Showing papers on "Phase correlation published in 2023"

Fourier Analysis and Its Application

Abstract: In this article, mathematicianswill give a general summary of Fourier analysis and some of its applications. The Fourier transform will be organized in a developing order, the discrete Fourier transform, Fourier transform on the unit circle and Fourier transform on the real line. Some theorems about the convergence of Fourier series in different forms will be proved in detail. Finally, an estimation of the upper bound of Fourier transform will be discussed. Mathematicianscan prove that it is controlled by the 1-norm of the derivative of f.

Convolution, Correlation and Uncertainty Principle in the One-Dimensional Quaternion Quadratic-Phase Fourier Transform Domain

Abstract: In this paper, we present a novel integral transform known as the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT). We first define the one-dimensional quaternion quadratic-phase Fourier transform (1D-QQPFT) of integrable (and square integrable) functions on R. Later on, we show that 1D-QQPFT satisfies all the respective properties such as inversion formula, linearity, Moyal’s formula, convolution theorem, correlation theorem and uncertainty principle. Moreover, we use the proposed transform to obtain an inversion formula for two-dimensional quaternion quadratic-phase Fourier transform. Finally, we highlight our paper with some possible applications.

Towards quaternion quadratic‐phase Fourier transform

Abstract: The quadratic-phase Fourier transform (QPFT) is a neoteric addition to the class of integral transforms and embodies a variety of signal processing tools like the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic-phase Fourier transform to quaternion-valued signals, known as the quaternion quadratic-phase Fourier transform (Q-QPFT). We initiate our investigation by studying the QPFT of 2D quaternionic signals, and later on, we introduce the Q-QPFT of 2D quaternionic signals. Using the fundamental relationship between the Q-QPFT and quaternion Fourier transform (QFT), we derive the inversion, Parseval's, and Plancherel's formulae associated with the Q-QPFT. Some other properties including linearity, shift, and modulation of the Q-QPFT are also studied. Finally, we formulate several classes of uncertainty principles (UPs) for the Q-QPFT like Heisenberg-type UP, logarithmic UP, Hardy's UP, Beurling's UP, and Donoho–Stark's UP. This study can be regarded as the first step in the applications of the Q-QPFT in the real world.

Enhancing Phase Measurement by a Factor of Two in the Stokes Correlation

Abstract: Phase loss is a typical problem in the optical domain, and optical detectors only measure the amplitude distribution of a signal without its phase. However, an optimal phase is desired in a variety of practical applications, such as optical metrology, nondestructive testing, and quantitative microscopy. Several methods have been proposed to quantitatively measure phase, among which interferometry is one of the most commonly used. An intensity interferometer has also been used to recover phase and enhance the phase difference measurement via the intensity correlation. In this paper, we present and examine another technique based on the Stokes correlation for enhancing phase measurement by a factor of two. The enhancement in phase measurement is accomplished through an evaluation of the correlation between two points of Stokes fluctuations of randomly scattered light and by recovering the enhanced phase of the object by using three-step phase shifting along with the Stokes correlations. This technique is expected to be useful for imaging and the experimental measurement of the phase of a weak signal.