Discrete integer Fourier transform in real space: elliptic Fourier transform

2009 ◽  
Author(s):  
Artyom M. Grigoryan ◽  
Merughan M. Grigoryan
Keyword(s):  
Fourier Transform ◽  
Real Space ◽  
Elliptic Fourier Transform

High Resolution Transmission Electron Microscopy

MRS Bulletin ◽  
1991 ◽  
Vol 16 (3) ◽  
pp. 27-33 ◽  
Author(s):  
J.M. Gibson
Keyword(s):  
Fourier Transform ◽  
Focal Plane ◽  
Materials Science ◽  
Image Plane ◽  
Real Space ◽  
Dark Field ◽  
Dark Field Imaging ◽  
Wave Nature ◽  
Transmission Electron ◽  
Field Imaging

The transmission electron microscope (TEM) has had a major impact on materials science in the last five decades, despite the fact that it is necessary to prepare thin samples in order to use the technique. The primary reason for this effectiveness is the ability to access both real space and diffraction data in the same instrument, and to filter in one and observe the effect in the other. This is possible because of the wave nature of electrons and the existence of effective magnetic lenses for focusing. Abbe showed that any lens has the ability to Fourier transform its input wavefield in its focal plane, and to provide a second Fourier transform in the image plane. This is schematically shown in Figure 1. A crystalline object will diffract only in certain directions, with Bragg angles (θB) depending on the inverse of the interplanar spacing. The diffraction pattern is a series of spots in the Fourier, or focal, plane of the lens. A filter placed in the focal plane serves to limit the resolution by limiting the bandwidth of the image, but it also can serve to select certain parts of the Fourier spectrum in the image. The simplest examples of this, as used in optical microscopy, are bright-field and dark-field imaging. In the former the un-scattered beam is allowed to reach the image, in the latter it is not.

Treatment field shape verification using elliptic Fourier transform

1999 ◽  
Vol 26 (11) ◽  
pp. 2415-2421 ◽  
Author(s):  
Qinghuai Gao ◽  
Fang-Fang Yin ◽  
Kaiwen Nie
Keyword(s):  
Fourier Transform ◽  
Elliptic Fourier Transform

scholarly journals DF-Fit: A Robust Algorithm for Detection of Crystallographic Information in Atom Probe Tomography Data

2019 ◽  
Vol 25 (2) ◽  
pp. 331-337
Author(s):  
Daniel Haley ◽  
Paul A. J. Bagot ◽  
Michael P. Moody
Keyword(s):  
Fourier Transform ◽  
Spatial Data ◽  
Atom Probe Tomography ◽  
Computational Cost ◽  
Three Dimensional ◽  
Atom Probe ◽  
Real Space ◽  
Distribution Functions ◽  
Quality Data ◽  
Error Metric

AbstractWe report on a new algorithm for the detection of crystallographic information in three-dimensional, as retained in atom probe tomography (APT), with improved robustness and signal detection performance. The algorithm is underpinned by one-dimensional distribution functions (DFs), as per existing algorithms, but eliminates an unnecessary parameter as compared to current methods.By examining traditional DFs in an automated fashion in real space, rather than using Fourier transform approaches, we utilize an error metric based upon the expected value for a spatially random distribution for detecting crystallography. We show cases where the metric is able to successfully obtain orientation information, and show that it can function with high levels of additive and displacive background noise. We additionally compare this metric to Fourier transform methods, showing fewer artifacts when examining simulated datasets. An extension of the approach is used to aid the automatic detection of high-quality data regions within an entire dataset, albeit with a large increase in computational cost.This extension is demonstrated on acquired aluminum and tungsten APT datasets, and shown to be able to discern regions of the data which have relatively improved spatial data quality. Finally, this program has been made available for use in other laboratories undertaking their own analyses.

Low Energy Photoelectron Holography on Gaas

1992 ◽  
Vol 295 ◽  
Author(s):  
R. Denecke ◽  
R. Eckstein ◽  
L. Ley ◽  
A. Bocquet ◽  
J. Riley ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Kinetic Energy ◽  
Single Scattering ◽  
Real Space ◽  
Low Energy ◽  
Space Images ◽  
Photoelectron Diffraction ◽  
Diffraction Patterns

AbstractFor GaAs (001) and (III) we have measured the photoelectron diffraction patterns at a kinetic energy of 86 eV. We applied an extended Fourier-transform algorithm to the (001) data to obtain real space images. The origin of structures in these images not representing atomic positions is investigated with the help of single scattering calculations

scholarly journals Fourier-spectral-method implementation of deformation in the phase-field crystal model

2018 ◽  
Vol 192 ◽  
pp. 01028
Author(s):  
Nirand Pisutha-Arnond
Keyword(s):  
Fourier Transform ◽  
Spectral Method ◽  
Phase Field ◽  
Real Space ◽  
Fourier Spectral Method ◽  
Crystal Model ◽  
Space Coordinate ◽  
Diffusive Time ◽  
Body Correlation ◽  
Phase Field Crystal

