The use of the Hartley transform in geophysical applications

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1488-1495 ◽  
Author(s):  
R. Saatcilar ◽  
S. Ergintav ◽  
N. Canitez
Keyword(s):  
Fourier Transform ◽  
Seismic Data ◽  
Integral Transform ◽  
Complex Multiplication ◽  
One Dimensional ◽  
Hartley Transform ◽  
And Migration ◽  
The Fourier Transform ◽  
Processing Techniques ◽  
Phase Spectra

The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.

scholarly journals Two Dimensional Short Time Hartley Transforms

2016 ◽  
Vol 21 (1) ◽  
pp. 41 ◽  
Author(s):  
Narasimman Sundararajan ◽  
A. Ebrahimi ◽  
Nannappa Vasudha
Keyword(s):  
Fourier Transform ◽  
Seismic Data ◽  
Statistical Properties ◽  
Two Dimensional ◽  
Short Time Fourier Transform ◽  
Hartley Transform ◽  
Hartley Transforms ◽  
The Fourier Transform ◽  
Definition Of ◽  
Short Time

The Hartley transform, as in the case of the Fourier transform, is not suitably applicable to non-stationary representations of signals whose statistical properties change as a function of time. Hence, different versions of 2-D short time Hartley transforms (STHT) are given in comparison with the short time Fourier transform (STFT). Although the two different versions of STHT defined here with their inverses are equally applicable, one of them is mathematically incorrect/incompatible due to the incorrect definition of the 2-D Hartley transform in literature. These definitions of STHTs can easily be extended to multi-dimensions. Computations of the STFT and the two versions of STHTs are illustrated based on 32 channels (traces) of synthetic seismic data consisting of 256 samples in each trace. Salient features of STHTs are incorporated. 

The Hartley transform in seismic imaging

Geophysics ◽  
2001 ◽  
Vol 66 (4) ◽  
pp. 1251-1257 ◽  
Author(s):  
Henning Kühl ◽  
Maurico D. Sacchi ◽  
Jürgen Fertig
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Shift Operator ◽  
Three Dimensions ◽  
Lateral Velocity ◽  
Data Set ◽  
Prestack Migration ◽  
Hartley Transform ◽  
Alternative Means ◽  
The Fourier Transform

Phase‐shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast Fourier transform (FFT). Generally, the Hermitian symmetry of the complex‐valued Fourier transform causes computational redundancies in terms of the number of operations and memory requirements. In practice a combination of the FFT with the well‐known real‐to‐complex Fourier transform is often used to avoid such complications. As an alternative means to the Fourier transform, we introduce the inherently real‐valued, non‐symmetric Hartley transform into phase‐shift migration techniques. By this we automatically avoid the Hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real‐to‐complex FFT. We derive the phase‐shift operator in the Hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation, split‐step migration, and split‐step double‐square‐root prestack migration in terms of the Hartley transform as examples. We test the Hartley phase‐shift operator for poststack and prestack migration using the SEG/EAGE salt model and the Marmousi data set, respectively.

scholarly journals Algorithm for B-scan Image Reconstruction in Optical Coherence Tomography

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kranti Patil ◽  
Anurag Mahajan ◽  
Balamurugan Subramani ◽  
Arulmozhivarman Pachiyappan ◽  
Roshan Makkar
Keyword(s):  
Optical Coherence Tomography ◽  
Fourier Transform ◽  
Optical Coherence ◽  
Cross Sectional ◽  
Image Construction ◽  
Data Points ◽  
Maximum Penetration ◽  
The Fourier Transform ◽  
Processing Techniques

Optical coherence tomography (OCT) is an evolving medical imaging technology that offers in vivo cross-sectional, sub-surface images in real-time. OCT has become popular in the medical as well as non-medical fields. The technique extensively uses for food industry, dentistry, dermatology, and ophthalmology. The technique is non-invasive and works on the Michelson interferometry principle, i.e., dependent on back reflections of the signal and its interference. The objective is to develop an algorithm for signal processing to construct an OCT image and then to enhance the quality of the image using image processing techniques like filtering. The image construction was primarily based on the Fourier transform (FT) of the dataset obtained by data acquisition. This FT could be performed rapidly with the extensively used algorithm of fast Fourier transform (FFT). The depth-wise information could be extracted from each A-scan, i.e., axial scan and also the B-scan was obtained from the A-scan to see the structure of sample. The maximum penetration depth achieved with proposed system was 2.82mm for 1024 data points. First and second layer of leaf were getting at thickness of 1mm and 1.6mm, respectively. A-scans for Human fingertip gave its first, second and third layer was at a thickness of 0.75mm, 0.9mm and 1.6mm, respectively. A-scans for foam sheet gave its first, second and third layer was at a thickness of 0.6mm, 0.75mm, and 0.85mm, respectively.

