Fourier Filtration Research Articles

Experimental and Theoretical Investigation of an Oscillator with Collisions

The oscillations of a nonlinear electromechanical oscillator with elastic collisions have been studied. A theoretical model has been built around the equation of motion including an elastic force that is calculated in terms of the Hertz contact theory. It has been shown that an oscillator with collisions may exhibit bifurcations and stable-focus and limit-cycle attractors. Fourier filtration has been applied to analyze the spectral characteristics of signals.

Improved Chandrayaan-1 M3 data: A northwest portion of the Aristarchus Plateau and contiguous maria

We provide and test a method to obtain significant improvement of available Chandrayaan-1 M3 data. The advance is achieved using the Gaussian λ-convolution of spectra and Fourier filtration of images. The main result is imagery of the reflectance across different wavelengths as well as parameters of 1 µm and 2 µm absorption bands with unprecedented quality. This approach can be particularly useful for further investigations using M3 data, since it produces improved imagery of various lunar surface characteristics. We studied a region comprising a portion of the Aristarchus Plateau, Montes Agricola, and a small part of the mare surface in Ocean Procellarum to the north of Montes Agricola. We found that the lava flows in the area between the Aristarchus Plateau and Montes Agricola have a chemical/mineral composition different in comparison with mare areas to the northwest of the ridge Montes Agricola. We also identified distinct spectral properties of morphologically young craters located on the plateau and mare surface. A correlation diagram for positions of the minima of the 1 µm and 2 µm bands allows a cluster analysis of the region, and we map areas associated with a cluster corresponding to pyroclastic glasses. Relationships between geologic and spectral parameter maps were established.

Windowed Fourier transform as an essential digital interferometry tool to study coupled heat and mass transfer

Abstract Series of thermodiffusion experiments using optical digital interferometry (ODI) have been conducted onboard the International Space Station. Conventionally, the two-dimensional (2D) fast Fourier transform (FFT) fringe analysis technique has been applied as a fast and reliable technique to extract data. In this study, for the first time, the windowed Fourier transform (WFT) method is used to analyze the same experiments. In this method, a Fourier transformation is applied on the fringes at two different stages: initially, during the filtration of the non-zero peaks and then on the wrapped phase image. We provide a detailed comparison between FFT and WFT results of binary and ternary mixtures for ODI thermodiffusion experiments. The substantial enhancements of this method are presented and discussed for different experiments conducted for both binary and ternary mixtures. We show that while disturbances in the phase fringe pattern can cause significant error in FFT techniques, if the windowed Fourier filtration (WFF) parameters are properly chosen this type of noise can be eliminated during WFF analysis. The importance of replacing the FFT method becomes more pronounced for the ternary system, as this method fails to reconcile reliable concentration profiles. The results of this work can show that the application of the windowed Fourier transform in optical digital interferometry investigations show improved results over the same experiments analyzed using FFT methods, especially for experiments involving very small heat and mass fluxes such as the Soret effect in multicomponent mixtures.

RADIO VARIABILITY OF THE QUASAR 3C 273 ON THE CENTIMETRIC WAVES -WAVELET-ANALYSIS

C 273- has been intensively investigated for many years, since opening of quasars in 1963. Since 1965 on radio telescope RT-26 of Michigan University on frequencies 14.5, 8 and 4.8 GHz long monitoring of this radio source have been carried out. Flux variability of a radio emission on studied frequencies consists a trend on which fast flux changes with characteristic time from 1 to 5 years are imposed. Fourier's methods and the wavelet-analysis that allowed investigating in details changes harmonious component of signals over time are applied. On a trend component the main period makes 8 years. With Fourier filtration have been received «O - C» data, for allocation high-frequency component in studied signals. By the results of calculations of wavelet-spectrums the periods of 3.5 and 2.3 years are revealed. On the basis of calculation of integrated wavelet-spectrums in a frequency range on «O - C» this «spectrum of the periods» characterizing main phases of activity of a source are defined. On the basis of the program written on IDL, supplementing the wavelet-analysis, delay change between fluxes on separate studied frequencies for each of periodic components has been defined eventually. The average delay for the 8 years periodic component in the range of frequencies of 4.8-8 GHz is about 1 year. In the range of frequencies 8 - to 14.5 GHz the average size of a delay was about 0.5 years. The average delay for all intervals of frequencies for the 3 years periodic components has appeared equal 0.3 years.

