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Three methods have been described for computing the two-dimensional discrete Hartley transform. Two of these employ a separable transform, the third method, the vector-radix algorithm, does not require separability. In-place computation of the vector-radix method is described. Operation counts and execution times indicate that the vector-radix method is fastest.
It is possible to calculate expectation values and transition probabilities from the Wigner phase-space distribution function. Based on the canonical transformation properties of the Wigner function, an algorithm is developed for calculating these quantities in quantum optics for coherent and squeezed states. It is shown that the expectation value of a dynamical variable can be written in terms of its vacuum expectation value of the canonically transformed variable. Parallel-axis theorems are established for the photon number and its variant. It is also shown that the transition probability between two squeezed states can be reduced to that of the transition from one squeezed state to vacuum.
This paper develops techniques to evaluate the discrete Fourier transform (DFT), the autocorrelation function (ACF), and the cross-correlation function (CCF) of time series which are not evenly sampled. The series may consist of quantized point data (e.g., yes/no processes such as photon arrival). The DFT, which can be inverted to recover the original data and the sampling, is used to compute correlation functions by means of a procedure which is effectively, but not explicitly, an interpolation. The CCF can be computed for two time series not even sampled at the same set of times. Techniques for removing the distortion of the correlation functions caused by the sampling, determining the value of a constant component to the data, and treating unequally weighted data are also discussed. FORTRAN code for the Fourier transform algorithm and numerical examples of the techniques are given.
New algorithms were derived for computing terrain corrections, all components of the attraction of the topography at the topographic surface and the gradients of these attractions. These algoriithms utilize fast Fourier transforms, but, in contrast to methods currently in use, all divergences of the integrals are removed during the analysis. Sequential methods employing a smooth intermediate reference surface were developed to avoid the very large transforms necessary when making computations at high resolution over a wide area. A new method for the numerical solution of Molodensky's problem was developed to mitigate the convergence difficulties that occur at short wavelengths with methods based on a Taylor series expansion. A trial field on a level surface is continued analytically to the topographic surface, and compared with that predicted from gravity observations. The difference is used to compute a correction to the trial field and the process iterated. Special techniques are employed to speed convergence and prevent oscillations. Three different spectral methods for fitting a point-mass set to a gravity field given on a regular grid at constant elevation are described. Two of the methods differ in the way that the spectrum of the point-mass set, which extends to infinite wave number, is matched to that of the gravity field which is band-limited. The third method is essentially a space-domain technique in which Fourier methods are used to solve a set of simultaneous equations.
Performance in a discrete aiming task was compared under several transformed visual/motor mappings: rotations by 45, 90, 135, and 180 deg and reflections about the horizontal and the vertical midlines. Eight aiming targets were used, corresponding to eight directions of movement: up, down, right, left, up-right, down-left, up-left, and down-right. Direction of movements were characterized in terms of separable visual and motor components, and two kinds of direction of movement effects were considered. First, a direction of movement effect paralleling that seen in rapid aiming under the usual nontransformed mapping. Second, because rotations, but not reflections, are physically realizable 2-D transformations, a visual/motor control system which is sensitive to physical constraints should perform reflections, but not rotations, in a piecemeal fashion. Results supported the hypothesis that a motor factor having to do with complexity of limb movement accounts for differences in movement accuracy between right and left oblique directions. Direction of movement effects were more evident in reflections than in rotations, and were consistent with the hypothesis that the visual/motor-control system seeks a physically realizable 2-D rotation solution to reflections. Results also suggested that reversal of two orthogonal basis dimensions is far less difficult than reversing only one and leaving the other intact.
The number of component part (nodes, struts and panels) termed part count, is an important factor in the design, manufacture, and assembly of modular space structures. Part count expressions are presented for a variety of profiles derived from the layered octahedral-tetrahedral truss configuration. Referred to as the tetrahedral truss in the NASA projects, this specific geometry has been used in several missions. The general expressions presented here transforms to others as one profile changes to another. Such transformational part count relations provide a measure of flexibility and generality, and may be useful when dealing with a wider range of geometric configurations.
Phase transformation kinetics that occur by a nucleation and growth process are investigated. A simple discrete space and time model is used for the dynamics and analytical results are obtained for the volume fraction of the material transformed for both finite systems and a special example of an inhomogeneously nucleated system. The theory is developed for two cases, initial nucleation, and continuous nucleation. The results are compared with simulations of the model.
Off-center vector magnetograms which use all three components of the measured field provide the maximum information content from the photospheric field and can provide the most consistent potential field independent of the viewing angle by defining the normal component of the field. The required transformations of the magnetic field vector and the geometric mapping of the observed field in the image plane into the heliographic plane have been described. Here we discuss the total transformation of specific vector magnetograms to detail the problems and procedures that one should be aware of in analyzing observational magnetograms. The effect of the 180-deg ambiguity of the observed transverse field is considered as well as the effect of curvature of the photosphere. Specific results for active regions AR 2684 (September 23, 1980) and AR 4474 (April 26, 1984) from the Marshall Space Flight Center Vector magnetograph are described which point to the need for the heliographic projection in determining the field structure of an active region.
