Showing papers on "Fourier transform published in 1972"
01 Jan 1972
TL;DR: In this article, the probability density, Fourier transforms and characteristic functions, joint statistics and statistical independence, Correlation functions and spectra, the central limit theorem, and the relation functions are discussed.
Abstract: This chapter contains sections titled: The probability density, Fourier transforms and characteristic functions, Joint statistics and statistical independence, Correlation functions and spectra, The central limit theorem
3,260 citations
TL;DR: It is established that the Fourier series expansion is optimal and unique with respect to obtaining coefficients insensitive to starting point and the amplitudes are pure form invariants as well as are certain simple functions of phase angles.
Abstract: A method for the analysis and synthesis of closed curves in the plane is developed using the Fourier descriptors FD's of Cosgriff [1]. A curve is represented parametrically as a function of arc length by the accumulated change in direction of the curve since the starting point. This function is expanded in a Fourier series and the coefficients are arranged in the amplitude/phase-angle form. It is shown that the amplitudes are pure form invariants as well as are certain simple functions of phase angles. Rotational and axial symmetry are related directly to simple properties of the Fourier descriptors. An analysis of shape similarity or symmetry can be based on these relationships; also closed symmetric curves can be synthesized from almost arbitrary Fourier descriptors. It is established that the Fourier series expansion is optimal and unique with respect to obtaining coefficients insensitive to starting point. Several examples are provided to indicate the usefulness of Fourier descriptors as features for shape discrimination and a number of interesting symmetric curves are generated by computer and plotted out.
1,973 citations
TL;DR: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described and some considerations of the technical realizability of a fast preprocessing system for reading printed text are included.
Abstract: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described. The ordinary Fourier coefficients are difficult to use as input to categorizers because they contain factors dependent upon size and rotation as well as an arbitrary phase angle. From these Fourier coefficients, however, other more useful features can easily be derived. By using these derived property constants, a distinction can be made between genuine shape constants and constants representing size, location, and orientation. The usefulness of the method has been tested with a computer program that was used to classify 175 samples of handprinted letters, e.g., 7 sets of the 25 letters A to Z. In this test, 98 percent were correctly recognized when a simple nonoptimized decision method was used. The last section contains some considerations of the technical realizability of a fast preprocessing system for reading printed text.
649 citations
Book•
01 Jan 1972
TL;DR: Fourier Integrals, Fourier Series, and Integrals on Groups: A Historical Introduction.
Abstract: Historical Introduction. Fourier Series. Fourier Integrals. Fourier Integrals and Complex Function Theory. Fourier Series and Integrals on Groups. Additional Reading. Bibliography.
637 citations
299 citations
TL;DR: The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} and an eigenvector basis is constructed for the DFT.
Abstract: The principal results of this paper are listed as follows. 1) The eigenvalues of a suitably normalized version of the discrete Fourier transform (DFT) are {1, -1,j, -j} . 2) An eigenvector basis is constructed for the DFT. 3) The multiplicities of the eigenvalues are summarized for an N×N transform as follows.
232 citations
TL;DR: In this paper, an efficient and practical method of simulating stationary and non-stationary random envelope processes is presented, in which the stationary envelope processes are simulated by using the fast Fourier transform while the nonstationary envelope process is simulated as the square root of the sum of a series of cosine functions with random phase angles.
195 citations
TL;DR: This direct method is shown to be equivalent to the Fourier method and also to certain methods of reconstruction developed for the solution of analogous problems in other fields, and to be very suitable for reconstructing the density of objects with rotational or helical symmetry.
Abstract: Under certain conditions it is possible to obtain the approximate projected density of a specimen from an electron micrograph, and by taking electron micrographs in different directions a number of different projections of a specimen can be determined. Several methods have been proposed for reconstructing the three-dimensional density distribution of an object from a series of projections whose planes are equally spaced in angle about an axis. One of these methods uses the Fourier transforms of the projections, while other methods operate directly on the projection data. The limitations of the direct method of 'back-projection' are discussed, and it is shown how a valid reconstruction may be achieved by back-projection from modified projected densities. This direct method is shown to be equivalent to the Fourier method and also to certain methods of reconstruction developed for the solution of analogous problems in other fields. An expression is given for the resolution of the reconstruction obtainable from a finite number of projections. Because of its computational simplicity this method is very suitable for reconstructing the density of objects with rotational or helical symmetry.
194 citations
TL;DR: In this article, the angular correlation function of a stationary optical field is introduced, which characterizes the correlation that exists between the complex amplitudes of any two plane waves in the angular spectrum description of the statistical ensemble that represents the field.
