DIRECTIONAL AND WIDE‐BAND VELOCITY FILTERING

Geophysics ◽  
1965 ◽  
Vol 30 (2) ◽  
pp. 279-280
Author(s):  
Philip L. Jackson
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Wide Band ◽  
Dimensional Structure ◽  
Two Dimensional ◽  
Contour Map ◽  
Directional Filtering ◽  
Velocity Filtering ◽  
Monochromatic Illumination

The purpose of this note is to point out that the condition of monochromatic illumination in obtaining a two‐dimensional Fourier transform by optical means may be relaxed for wide‐band velocity filtering, and for the directional filtering of any two‐dimensional structure such as a contour map. Similar considerations hold for two‐dimensional Fourier transforms obtained by any means.

Possible Models of Murein and Their Fourier Transforms

1982 ◽  
Vol 37 (3-4) ◽  
pp. 226-235 ◽  
Author(s):  
Helmut Formanek
Keyword(s):  
Fourier Transform ◽  
Cell Walls ◽  
Fourier Transforms ◽  
Elementary Cell ◽  
Two Dimensional ◽  
Density Measurements ◽  
X Ray ◽  
Rigid Layer ◽  
The Fourier Transform ◽  
Almost All

Abstract Murein, Models, Fourier Transforms Murein, the rigid layer of the cell walls of almost all bacteria can be regarded as derivative of chitin. Within the sterically allowed region its polysaccharide chain can perform conformations with two-to threefold screw axes. Two dimensional Fourier transforms calculated from different possible conformations have been compared with data of density measurements, X-ray and electron diffraction. The Fourier transform of murein with a chitin-like conformation of the poly­ saccharide chain and an elementary cell of 4.5 × 10.4 × 21.5 Å3 provides the best agreement with the experimental results.

scholarly journals Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples

2021 ◽  
pp. 55-65
Author(s):  
Olga Ponomareva ◽  
Aleksey Ponomarev
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Information Control ◽  
Two Dimensional ◽  
Negative Effects ◽  
Variable Parameters ◽  
Discrete Fourier Transforms ◽  
Fourier Matrix

Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

APPLICATION OF FOURIER TRANSFORMS TO ANTENNA PATTERN SYNTHESIS

1961 ◽  
Vol 39 (9) ◽  
pp. 1347-1356
Author(s):  
Alfred T. Villeneuve
Keyword(s):  
Fourier Transform ◽  
Field Distribution ◽  
Fourier Transforms ◽  
Antenna Pattern ◽  
Two Dimensional ◽  
Pattern Synthesis

The two-dimensional Fourier transform of the field quantities is employed to synthesize source distributions over plane rectangular apertures when a field distribution is prescribed at an arbitrary distance from the source. Several examples of application of the technique are included. The approximations involved are discussed.

scholarly journals Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Heat Equation ◽  
Quaternion Algebra ◽  
Fourier Transforms ◽  
Two Dimensional ◽  
Quaternion Fourier Transform

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

Parallelisation of a 2-D Fast Fourier Transform Algorithm

1991 ◽  
Vol 02 (01) ◽  
pp. 363-366 ◽  
Author(s):  
ANDREW HAMMERSLEY
Keyword(s):  
Fourier Transform ◽  
Data Analysis ◽  
Fast Fourier Transform ◽  
Fourier Transforms ◽  
Computational Physics ◽  
Fast Fourier Transforms ◽  
Two Dimensional ◽  
Fast Fourier Transform Algorithm ◽  
Higher Dimension ◽  
Parallel Network

The calculation of two and higher-dimension Fast Fourier Transforms (FFT’s) are of great importance in many areas of data analysis and computational physics. The two-dimensional FFT is implemented for a parallel network using a master-slave approach. In-place performance is good, but the use of this technique as an “accelerator” is limited by the communications time between the host and the network. The total time is reduced by performing the host-master communications in parallel with the master-slave communications. Results for the calculation of the two-dimensional FFT of real-valued datasets are presented.

The principle of complex frequency scaling—Applicability in inclined continuation of potential fields

Geophysics ◽  
1984 ◽  
Vol 49 (11) ◽  
pp. 2019-2023 ◽  
Author(s):  
R. Nagendra ◽  
N. Laxminarayana
Keyword(s):  
Fourier Transforms ◽  
Potential Fields ◽  
Dimensional Structure ◽  
Complex Frequency ◽  
Two Dimensional ◽  
Complex Scaling ◽  
Frequency Scaling ◽  
Space Frequency ◽  
Scaling Principle ◽  
Inclined Planes

A remarkable property of Fourier transforms, especially applicable to inclined continuation of potential geophysical fields, is the principle of “complex frequency scaling.” Briefly stated, let [Formula: see text] be the (gravity/magnetic) field due to a two‐dimensional structure along a principal profile and F(ω) be its Fourier transform. The field along a profile passing through the same reference origin and tilted by an angle ϕ in the counter‐clockwise direction (+X to −Z) is obtained by inverse Fourier transforming F[ωexp(−iϕ)] for positive ω. The complex scaling property and proof of the resulting space‐frequency domain relationship are presented, introducing the total field as the analytic signal of the horizontal component. The applicability of the complex scaling principle is illustrated by considering selected geometric models. This principle can be advantageously applied for continuation of two‐dimensional potential fields onto inclined planes.

Directional spectral analysis and filtering of geophysical maps

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1587-1591 ◽  
Author(s):  
Freyr Thorarinsson ◽  
Stefan G. Magnusson ◽  
Axel Bjornsson
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Power Spectrum ◽  
Two Dimensional ◽  
Directional Filtering ◽  
Gravity And Magnetic ◽  
The Fourier Transform ◽  
Map Data ◽  
Tectonic Features

The detection of linear anomalies in map data is facilitated by studying the two‐dimensional power spectrum, because the directivity of the energy in the map is preserved in the Fourier transform. The lineaments associated with individual peaks in the spectrum are then separated from the map data by directional filtering and studied independently of other map features. Gravity and magnetic maps from an active rift area in southwestern Iceland are analyzed in this manner. The agreement between the filtered maps is good and they fit the observed tectonic features quite well.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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