these measurements to dynamically adjust its window. Our design philosophy is quasi-stationary: we first obtain, for a complete range of parameterized stationary conditions, the relation, called the “design equation”, that exists between the window and the mean response time in asymptotically optimal designs; the adaptation algorithm is simply an iterative algorithm for tracking the root of the design equation as conditions change in a non-stationary environment. A report is given of extensive simulations of networks with data rates of 45 Mbps and propagation delays of up to 47 msecs. The simulations generally confirm that the realizations of the adaptive algorithms give stable, efficient performance and are close to theoretical expectations when these exist.

Abstract: This paper describes a control-volume, finite-element technique for coupling coarse grids with local fine meshes. The pressure is treated in a finite-element manner, while the mobility terms are upstream weighted in the usual way. This requires identification of the cell volume and edges that are consistent with the linear finite-element discretization of the pressure. To ensure that the pressure equation yields an M matrix, various conditions are required for the type of triangulation allowed. Because the form of the equations is similar to the usual finite-difference discretization, standard techniques can be used to solve the Jacobian. The local mesh-refinement method is demonstrated on some thermal reservoir simulation problems, and computational results are presented. Significant savings in execution times are obtained while predictions similar to global fine-mesh runs are given.

Abstract: A different approach to hardware evaluation of elementary functions for high-precision floating-point numbers (in particular, the extended double precision format of the IEEE standard P754) is examined. The evaluation is based on rational approximations of the elementary functions, a method which is commonly used in scientific software packages. A hardware model is presented of a floating-point numeric coprocessor consisting of a fast adder and a fast multiplier, and the minimum hardware required for evaluation of the elementary functions is added to it. Next, rational approximations for evaluating the elementary functions and testing the accuracy of the results are derived. The calculation time of these approximations in the proposed numeric processor is then estimated. It is concluded that rational approximations can successfully complete with previously used methods when execution time and silicon area are considered. >

Abstract: This paper describes implementations of Cuppen's method, bisection, and multisection for the computation of all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix on a distributed-memory hypercube multiprocessor. Numerical results and timings for Intel's iPSC-1 are presented. Cuppen's method is found to be the numerically most accurate of three methods, while bisection with inverse iteration is observed experimentally to be the fastest method.

Abstract: We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip- tic partial differential equations. A prototype is the convection-diffusion equa- tion. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, com- bined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.

Abstract: The authors develop an iterative, adaptive, space-variant restoration incorporating both regularization and artifact-suppression constraints. The algorithm is formulated using the method of projections onto convex sets (POCS). They introduce a closed-convex regularization constraint set called the partial Wiener solution set. Projection onto this set forces the solution to be equal to the Wiener solution over a predetermined set of frequencies that lie outside the neighborhoods of the zeros of the degradation transfer function. In the neighborhoods of the zeros, the Wiener solution contains significant errors that contribute to artifacts. Hence, in these neighborhoods the Wiener solution is discarded, and the missing frequency components are determined so that the entire solution is consistent with the artifact-suppression constraints and other a priori information. The proposed artifact-suppression constraints bound the norm of the variation of the image from its local mean using regionally adaptive bounds. The high quality of the resulting restorations is noteworthy. >

Abstract: This thesis is concerned with Electrical Impedance Tomogaphy (EIT), a medical imaging technique in which pictures of the electrical conductivity distribution of the body are formed from current and voltage data taken on the body surface. The focus of the thesis is on the mathematical aspects of reconstructing the conductivity image from the measured data (the reconstruction problem). The reconstruction problem is particularly difficult and in this thesis it is investigated analytically and numerically. The aim of this investigation is to understand why the problem is difficult and to find numerical solution methods which respect the difficulties encountered. The analytical investigation of this non-linear inverse problem for an elliptic partial differential equation shows that while the forward mapping is analytic the inverse mapping is discontinuous. A rigorous treatment of the linearisation of the problem is given, including proofs of forms of linearisation assumed by previous authors. It is shown that the derivative of the forward problem is compact. Numerical calculations of the singular value decomposition (SVD) are given including plots of singular values and images of the singular functions. The SVD is used to settle a controversy concerning current drive patterns. Reconstruction algorithms are investigated and use of Regularised Newton methods is suggested. A formula for the second derivative of the forward mapping is derived which proves too computationally expensive to calculate. Use of Tychonov regularisation as well as filtered SVD and iterative methods are discussed. The similarities, and differences, between EIT and X-Ray Computed Tomography (X-Ray CT) are illuminated. This leads to an explanation of methods used by other authors for EIT reconstuction based on X-Ray CT. Details of the author's own implementation of a regularised Newton method are given. Finally the idea of adaptive current patterns is investigated. An algorithm is given for the experimental determination of optimal current patterns and the integration of this technique with regularised Newton methods is explored. Promising numerical results from this technique are given. The thesis concludes with a discussion of some outstanding problems in EIT and points to possible routes for their solution. An appendix gives brief details of the design and development of the Oxford Polytechnic Adaptive Current Tomograph.

