scholarly journals Convolution Theorems for Quaternion Fourier Transform: Properties and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Heat Equation ◽  
Quaternion Algebra ◽  
Fourier Transforms ◽  
Two Dimensional ◽  
Quaternion Fourier Transform

General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

Continuous quaternion fourier and wavelet transforms

2014 ◽  
Vol 12 (04) ◽  
pp. 1460003 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Wavelet Transforms ◽  
Quaternion Algebra ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Fundamental Properties

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel [Formula: see text] is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

scholarly journals A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Mawardi Bahri ◽  
Resnawati ◽  
Selvy Musdalifah
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Young Inequality ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Research Papers ◽  
Quaternion Fourier Transform ◽  
Quaternion Linear Canonical Transform ◽  
The Relationship

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

scholarly journals Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Gaussian Functions ◽  
The Fourier Transform ◽  
Definition Of ◽  
Quaternion Linear Canonical Transform

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

Possible Models of Murein and Their Fourier Transforms

1982 ◽  
Vol 37 (3-4) ◽  
pp. 226-235 ◽  
Author(s):  
Helmut Formanek
Keyword(s):  
Fourier Transform ◽  
Cell Walls ◽  
Fourier Transforms ◽  
Elementary Cell ◽  
Two Dimensional ◽  
Density Measurements ◽  
X Ray ◽  
Rigid Layer ◽  
The Fourier Transform ◽  
Almost All

Abstract Murein, Models, Fourier Transforms Murein, the rigid layer of the cell walls of almost all bacteria can be regarded as derivative of chitin. Within the sterically allowed region its polysaccharide chain can perform conformations with two-to threefold screw axes. Two dimensional Fourier transforms calculated from different possible conformations have been compared with data of density measurements, X-ray and electron diffraction. The Fourier transform of murein with a chitin-like conformation of the poly­ saccharide chain and an elementary cell of 4.5 × 10.4 × 21.5 Å3 provides the best agreement with the experimental results.

DIRECTIONAL AND WIDE‐BAND VELOCITY FILTERING

Geophysics ◽  
1965 ◽  
Vol 30 (2) ◽  
pp. 279-280
Author(s):  
Philip L. Jackson
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Wide Band ◽  
Dimensional Structure ◽  
Two Dimensional ◽  
Contour Map ◽  
Directional Filtering ◽  
Velocity Filtering ◽  
Monochromatic Illumination

The purpose of this note is to point out that the condition of monochromatic illumination in obtaining a two‐dimensional Fourier transform by optical means may be relaxed for wide‐band velocity filtering, and for the directional filtering of any two‐dimensional structure such as a contour map. Similar considerations hold for two‐dimensional Fourier transforms obtained by any means.

scholarly journals Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples

2021 ◽  
pp. 55-65
Author(s):  
Olga Ponomareva ◽  
Aleksey Ponomarev
Keyword(s):  
Fourier Transform ◽  
Fourier Analysis ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Information Control ◽  
Two Dimensional ◽  
Negative Effects ◽  
Variable Parameters ◽  
Discrete Fourier Transforms ◽  
Fourier Matrix

Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

scholarly journals TEOREMA KONVOLUSI PADA TRANSFORMASI FOURIER FRAKSIONAL QUATERNION SISI KIRI DAN SIFAT-SIFATNYA

2019 ◽  
Vol 16 (1) ◽  
pp. 1-12
Author(s):  
N H Wibowo ◽  
S Musdalifah ◽  
Resnawati Resnawati
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Quaternion Algebra ◽  
Fractional Fourier Transform ◽  
Convolution Theorem ◽  
Quaternion Fourier Transform

Fourier transform (FT) is growing very rapidly and applying in various fields such as analyzing and decomposing signals in the frequency domain. FT has been extended to quaternion algebra known as the Quaternion Fourier Transform (QFT). The purpose of this paper are to formulate the definition and properties of the left sided Quaternion Fractional Fourier Transform (QFFT), to formulate the definition and convolution theorem for left sided QFFT. Firstly, the results showed the formulation of the left sided QFFT definition and some of the properties such as linearity, translation, modulation and scalar. Secondly, its showed the formulation of convolution theorem for left sided QFFT and also the left sided QFFT of conjugate and translation convolution.

APPLICATION OF FOURIER TRANSFORMS TO ANTENNA PATTERN SYNTHESIS

1961 ◽  
Vol 39 (9) ◽  
pp. 1347-1356
Author(s):  
Alfred T. Villeneuve
Keyword(s):  
Fourier Transform ◽  
Field Distribution ◽  
Fourier Transforms ◽  
Antenna Pattern ◽  
Two Dimensional ◽  
Pattern Synthesis

The two-dimensional Fourier transform of the field quantities is employed to synthesize source distributions over plane rectangular apertures when a field distribution is prescribed at an arbitrary distance from the source. Several examples of application of the technique are included. The approximations involved are discussed.

scholarly journals Color image watermarking using multidimensional Fourier transforms

2021 ◽  
Author(s):  
Tsz Kin Tsui
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Discrete Fourier Transform ◽  
Fourier Transforms ◽  
Color Image ◽  
Image Watermarking ◽  
Color Image Watermarking ◽  
Quaternion Fourier Transform ◽  
Just Noticeable Distortion ◽  
Human Eyes

This thesis presents two vector watermarking schemes that are based on the use of complex and quaternion Fourier transforms and demonstrates, for the first time, how to embed watermarks into the coefficients consistent with our human visual systems (HVS). Watermark casting is performed by estimating the Just-Noticeable distortion (JND) of the images, to ensure watermark invisibility. The first method encodes the chromatic content of a color image as CIE a*b* chromaticity coordinates whereas the achromatic content is encoded as CIE L tristimulus value. Color watermarks (yellow and blue) are embedded in the frequency domain of the chromatic channels by using Spatio Chromatic Discrete Fourier Transform (SCDFT). It first encodes a* and b* as complex values, followed by a single discrete Fourier Transform. The most interesting characteristic of the scheme is the possibility of performing watermarking in the frequency domain of chromatic components. The second method encodes the L*a*b* components of color images and color watermarks are embedded as vectors in the frequency domain of the channels by using the Quaternion Fourier Transform (QFT). The idea is twofold: Robustness is achieved by embedding a color watermark in the coefficient with positive frequency, which spreads it to all components in the spatial domain. On the other hand, invisibility is satisfied by modifying the coefficient with negative frequency, such that the combined effects of the two are insensitive to human eyes

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