Showing papers on "Fourier transform published in 1978"

Reconstruction of an object from the modulus of its Fourier transform.

Abstract: We present a digital method for solving the phase-retrieval problem of optical-coherence theory: the reconstruction of a general object from the modulus of its Fourier transform. This technique should be useful for obtaining high-resolution imagery from interferometer data.

Migration by fourier transform

Abstract: Wave equation migration is known to be simpler in principle when the horizontal coordinate or coordinates are replaced by their Fourier conjugates. Two practical migration schemes utilizing this concept are developed in this paper. One scheme extends the Claerbout finite difference method, greatly reducing dispersion problems usually associated with this method at higher dips and frequencies. The second scheme effects a Fourier transform in both space and time; by using the full scalar wave equation in the conjugate space, the method eliminates (up to the aliasing frequency) dispersion altogether. The second method in particular appears adaptable to three‐dimensional migration and migration before stack.

Linear systems, Fourier transforms, and optics

Abstract: Representation of Physical Quantities by Mathematical Functions. Special Functions. Harmonic Analysis. Mathematical Operators and Physical Systems. Convolution. The Fourier Transform. Characteristics and Applications of Linear Filters. Two-Dimensional Convolution and Fourier Transformation. The Propagation and Diffraction of Optical Wave Fields. Image-Forming Systems. Appendices. Index.

A numerical and theoretical study of certain nonlinear wave phenomena

Abstract: An efficient numerical method is developed for solving nonlinear wave equations typified by the Korteweg-de Vries equation and its generalizations. The method uses a pseudospectral (Fourier transform) treatment of the space dependence together with a leap-frog scheme in time. It is combined with theoretical discussions in the study of a variety of problems including solitary wave interactions, wave breaking, the resolution of initial steps and wells, and the development of nonlinear wavetrain instabilities.

A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena

Abstract: An efficient numerical method is developed for solving nonlinear wave equations typified by the Korteweg-de Vries equation and its generalizations. The method uses a pseudospectral (Fourier transform) treatment of the space dependence together with a leap-frog scheme in time. It is combined with theoretical discussions in the study of a variety of problems including solitary wave interactions, wave breaking, the resolution of initial steps and wells, and the development of nonlinear wavetrain instabilities.

Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal.

Abstract: All 16 elements of the Mueller matrix of an optical system (sample) can be encoded onto, hence can be retrieved from, a single detected signal using a class of photopolarimeters with modulated polarizing and analyzing optics. The general theory of operation of such polarimeters is presented. We also propose a specific new photopolarimeter whose polarizing and analyzing optics are modulated by synchronously rotating two quarter-wave retarders at angular speeds ω and 5ω. When the light flux leaving such polarimeter is linearly detected, a periodic signal J=a0+∑n=112(an cos nωft+bn sin nωft) is generated, with fundamental frequency ωf = 2ω. From the Fourier amplitudes a0, an, bn, to be measured by performing a discrete Fourier transform (DFT) of the signal ℐ, the 16 elements of the Mueller matrix are simply determined.

Fourier transform multiple quantum nuclear magnetic resonance

Abstract: The excitation and detection of multiple quantum transitions in systems of coupled spins offers, among other advantages, an increase in resolution over single quantum n.m.r. since the number of lines decreases as the order of the transition increases. This paper reviews the motivation for detecting multiple quantum transitions by a Fourier transform experiment and describes an experimental approach to high resolution multiple quantum spectra in dipolar systems along with results on some protonated liquid crystal systems. A simple operator formalism for the essential features of the time development is presented and some applications in progress are discussed.

Computer-generated double-phase holograms

Abstract: A method of computer generating binary holograms based on the decomposition of a complex value into two phase quantities is described. Each Fourier transform cell is divided into subcells, and phase quantities are encoded by the detour phase technique. Noise due to the displacement of the subcells and the phase coding is discussed. Methods of suppressing this noise are also included.

Optical thin film synthesis program based on the use of Fourier transforms.

Abstract: In Sossi's formulation of the Fourier transform method of optical multilayer design the refractive-index profile is derived for an inhomogeneous layer of infinite extent having the desired spectral transmittance. This layer is then approximated by a finite system of discrete homogeneous layers. Because it does not make any assumptions about the refractive indices, thicknesses, or number of layers, it is the most powerful analytical method proposed so far. The method has been programmed for a computer and combined with other numerical design procedures. With the program it is possible to design filters with almost any desired transmittance characteristics using realistic refractive indices.

