Generalized Inverse Of Matrix Research Articles

A Fast Reconstruction Algorithm for Fluorescence Optical Diffusion Tomography Based on Preiteration

Fluorescence optical diffusion tomography in the near-infrared (NIR) bandwidth is considered to be one of the most promising ways for noninvasive molecular-based imaging. Many reconstructive approaches to it utilize iterative methods for data inversion. However, they are time-consuming and they are far from meeting the real-time imaging demands. In this work, a fast preiteration algorithm based on the generalized inverse matrix is proposed. This method needs only one step of matrix-vector multiplication online, by pushing the iteration process to be executed offline. In the preiteration process, the second-order iterative format is employed to exponentially accelerate the convergence. Simulations based on an analytical diffusion model show that the distribution of fluorescent yield can be well estimated by this algorithm and the reconstructed speed is remarkably increased.

Recent Advances in Molecular Imaging

Molecular imaging is a newly emerging and rapidly developing biomedical imaging field in which the modern tools are being married to depict noninvasive in vivo cellular and molecular processes sensitively and specifically, such as monitoring multiple molecular events, cell trafficking and targeting. The goals of this field are to develop technologies and instruments for studying biological and medical processes as well as diagnosing and managing diseases better. Although rapid progress in the fundamentals and applications make molecular imaging become an important tool for biomedical research in recent years, many difficult problems and challenges remain. Discussing the problems and challenges in detail and illustrating recent progress and future direction, the special issue collects the high-quality, peer-reviewed, original research papers in the area of molecular imaging. Novel molecular imaging theories and algorithms, new molecular probes, multimodality molecular imaging prototype systems and experiments, and final clinical applications are introduced mainly in this special issue. In molecular imaging theories and algorithms, genetic algorithm-based optimization tool is used to improve the accuracy of the diffusion model in strongly absorbing media by adjusting the optical parameters. Furthermore, a penalized linear and nonlinear combined conjugate gradient method, a fast preiteration algorithm based on the generalized inverse matrix, and a Monte-Carlo-based network method are also proposed for light source reconstruction in optical tomography. Considering the importance of molecular probes, synthesis and bioconjugation of gold nanoparticles as potential molecular probes for light-based imaging techniques are described in this special issue, and the nitroimidazole-based thioflavin-T derivatives as cerebral ischemia markers are evaluated in vivo. In order to test the feasibility and effectiveness of imaging theories, algorithms and probes, molecular imaging prototype systems should be designed, constructed and employed for small animal or phantom imaging. Thus, multimodality fusion near-infrared optical tomography Systems with highly sensitive CCD camera and photomultiplier tube can be consulted respectively in this special issue. Furthermore, a novel confocal optical system design and a dual laser confocal scanner have been developed for molecular imaging applications of biochip. In clinical applications, the advantage of PET and CT integration in examination of lung tumors is analyzed. Moreover, several innovative processing methods of molecular image are also presented in this special issue. In conclusion, this special issue covers recent important advances in molecular imaging field.

Uncertainties in source distribution temperature and correlated colour temperature

Uncertainties in the distribution temperature (DT) and correlated colour temperature are estimated for common sources and typical measurement uncertainties of a spectral transfer from a reference lamp. Uncertainty sensitivity coefficients for both parameters in terms of measured values of spectral irradiance are derived using a generalized matrix inverse. The uncertainty values are compared with differences in the source temperature parameters. DT is calculated using the CIE definition; shifts in DT due to the alternative of a direct fit of Planck's distribution, and to including weights in the process, are compared with the estimated uncertainties.

SLIDING MODE CONTROL OF LINEAR SYNCHRONOUS MOTOR SERVODRIVE

This paper proposes a method to design sliding mode control (SMC) for the position tracking control of a permanent magnet linear synchronous motor (PMLSM). The controller is designed to achieve accurate performance in the presence of unknown disturbance and parameter uncertainties. The components of the control law, the equivalent component and the robust component, are designed such that the nominal system exhibits desirable dynamics and the reaching condition is guaranteed. Switching surface is designed based on pole placement and generalized matrix inverse. Simulation results show that the suggested procedure provides high performance dynamic characteristics and robust with regard to external disturbance and parameter variations.