Narrow frequency analysis for periodic broadband noisy signals.

The Wigner–Ville transform was used twice for analysis of periodic broadband noisy signals. The first time use is for an extraction of informative part (part in future) from the signal when time window is sliding across a signal from a beginning to the end. Time window overlap N% is applied. This corresponds to N% time of existing window for a part of analyzed signal. The time window size is equal to the part duration time T1. That allows increasing the signal to noise ratio for this part within the window. Then the narrow Fourier filtration (which corresponds to the most informative part of the signal) is applied for the output of time window. Broadband signal becomes narrow band after that. Second time the Wigner–Ville approach is used to extract a periodicity in filtered signal. In this case second window was applied. This window size is at least two times more parts’ repetition time T2. Overlap is M%. This signal processing was successfully used for acoustic and electromagnetic signals of human motion due to footsteps and for underwater clicks of beaked whales in passive acoustic identification. [This work was supported by Army Armament Research, Development, and Engineering Center Contract No. W15QKN-09-C-0163.]

How to measure positions of elements of intraocular optics: Methods for improving resolution

The previously proposed method for noninvasively measuring the positions of elements in the human eye is further developed. A procedure for increasing the resolution of the method based on decreasing the discreteness of measurements is proposed and implemented in numerical experiments. For a system of two closely located phase screens, the resolution was increased 16-fold, which made it possible to determine the positions of the two screens. In order to resolve the positions of all optical elements, for the Gullstrand eye model, it was sufficient to increase the resolution and the Fourier filtration by a factor of four.

An electron microscopic study of the archaeal feast/famine regulatory protein 5. Fourier filtration of images.

Microcrystals of the feast/famine regulatory protein (FFRP) pot0434017 (FL11) were prepared by sonicating larger crystals. Using the microcrystals a cryo-electron micrograph was obtained, which showed a hexagonal packing of cylinder-like assemblies of FL11. This micrograph was processed by selecting, in the Fourier space, spots reflecting the crystal lattice, thereby removing the noise. The microcrystal was not totally free from distortion, and cylinders in local clusters adopted slightly different orientations. Thus, 25 hexagonal units closest to the ideal, each containing a cylinder at the center surrounded by six others, were manually selected. The averaged image was further processed to yield a perfect six-fold symmetry. These processed images, and some of the original images too, show bridges connecting cylinders, each corresponding to two pairs of N-domains, protruding from the two cylinders and contacting between them in the X-ray structure. (Communicated by Masanori OTSUKA, M.J.A.)

Fourier filtration of XANES as a source of quantitative information of interatomic distances and coordination numbers in crystalline minerals and amorphous compounds

The method for quantitative determination of interatomic distances and coordination numbers (CN) in minerals and amorphous compounds by their x-ray absorption near edge structure (XANES) is proposed. The generation of themethod and the proof of its main points are based on the revealed theoretical description of XANES for crystalline compounds and minerals as a sum of different photoelectron scattering processes on two- and approximately linear three-atoms chains, originated at the absorbing atom. The method consists of the Fourier filtration (FF) of the experimental XANES within the short range of photoelectron's wave numbers and the following fitting, with three varied parameters, of the first shell term extracted by the FF procedure. The accuracy of the obtained local structural parameters is illustrated for the reference crystalline minerals, which are used also to simulate the XANES formation in glasses. The application of the method to Mg XANES in diopside-glass and its crystalline equivalent, Diopside, permits one to determine the Mg-O distance and CN for the first oxygen shell around the Mg atom and to reveal the change of these parameters between the glass and the crystal with similar chemical composition.