Geometric image transformations are of interest to pattern recognition algorithms for their use in simplifying some aspects of the pattern recognition process. Examples include reducing sensitivity to rotation, scale, and perspective of the object being recognized. The NASA Programmable Remapper can perform a wide variety of geometric transforms at full video rate. An architecture is proposed that extends its abilities and alleviates many of the first version's shortcomings. The need for the improvements are discussed in the context of the initial Programmable Remapper and the benefits and limitations it has delivered. The implementation and capabilities of the proposed architecture are discussed.
A coordinate time, t(A), with absolute synchronization is defined as an alternative fourth spatial coordinate for model universes with flat space-time, and the theoretical implications of t(A) are explored in detail. Particular attention is given to a t(A)-based reformulation of the Lorentz transformation, the generalized Galilean transformation, which is found to offer significant advantages in understanding special-relativistic phenomena such as length contraction, time dilation, and the interaction of objects with the physical vacuum. With respect to astrophysical observations of superluminal motion, it is shown that the problem of causality violation can be avoided; the theory also predicts that weak anisotropic effects may be detectable in the earth reference frame.
The kinetics of the olivine-spinel transformation in Ni2SiO4 were investigated in an in situ X-ray diffraction experiments in which synchrotron radiation was used as an X-ray source. The starting material was Ni2SO4 olivine which was hot-pressed in situ at 980 C and 2.5 GPa; during the transformation, X-ray diffraction patterns were collected at intervals of 30 or 120 sec. The kinetic data were analyzed using Cahn's (1956) model. The activation energy for growth at 3.6-3.7 GPa was estimated as 438 + or - 199 kJ/mol. It is shown that, in order to make significant extrapolations of the kinetic data to a geological scale, the dependence of the rates of both nucleation and growth on temperature and pressure must be evaluated separately.
Report describes experimental study of phase transformations occurring in monophasic and diphasic xerogels of overall stoichiometric mullite composition during thermal treatment. Mullite valued as refractory material because of relatively high mechanical strength, excellent resistance to thermal shock, high resistance to creep, low density, low thermal conductivity, and stability at high temperature. Part of continuing effort to develop precursor materials transformed into mullite and sintered at temperatures low enough to avoid degradation of non-mullite reinforcing fibers in manufacture of mullite/fiber composites.
Transformations based on polynomial matrices of finite degree developed for use in computing functions for compensation, inversion, and approximation of discrete-time, multivariable, linear systems. Method derived from z-transform transfer-function form of matrices. Applicable to cascade-compensation problems in design of control systems.
Modified versions of the Hilbert transform pair and the Kramers-Kronig relations are derived for the complex permittivity of a plasma/dielectric medium which is singular at the frequency of the applied electric field equal to 0. Such a complex permittivity exists when the plasma/dielectric model allows a loss term but no restoring term. Permittivity, in which both loss and restoring terms are included, is shown to satisfy the standard Hilbert transform pair and, thus, the Kramers-Kronig relations.
The Long Duration Exposure Facility's impact signature record and the size frequency distribution of craters and perforations offers a unique record of environmental data referenced conveniently to the geocentric reference frame. Chemical analysis of residues can offer only limited assistance. Hence, flux modelling has been developed to transform from both geocentric orbital distributions and geocentrically unbounded interplanetary source distributions. This is applied to the foil and crater penetration records in the Ram (E), Trailing (W), and Space Pointing directions to offer the means of decoding the records. It shows that the mix of the components is size dependent. Parametric forms of the modelling transformations are presented for the orbital and unbound populations.
Geodetic VLBI measurements were used to test whether the Pacific-North American plate velocity averaged over several years of direct observation (1984-1987) equals that averaged over millions of years. It was also tested whether this velocity parallels the San Andreas fault, transform faults and earthquake slip vectors in the Gulf of California, and earthquake slip vectors along the Queen Charlotte fault, along the Alaskan peninsula, and along the Kamchatkan peninsula. The VLBI data provide an estimate of the direction of plate motion that is independent of estimates from fault azimuths and earthquake slip vectors. The Euler vector determined from VLBI was found to be nearly identical to the Euler vector of plate motion model NUVEL-1, which is based on the trends of transform faults, earthquake slip vectors, and spreading rates from marine magnetic anomalies that average motion since 3 Ma. The velocity between the Pacific and North American plates averaged over the past several years equals or nearly equals its velocity averaged over the past several million years, the difference along their boundary nowhere exceeding 4 + or - 7 mm/yr.
Laboratory investigations have been conducted on the effects of variations in sulfur sample histories on their solid-state transformation rate and the corresponding spectral variation of freshly frozen sulfur. The temporal variations in question may be due to differences in the amount and type of metastable allotropes present in the sulfur after solidification, as well as to the physics of the phase-transformation process itself. The results obtained are pertinent to the physical behavior and spectral variation of such freshly solidified sulfur as may exist on the Jupiter moon Io; this would initially solidify into a glassy solid or monoclinic crystalline lattice, then approach ambient dayside temperatures. Laboratory results imply that the monoclinic or polymeric allotropes can in these circumstances be maintained, and will take years to convert to the stable orthorhombic crystalline form.
A technique using data association target tracking in a motion sequence via an adaptive joint transform correlator is presented. The massive data in the field of view can be reduced to a few correlation peaks. The average velocity of a target during the tracking cycle is then determined from the location of the correlation peak. A data-association algorithm is used for the analysis of these correlation signals, for which multiple targets can be tracked. A phase-mostly liquid-crystal TV is used in the hybrid joint transform correlation system, and simultaneous tracking of three targets is demonstrated.