Optimization of multiple-impulse minimum-time rendezvous with impulse constraints using a hybrid genetic algorithm

The minimum-time multiple-impulse rendezvous with impulse constraints is investigated in this paper. Based on the Clohessy–Wiltshire (C–W) equations, an optimization model including several different kinds of impulse constraints such as the maximum impulse magnitude, the total velocity change magnitude and the time of imposing impulse for multiple-impulse minimum-time rendezvous is established. A generalized inverse matrix solution for linear equation is applied to avoid handling the terminal equality constraints. In order to obtain the global solution efficiently, a hybrid optimizer combining the advantages of a floating-coded genetic algorithm and simplex method is employed. A low-earth orbit multiple-impulse rendezvous problem is used as an example. The influence of the number of impulses, the optimization variables and the constraints on the solution is analyzed, and the different optimization algorithms are compared. Results indicate our proposed model and approach is effective in designing linearized minimum-time rendezvous trajectory with impulse constraints.

First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors

The equations are derived which relate first-order perturbations of a mirror surface to the optical path difference (OPD) function observed during interferometric centre-of-curvature null tests. Rigid body displacements in particular and their effects on the Fourier coefficients of the OPD function are then considered. Next, the linearity of the rigid body component of the OPD is shown to lead to a method, based on the generalized matrix inverse, for aligning mirrors. Finally, the limits of the validity, in terms of the magnitude of local displacements, of our linear model are analysed. The result of the analysis is a concise, conceptually appealing treatment of general surface deformations in the interferometric regime and a matrix formulation for the particular case of rigid body displacements in the centre-of-curvature null interferometer configuration. This robust formulation treats interferometrically measured OPD functions as state vectors and small deformations as linear transformations of these vectors.

Stationary distributions and mean first passage times of perturbed Markov chains

For finite irreducible discrete time Markov chains, whose transition probabilities are subjected to a perturbation, it is shown that the mean first passage times play an important role in determining the differences between the stationary probabilities in the perturbed and unperturbed situations. The derivation of the interconnection, under the updating procedure, is explored through the use of generalized matrix inverses. New improved bounds for the relative and absolute differences between the stationary probabilities are derived. Some interesting qualitative results regarding the nature of the relative and absolute changes to the stationary probabilities are also obtained. Similar procedures are used to establish an updating procedure for mean first passage times under perturbations.

A simple analysis of thermodynamic properties for classical plasmas: I. Theory

By eliminating the short range negative divergence of the Debye–Hückel pair distributionfunction, but retaining the exponential charge screening known to operate at largeinterparticle separation, the thermodynamic properties of one-component plasmas of pointions or charged hard spheres can be well represented even in the strong coupling regime.Predicted electrostatic free energies agree within 5% of simulation data for typicalCoulomb interactions up to a factor of 10 times the average kinetic energy. Here, thisidea is extended to the general case of a uniform ionic mixture, comprising anarbitrary number of components, embedded in a rigid neutralizing background.The new theory is implemented in two ways: (i) by an unambiguous iterativealgorithm that requires numerical methods and breaks the symmetry of crosscorrelation functions; and (ii) by invoking generalized matrix inverses that maintainsymmetry and yield completely analytic solutions, but which are not uniquelydetermined. The extreme computational simplicity of the theory is attractive whenconsidering applications to complex inhomogeneous fluids of charged particles.

A note on the determinant formulas computation of generalized inverse matrix Padé approximation

In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Padé approximation of type [ n/2 k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem.

