Poisson and Fourier Transforms for Tensor Products and an Overalgebra

2015 ◽  
pp. 195-203 ◽  
Author(s):  
Vladimir F. Molchanov
Keyword(s):  
Fourier Transforms ◽  
Tensor Products ◽  
Poisson And Fourier Transforms

Generalized Maps

2020 ◽  
pp. 157-170
Author(s):  
Daniel Canarutto
Keyword(s):  
Hilbert Space ◽  
Fourier Transforms ◽  
Tensor Products ◽  
Hermitian Structure ◽  
Quantum Geometry ◽  
Basic Notion ◽  
Rigged Hilbert Space ◽  
Fundamental Properties ◽  
Special Cases ◽  
Generalized Maps

Spaces of generalised sections (also called section-distributions) are introduced, and their fundamental properties are described. Several special cases are considered, with particular attention to the case of semi-densities; when a Hermitian structure on the underlying classical bundle is given, these determine a rigged Hilbert space, which can be regarded as a basic notion in quantum geometry. The essentials of tensor products in distributional spaces, kernels and Fourier transforms are exposed.

Computer processing of high-resolution images of periodic specimens

1986 ◽  
Vol 44 ◽  
pp. 2-5
Author(s):  
W. Chiu ◽  
M.F. Schmid ◽  
T.-W. Jeng
Keyword(s):  
High Resolution ◽  
Fourier Transforms ◽  
Complex Structure ◽  
Distribution Functions ◽  
Multiple Image ◽  
Cryo Electron Microscopy ◽  
Structure Factors ◽  
Unit Cells ◽  
High Resolution Images ◽  
Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

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.

Observations of Frozen-Hydrated Microtubules

1990 ◽  
Vol 48 (1) ◽  
pp. 504-505
Author(s):  
D. Chrétien ◽  
D. Job ◽  
R.H. Wade
Keyword(s):  
Fourier Transforms ◽  
Structural Complexity ◽  
Image Contrast ◽  
Phase Behaviour ◽  
Experimental Conditions ◽  
Thin Sections ◽  
Surface Lattice ◽  
Cilia And Flagella ◽  
Outstanding Question

Microtubules are filamentary structures found in the cytoplasm of eukaryotic cells, where, together with actin and intermediate filaments, they form the components of the cytoskeleton. They have many functions and show various levels of structural complexity as witnessed by the singlet, doublet and triplet structures involved in the architecture of centrioles, basal bodies, cilia and flagella. The accepted microtubule model consists of a 25 nm diameter hollow tube with a wall made up of 13 paraxial protofilaments (pf). Each pf is a string of aligned tubulin dimers. Some results have suggested that the pfs follow a superhelix. To understand how microtubules function in the cell an accurate model of the surface lattice is one of the requirements. For example the 9x2 architecture of the axoneme will depend on the organisation of its component microtubules. We should also note that microtubules with different numbers of pfs have been observed in thin sections of cellular and of in-vitro material. An outstanding question is how does the surface lattice adjust to these different pf numbers?We have been using cryo-electron microscopy of frozen-hydrated samples to study in-vitro assembled microtubules. The experimental conditions are described in detail in this reference. The results obtained in conjunction with thin sections of similar specimens and with axoneme outer doublet fragments have already allowed us to characterise the image contrast of 13, 14 and 15 pf microtubules on the basis of the measured image widths, of the the image contrast symmetry and of the amplitude and phase behaviour along the equator in the computed Fourier transforms. The contrast variations along individual microtubule images can be interpreted in terms of the geometry of the microtubule surface lattice. We can extend these results and make some reasonable predictions about the probable surface lattices in the case of other pf numbers, see Table 1. Figure 1 shows observed images with which these predictions can be compared.

ON THE NATURE OF NONSTRUCTURAL LOW-R PEAK IN FOURIER TRANSFORMS OF SOME EXAFS-SPECTRA

1986 ◽  
Vol 47 (C8) ◽  
pp. C8-83-C8-87 ◽  
Author(s):  
A. G. KOCHUR ◽  
A. M. NADOLINSKY ◽  
V. F. DEMEKHIN
Keyword(s):  
Fourier Transforms ◽  
Exafs Spectra

Pattern Recognition for Classification and Matching of Car Tires

2005 ◽  
Vol 33 (1) ◽  
pp. 2-17 ◽  
Author(s):  
D. Colbry ◽  
D. Cherba ◽  
J. Luchini
Keyword(s):  
Computer Vision ◽  
Feature Extraction ◽  
Pattern Matching ◽  
Fourier Transforms ◽  
Two Dimensions ◽  
Multiple Sources ◽  
Matching Process ◽  
Visual Searches ◽  
New Feature ◽  
2D Fft

Abstract Commercial databases containing images of tire tread patterns are currently used by product designers, forensic specialists and product application personnel to identify whether a given tread pattern matches an existing tire. Currently, this pattern matching process is almost entirely manual, requiring visual searches of extensive libraries of tire tread patterns. Our work explores a first step toward automating this pattern matching process by building on feature analysis techniques from computer vision and image processing to develop a new method for extracting and classifying features from tire tread patterns and automatically locating candidate matches from a database of existing tread pattern images. Our method begins with a selection of tire tread images obtained from multiple sources (including manufacturers' literature, Web site images, and Tire Guides, Inc.), which are preprocessed and normalized using Two-Dimensional Fast Fourier Transforms (2D-FFT). The results of this preprocessing are feature-rich images that are further analyzed using feature extraction algorithms drawn from research in computer vision. A new, feature extraction algorithm is developed based on the geometry of the 2D-FFT images of the tire. The resulting FFT-based analysis allows independent classification of the tire images along two dimensions, specifically by separating “rib” and “lug” features of the tread pattern. Dimensionality of (0,0) indicates a smooth treaded tire with no pattern; dimensionality of (1,0) and (0,1) are purely rib and lug tires; and dimensionality of (1,1) is an all-season pattern. This analysis technique allows a candidate tire to be classified according to the features of its tread pattern, and other tires with similar features and tread pattern classifications can be automatically retrieved from the database.

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