Self-marking phase-stepping electronic speckle pattern interfometry (ESPI) for 3D displacement measurement on cathode ray tube (CRT)-panels

This study demonstrates the feasibility of applying phase-shifting electronic speckle pattern interfometry to measure the deformation field of the front panel of a cathode ray tube, to support analysis to enhance the implosion-resistance capacity under violent collapse. Two effects, the air exhaustion and shrink band constraint effects, are comprehensively investigated. The angle of an adjustable mirror is switched, to provide three sensitivity vectors that are required in 3D-displacement measurement. A Fourier filtration is employed to remove speckle noise and establish a noise-free phase map. Inconsistent points are identified and masked to prevent any possible divergence during phase unwrapping. The results show that the accuracy of this method is satisfactory.

Image processing and fine structure of hydroxyapatite particles

Interpretation of high resolution electron microscope (HREM) images of small hydroxyapatite (OHAp) crystals with image processing using Fourier filtration techniques has been performed. The effects of digitizing filter parameters in relation to the extent of possible artefacts — like crystal bleeding — which limit the correct image interpretation have been tested. Image processing for OHAp crystallites was applied in two different cases: (i) crystal inhomogeneities (void boundaries) where image provides a good test for the extension of the bleeding in reconstructed images; (ii) beam-induced domains with different crystal structure (or different orientation) than the OHAp matrix where processing provides a test on the precise location and contrast enhancement of those secondary structures.

Two-dimensional Fourier filtration of acoustic quantification echocardiographic images: Improved reproducibility and accuracy of automated measurements of left ventricular performance

To determine the accuracy of Fourier filtration in removing the high-frequency component of noise from acoustic quantification (AQ) echocardiographic images, we processed 800 parasternal short-axis images obtained from 10 study subjects. M-mode tracings were also obtained and used as gold standard for correlating the results from raw AQ and Fourier-filtered images. Left ventricular short-axis diameters measured from the raw AQ and Fourier-filtered data were compared with the M-mode diameters ( r = 0.91, p < 0.001 for raw AQ; and r = 0.96, p < 0.001, for Fourier filtered images). Fractional shortening showed better correlation between Fourier-filtered images and M-mode ( r = 0.79, p < 0.03) versus raw AQ and M-mode ( r = 0.33, p = 0.46). Beat-to-beat reproducibility was also found to be better for fractional area change ( r = 0.82, p = 0.01 versus r = 0.66, p = 0.07), peak area filling rate ( r = 0.87, p = 0.004 versus r = 0.62, p = 0.1), and peak area emptying rate ( r = 0.99, p < 0.0001 versus r = 0.19, p < 0.7) for Fourier filtered versus raw AQ. Our results indicate that Fourier filtration of AQ data results in more accurate representation of the true endocardial borders. (J Am Soc Echocardiogr 1997;10:310-9.)

Automatic analysis of three-dimensional displacement field using ESPI

Electronic speckle pattern interferometry (ESPI) has been used to study the in-plane and out-of-plane displacements of the object. In order to improve the accuracy, a Fourier filtration algorithm has been used to remove the speckle noise and get the holographic-quality ESPI fringe pattern. The processing steps of this method are described in detail in this paper. In addition, a phase shifter, which is easily used with simplified structure and high stability, is also presented. It can be applied to various coherent arrangements in experiments to obtain phase shifted fringe patterns. Experiments of determining the 3D displacement field of a circular fixed plate with a uniform load have been carried out using these methods. The results presented in this paper indicate that the accuracy of this method is satisfactory.

Prospects for the direct electron crystallographic determination of zeolite structures.

Recently two successful zeolite structures based on experimental electron crystallographic data have been published. Diffraction and image data based on the silicate portion of the zeolite, mordenite, which are perturbed by dynamical (as well as secondary) scattering, have been simulated by a multiple-beam dynamical scattering program. Structure analyses with these data show that the above claims are not unreasonable, given a high enough accelerating voltage for the electron beam. If, for example, 2.9 A resolution micrographs are taken from a 120 A thick crystal in a 200 or 400 kV electron microscope, the crystallographic phases found by image analysis (Fourier filtration) are accurate enough to be extended by the Sayre equation to the (atomic) resolution limit of the electron diffraction pattern (for example from a 105 A thick crystal illuminated by a 1,200 kV electron source). The resultant potential map can be interpreted to find most of the atomic positions and the remaining ones will appear during the progress of a Fourier refinement.

