Inverse of the Vandermonde matrix with applications
Inverse of Vandermonde matrix with applications
MATRIX ALGEBRA↗
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On the computation of a generalized inverse of a matrix
Noniterative algorithm for generalized inverse of arbitrary rectangular matrix computation
La Budde, C. D.↗
Inversion of imaging spectrometry data using singular value decomposition
The use of imaging spectrometers, which acquire data that are both spectrally contiguous images and spatially contiguous spectra, for quantitative remote sensing of the earth is addressed. Such data sets cannot be analyzed fully using either existing spectroscopic or image techniques. Singular value decomposition (SVD) is used here for spectral unmixing and determination of the spatial scales of mixing. It is shown that when it is used to invert the mixing endmember library, SVD allows more insight into library characteristics and more control of the inversion process than other commonly used matrix inversion techniques.
Boardman, Joe W.↗
Spectral response from blackbody measurements
Far-infrared and submillimeter detector responsivity and spectral response measurements can be performed simultaneously by sweeping the temperature of a cooled blackbody. Such measurements yield n simultaneous linear equations for n blackbody temperatures. Matrix inversion solutions are observed to fail due to a matrix ill-conditioned for inversion. However, an unconditionally convergent iterative solution can be performed. Results for a gallium-doped germanium detector are described.
Moore, W. J.↗
Inversion Of Jacobian Matrix For Robot Manipulators
Report discusses inversion of Jacobian matrix for class of six-degree-of-freedom arms with spherical wrist, i.e., with last three joints intersecting. Shows by taking advantage of simple geometry of such arms, closed-form solution of Q=J-1X, which represents linear transformation from task space to joint space, obtained efficiently. Presents solutions for PUMA arm, JPL/Stanford arm, and six-revolute-joint coplanar arm along with all singular points. Main contribution of paper shows simple geometry of this type of arms exploited in performing inverse transformation without any need to compute Jacobian or its inverse explicitly. Implication of this computational efficiency advanced task-space control schemes for spherical-wrist arms implemented more efficiently.
Fijany, Amir↗
On the concept of a p-q generalized inverse of a matrix.
P-Q norms generalized inverse concept of matrix derived from extension of Penrose best approximate solution of linear equations
Newman, T. G.↗
A special form of a generalized inverse of an arbitrary complex matrix
Equations for inversion of singular matrix - special form of generalized inverse of arbitrary complex matrix
INVERSION↗
On the use of generalized inverses in function minimization
Minimization algorithms unified derivation, using Hessian matrix inverse
Tewarson, R. P.↗
On the inverses of certain matrices.
Recursive algorithm for identification of time varying parameters of system, noting use for matrix inversion
TIME FUNCTION↗
Special computational procedures for differential equations
Analytical study of ordinary and partial differential equations, illustrating initial value problem and large matrix inversion
INITIAL VALUE PROBLEM↗
Convergence of the SMI and the diagonally loaded SMI algorithms with weak interference
Approximations for the power levels at the output of an adaptive array that uses the diagonally loaded sample-matrix inversion (SMI) algorithm are derived. Diagonal loading is a technique where the diagonal of the covariance matrix is augmented with a positive or negative constant prior to inversion. This paper examines how the signal-to-interference-plus-noise ratio (SINR) and signal-to-interference ratio (SIR) at the array output vary with the number of samples taken when the input signals are CW. It is shown that positive loading produces more rapid convergence with a reduction in output SIR. Negative loading provides an improved SIR level, but positive loading produces more rapid convergence. Negative loading provides an improved SIR level, but the output power levels are erratic and slow to converge. Simulation results which verify the theoretical procedure are given.
Ganz, Matthew W.↗
Effects of the oceans on polar motion: Extended investigations
A method was found for expressing the tide current velocities in terms of the tide height (with all variables expanded in spherical harmonics). All time equations were then combined into a single, nondifferential matrix equation involving only the unknown tide height. The pole tide was constrained so that no tidewater flows across continental boundaries. The constraint was derived for the case of turbulent oceans; with the tide velocities expressed in terms of the tide height. The two matrix equations were combined. Simple matrix inversion then yielded the constrained solution. Programs to construct and invert the matrix equations were written. Preliminary results were obtained and are discussed.
Dickman, Steven R.↗
On the fixed point probability vector of regular or ergodic transition matrices.
Fixed point probability row vector of regular or ergodic transition matrices simplified using generalized matrix inversion theory
MATRIX ANALYSIS↗
Time-resolved diffusion tomographic 2D and 3D imaging in highly scattering turbid media
A method for imaging objects in highly scattering turbid media. According to one embodiment of the invention, the method involves using a plurality of intersecting source/detectors sets and time-resolving equipment to generate a plurality of time-resolved intensity curves for the diffusive component of light emergent from the medium. For each of the curves, the intensities at a plurality of times are then inputted into the following inverse reconstruction algorithm to form an image of the medium: wherein W is a matrix relating output at source and detector positions r.sub.s and r.sub.d, at time t, to position r, .LAMBDA. is a regularization matrix, chosen for convenience to be diagonal, but selected in a way related to the ratio of the noise, to fluctuations in the absorption (or diffusion) X.sub.j that we are trying to determine: .LAMBDA..sub.ij =.lambda..sub.j .delta..sub.ij with .lambda..sub.j = /<.DELTA.Xj.DELTA.Xj> Y is the data collected at the detectors, and X.sup.k is the kth iterate toward the desired absorption information. An algorithm, which combines a two dimensional (2D) matrix inversion with a one-dimensional (1D) Fourier transform inversion is used to obtain images of three dimensional hidden objects in turbid scattering media.