The phase-field crystal (PFC) method is a promising computational model with atomistic resolution and diffusive time scale. In this work, the Fourier-spectral-method (FSM) scheme was developed for evaluating the PFC free energy of a system subjected homogeneous deformation. This scheme addresses the complication where, in numerical implementation of FSM using discrete Fourier transform (DFT), the discretized data may no longer lie along the directions of the Cartesian basis due to deformation. In this scheme, the real-space coordinate transformation is employed so that the (continuous) Fourier transform is performed on the function of the undeformed coordinates. This transformation allows straightforward DFT implementation because the sampling at the undeformed configuration is unaffected by the deformation. This scheme is also shown to be applicable to both the original PFC model and a “CDFT”-type PFC model containing a two-body correlation function.

scholarly journals A GEOMETRIC MORPHOMETRIC APPROACH TO THE ANALYSIS OF THE SHAPE VARIABILITY OF THE HAPTORAL ATTACHMENT STRUCTURES OF LIGOPHORUS SPECIES (PLATYHELMINTHES: MONOGENEA)

2017 ◽  
Vol 14 ◽  
pp. 92-101
Author(s):  
Anton Lyakh ◽  
Evgenija Dmitrieva ◽  
Maryana Popyuk ◽  
Olga Shikhat ◽  
Alexandr Melnik
Keyword(s):  
Fourier Transform ◽  
Numerical Data ◽  
The Black Sea ◽  
Geometric Morphometric ◽  
Interspecific Variability ◽  
Attachment Structures ◽  
Average Form ◽  
Attachment Organ ◽  
Outline Images ◽  
Elliptic Fourier Transform

The taxonomy of Ligophorus Euzet & Suriano, 1977, like the most of monopisthocotylean monogeneans, relies heavily on the morphology of sclerites of the posterior attachment organ (haptor). Geometric morphometric approach is used to analyse variability and compare the shapes of haptoral structures of these monogeneans. We outline the shapes of the sclerities by cubic Bezier curves and store results in SVG files. Every SVG outline is reduced to a set of harmonics of Elliptic Fourier transform using ElFourier program. Harmonics are the sequence of unique numbers that describe the shape of structures and are invariant to their sizes, rotation, and orientation. They allow reconstructing source outline images, finding their average form, analyzing variability and comparing shapes in combination with other numerical data like dimensions. We use that approach to investigate intra- and interspecific variability of 400 haptoral structures of seven representatives of Ligophorus, parasitising four mullet species from the Black Sea, and to discriminate these monogeneans. This method is perspective for the creation of semiautomatic key for identification of helminthes, which are mainly distinguished by the shape and dimensions of the attachment organs. The obtained results and method prospects are discussed.

High-resolution structure determination by ALCHEMI

1983 ◽  
Vol 41 ◽  
pp. 178-181 ◽  
Author(s):  
Peter G. Self ◽  
Peter R. Buseck
Keyword(s):  
Site Occupancy ◽  
Real Space ◽  
Unit Cell ◽  
Bragg Angle ◽  
Electron Current ◽  
High Resolution Structure ◽  
Beam Incident ◽  
Crystallographic Planes ◽  
Electron Beam Incident ◽  
Resolution Structure

ALCHEMI (Atom Location by CHanneling Enhanced Microanalysis) enables the site occupancy of atoms in single crystals to be determined. In this article the fundamentals of the method for both EDS and EELS will be discussed. Unlike HRTEM, ALCHEMI does not place stringent resolution requirements on the microscope and, because EDS clearly distinguishes between elements of similar atomic number, it can offer some advantages over HRTEM. It does however, place certain constraints on the crystal. These constraints are: a) the sites of interest must lie on alternate crystallographic planes, b) the projected charge density on the alternate planes must be significantly different, and c) there must be at least one atomic species that lies solely on one of the planes.An electron beam incident on a crystal undergoes elastic scattering; in reciprocal space this is seen as a diffraction pattern and in real space this is a modulation of the electron current across the unit cell. When diffraction is strong (i.e., when the crystal is oriented near to the Bragg angle of a low-order reflection) the electron current at one point in the unit cell will differ significantly from that at another point.

An Image Reconstruction System Applied to High Voltage Microscopy

1975 ◽  
Vol 33 ◽  
pp. 292-293
Author(s):  
D. E. Johnson
Keyword(s):  
High Voltage ◽  
Prior Information ◽  
Block Diagram ◽  
Three Dimensional ◽  
Real Space ◽  
Two Dimensional ◽  
Efficient Manner ◽  
Fourier Methods ◽  
Tilt Series ◽  
Methods Of Reconstruction

Increased specimen penetration; the principle advantage of high voltage microscopy, is accompanied by an increased need to utilize information on three dimensional specimen structure available in the form of two dimensional projections (i.e. micrographs). We are engaged in a program to develop methods which allow the maximum use of information contained in a through tilt series of micrographs to determine three dimensional speciman structure.In general, we are dealing with structures lacking in symmetry and with projections available from only a limited span of angles (±60°). For these reasons, we must make maximum use of any prior information available about the specimen. To do this in the most efficient manner, we have concentrated on iterative, real space methods rather than Fourier methods of reconstruction. The particular iterative algorithm we have developed is given in detail in ref. 3. A block diagram of the complete reconstruction system is shown in fig. 1.

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