950-95 Enhanced Detection of Endocardial Motion by Spatiotemporal Fourier Filtration of 2-D Echocardiographic Acoustic Quantification Images

Automated detection of endocardial borders by 2-D echocardiography can be useful for the assessment of regional wall motion. Although Acoustic Quantification (AQ) currently provides on-line endocardial edge detection, the presence of high frequency noise in AQ images limits its utility for this purpose. We propose that spatiotemporal Fourierfiltration (FF) of AQ images may improve signal-to-noise ratio and enhance accuracy of analysis of endocardial motion. We stored digitally the 2-D AQ echocardiographic images obtained from the parasternal short axis view in 10 study subjects. These were processed using a stepwise computer algorithm that included smoothing using FF with spatial and temporal cutoff of 10 and 5 harmonics, respectively. We tested the accuracy of the FF against the raw AQ images by 1) comparing the vertical axis diameters of the raw AQ and FF images with the measurements obtained from digitized M-Mode tracings, and 2) counting the number of LV internal diameter reversals during each cardiac cycle along the vertical and horizontal axis. Five successive frames of endocardial borders from a loop of raw AQ and after FF are superimposed below. 1) The correlation between M-mode and raw AQ was 0.86, vs. 0.95 between M-mode and FF (p &lt; 0.001). 2) The number of reversals per cycle was 13 ± 3.5 for AQ vS. 3.1 ± 3 for FF (p &lt; 0.001). By eliminating high frequency noise from the echocardiographic AQ images, spatiotemporal Fourier filtration provides a more accurate representation of the endocardial edge motion.

Laminogram reconstruction through regularizing Fourier filtration

Abstract Coplanar rotational laminography (CRL) is a special case of laminography which is a tomographic technique used to image cross-sectional views through solid objects. In the case of CRL the film and object rotate synchronously, while the radiation source remains stationary. The image formed on the exposed film is highly blurred. This blurring may be described as a convolution of the ideal image with a radially symmetric point spread function (PSF). To deblur the laminographic image (laminogram) one has to perform a deconvolution of this image based on a laminographic PSF. The deconvolution is an ill-posed problem and therefore special methods are required for its realization. One of the possible methods, regularizing Fourier filtration, was tried and its most successful version was selected. A simple technique for the choice of filter parameters is proposed, based on an image quality evaluation method which was derived. Several laminograms (both simulated and experimental) were successfully reconstructed by this method.

Surface topography and molecular stoichiometry of the mitochondrial channel, VDAC, in crystalline arrays

The mitochondrial outer membrane contains a protein, called VDAC, that forms large aqueous pores. In Neurospora crassa outer membranes, VDAC forms two-dimensional crystalline arrays whose size and frequency can be greatly augmented by lipase treatment of these membranes (C. Mannella, Science 224, 165, 1984). Fourier filtration and surface reconstruction of freeze-dried/shadowed (45°) arrays produced detailed images of two populations of crystals, whose lattices are mirror images of each other. Most likely, this technique has revealed both surfaces of the same two-dimensional crystal with lattice parameters: a = 12.3 ± 0.1 nm, b = 11.2 ± 0.1 nm, and θ = 109 ± 1°. Three-dimensional reconstructions of the surface reliefs on both sides of the crystal show them to be very similar. The majority of the protein forming the channel appears to be at or below the level of the membrane. To address the issue of the number of 30-kDa polypeptides that form a VDAC channel, measurements of mass per unit area were carried out by analyzing scanning transmission electron micrographs of unstained, freeze-dried arrays. The crystal form used for mass analysis contained the same motif of six stain-accumulating centers per unit cell, with p2 symmetry as in the oblique configuration, but it had a different orientation relative to the lattice lines. These data yielded a surface density of 1.9 ± 0.2 kDa/nm 2, indicating that there is a one-to-one ratio between VDAC polypeptides and the channels visualized in filtered electron micrographs, and that VDAC membrane crystals contain 68% protein and 32% lipid by mass.

