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NASA NTRS1973

Application of Fourier analysis to multispectral/spatial recognition

One approach for investigating spectral response from materials is to consider spatial features of the response. This might be accomplished by considering the Fourier spectrum of the spatial response. The Fourier Transform may be used in a one-dimensional to multidimensional analysis of more than one channel of data. The two-dimensional transform represents the Fraunhofer diffraction pattern of the image in optics and has certain invariant features. Physically the diffraction pattern contains spatial features which are possibly unique to a given configuration or classification type. Different sampling strategies may be used to either enhance geometrical differences or extract additional features.

Hornung, R. J.↗

NASA NTRS1975

A study to analyze six band multispectral images and fabricate a Fourier transform detector

An automatic Fourier transform diffraction pattern sampling system, used to investigate techniques for forestry classification of six band multispectral aerial photography is presented. Photographs and diagrams of the design, development and fabrication of a hybrid optical-digital Fourier transform detector are shown. The detector was designed around a concentric ring fiber optic array. This array was formed from many optical fibers which were sorted into concentric rings about a single fiber. All the fibers in each ring were collected into a bundle and terminated into a single photodetector. An optical/digital interface unit consisting of a high level multiplexer, and an analog-to-digital amplifier was also constructed and is described.

Shackelford, R. G.↗

NASA NTRS1975

Application of Fourier transforms for microwave radiometric inversions

Existing microwave radiometer technology now provides a suitable method for remote determination of the ocean surface's absolute brightness temperature. To extract the brightness temperature of the water from the antenna temperature, an unstable Fredholm integral equation of the first kind is solved. Fourier transform techniques are used to invert the integral after it is placed into a cross correlation form. Application and verification of the methods to a two-dimensional modeling of a laboratory wave tank system are included. The instability of the ill-posed Fredholm equation is examined and a restoration procedure is included which smooths the resulting oscillations. With the recent availability and advances of fast Fourier transform (FFT) techniques, the method presented becomes very attractive in the evaluation of large quantities of data.

Holmes, J. J.↗

NASA NTRS1977

A Fourier approach to cloud motion estimation

A Fourier technique is described for estimating cloud motion from pairs of pictures using the phase of the cross spectral density. The method allows motion estimates to be made for individual spatial frequencies, which are related to cloud pattern dimensions. Results obtained are presented and compared with the results of a Fourier domain cross correlation scheme. Using both artificial and real cloud data show that the technique is relatively sensitive to the presence of mixtures of motions, changes in cloud shape, and edge effects.

Arking, A.↗

NASA NTRS1978

A new hybrid algorithm for computing a fast discrete Fourier transform

For certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform is extended considerably. This is accomplished by performing the cyclic convolution, required by Winograd's method, with the Mersenne-prime number theoretic transform. This new algorithm requires fewer multiplications than either the standard fast Fourier transform or Winograd's more conventional algorithm.

Reed, I. S.↗

NASA NTRS1978

Far-field radiation patterns of aperture antennas by the Winograd Fourier transform algorithm

A more time-efficient algorithm for computing the discrete Fourier transform, the Winograd Fourier transform (WFT), is described. The WFT algorithm is compared with other transform algorithms. Results indicate that the WFT algorithm in antenna analysis appears to be a very successful application. Significant savings in cpu time will improve the computer turn around time and circumvent the need to resort to weekend runs.

Heisler, R.↗

NASA NTRS1979

A new hybrid algorithm for computing a fast discrete Fourier transform

In this paper for certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform (DFT) is extended considerably. This is accomplished by performing the cyclic convolution, required by Winograd's method, with the Mersenne prime number-theoretic transform developed originally by Rader. This new algorithm requires fewer multiplications than either the standard fast Fourier transform (FFT) or Winograd's more conventional algorithm. However, more additions are required.

Reed, I. S.↗

NASA NTRS1980

Infrared Fourier spectrometer for airborne and ground-based astronomy

A unique high resolving power near-infrared, astronomical Fourier spectrometer has been constructed for use at both ground-based and airborne telescopes. Its capabilities include a limiting resolution of 0.01/cm and spectral coverage over the InSb detector sensitivity region (0.8-5.6 micron). The instrument is optimized for operation in environments inaccessible to other high-resolution Fourier spectrometers, namely, the Cassegrain focuses of conventional telescopes and the NASA Kuiper Airborne Observatory's 91-cm telescope. Details of the spectrometer are discussed, and various astronomical and laboratory spectra are presented.

Davis, D. S.↗

NASA NTRS1981

Products of multiple Fourier series with application to the multiblade transformation

A relatively simple and systematic method for forming the products of multiple Fourier series using tensor like operations is demonstrated. This symbolic multiplication can be performed for any arbitrary number of series, and the coefficients of a set of linear differential equations with periodic coefficients from a rotating coordinate system to a nonrotating system is also demonstrated. It is shown that using Fourier operations to perform this transformation make it easily understood, simple to apply, and generally applicable.

