Constrained Conjugate Gradient Research Articles

Spectral Three-Term Constrained Conjugate Gradient Algorithm for Function Minimizations

In this work, we tend to deal within the field of the constrained optimization methods of three-term Conjugate Gradient (CG) technique which is primarily based on Dai–Liao (DL) formula. The new proposed technique satisfies the conjugacy property and the descent conditions of Karush–Kuhn–Tucker (K.K.T.). Our planned constrained technique uses the robust Wolfe line search condition with some assumptions. We tend to prove the global convergence property of the new planned technique. Numeral comparisons for (30-thirty) constrained optimization issues make sure the effectiveness of the new planned formula.

X-ray radiographs reconstruction based on nonlinear least squares with deconvolution

Abstract High energy X-ray radiographs reconstruction is a typical nonlinear process. However, most of the current reconstruction methods are based on a linear approximation model, which uses optical depth images for reconstruction. As radiographs are convolved by the system blur, images reconstructions with these algorithms are blurred, especially at edges. In this paper, a nonlinear least squares reconstruction model for transmittance images is established and optimized by the Levenberg–Marquardt algorithm assuming that the system blur is known. The numerical experiments show that the reconstruction error of this approach is greatly reduced and the edges sharpness is significantly improved, compared with the traditional constrained conjugate gradient method and the trust region reflective method. For images contaminated by blur and low noise, the reconstruction is still improved and acceptable even when an inaccurate system blur is used. Furthermore, we propose an idea of rectifying the system blur measured by experiments. Our reconstruction method is practical in deconvolution and shows a promising prospect in high energy X-ray radiographs reconstruction.

Image regularization by nonnegatively constrained Conjugate Gradient

In the image reconstruction context the nonnegativity of the computed solution is often required. Conjugate Gradient (CG), used as a reliable regularization tool, may give solutions with negative entries, particularly when large nearly zero plateaus are present. The active constraints set, detected by projection onto the nonnegative orthant, turns out to be largely incomplete leading to poor effects on the accuracy of the reconstructed image. In this paper an inner-outer method based on CG is proposed to compute nonnegative reconstructed images with a strategy which enlarges subsequently the active constraints set. This method appears to be especially suitable for the reconstruction of images having large nearly zero backgrounds. The numerical experimentation validates the effectiveness of the proposed method when compared to other strategies for nonnegative reconstruction.

Forward model with space-variant of source size for reconstruction on X-ray radiographic image

The Forward Imaging Technique is a method to solve the inverse problem of density reconstruction in radiographic imaging. In this paper, we introduce the forward projection equation (IFP model) for the radiographic system with areal source blur and detector blur. Our forward projection equation, based on X-ray tracing, is combined with the Constrained Conjugate Gradient method to form a new method for density reconstruction. We demonstrate the effectiveness of the new technique by reconstructing density distributions from simulated and experimental images. We show that for radiographic systems with source sizes larger than the pixel size, the effect of blur on the density reconstruction is reduced through our method and can be controlled within one or two pixels. The method is also suitable for reconstruction of non-homogeneousobjects.

Research and application of forward imaging technology with systemic blur

Forward imaging technique is the base of the combined method for density reconstruction with forward calculation and inverse problem solution. This paper proposes the idea of girding the areal source as many point sources. Using the blurring window as the affecting factor of the detector blur, the projecting matrix from any point source to any detector pixel with X-ray trace technique is obtained, the projection equation for the radiographic system with areal source blur and detector blur is introduced. The projection equation is used to gain the same deviation information about the object edge as the experimental image. The forward projection equation is combined with the constrained conjugate gradient method to form a full procedure of density reconstruction, which is applied on a simulated image of French Test Object and an experimental image. The results show that using the projection equation, the affecting range of the blur is decreased and can be controlled in one or two pixels.

Frequency-domain beamformers using conjugate gradient techniques for speech enhancement.

