Quaternion Fourier Transform Research Articles

PSO-based Quaternion Fourier Transform steganography: Enhancing imperceptibility and robustness through multi-dimensional frequency embedding

This paper presents a novel steganography technique using the Quaternion Fourier Transform (QFT) in the 4D frequency domain to enhance imperceptibility and robustness in digital image embedding. Steganography, the art of hiding information within media, faces challenges in balancing security, imperceptibility, and robustness. To address this, we leverage the multi-dimensional properties of quaternions, enabling the embedding of secret data in both grayscale and color images. For grayscale images, two quaternion dimensions are utilized for intensity and secret data, while for color images, all four dimensions are employed with one reserved for metadata. The research question centers on how to maximize spatial dispersion and color similarity while maintaining high imperceptibility and robustness against attacks. Experimental results show that the proposed method improves visual imperceptibility by more than 4 % and exhibits a 13 % increase in robustness against common steganalysis attacks compared to the best state-of-the-art existing technique. These advancements highlight the potential of this method for applications in secure communication, digital watermarking, and copyright protection. By combining the quaternion mathematical framework with a novel optimization strategy, this approach significantly improves upon traditional steganography methods.

Ellipsometry of surface acoustic waves using 3D vibrometry for viscoelastic material characterization by the estimation of complex Lamé coefficients versus the frequency

Surface acoustic waves (SAW) are adequate regarding material characterization because they have low geometric attenuation compared to bulk waves. SAW can be generated easily by normal excitation using contact transducers or power lasers and have also a unique elliptic polarization, characterized by two parameters: the ellipticity (H/V) ratio between the horizontal and the vertical components of the elliptic motions and the orientation angle (θ) between the horizontal axis of the ellipse and the surface. In the case of a viscoelastic isotropic material, a complete characterization is achieved by the association of the quantitative measurement of the polarization and the propagative characteristics, the complex wavenumber, of the SAW. In practice, this operation is performed using 3D lased vibrometry for propagation monitoring in space and time. The post-processing is carried out by Quaternion Fourier Transform, the Prony algorithm and the complex Lamé coefficients identification for the theoretical model of propagation on the material. Good agreement is observed between the obtained results and the ones of the pulse-echo method.

Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform

In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.

Hyperspectral Anomaly Detection Using Reconstruction Fusion of Quaternion Frequency Domain Analysis.

Most existing techniques consider hyperspectral anomaly detection (HAD) as background modeling and anomaly search problems in the spatial domain. In this article, we model the background in the frequency domain and treat anomaly detection as a frequency-domain analysis problem. We illustrate that spikes in the amplitude spectrum correspond to the background, and a Gaussian low-pass filter performing on the amplitude spectrum is equivalent to an anomaly detector. The initial anomaly detection map is obtained by the reconstruction with the filtered amplitude and the raw phase spectrum. To further suppress the nonanomaly high-frequency detailed information, we illustrate that the phase spectrum is critical information to perceive the spatial saliency of anomalies. The saliency-aware map obtained by phase-only reconstruction (POR) is used to enhance the initial anomaly map, which realizes a significant improvement in background suppression. In addition to the standard Fourier transform (FT), we adopt the quaternion FT (QFT) for conducting multiscale and multifeature processing in a parallel way, to obtain the frequency domain representation of the hyperspectral images (HSIs). This helps with robust detection performance. Experimental results on four real HSIs validate the remarkable detection performance and excellent time efficiency of our proposed approach when compared to some state-of-the-art anomaly detection methods.

Boas Type Results for Two-Sided Quaternion Fourier Transform and Uniform Lipschitz Spaces

Boas Type Results for Two-Sided Quaternion Fourier Transform and Uniform Lipschitz Spaces

Robust watermarking method for securing color medical images using Slant-SVD-QFT transforms and OTP encryption

This paper proposes a new robust watermarking method for securing color medical images, where the proposed method relies on combining Slant, Singular Value Decomposition (SVD), and quaternion Fourier-Transform (QFT). The stimulus behind this combination is to improve the invisibility and durability of the proposed method. First, the input cover image is split into four equal parts for fast computation and then encrypted using one-time padding (OTP) encryption to increase robustness. Slant transform is applied to encrypted blocks to increase image compaction. The SVD is applied to the transformed image blocks to preserve quality during the embedding process. Finally, the QFT is applied to increase visual imperceptibility. The binary watermark is first compressed using SVD, scrambled using Arnold encryption to raise security, and then embedded in the modified QFT coefficients. The extraction procedures are the inverse operations of the embedding procedures. The proposed method achieves a good tradeoff between invisibility and robustness compared to existing schemes versus many geometrical, processing, and hybrid attacks. Also, the proposed method attains high visual imperceptibility. The extracted watermark seems to be the original watermark with minimum BER and high NC values. Empirical findings indicate that the proposed method is giving better invisibility and robustness results while maintaining a high capacity compared to other existing watermarking methods.

Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters.

One-Dimensional Quaternion Fourier Transform with Application to Probability Theory

The Fourier transform occupies a central place in applied mathematics, statistics, computer sciences, and engineering. In this work, we introduce the one-dimensional quaternion Fourier transform, which is a generalization of the Fourier transform. We derive the conjugate symmetry of the one-dimensional quaternion Fourier transform for a real signal. We also collect other properties, such as the derivative and Parseval’s formula. We finally study the application of this transformation in probability theory.

Towards quaternion quadratic‐phase Fourier transform

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.

The Lorentz Group Using Conformal Geometric Algebra and Split Quaternions for Color Image Processing: Theory and Practice

The processing of color images is of great interest, because the human perception of color is a very complex process, still not well understood. In this article, firstly the authors present an analysis of the well-known mathematical methods used to model color as a phenomenon and to process color images. We propose an adequate metric to process color images using the Minkowski or space-time metric. We propose that the processing of HSV images can be done using the HSV cone in the quaternion split algebra or the conformal geometric algebra frameworks. We show that the processing of RGB images using the Euclidean metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> doesn’t yield good results. The colors of images change by daylight, we understand this phenomenon as the color change follows the action of the Lorentz Lie group. Consequently, we formulate a new model for representing and processing color using split quaternions and space-time conformal geometric algebra. We propose new algorithms for practical color image processing: we formulate the novel Split Quaternion Fourier Transform for color image processing and we interpolate color images using Split Motors which belong to the space-time conformal geometric algebra. The experimental part proves that real color images have to be processed in the HSV cone. We show successful applications of the Quaternion Split Fourier Transform and the interpolation processing. We compare these computations using the RGB metric in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{3}$ </tex-math></inline-formula> space and using the Minkowski metric in the HSV cone. This comparison shows clearly that the color image processing using the Minkowski metric in the HSV cone performs better.

The Ortogonal Property of Directional short-time quaternion Fourier transform

This study will examine the orthogonal properties of Directional Short-time quaternion Fourier transform (DSTQFT) which is a further study of DSTFT with an expansion in the form of a function that has a Quaternion value. The orthogonal properties of DSTQFT are obtained by combining the orthogonal properties of DSTFT and the Fourier quaternion transform (QFT). Based on the results of the study, it was found that the orthogonal nature of the Directional Short-Time quaternion Fourier Transform (DSTQFT) is different from the orthogonal nature of the Directional Short-Time Fourier Transform (DSTFT). In addition, there are three important consequences of orthogonality in TFQSTB as a result of the quaternion-valued function.

Uncertainty Principles for the Two-Sided Quaternion Windowed Quadratic-Phase Fourier Transform

A recent addition to the class of integral transforms is the quaternion quadratic-phase Fourier transform (Q-QPFT), which generalizes various signal and image processing tools. However, this transform is insufficient for addressing the quadratic-phase spectrum of non-stationary signals in the quaternion domain. To address this problem, we, in this paper, study the (two sided) quaternion windowed quadratic-phase Fourier transform (QWQPFT) and investigate the uncertainty principles associated with the QWQPFT. We first propose the definition of QWQPFT and establish its relation with quaternion Fourier transform (QFT); then, we investigate several properties of QWQPFT which includes inversion and the Plancherel theorem. Moreover, we study different kinds of uncertainty principles for QWQPFT such as Hardy’s uncertainty principle, Beurling’s uncertainty principle, Donoho–Stark’s uncertainty principle, the logarithmic uncertainty principle, the local uncertainty principle, and Pitt’s inequality.

Small and dim infrared moving target detection based on spatial–temporal saliency

Infrared target detection is a significant work in military, high accuracy and high speed are the essential capabilities for precise guidance. With limited conditions, making full use of spatial–temporal information can really improve performance. This article studied small and dim infrared moving target detection based on sequence images. Quaternion Fourier Transform is an efficient method of visual saliency. To make this algorithm more accurate, we introduced four preprocessing feature map as input channels for quaternion. They are selective inter frame difference, fuzzy-based morphology filter, joint local gradient and gray contrast measure and temporal-constrained Gaussian curvature filter. All of them aim to highlight small-dim target and help to construct saliency map. Simultaneously, a trajectory constraint is introduced to reduce false alarm rate. Compared with single-frame based methods and multi-frames based methods, our novel algorithm performed better on balancing time consuming and detection precision.