Abstract: In the first part of this paper, the concept of the angular correlation function of a stationary optical field is introduced. This function characterizes the correlation that exists between the complex amplitudes of any two plane waves in the angular spectrum description of the statistical ensemble that represents the field. Relations between this function and the more commonly known correlation functions are derived. In particular, it is shown that the angular correlation function is essentially the four-dimensional spatial Fourier transform of the cross-spectral density function of the source. The angular correlation function is shown to characterize completely the second-order coherence properties of the far field. An expression for the intensity distribution in the far zone of a field generated by a source of any state of coherence is deduced. Some generalizations of the far-zone form of the Van Cittert–Zernike theorem are also obtained.
TL;DR: In this paper, a standard [ π - HSP - t - π /2 - T ] n inversion recovery pulse sequence can be used to eliminate the residual HDO resonance from the spectrum.
TL;DR: In this article, the Fourier transform technique was used to obtain high resolution spectra and/or relaxation of chemically shifted nuclei under these extreme conditions, at pressures up to 5 kilobar and temperatures from −50 to 350°C.
Abstract: Instrumentation for the measurement of spin lattice relaxation times in liquids at pressures up to 5 kilobar and temperatures from −50 to 350°C is described. The experimental setup allows the use of the Fourier transform technique to obtain high resolution spectra and/or relaxation of chemically shifted nuclei under these extreme conditions.
TL;DR: In this paper, two simple opening-mode finite strip problems are discussed and a dynamic steady-state solution within the realms of the classical theory of elasticity is provided. But the complete stress and displacement distributions are difficult to obtain, and no attempt is made to arrive at this.
Abstract: Two simple opening-mode finite strip problems are discussed. The discussion is limited to a dynamic steady-state solution within the realms of the classical theory of elasticity. It is shown that by using Fourier transform methods, the problems are reduced to equations of the Wiener-Hopf type. The complete stress and displacement distributions are difficult to obtain, and no attempt is made to arrive at this. By application of the asymptotic properties of the Fourier transform the stress-intensity factor is, instead, derived.
TL;DR: In this article, a time-dependent conditional phase-space distribution function for rigid ensembles of rigid molecules undergoing collision-interrupted free rotation is derived for the J•diffusion and the M•Diffusion models.
Abstract: Time‐dependent conditional phase‐space distribution functions are derived for classical ensembles of rigid molecules undergoing collision‐interrupted free rotation. The J‐diffusion and the M‐diffusion models proposed by Gordon, [J. Chem. Phys. 44, 1830 (1966)] are explored in detail. The expressions for the conditional distribution functions are evaluated for these models in terms of multiple time integrals. It is then shown that Fourier transform techniques can be used to express the integrals as convolutions which are analogous to Fixman and Rider's expressions [J. Chem. Phys. 51, 2425 (1969)]. A number of numerical results are presented.
TL;DR: The Walsh power spectrum of a sequence of random samples is defined as the Walsh transform of the logical autocorrelation function of the random sequence and the Fourier power spectrum can be obtained from the Walsh power Spectrum by a linear transformation.
Abstract: The Walsh power spectrum of a sequence of random samples is defined as the Walsh transform of the logical autocorrelation function of the random sequence. The "logical" autocorrelation function is defined in a similar form as the "arithmetic" autocorrelation function. The Fourier power spectrum, which is defined as the Fourier transform of the arithmetic autocorrelation function, can be obtained from the Walsh power spectrum by a linear transformation. The recursive relations between the logical and arithmetic auto-correlation functions are derived in this paper. For a given process with computed or modeled autocorrelation function the Fourier and Walsh power spectra are computed by using the fast Fourier and Walsh transforms, respectively. Examples are given from the speech and imagery data.
TL;DR: In this article, a self-consistent decoupling procedure analogous to the Hartree-Fock decoupled for the electron propagator was proposed for finding the particle-hole propagator.
Abstract: A formal scheme is developed for finding the particle‐hole propagator by means of a self‐consistent decoupling procedure analogous to the Hartree‐Fock decoupling for the electron propagator. The self‐consistency comes from a contour integration of the Fourier transform of the propagator, utilizing a method introduced by Coulson. The method yields oscillator strengths and excitation energies, as well as the two‐particle density matrix, which allows any one‐ or two‐particle operator expectation value to be evaluated. The stability of the scheme is discussed, and comparisons with other, related, approximations are made.