A multiple-iteration constrained conjugate gradient (MICCG) algorithm and a single-iteration constrained conjugate gradient (SICCG) algorithm are proposed to realize the widely used frequency-domain minimum-variance-distortionless-response (MVDR) beamformers and the resulting algorithms are applied to speech enhancement. The algorithms are derived based on the Lagrange method and the conjugate gradient techniques. The implementations of the algorithms avoid any form of explicit or implicit autocorrelation matrix inversion. Theoretical analysis establishes formal convergence of the algorithms. Specifically, the MICCG algorithm is developed based on a block adaptation approach and it generates a finite sequence of estimates that converge to the MVDR solution. For limited data records, the estimates of the MICCG algorithm are better than the conventional estimators and equivalent to the auxiliary vector algorithms. The SICCG algorithm is developed based on a continuous adaptation approach with a sample-by-sample updating procedure and the estimates asymptotically converge to the MVDR solution. An illustrative example using synthetic data from a uniform linear array is studied and an evaluation on real data recorded by an acoustic vector sensor array is demonstrated. Performance of the MICCG algorithm and the SICCG algorithm are compared with the state-of-the-art approaches.

Set-membership constrained conjugate gradient adaptive algorithm for beamforming

In this work, a constrained adaptive filtering strategy based on conjugate gradient (CG) and set-membership techniques is presented for adaptive beamforming. A constraint on the magnitude of the array output is imposed to derive an adaptive algorithm that performs data-selective updates when calculating the beamformer's parameters. A linearly constrained minimum variance optimisation problem is consider with the bounded constraint based on this strategy and propose a CG-type algorithm for implementation. The proposed algorithm has data-selective updates, a variable forgetting factor and performs one iteration per update to reduce the computational complexity. The updated parameters construct a space of feasible solutions that enforce the constraints. The authors also introduce two time-varying bounding schemes to measure the quality of the parameters that could be included in the parameter space. A comprehensive complexity and performance analysis between the proposed and existing algorithms are provided. Simulations are performed to show the enhanced convergence and tracking performance of the proposed algorithm as compared with existing techniques.

改进的约束共轭梯度闪光照相图像重建算法

改进的约束共轭梯度闪光照相图像重建算法

Reduced-Rank STAP Schemes for Airborne Radar Based on Switched Joint Interpolation, Decimation and Filtering Algorithm

In this paper, we propose a reduced-rank space-time adaptive processing (STAP) technique for airborne phased array radar applications. The proposed STAP method performs dimensionality reduction by using a reduced-rank switched joint interpolation, decimation and filtering algorithm (RR-SJIDF). In this scheme, a multiple-processing-branch (MPB) framework, which contains a set of jointly optimized interpolation, decimation and filtering units, is proposed to adaptively process the observations and suppress jammers and clutter. The output is switched to the branch with the best performance according to the minimum variance criterion. In order to design the decimation unit, we present an optimal decimation scheme and a low-complexity decimation scheme. We also develop two adaptive implementations for the proposed scheme, one based on a recursive least squares (RLS) algorithm and the other on a constrained conjugate gradient (CCG) algorithm. The proposed adaptive algorithms are tested with simulated radar data. The simulation results show that the proposed RR-SJIDF STAP schemes with both the RLS and the CCG algorithms converge at a very fast speed and provide a considerable SINR improvement over the state-of-the-art reduced-rank schemes.

AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data

We describe an adaptive image deconvolution algorithm (AIDA) for myopic deconvolution of multi-frame and three-dimensional data acquired through astronomical and microscopic imaging. AIDA is a reimplementation and extension of the MISTRAL method developed by Mugnier and co-workers and shown to yield object reconstructions with excellent edge preservation and photometric precision [J. Opt. Soc. Am. A21, 1841 (2004)]. Written in Numerical Python with calls to a robust constrained conjugate gradient method, AIDA has significantly improved run times over the original MISTRAL implementation. Included in AIDA is a scheme to automatically balance maximum-likelihood estimation and object regularization, which significantly decreases the amount of time and effort needed to generate satisfactory reconstructions. We validated AIDA using synthetic data spanning a broad range of signal-to-noise ratios and image types and demonstrated the algorithm to be effective for experimental data from adaptive optics-equipped telescope systems and wide-field microscopy.

A bound constrained conjugate gradient solution method as applied to crystallographic refinement problems

Small modifications to the conjugate gradient method for solving symmetric positive definite systems have resulted in an increase in performance over LU decomposition by a factor of around 84 for solving a dense system of 1325 unknowns. Performance is further increased in the case of applying upper- and lower-bound parameter constraints. For structure solution employing simulated annealing and the Newton–Raphson method of non-linear least squares, the overall performance gain can be a factor of four, depending on the applied constraints. In addition, the new algorithm with bounding constraints often seeks out lower minima than would otherwise be attainable without constraints. The behaviour of the new algorithm has been tested against the crystallographic problems of Pawley refinement, rigid-body and general crystal structure refinement.