Abstract: The effectiveness of several iterative methods of item response theory (IRT) item bias detection was examined in a simulation study. The situations employed were based on biased items created using a two-dimensional IRT model. Previous research demonstrated that the non-iterative application of some IRT parameter linking procedures produced unsatisfactory results in a simulation study involving unidirectional item bias. A modified form of Drasgow’s iterative item parameter linking method and an adaptation of Lord’s test purification procedure were examined in conditions that simulated unidirectional and mixed-directional forms of item bias. The results illustrate that iterative linking holds promise for differentiating biased from unbiased items under several item bias conditions. In addition, a combination of methods, involving cycles of iterative linking followed by ability scale purification, was found to be even more effective than iterative linking alone. This combination of procedures totally eliminated false positive misidentifications for the most pervasive item bias condition, and false negative misidentifications were also reduced. Combining iterative linking with ability scale purification appears to be a viable method for analyzing multidimensional IRT data with unidimensional IRT item-bias methods. Index terms: ability scale purification, item bias, item response theory, iterative linking, iterative methods, metric linking, multidimensional IRT model.

Abstract: Extracting surface orientation and surface depth from a shaded image is one of the classic problems in computer vision. Many previous algorithms either violate integrability, i.e., the surface normals do not correspond to a feasible surface, or use regularization, which biases the solution away from the true answer. A recent iterative algorithm proposed by Horn overcomes both of these limitations but converges slowly. This paper uses a new algorithm, hierarchical basis conjugate gradient descent, to provide a faster solution to the same problem. This approach is similar to the multigrid techniques which have previously been used to speed the convergence, but it does not require heuristic approximations to the true irradiance equation. The paper compares the accuracy and the convergence rates of the new techniques to previous algorithms.

Abstract: A fast approach to including attenuation in iterative maximum-likelihood and least-squares algorithms for single-photon-emission computed tomography (SPECT) is presented. Ray-tracing and summing of attenuation coefficients are replaced by the use of two lookup tables, one to compute attenuated ray path integrals based on a set of polar grid points and one to perform polar-to-rectangular transformations. The resulting algorithm implements a spatial average which is comparable in accuracy to ray-tracing with rectangular pixels, yet requires less than one sixteenth the CPU time. >

Abstract: Some finite-element approximation procedures are presented for a model proposed by Ladyzhenskaya for stationary incompressible viscous flow. The approximate problems are proved to be well posed and stable under standard assumptions on the finite-element families. The solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions. Some error estimates are derived. The optimal order of accuracy is assured with, or even without, using exact integration rules in the approximation procedure. Iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided. Special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary Navier–Stokes equations.

Abstract: Parallel iterative methods are studied, and the focus is on linear algebraic systems whose matrix is symmetric and positive definite. The set of unknowns may be viewed as a union of subsets of unknowns (possibly with overlap). The parallel iteration matrix is then formed by a weighted sum of iteration matrices that are associated with splittings of the matrix corresponding to the blocks. When the blocks are from a matrix in dissection form, it can be shown under suitable conditions that the parallel algorithm is convergent. When the multisplitting version of successive over-relaxation (SOR) is used, the SOR parameter is required to be less than $\omega _0 < 2.0$. Calculations done on the Alliant FX/8 multiprocessing/vector computer indicate speedups of nine to ten.

Abstract: We analyze sequential quadratic programming (SQP) methods to solve nonlinear constrained optimization problems that are more flexible in their definition than standard SQP methods. The type of flexibility introduced is motivated by the necessity to deviate from the standard approach when solving large problems. Specifically we no longer require a minimizes of the QP subproblem to be determined or particular Lagrange multiplier estimates to be used. Our main focus is on an SQP algorithm that uses a particular augmented Lagrangian merit function. New results are derived for this algorithm under weaker conditions than previously assumed; in particular, it is not assumed that the iterates lie on a compact set.