IBM1
TL;DR: The requirements for the achromatization of an irradiance distribution which is a function of a space coordinate multiplied by wavelength raised to a power are shown and the theory is applied to the following problems: achrom atization of Newton's ring patterns and Fraunhofer diffraction patterns and frequency plane.
Abstract: This paper shows the requirements for the achromatization of an irradiance distribution which is a function of a space coordinate multiplied by wavelength raised to a power. The particular requirements for achromatic Fourier transformation are then presented. The theory is applied to the following problems: achromatization of Newton’s ring patterns and Fraunhofer diffraction patterns and frequency plane filtering of an object illuminated with a broadband light source. Experimental systems that perform these functions are presented.
TL;DR: It is shown that by adopting the random phase shifters of the phase quantization levels beyond 2, the effectiveness becomes about twice that of 2, and that the reduction of the effectiveness by the coincidence between a pattern of the information and a phase shifter can be withdrawn under the probability of 10(-10).
Abstract: The random phase shifter method for Fourier transformed holograms is discussed. It is shown that by adopting the random phase shifters of the phase quantization levels beyond 2 (4, for example), the effectiveness becomes about twice that of 2, and that the reduction of the effectiveness by the coincidence between a pattern of the information and a phase shifter can be withdrawn under the probability of 10(-10). Hologram memories of information storage density of 10(5) bits/mm(2) and 2.0 x 10(3) characters/ mm(2) are demonstrated.
TL;DR: The Driven Equilibrium Fourier Transform (DEFT) technique for signal enhancement in pulsed 13C magnetic resonance spectroscopy has been investigated for several small 60-enriched molecules.
TL;DR: The technique is well-suited for routine analysis on a computer, and several applications are described; the examples are myelin and a variety of lipid-water phases, all centrosymmetric structures which display a crystalline order in one, two or three dimensions.
TL;DR: In this article, the n-beam dynamical theory of high-energy electrons is used in transmission for accurate determination of the Fourier components of the crystal potential, and a model has been developed which incorporates these changes into the theory.
Abstract: The n-beam dynamical theory of high-energy electrons is currently used in transmission (Laue case) for accurate determination of the Fourier components of the crystal potential. The same theory is expected to provide information about the surface potential when used to interpret diffraction patterns in reflection at glancing incidence (Bragg case). Some peculiar aspects are elucidated in detail, insofar that they are different from the transmission case, particularly the boundary conditions. Inelastic scattering effects are introduced by means of a complex potential. Changes of the Fourier components of the potential near the surface are considered, and a model has been developed which incorporates these changes into the theory. A slice treatment developed in the frame of Bethe's theory is presented.
TL;DR: In this paper, a simple closed form for the Fourier transform of the correlation function of an order-parameter correlation function was obtained and its dependence on wave vector and the reciprocal correlation length was examined.
Abstract: A central problem in the theory of phase transitions is the calculation of the order-parameter correlation function. Here we study the correlation function of an $n$-component order parameter whose configuration energy is determined by the usual Landau functional. Using a screening approximation, we obtain a simple closed form for the Fourier transform of the correlation function and examine its dependence on wave vector $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and the reciprocal correlation length $k$.
TL;DR: The application of IR interferometry or of NMR pulse methods, followed by Fourier transformation of the resultant interferogram, permits acquisition of spectral data in a time that is orders of magnitude less than by conventional spectroscopic methods.
Abstract: The recent introduction of Fourier transform methods is revolutionizing IR and NMR spectroscopy. The application of IR interferometry or of NMR pulse methods, followed by Fourier transformation of the resultant interferogram, permits acquisition of spectral data in a time that is orders of magnitude less than by conventional spectroscopic methods. This reduction in time permits the study of transient species, or by "time-averaging" procedures S/N may be improved without the expenditure of inordinate amounts of time. The FT methods finds especially important application in the study of NMR spectra of nuclei of low sensitivity and low abundance, such as (13)C.
TL;DR: This paper develops performance criteria for evaluating transform data coding schemes under computational constraints characterized by a rate-distortion relation R(D) similar in form to the theoretical rate- Distortion function.
Abstract: This paper develops performance criteria for evaluating transform data coding schemes under computational constraints. Computational constraints that conform with the proposed basis-restricted model give rise to suboptimal coding efficiency characterized by a rate-distortion relation R(D) similar in form to the theoretical rate-distortion function. Numerical examples of this performance measure are presented for Fourier, Walsh, Haar, and Karhunen-Loeve transforms.