A method of multichannel PID control of two‐degree‐of‐freedom wrist joint movements by functional electrical stimulation

AbstractIn restoring motor functions of paralyzed extremities by functional electrical stimulation (FES), determination of stimulus intensities of many muscles in multichannel control is an ill‐posed problem because of redundancy in the input (stimulus intensity)‐output (joint angle) relationship of the musculoskeletal system. In this paper, we use a multi‐input and multi‐output PID controller and propose a parameter determination method for the controller, which can solve the ill‐posed problem in closed‐loop control. In the parameter determination process, the stimulus intensity‐joint angle characteristics of all muscles controlled were measured first. The elements of the matrix that transforms stimulus intensities into joint angles were determined by linear approximation of the measured characteristics. Then a generalized inverse matrix of the transformation matrix was calculated. The generalized inverse matrix and an expanded CHR method were used to determine the parameters of the PID controller. The PID controller was examined in tracking control on several trajectories of two‐degree‐of‐freedom movement of the wrist joint with neurologically intact subjects. Electrical stimulation was applied to four muscles relating to the wrist joint movements through surface electrodes. The tracking control was achieved generally with good performance under different conditions of the gravitational effect. The new method proposed in this paper was found to provide a solution of the ill‐posed problem. Multichannel closed‐loop FES control of the wrist joint could be realized with this method. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(5): 25–36, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10298

Inverse analysis method using spectral decomposition of Green's function

SUMMARY An inverse analysis method using the spectral decomposition of Green's function is proposed. For linear inverse problems of identifying inner sources from surface responses, Green's function, relates the sources to the responses. Since this function behaves as a compact operator, it admits the spectral decomposition and is expressed as a sum of distinct eigen values and eigen functions. A suitable inverse operator that maps the responses to the sources is then determined. The proposed method consists of the following three procedures: (1) numerically computing the spectral decomposition and determining the inverse operator; (2) estimating a response function using a set of measured data; and (3) predicting a source function from the response function using inverse operator. A simple inverse analysis method, which uses a pointwise discretization of Green's function and computes a generalized inverse matrix applying the singular-value decomposition, is regarded as an approximation to compute the inverse operator. The accuracy, however, is much lower than the proposed method, because of the pointwise discretization and the less accurate computation of the spectral decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method. Errors due to improper calculation of the inverse operator of Green’s function are shown.

Development of a Decision Method for Optimum Ferromagnetic Board Arrangements to Reduce Toroidal Magnetic Field Ripple

A new decision method for ferromagnetic board (FB) arrangements was developed to obtain the desired toroidal magnetic field (TF) distribution. An approximation, in which the magnetization vector is only pointed in the toroidal direction and its magnitude is magnetically saturated, was introduced to allow treatment as a linear problem. The desired thickness distribution of FBs could be obtained by calculating a generalized inverse matrix which transforms from the thickness distribution to the TF distribution due to FBs. We employed a singular value decomposition (SVD) method to calculate the generalized inverse matrix. The final TF distribution was confirmed by the general-purpose 3D non-linear magnetostatic field analysis code. This method was applied to the TF coils of JFT-2M. The maximum TF ripple in the whole plasma region could be reduced from 2.1% to 0.1%.

HYPOSAT – An Enhanced Routine to Locate Seismic Events

A program package, called HYPOSAT, has been under development that attempts to use the maximum information possible to estimate the hypocenter of a seismic source. The standard input parameters can be used: arrival times of first and later onsets with backazimuths and ray parameters (or apparent velocities). In addition, travel-time differences between different phases observed at the same station can be optionally used. The observed standard deviations are used to weight all input parameters and the inversion is done with a generalized matrix inversion code.

Generalized inverse matrix Padé approximation on the basis of scalar products

A new type of generalized matrix inverse is used to define the generalized inverse matrix Padé approximants (GMPA). GMPA is introduced on the basis of scalar product of matrices, with the form of matrix numerator and scalar denominator. It is different from the existing matrix Padé approximants in that it does not need multiplication of matrices in the construction process. Some algebraic properties are discussed. The representations of GMPA are provided with the following three forms: (i) the explicit determinantal formulas for the denominator scalar polynomials and the numerator matrix polynomials; (ii) ε-algorithm expression; (iii) Thiele-type continued fraction expression. The equivalence relations above three representations are proposed.