The cell envelope of the hyperthermophilic archaebacterium Pyrobaculum organotrophum consists of two regularly arrayed protein layers: three‐dimensional structure of the outer layer

The cell envelope of the hyperthermophilic sulphur-reducing archaebacterium Pyrobaculum organotrophum H10 was found to be composed of two distinct hexagonally arranged crystalline protein arrays. Electron microscopic analysis of freeze-etched cells and isolated envelopes in conjunction with image processing showed that the inner layer (lattice centre-to-centre spacing 27.9 nm) is essentially identical to the protein array of Pyrobaculum islandicum GEO3, a complex, rigid structure implicated in the maintenance of cell shape. The outer layer has clear p6 symmetry and a lattice spacing of 20.6 nm. Its three-dimensional structure was reconstructed from a negative stain tilt series of an intact double-layered envelope using Fourier filtration to separate the desired information from the other lattices present. The outer layer is a unique, porous network of blocklike dimers disposed around six-fold axes, and exhibits minimal asymmetry between its inner and outer faces. It appears to be rather loosely associated with the outer surface of the inner layer. In most H10 envelopes, the inner layer is orientated with one base vector exactly perpendicular to the long axis of the cell, so that the cylindrical portion is composed of a series of parallel cell-girdling hoops of hexameric morphological units. All the other known Pyrobaculum strains were found to have a GEO3-type envelope structure, consisting of a single rigid protein array and a fibrous capsule. Although H10 does not possess a capsule, fibrils appear to be sandwiched between the two protein layers.

Unbending: Characterising Distortion and Correcting for IT

Unbending is becoming the fashionable alternative to simple correlation averaging, just as c-av earlier supplanted Fourier filtration or profile fitting. We note the circumstances in which it may be useful; and we recommend local real space image averaging in preference to Fourier methods.The displacement field in an image can be estimated reasonably closely from the unit cell positions indicated by cross-correlation with a ‘reference’ image patch. Correlation averaging and unbending both compensate for the displacements of unit cells as a whole, but in their standard forms do not address distortion of individual cells at all. Obviously, both can be easily adapted to do so; some such work has been reported and more is in hand. Experience suggests that correction of the displacements is indeed the most important step: these accumulate from cell to cell across the crystal.

Image processing of partially periodic lattice images of polymers: The study of crystal defects

High resolution electron microscopy (HREM) is becoming an important technique for the study of crystal defects in polymers. Because of radiation damage, lattice images of polymer crystals suffer from a lack of contrast and a low signal-to-noise ratio that can be improved by digital processing. We investigate the limits and benefits of such methods based on manipulation in Fourier space where periodic features contained in the image can be recognized and separated from the noise. We have calculated the effects of geometrical factors describing the shape and extent of lattice domain(s) present in the image on its discrete Fourier transform by taking into account digitization parameters. Lattice images were obtained from polyparaxylylene (PPX) single crystals which, under the growth conditions we used, contain defects as twin boundaries. Also, a crystal edge and a pair of edge dislocations have been imaged. Image processing methods used were filtration and truncation in Fourier space. The dislocation dipole is well restored by Fourier filtration but whenever the image contains different lattice domains such as rotated parts (twin) or noncrystalline zones, Fourier filtration introduces artefacts of two types: (1) lattice bleeding due to averaging effects results in a loss of crystal/crystal boundary or crystal edge definition, and we show that in these cases better results are obtained by Fourier truncation; (2) a ghost lattice related to the noise components of the image is created by the filter and appears on the reconstructed image at every place where the crystal lattice information is weak or nonexistent.

Artifacts on lung CT scans: removal with Fourier filtration.

A preliminary study was done of Fourier analysis of high-resolution computed tomographic (CT) images of normal pulmonary parenchyma. A two-dimensional fast Fourier transform was applied to 20 regions of interest (64 x 64 pixels) obtained from CT images of six lungs. A qualitative analysis of the spectra and the application of filters in the frequency domain allowed the characterization and removal of reconstruction artifacts. Selective application of a low-pass filter removed these parasitic frequencies and clearly enhanced the quality of the image.