Performance evaluation of pulsed photothermal profiling of port wine stain in human skin

Treatment of port wine stain (PWS) birthmarks in human skin by pulsed laser irradiation requires the knowledge of the maximum epidermal temperature rise and PWS depth for an attending physician to select the optimal light dosage, irradiation wavelength, and cryogen spray cooling spurt duration on an individual patient basis. Pulsed photothermal radiometry (PPTR) is a promising technique to provide such information. In this article, computer simulations are performed to evaluate the performance of PPTR depth profiling of the laser-induced temperature rise in PWS. An iterative, non-negatively constrained conjugate gradient algorithm is used to reconstruct the laser-induced temperature profile from simulated PPTR signals. Human skin is assumed to contain an epidermal melanin layer and a single homogeneous PWS layer in the dermis. The influence of structural, experimental, and algorithm parameters on the temperature profile reconstruction are discussed. Accuracy of the maximum epidermal temperature rise and PWS depth determined from the reconstructed profiles is statistically analyzed. The simulations show that when the melanin and PWS layers are physically discrete, a good reconstruction can be obtained and the maximum epidermal temperature rise and PWS depth can be determined with accuracy sufficient for the intended clinical application. Measurements and reconstructions from PWS patients are performed and the results are in agreement with the simulations.

A constrained conjugate gradient method for solving the magnetic field boundary integral equation

It is well-known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer. In this paper a simple and efficient computational method is developed, in which the interior integral representations, required to hold on an interior closed surface, are used as a sufficient constraint to restore uniqueness. We use the interior equations together with the second kind magnetic field integral equation, so that the ill-posedness of the interior equations does not give a problem. We develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to the magnetic field boundary integral equation, while the second term is the error norm with respect to the interior equations over a closed interior surface, which is chosen as small as possible. Some numerical examples show the robustness and efficiency of the pertaining computational procedure.

Growth of (110) diamond using pure dicarbon

We use a density-functional-based tight-binding method to study diamond growth steps by depositing dicarbon species onto a hydrogen-free diamond (110) surface. Subsequent ${\mathrm{C}}_{2}$ molecules are deposited on an initially clean surface, in the vicinity of a growing adsorbate cluster, and finally near vacancies just before completion of a full new monolayer. The preferred growth stages arise from ${\mathrm{C}}_{2n}$ clusters in near ideal lattice positions forming zigzag chains running along the $[1\ifmmode\bar\else\textasciimacron\fi{}10]$ direction parallel to the surface. The adsorption energies are consistently exothermic by 8--10 eV per ${\mathrm{C}}_{2},$ depending on the size of the cluster. The deposition barriers for these processes are in the range of 0.0--0.6 eV. For deposition sites above ${\mathrm{C}}_{2n}$ clusters the adsorption energies are smaller by 3 eV, but diffusion to more stable positions is feasible. We also perform simulations of the diffusion of ${\mathrm{C}}_{2}$ molecules on the surface in the vicinity of existing adsorbate clusters using a constrained conjugate gradient method. We find migration barriers in excess of 3 eV on the clean surface, and 0.6--1.0 eV on top of graphenelike adsorbates. The barrier heights and pathways indicate that the growth from gaseous dicarbon proceeds either by direct adsorption onto clean sites or after migration on top of the existing ${\mathrm{C}}_{2n}$ chains.

The constrained conjugate gradient algorithm

Based on the condition for equivalence between linearly constrained minimum-variance (LCMV) filters and their generalized sidelobe canceler (GSC) implementations, we derive the new constrained conjugate gradient (CCG) algorithm. We discuss the use of orthogonal and nonorthogonal blocking matrices for the GSC structure and how the choice of this matrix may affect the relationship with the LCMV counterpart. The newly derived algorithm was tested in a computer experiment for adaptive multiuser detection and showed excellent results.