Kunz, D. L.↗

NASA NTRS1981

Inference of sea surface temperature, near surface wind, and atmospheric water by Fourier analysis of Scanning Multichannel Microwave Radiometer data

The Scanning Multichannel Microwave Radiometer measures thermal microwave emission from the earth in both polarizations at wavelengths of 0.8, 1.4, 1.7, 2.8 and 4.6 cm. Similar instruments were launched on Nimbus 7 and Seasat. Both spatial resolution on the earth and relative sensitivity to different geophysical parameters change with wavelength. Therefore, spatial Fourier components of geophysical parameters are inferred from the corresponding Fourier components of the radiometer measurements, taking into account the different dependence of signal-to-noise ratio on spatial frequency for each radiometer wavelength. The geophysical parameters are sea surface temperature, near-surface wind speed, integrated water vapor mass, integrated liquid water mass, and the product of rainfall rate with height of the rain layer. The capabilities and limitations of the inversion method are illustrated by means of data from the North Atlantic and from tropical storms.

Rosenkranz, P. W.↗

NASA NTRS1981

A time Fourier analysis of zonal averaged ozone heating rates

A time-Fourier analysis is presented for the yearly variation of the zonal averaged ozone heating rates in the middle atmosphere based on a model study. The ozone heating rates are determined by utilizing two-dimensional ozone distributions, the altitude and latitude, and by including the effect of the curved earth's atmosphere. In addition, assumptions are introduced to the yearly variations of the ozone distributions due to the lack of sufficient existing ozone data. Among other results, it is shown that the first harmonic component indicates that the heating rates are completely out of phase between the northern and southern hemispheres. The second Fourier component shows a symmetric pattern with respect to the equator, as well as five distinct local extreme values of the ozone heating rate. The third harmonic component shows a pattern close to that of the first component except in the regions above 70 deg between 45-95 km in both hemispheres.

Wang, P.-H.↗

NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

Hussaini, M. Y.↗

NASA NTRS1984

Geometric interpretations of the Discrete Fourier Transform (DFT)

One, two, and three dimensional Discrete Fourier Transforms (DFT) and geometric interpretations of their periodicities are presented. These operators are examined for their relationship with the two sided, continuous Fourier transform. Discrete or continuous transforms of real functions have certain symmetry properties. The symmetries are examined for the one, two, and three dimensional cases. Extension to higher dimension is straight forward.

Campbell, C. W.↗

NASA NTRS1984

Analysis and application of Fourier transform spectroscopy in atmospheric remote sensing

An analysis method for Fourier transform spectroscopy is summarized with applications to various types of distortion in atmospheric absorption spectra. This analysis method includes the fast Fourier transform method for simulating the interferometric spectrum and the nonlinear least-squares method for retrieving the information from a measured spectrum. It is shown that spectral distortions can be simulated quite well and that the correct information can be retrieved from a distorted spectrum by this analysis technique.

Park, J. H.↗

NASA NTRS1985

The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx

It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation.

Tal-Ezer, H.↗

NASA NTRS1985

Spectral methods for the Euler equations. I - Fourier methods and shock capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collociation methods are put into historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock-capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem.

Hussaini, M. Y.↗

NASA NTRS1985

Three-dimensional radiative transfer using a Fourier-transform matrix-operator method

The three-dimensional equation of transfer for a scattering medium with planar geometry is solved by using a spatial Fourier transform and extending matrix-operator techniques developed previously for the one-dimensional equation. Doubling and adding algorithms were derived by means of an interaction principle for computing the Fourier-transformed radiation field. The resulting expressions fully describe the radiative transfer process in a scattering medium, inhomogeneous in the x-, y- and z-directions, illuminated from above by an arbitrarily general intensity field and bounded from below by a surface with completely general reflection properties.

Martonchik, J. V.↗

NASA NTRS1986

Fourier transform techniques for the evaluation of the Thematic Mapper line spread function

An endeavor was made to use Fourier transform techniques to estimate the line spread function (LSF) of the Thematic Mapper (TM) from processed data. Data near the boundary of small fields from TM scenes were sampled, fitted to a theoretical curve, lowpass filtered, and then deconvolved to obtain the LSF. Lowpass filtering improved the truncation effect and eliminated the discontinuity at both ends of the data sequence.The selection of the cutoff frequency and the order of a lowpass filter is based on a newly defined criterion which tests the distortion of the processed edge by comparing it with the original edge. A parameter of the LSF, the equivalent width, is used to compare these LSF's deconvolved from sampled edge data with those evaluated by the derivative techniques. The calculated equivalent widths using the Fourier transform techniques for infrared bands are very close to TM design specifications.

Carnahan, W. H.↗