The ε-algorithm and η-algorithm for generalized inverse rectangular matrix pade approximants

An axiomatic definition for the generalized inverse matrix Pade approximation (GMPA) is introduced. The matrix rational approximants are of the form of the matrix-valued numerator and the scalar denominator. By means of generalized inverse for matrices, the e-algorithm for the computation of GMPA is established. The well-known Wynn identity for GMPA is proved on the basis of e-algorithm. The η-algorithm is defined in a similar way. The equivalence relation between e-algorithm and η-algorithm is proposed. Some common examples and a numerical example are given to illustrate the methods in this paper.

Thiele-type and Lagrange-type generalized inverse rational interpolation for rectangular complex matrices

A variety of matrix rational interpolation problems include the partial realization problem for matrix power series and the minimal rational interpolation problem for general matrix functions. Different from the previous work, in this paper we consider a new method of matrix rational interpolation, with rectangular real or complex interpolated matrices and distinct real or complex interpolation points. Based on an axiomatic definition for the generalized inverse matrix rational interpolants (GMRI), GMRI are constructed in the following two forms: (i) Thiele-type continued fraction expression; (ii) an explicit determinantal formula for the denominator scalar polynomials and for the numerator matrix polynomials, which are of Lagrange-type expression. As a direct application of GMRI, a matrix rational extrapolation is introduced.

Optimal truncation criterion for application of singular value decomposition to ionospheric tomography

In this paper, we present a generic method to solve the subspace‐oriented estimation problem. We have optimized our approach by taking account of the a priori and a posteriori covariances of both the data and the model parameters in general linear inverse notations. In our work, singular value decomposition (SVD) was employed to provide a robust optimal solution. In computation of the generalized matrix inverse, a very simple truncation criterion on the singular value (SV) spectrum was set up which guarantees the minimal variance of the estimate. This algorithm based on SVD produces an optimal estimate independent of computing resources. Specifically, we applied this method to studies of ionospheric tomography by inverting total electron content (TEC), which may be measured by means of satellite beacons. We processed two simulated cases. The residual variances of the a posteriori covariances of the model parameters were used as the measure to evaluate the uncertainties of the estimates. Our examples indicate that this algorithm can resolve about 60% of the a priori variance while achieving a significant decrease of the computation time by truncation of the SV spectrum.

Comment: Key agreement scheme based on generalised inverses of matrices

For original paper see E. Dawson and C.-K. Wu, ibid., vol.33, no.14, pp.1210-11 (1997). Dawson and Wu proposed a key agreement scheme based on generalised matrix inverses. They suggest that for a key length of kn, using matrices of size k/spl times/m and m/spl times/n, with k/spl les/m/spl les/n the security parameter should be at least 2/sup (m-k)n/. We give an attack for which the security parameter is at most 2/sup k2/, and a second attack which applies in at least 28% of cases and reduces the original 2/sup (m-k)n/ to 2/sup (m-k)m/ m/sup 3/.

Analytical inversion of general tridiagonal matrices

In this paper we give a complete analysis for general tridiagonal matrix inversion for both non-block and block cases, and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridiagonal matrices.

ゴルフクラブシャフトの最適設計手法に関する研究

An inverse analysis program using a sensitivity method has been developed to optimize the natural frequency of golf club shafts. In this program, a composite cylinder which has various diameters and thicknesses laminated in various fiber orientations is modeled by a single beam element to decrease the degrees of freedom of the vibration problems. Therefore, eigenvalues of laminated composite cylinders can be analyzed without influence from the aspect ratio of the element. In the sensitivity analysis, a generalized inverse matrix is utilized. As a numerical example, a CFRP golf club shaft is designed to have higher frequencies in its 2nd and 3rdmodes.