Fourier Transform Infrared Spectroscopy: Applications to Chemical Systems

Abstract: P.C. Gillette, J.B. Lando, and J.L. Koenig, A Survey of Infrared Spectral Data Processing Techniques. P.L. Polavarapu, Fourier Transform Infrared Vibrational Circular Dichroism. K. Krishnan, Advances in Capillary Gas Chromatography*b1Fourier Transform Interferometry. A.G. Nerheim, Applications of Spectral Techniques to Thermal Analysis. P. Painter, M. Starsinic, and M. Coleman, Determination of Functional Groups in Coal by Fourier Transform Interferometry. J.R. Ferraro and A.J. Rein, Applications of Diffuse Reflectance Spectroscopy in the Far-Infrared Region. J.D. Swalen and J.F. Rabolt, Characterization of Orientation and Lateral Order in Thin Films by Fourier Transform Infrared Spectroscopy. W.G. Golden, Fourier Transform Infrared Reflection*b1Absorption Spectroscopy. J.A. Graham, W.M. Grim III, and W.G. Fateley, Fourier Transform Infrared Photoacoustic Spectroscopy of Condensed-Phase Samples. Each chapter includes references. Index.

The Fourier method for nonsmooth initial data

Abstract: Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing. Introduction. In recent years the Fourier method for the numerical approximation of solutions to hyperbolic initial value problems has been used quite successfully. In fact, if the initial function is C°° and the coefficients of the equation are constant the method converges arbitrarily fast, i.e. is limited in practice only by the method of time discretization which is chosen. This is the reason that the Fourier method is caled "infinite order" accurate. However, the situation is drastically different when the initial function is not smooth. We take as a model the one space dimension scalar problem ut = ux to be solved for 2ir periodic u on the interval n < x < n with initial values

Fourier transform infrared studies on the thermal degradation of polyacrylonitrile

Abstract: Fourier transform infrared studies of the thermal degradation of polyacrylonitrile (PAN) conducted at 200°C in air and under reduced pressure are presented. The spectra are markedly superior to those published previously. A mechanism consistent with the infrared results obtained under reduced pressure is advanced, and assignments for the infrared bands occuring in the spectrum of both the reduced pressure and air degraded PAN are given.

Analysis of the vibratory excitation of gear systems: Basic theory

Abstract: Formulation of the equations of motion of a generic gear system in the frequency domain is shown to require the Fourier‐series coefficients of the components of vibration excitation; these components are the static transmission errors of the individual pairs of meshing gears in the system. A general expression for the static transmission error is derived and decomposed into components attributable to elastic tooth deformations and to deviations of tooth faces from perfect involute surfaces with uniform lead and spacing. The component due to tooth‐face deviations is further decomposed into appropriately defined mean and random components. The harmonic components of the static transmission error that occur at integral multiples of the tooth‐meshing frequency are shown to be caused by tooth deformations and mean deviations of the tooth faces from perfect involute surfaces. Harmonic components that occur at the remaining multiples of gear‐rotation frequencies are shown to be caused by the random components of the tooth‐face deviations. Expressions for the Fourier‐series coefficients of all components of the static transmission error are derived in terms of two‐dimensional Fourier transforms of local tooth‐pair stiffnesses and stiffness‐weighted weighted deviations of tooth faces from perfect involute surfaces. Results are valid for arbitrary, specified tooth‐face contact regions and include spur gears as the special case of helical gears with zero helix angle.

Partial differential equations and finite difference methods in image processing--Part II: Image restoration

Abstract: Application of Partial Differential Equation (PDE) models for restoration of noisy images is considered. The hyperbolic, parabolic, and elliptic classes of PDE's yield recursive, semirecursive, and nonrecursive filtering algorithms. The two-dimensional recursive filter is equivalent to solving two sets of filtering equations, one along the horizontal direction and other along the vertical direction. The semirecursive filter can be implemented by first transforming the image data along one of its dimensions, say Column, and then recursive filtering along each row independently. The nonrecursive filter leads to Fourier domain Wiener filtering type transform domain algorithm. Comparisons of the different PDE model filters are made by implementing them on actual image data. Performances of these filters are also compared with Fourier Wiener filtering and spatial averaging methods. Performance bounds based on PDE model theory are calculated and implementation tradeoffs of different algorithms are discussed.

The theory of structurally incommensurate systems. III. The fluctuation spectrum of incommensurate phases

Abstract: For pt.II see ibid., vol.11, no.17, p.3591 (1978). The authors discussed the nature of the excitation spectrum of a structurally incommensurate phase, both when the displacement field is describable by a single Fourier component and when it may be described in terms of an array of phase solitons. In each case they established the character of the Goldstone phase mode, and in the latter case they found an additional phase excitation whose properties merge into those of a phase mode in the commensurate structure as the 'lock-in' phase boundary is approached. They examined phase mode-acoustic mode interaction and give a brief discussion of phase mode pinning.