Alfano, Robert R.↗
Time-resolved diffusion tomographic 2D and 3D imaging in highly scattering turbid media
A method for imaging objects in highly scattering turbid media. According to one embodiment of the invention, the method involves using a plurality of intersecting source/detectors sets and time-resolving equipment to generate a plurality of time-resolved intensity curves for the diffusive component of light emergent from the medium. For each of the curves, the intensities at a plurality of times are then inputted into the following inverse reconstruction algorithm to form an image of the medium: ##EQU1## wherein W is a matrix relating output at source and detector positions r.sub.s and r.sub.d, at time t, to position r, .LAMBDA. is a regularization matrix, chosen for convenience to be diagonal, but selected in a way related to the ratio of the noise, to fluctuations in the absorption (or diffusion) X.sub.j that we are trying to determine: .LAMBDA..sub.ij =.lambda..sub.j .delta..sub.ij with .lambda..sub.j = /<.DELTA.Xj.DELTA.Xj> Y is the data collected at the detectors, and X.sup.k is the kth iterate toward the desired absoption information. An algorithm, which combines a two dimensional (2D) matrix inversion with a one-dimensional (1D) Fourier transform inversion is used to obtain images of three dimensional hidden objects in turbid scattering media.
Alfano, Robert R.↗
Elastic lithosphere thickness on the moon from mare tectonic features - A formal inversion
The thickness (T) of the lunar elastic lithosphere at the time (3.6 to 3.8 billion years ago) of the earliest preserved basalt flows in circular mare basins can be estimated by inverting the observed locations of extensional tectonic features in and surrounding the maria. In performing the inversion, the lithosphere is modeled as an elastic shell with a liquid interior, and the basalt load for each mare is approximated by a set of concentric cylinders. To permit solving the forward problem of placing radial limits on the positions of the rilles around a given mare, an additional parameter F, the ratio of the radial stress at the radial limits to the maximum radial stress, is introduced. T and F are chosen to give the best weighted-squares fit of the radial limits to the observations, and are used as the initial values in a linearized matrix inversion to check the resolution and estimate errors. The application of the procedure to three maria with prominent extensional features, Humorum, Orientale, and Serenitatis, gives values of T from about 40 + or - 10 to 50 + or - 10 km, and in each case the linearized matrix equation has an exact inverse.
Comer, R. P.↗
The OMPS Limb Profiler Instrument: Two-Dimensional Retrieval Algorithm
The upcoming Ozone Mapper and Profiler Suite (OMPS), which will be launched on the NPOESS Preparatory Project (NPP) platform in early 2011, will continue monitoring the global distribution of the Earth's middle atmosphere ozone and aerosol. OMPS is composed of three instruments, namely the Total Column Mapper (heritage: TOMS, OMI), the Nadir Profiler (heritage: SBUV) and the Limb Profiler (heritage: SOLSE/LORE, OSIRIS, SCIAMACHY, SAGE III). The ultimate goal of the mission is to better understand and quantify the rate of stratospheric ozone recovery. The focus of the paper will be on the Limb Profiler (LP) instrument. The LP instrument will measure the Earth's limb radiance (which is due to the scattering of solar photons by air molecules, aerosol and Earth surface) in the ultra-violet (UV), visible and near infrared, from 285 to 1000 nm. The LP simultaneously images the whole vertical extent of the Earth's limb through three vertical slits, each covering a vertical tangent height range of 100 km and each horizontally spaced by 250 km in the cross-track direction. Measurements are made every 19 seconds along the orbit track, which corresponds to a distance of about 150km. Several data analysis tools are presently being constructed and tested to retrieve ozone and aerosol vertical distribution from limb radiance measurements. The primary NASA algorithm is based on earlier algorithms developed for the SOLSE/LORE and SAGE III limb scatter missions. All the existing retrieval algorithms rely on a spherical symmetry assumption for the atmosphere structure. While this assumption is reasonable in most of the stratosphere, it is no longer valid in regions of prime scientific interest, such as polar vortex and UTLS regions. The paper will describe a two-dimensional retrieval algorithm whereby the ozone distribution is simultaneously retrieved vertically and horizontally for a whole orbit. The retrieval code relies on (1) a forward 2D Radiative Transfer code (to model limb radiances within a non-uniform atmosphere and evaluate 2D analytical partial derivatives) and (2) an optimal estimator inversion routine. The algorithm uses the typically sparse nature of the kernel matrices as well as fast matrix inversion techniques to allow for fast inversion of limb data with efficient memory management (as was done for MIPAS data processing). While the method has so far only been developed in the context of Single Scatter, the paper will show how the CPU intensive Multiple Scatter modeling can be implemented using parallel CPU processing. Initial results will be presented in terms of retrieved ozone profiles and code performance.
Rault, Didier F.↗