Deconvolution analysis in membrane diffraction using the modulus of the continuous Fourier transform

1981 ◽  
Vol 14 (6) ◽  
pp. 383-386 ◽  
Author(s):  
C. R. Worthington
Keyword(s):  
Fourier Transform ◽  
Autocorrelation Function ◽  
Sampling Theorem ◽  
Unit Cell ◽  
Good Representation ◽  
Missing Information ◽  
Interference Function ◽  
Deconvolution Analysis ◽  
Unit Cells ◽  
The Fourier Transform

A method of deconvolution using the modulus of the continuous Fourier transform of the unit cell is described. This method differs from previous deconvolution methods in membrane diffraction in that calculations are carried out in reciprocal space. The modulus profile is obtained from the continuous intensity transform which is itself the Fourier transform of the autocorrelation function. Sampling theorem methods are used to reconstruct the continuous Fourier transform of the unit cell. The various phase choices are examined and compared. In membrane diffraction, the autocorrelation function is derived in two distinct situations: when the interference function is broad as in the case of a few unit cells and when the membrane systems contain wide regions of constant electron density. It is concluded that, in the first situation, the derived autocorrelation function contains missing information and is incorrect. On the other hand, in the second situation, the derived autocorrelation function is a good representation of the true autocorrelation function.

Electron-density maps

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Electron Density ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Unit Cell ◽  
Density Maps ◽  
Wave Cycle ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Stable reduction to the pole at the magnetic equator

Geophysics ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 571-578 ◽  
Author(s):  
Yaoguo Li ◽  
Douglas W. Oldenburg
Keyword(s):  
Fourier Transform ◽  
Magnetic Equator ◽  
Magnetic Data ◽  
Good Representation ◽  
Wavenumber Domain ◽  
Power Spectral ◽  
Reduction To The Pole ◽  
The Fourier Transform ◽  
Stable Reconstruction ◽  
Estimated Error

The solution of reduction to the pole (RTP) of magnetic data in the wavenumber domain faces a long standing difficulty of instability when the observed data are acquired at low magnetic latitudes or at the equator. We develop a solution to this problem that allows stable reconstruction of the RTP field with a high fidelity even at the magnetic equator. The solution is obtained by inverting the Fourier transform of the observed magnetic data in the wavenumber domain with explicit regularization. The degree of regularization is chosen according to the estimated error level in the data. The Fourier transform of the RTP field is constructed as a model that is maximally smooth and, at the same time, has a power‐spectral decay common to all fields produced by the same source. The applied regularization alleviates the singularity associated with the wavenumber‐domain RTP operator, and the imposed power spectral decay ensures that the constructed RTP field has the correct spectral content. As a result, the algorithm can perform the reduction to the pole stably at any magnetic latitude, and the constructed RTP field yields a good representation of the true field at the pole even when the reduction is carried out at the equator.

Introduction

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Fourier Transform ◽  
Magnetic Resonance ◽  
Fourier Analysis ◽  
Fourier Integral ◽  
Sampling Theorem ◽  
Two Dimensional ◽  
The Fourier Transform ◽  
Array Imaging ◽  
Nuclear Magnetic

Jean Baptiste Joseph Fourier’s powerful idea of decomposition of a signal into sinusoidal components has found application in almost every engineering and science field. An incomplete list includes acoustics [1497], array imaging [1304], audio [1290], biology [826], biomedical engineering [1109], chemistry [438, 925], chromatography [1481], communications engineering [968], control theory [764], crystallography [316, 498, 499, 716], electromagnetics [250], imaging [151], image processing [1239] including segmentation [1448], nuclear magnetic resonance (NMR) [436, 1009], optics [492, 514, 517, 1344], polymer characterization [647], physics [262], radar [154, 1510], remote sensing [84], signal processing [41, 154], structural analysis [384], spectroscopy [84, 267, 724, 1220, 1293, 1481, 1496], time series [124], velocity measurement [1448], tomography [93, 1241, 1242, 1327, 1330, 1325, 1331], weather analysis [456], and X-ray diffraction [1378], Jean Baptiste Joseph Fourier’s last name has become an adjective in the terms like Fourier series [395], Fourier transform [41, 51, 149, 154, 160, 437, 447, 926, 968, 1009, 1496], Fourier analysis [151, 379, 606, 796, 1472, 1591], Fourier theory [1485], the Fourier integral [395, 187, 1399], Fourier inversion [1325], Fourier descriptors [826], Fourier coefficients [134], Fourier spectra [624, 625] Fourier reconstruction [1330], Fourier spectrometry [84, 355], Fourier spectroscopy [1220, 1293, 1438], Fourier array imaging [1304], Fourier transform nuclear magnetic resonance (NMR) [429, 1004], Fourier vision [1448], Fourier optics [419, 517, 1343], and Fourier acoustics [1496]. Applied Fourier analysis is ubiquitous simply because of the utility of its descriptive power. It is second only to the differential equation in the modelling of physical phenomena. In contrast with other linear transforms, the Fourier transform has a number of physical manifestations. Here is a short list of everyday occurrences as seen through the lens of the Fourier paradigm. • Diffracting coherent waves in sonar and optics in the far field are given by the two dimensional Fourier transform of the diffracting aperture. Remarkably, in free space, the physics of spreading light naturally forms a two dimensional Fourier transform. • The sampling theorem, born of Fourier analysis, tells us how fast to sample an audio waveform to make a discrete time CD or an image to make a DVD.