Expansions for spherical functions on noncompact symmetric spaces

Abstract: Section 0 Lot G be a connected noncompact semisimple Lie group with finite center and real rank one; fix a maximal compact subgroup K. Our concern in this paper is Fourier analysis on the Riemannian symmetric space G]K. We shall analyze the local and global behavior of spherical functions, the boundedness of multiplier operators, and the inversion of differential operators. The core of the paper, however, is an analysis of how the size of a function is controlled by the size of its Fourier transform. There is an extensive literature on such topics, centered about the Paley-Wiener and Plancherel theorems. Our work relies heavily on these earlier ideas and techniques, to which detailed reference will be made in the body of the paper. The problems we wish to solve, however, require estimates more precise and of a different nature than are necessary for the Plancherel or Paley-Wienor theorem. Thus the first two sections of this paper are devoted to the construction of various asymptotic expansions for spherical functions; in later sections we show how these expansions may be applied to the Fourier analysis of G/K.

Three-dimensional NMR imaging using selective excitation

Abstract: Using numerical solutions of the Bloch (1946) equations under boundary conditions representing the local external magnetic field, and with the aid of a simple model of the nuclear spin process, various effects occurring during selective excitation are described. In particular it is shown that in extending the technique for one to two dimensions there arises the possibility of spurious image response, and that this can be corrected by a suitable combination of signals from a set of three excitation sequences. Since two-dimensional selection activates a column of nuclei, the subsequent application of a lengthwise magnetic field gradient followed by Fourier transformation of the resulting spin signal will directly yield an image line. This provides a technique for full three-dimensional imaging which satisfies moderate data-processing requirements and yet retains (line by line) the advantage of Fourier methods.

Method for the calculation of partially coherent imagery.

Abstract: The tedious numerical computations associated with the calculation of partially coherent imagery are alleviated by a method which uses dimensionless coordinates and takes advantage of the properties of the Fourier transform. A 1-D periodic object function can model many objects of practical interest, including nonperiodic objects. The properties of a given optical system are described in terms of the transmission cross coefficient. For aberration-free systems with circular pupils, including annular sources (dark-field illumination), the cross coefficient can be calculated analytically. For aberrated or apodized systems, a 1-D approximation can be used. The effect of a convolving slit in the image plane of a scanning microscope can also be included.

Constant-Q signal analysis and synthesis

Abstract: A generalization of the short-time Fourier transform is presented which performs constant-percentage bandwidth analysis of time-domain signals. The transform is shown to exhibit frequency-dependent time and frequency resolution. A synthesis transform is also developed which provides an analysis-synthesis system which is an identity in the absence of spectral modification (given a mild analysis window constraint). The effect of stationary multiplicative modifications is discussed. Finally, similarities between the constant-Q spectral domain and the human auditory system are explored, and some implications for acoustic signal processing mentioned.

The Fourier-Mellin transform and mammalian hearing

Abstract: A combined Fourier–Mellin transform yields a representation of a signal that is independent of delay and scale change. Such a representation should be useful for speech analysis, where delay and scale differences degrade the performance of correlation operations or other similarity measures. At least two different versions of a combined Fourier–Mellin transform can be implemented. The simplest version (the ‖F‖2−‖M‖2 transform) completely eliminates spectral phase information, while a slightly more complicated version (the ?−? transform) preserves some phase information. Both versions can be synthesized with a Fourier transform and an exponential‐sampling algorithm. Exponential sampling produces a frequency scale distortion that is similar to the effect of the cochlea. The ‖F‖2−‖M‖2 transform can also be implemented with a bank of proportional bandwidth filters. If the relative phase between spectral components is preserved, then a Fourier–Mellin transformer can perform compression of linear‐period modulat...

The radiant intensity from planar sources of any state of coherence

Abstract: A new formula is derived for the radiant intensity from any steady, finite, primary or secondary planar source of any state of coherence. It expresses the radiant intensity as a two-dimensional spatial Fourier transform of a quantity that represents a correlation function of the field in the source plane, averaged over the area of the source. The formula may be regarded as a natural counterpart for fields generated by partially coherent sources to the well-known two-dimensional Fourier transform relation between the field distributions in the plane of a finite coherent source and in the far zone. Some implications of the new formula are discussed. An alternative expression is also obtained that is applicable when the the source is a primary one and it is shown to imply that the radiant intensity is then a boundary value on two real axes of an entire analytic function of two complex variables.