scholarly journals Microstructural analysis of non-woven fabrics using scanning electron microscopy and image processing. Part 1: Development and verification of the methods

2002 ◽  
Vol 216 (3) ◽  
pp. 199-207 ◽  
Author(s):  
E Ghassemieh ◽  
M Acar ◽  
H Versteeg
Keyword(s):  
Image Processing ◽  
Fourier Transform ◽  
Microstructural Analysis ◽  
Orientation Distribution ◽  
Woven Fabrics ◽  
Computational Time ◽  
Sem Images ◽  
The Fourier Transform ◽  
Processing Techniques ◽  
Scanning Electron

This paper reports image analysis methods that have been developed to study the microstructural changes of non-wovens made by the hydroentanglement process. The validity of the image processing techniques has been ascertained by applying them to test images with known properties. The parameters in preprocessing of the scanning electron microscope (SEM) images used in image processing have been tested and optimized. The fibre orientation distribution is estimated using fast Fourier transform (FFT) and Hough transform (HT) methods. The results obtained using these two methods are in good agreement. The HT method is more demanding in computational time compared with the Fourier transform (FT) method. However, the advantage of the HT method is that the actual orientation of the lines can be concluded directly from the result of the transform without the need for any further computation. The distribution of the length of the straight fibre segments of the fabrics is evaluated by the HT method. The effect of curl of the fibres on the result of this evaluation is shown.

scholarly journals Rotor Fault Detection in Induction Motors Based on Time-Frequency Analysis Using the Bispectrum and the Autocovariance of Stray Flux Signals

Energies ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 597 ◽  
Author(s):  
Miguel Iglesias-Martínez ◽  
Jose Antonino-Daviu ◽  
Pedro Fernández de Córdoba ◽  
J. Conejero
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Induction Motor ◽  
Mean Value ◽  
High Order ◽  
One Dimensional ◽  
Time Frequency ◽  
Temporal Domain ◽  
The Fourier Transform ◽  
The One

The aim of this work is to find out, through the analysis of the time and frequency domains, significant differences that lead us to obtain one or several variables that may result in an indicator that allows diagnosing the condition of the rotor in an induction motor from the processing of the stray flux signals. For this, the calculation of two indicators is proposed: the first is based on the frequency domain and it relies on the calculation of the sum of the mean value of the bispectrum of the flux signal. The use of high order spectral analysis is justified in that with the one-dimensional analysis resulting from the Fourier Transform, there may not always be solid differences at the spectral level that enable us to distinguish between healthy and faulty conditions. Also, based on the high-order spectral analysis, differences may arise that, with the classical analysis with the Fourier Transform, are not evident, since the high order spectra from the Bispectrum are immune to Gaussian noise, but not the results that can be obtained using the one-dimensional Fourier transform. On the other hand, a second indicator based on the temporal domain that is based on the calculation of the square value of the median of the autocovariance function of the signal is evaluated. The obtained results are satisfactory and let us conclude the affirmative hypothesis of using flux signals for determining the condition of the rotor of an induction motor.

scholarly journals Note on Crystallization for Alternating Particle Chains

2020 ◽  
Vol 181 (3) ◽  
pp. 803-815
Author(s):  
Laurent Bétermin ◽  
Hans Knüpfer ◽  
Florian Nolte
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Power Laws ◽  
High Density ◽  
Inverse Power ◽  
Necessary Condition ◽  
Neutral Systems ◽  
One Dimensional ◽  
The Fourier Transform ◽  
Particle Chains

Abstract We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum.

scholarly journals A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics

2013 ◽  
Vol 23 (3) ◽  
pp. 685-695 ◽  
Author(s):  
Navdeep Goel ◽  
Kulbir Singh
Keyword(s):  
Fourier Transform ◽  
Quantum Mechanics ◽  
Fractional Fourier Transform ◽  
Integral Transform ◽  
Linear Canonical Transform ◽  
Product Theorem ◽  
Special Cases ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Representation Transformation

Abstract The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.

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