A Variation on Inequality for Quaternion Fourier Transform, Modified Convolution and Correlation Theorems for General Quaternion Linear Canonical Transform

The quaternion linear canonical transform is an important tool in applied mathematics and it is closely related to the quaternion Fourier transform. In this work, using a symmetric form of the two-sided quaternion Fourier transform (QFT), we first derive a variation on the Heisenberg-type uncertainty principle related to this transformation. We then consider the general two-sided quaternion linear canonical transform. It may be considered as an extension of the two-sided quaternion linear canonical transform. Based on an orthogonal plane split, we develop the convolution theorem that associated with the general two-sided quaternion linear canonical transform and then derive its correlation theorem. We finally discuss how to apply general two-sided quaternion linear canonical transform to study the generalized swept-frequency filters.

New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain

The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB.

Generalized uncertainty principles associated with the quaternionic offset linear canonical transform

The quaternionic offset linear canonical transform (QOLCT) can be defined as a generalization of the quaternionic linear canonical transform (QLCT). In this paper, we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact, we prove the Rayleigh formula and some properties related to the QOLCT. Then, we generalize some different uncertainty principles (UPs), including Heisenberg-Weyl's UP, Hardy's UP, Beurling's UP, and logarithmic UP to the QOLCT domain.

Key Frame Extraction Based on Quaternion Fourier Transform with Multiple Features Fusion

Key Frame Extraction Based on Quaternion Fourier Transform with Multiple Features Fusion

Quaternion Fourier Transform spectral analysis of electrical currents for bearing faults detection and diagnosis

This paper suggests a new method for detecting and diagnosing bearing faults in three-phase asynchronous devices using the periodogram estimation of the Power Spectral Density of the three-phase electrical current space vector. Rather than a single single-phase spectral analysis or a space vector component spectral analysis using the classic Fourier Transform, The Quaternion Fourier Transform was used to do a full spectral analysis of the three-phase currents scheme. Spectral indicators are given from the Quaternion Power Spectral Density and their use suggested for accurate detection and diagnosis of bearing faults via the quaternion's formalism. To validate this approach, the proposed indicators were compared to fault indicators derived from a single-phase or inferred from the real and imaginary parts of the currents space vector. The results reveal that two of the new metrics outperform the classic ones derived from either single-phase spectral analysis or the three-phase space vector components spectral analysis (classification rates of 94.2 percent and 93.8 percent, respectively). These findings support the preference for three-phase approaches over single-phase approaches, as well as the ability to extend classical research methods developed for single-phase systems to three-phase systems using the Quaternion Fourier transform.

Uncertainty principles for the two‐sided offset quaternion linear canonical transform

The offset quaternion linear canonical transform (OQLCT) provides a more general framework for a number of linear integral transforms in signal processing and optics, such as quaternion Fourier transform (QFT), fractional quaternion Fourier transform (FrQFT), and linear canonical transform (QLCT). We devote this paper to various different of uncertainty principles (UPs) for the two‐sided OQLCT, which including logarithmic UP, Heisenberg‐type UP, Hardy's UP, Beurling's UP, Entropic UP, Donoho‐Stark's UP, and Local UP. Moreover, we also prove Lieb's UP for the two‐sided short‐time offset quaternion linear canonical transform (SOQLCT).

Advanced Color Edge Detection Using Clifford Algebra in Satellite Images

Edge detection is widely used for image processing to improve the detection and classification of objects, segmentation, and extraction of other features. Satellite images are rich in information about objects with different color intensity and have a large amount of noise, so it is difficult to achieve recognition, classification, and feature extraction of small objects through traditional edge detection algorithms. The colors in satellite images suffer from a large amount of overlap due to areas or weather conditions that generate a lot of noise. Edge detection provides detailed information about objects in an image by reducing unnecessary feature information. Edge detection in color images is more challenging than edge detection in gray-level images. This paper proposes a method for the edge detection of color images using Clifford algebra and its sub-algebra, quaternions. Quaternion-based Fourier transform is used to process red, green and blue (RGB) images separately in the vector field. A 3×3 quaternion mask is developed to filter out frequencies of the image in multiple directions and only provides details about the edges. The algorithm works on three channels individually; the output is then processed through quaternion Fourier transform (QFT) and inverse QFT with a 3×3 mask to filter high frequencies. The proposed algorithm is compared with traditional edge detection algorithms using a satellite image dataset that has different types of objects and detailed information. Results are validated through entropy, structure similarity, and noise error to prove that our proposed algorithm provides satisfactory performance on different remote sensed images.