scholarly journals Neutron Noise Analysis with Flash-Fourier Algorithm at the IBR-2M Reactor

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mihai O. Dima ◽  
Yuri N. Pepelyshev ◽  
Lachin Tayibov
Keyword(s):  
Fourier Transform ◽  
Autocorrelation Function ◽  
Fourier Transforms ◽  
Nuclear Reactors ◽  
Noise Analysis ◽  
Pulsed Reactor ◽  
Neutron Noise ◽  
The Fourier Transform ◽  
Fourier Algorithm ◽  
Noise Spectra

Neutron noise spectra in nuclear reactors are a convolution of multiple effects. For the IBR-2M pulsed reactor (JINR, Dubna), one part of these is represented by the reactivities induced by the two moving auxiliary reflectors and another part of these by other sources that are moderately stable. The study of neutron noise involves, foremostly, knowing its frequency spectral distribution, hence Fourier transforms of the noise. Traditional methods compute the Fourier transform of the autocorrelation function. We show in the present study that this is neither natural nor real-time adapted, for both the autocorrelation function and the Fourier transform are highly CPU intensive. We present flash algorithms for processing the Fourier-like transforms of the noise spectra.

scholarly journals On a sampling expansion with partial derivatives for functions of several variables

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3339-3347
Author(s):  
Saulius Norvidas
Keyword(s):  
Fourier Transform ◽  
Sampling Theorem ◽  
Partial Derivatives ◽  
Several Variables ◽  
Sampling Series ◽  
Sampling Points ◽  
The Fourier Transform

Let Bp?, 1 ? p < ?,? > 0, denote the space of all f ?Lp(R) such that the Fourier transform of f (in the sense of distributions) vanishes outside [-?,?]. The classical sampling theorem states that each f ? Bp? may be reconstructed exactly from its sample values at equispaced sampling points {?m=?}m?Z spaced by ?/?. Reconstruction is also possible from sample values at sampling points {??m/?}m with certain 1 < ? ? 2 if we know f(??m/?) and f?(??m/?), m ? Z. In this paper we present sampling series for functions of several variables. These series involves samples of functions and their partial derivatives.

Bayesian removal of noise for increased sensitivity in vector pattern recognition lattice imaging of interfaces

1993 ◽  
Vol 51 ◽  
pp. 994-995
Author(s):  
L. Fei ◽  
P. Fraundorf
Keyword(s):  
Pattern Recognition ◽  
Perfect Crystal ◽  
Unit Cell ◽  
Ion Milling ◽  
Beam Radiation ◽  
Major Interest ◽  
Unit Cells ◽  
Mapping Process ◽  
Increased Sensitivity ◽  
Electron Microscopes

Interface structure is of major interest in microscopy. With high resolution transmission electron microscopes (TEMs) and scanning probe microscopes, it is possible to reveal structure of interfaces in unit cells, in some cases with atomic resolution. A. Ourmazd et al. proposed quantifying such observations by using vector pattern recognition to map chemical composition changes across the interface in TEM images with unit cell resolution. The sensitivity of the mapping process, however, is limited by the repeatability of unit cell images of perfect crystal, and hence by the amount of delocalized noise, e.g. due to ion milling or beam radiation damage. Bayesian removal of noise, based on statistical inference, can be used to reduce the amount of non-periodic noise in images after acquisition. The basic principle of Bayesian phase-model background subtraction, according to our previous study, is that the optimum (rms error minimizing strategy) Fourier phases of the noise can be obtained provided the amplitudes of the noise is given, while the noise amplitude can often be estimated from the image itself.

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

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