Accuracy of subsurface temperature distributions computed from pulsed photothermal radiometry

Pulsed photothermal radiometry (PPTR) is a non-contact method for determining the temperature increase in subsurface chromophore layers immediately following pulsed laser irradiation. In this paper the inherent limitations of PPTR are identified. A time record of infrared emission from a test material due to laser heating of a subsurface chromophore layer is calculated and used as input data for a non-negatively constrained conjugate gradient algorithm. Position and magnitude of temperature increase in a model chromophore layer immediately following pulsed laser irradiation are computed. Differences between simulated and computed temperature increase are reported as a function of thickness, depth and signal-to-noise ratio (SNR). The average depth of the chromophore layer and integral of temperature increase in the test material are accurately predicted by the algorithm. When the thickness/depth ratio is less than 25%, the computed peak temperature increase is always significantly less than the true value. Moreover, the computed thickness of the chromophore layer is much larger than the true value. The accuracy of the computed subsurface temperature distribution is investigated with the singular value decomposition of the kernel matrix. The relatively small number of right singular vectors that may be used (8% of the rank of the kernel matrix) to represent the simulated temperature increase in the test material limits the accuracy of PPTR. We show that relative error between simulated and computed temperature increase is essentially constant for a particular thickness/depth ratio.

Dependence of Image Quality on Image Operator and Noise for Optical Diffusion Tomography

By applying linear perturbation theory to the radiation transport equation, the inverse problem of optical diffusion tomography can be reduced to a set of linear equations, Wμ=R, where W is the weight function, μ are the cross-section perturbations to be imaged, and R is the detector readings perturbations. We have studied the dependence of image quality on added systematic error and/or random noise in W and R. Tomographic data were collected from cylindrical phantoms, with and without added inclusions, using Monte Carlo methods. Image reconstruction was accomplished using a constrained conjugate gradient descent method. Results show that accurate images containing few artifacts are obtained when W is derived from a reference state whose optical thickness matches that of the unknown test medium. Comparable image quality was also obtained for unmatched W, but the location of the target becomes more inaccurate as the mismatch increases. Results of the noise study show that image quality is much more sensitive to noise in W than in R, and the impact of noise increases with the number of iterations. Images reconstructed after pure noise was substituted for R consistently contain large peaks clustered about the cylinder axis, which was an initially unexpected structure. In other words, random input produces a nonrandom output. This finding suggests that algorithms sensitive to the evolution of this feature could be developed to suppress noise effects. © 1998 Society of Photo-Optical Instrumentation Engineers.

Imaging laser heated subsurface chromophores in biological materials: determination of lateral physical dimensions

We describe a non-contact method using infrared radiometry to determine lateral physical dimensions of laser heated subsurface chromophores in biological materials. An imaging equation is derived that relates measured radiometric temperature change to the reduced two-dimensional temperature increase of laser heated chromophores. From measured images of radiometric temperature change, the lateral physical dimensions of chromophores positioned in an in vitro model of human skin are determined by deconvolution of the derived imaging equation using a non-negative constrained conjugate gradient algorithm. Conditions for optimum spatial resolution are found by analysis of a derived radiometric transfer function and correspond to superficial chromophores and/or weak infrared absorption in a laser irradiated biological material. Analysis indicates that if the infrared attenuation coefficient is sufficiently small (i.e., less than 10 ), infrared radiometry in combination with a deconvolution algorithm allows estimation of lateral physical dimensions of laser heated subsurface chromophores in human skin.

A constrained conjugate gradient method and the solution of linear equations

A conjugate gradient method for solving minimization problems subject to linear equality constraints is developed. The method can be considered as an extension of the unconstrained Fletcher and Reeves algorithm. Proofs of several theorems related to the method are given. An application of the method is to provide an alternative method to solve a real system of linear equations for varying right-hand sides. Numerical solutions can be obtained in one iteration which leads to the development of a new direct method to invert a square matrix. Since this direct method involves matrix-vector multiplications and outer products which are easy to parallelize, it has an advantage over existing well-known direct methods. Numerical examples are given.

Projection minimization techniques for orthogonal QMF filters with vanishing moments

In many applications, a cost function is available that determines how good the filters of a filter bank decomposition are at performing a particular job. For example, in signal compression, the filters of a quadrature mirror filter (QMF) bank could be rated on their accuracy in compression and reconstruction of a given input signal. Given a cost function and the constraint set, it would be useful to be able to use an iterative minimization algorithm, e.g., steepest descent, to minimize the cost function while satisfying all of the constraints. The resulting performance would exceed an implementation that was not application specific (i.e., fixed filters). A projection minimization approach is taken with the particulars of this application determining the projection space. A constrained conjugate gradient descent approach is taken in choosing a filter pair that minimizes a cost function, but the methodology could be easily modified and applied to a wide variety of iterative minimization techniques. >