Quaternion Discrete Fourier Transform Research Articles

One-Dimensional Quaternion Discrete Fourier Transform and an Approach to Its Fast Computation

This paper proposes a new method for calculating the quaternion discrete Fourier transform for one-dimensional data. Although the computational complexity of the proposed method still belongs to the O(Nlog2N) class, it allows us to reduce the total number of arithmetic operations required to perform it compared to other known methods for computing this transform. Moreover, compared to the method using symplectic decomposition, the presented method does not require changing the basis in the subspace of pure quaternions and, consequently, calculating the new basis vectors and change-of-basis matrix.

Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform

In recent years, quaternion algebra has been applied more and more in different applications. A primary motivation for investigating quaternion algebras is to extend the classical signal processing to quaternion signal processing theory. However, this algebra has non-commutative multiplication, leading to the uncertainty of such essential operations as the convolution and Fourier transform. As a result, it became difficult to process quaternion signals in the frequency domain. In this work, the arithmetic of quaternion numbers is analyzed using the Cayley-Dickson construction, but with an associative and commutative operation of multiplication. We call it the (2,2)-model of quaternions. This model's main properties of quaternions are described, including inverse, division, and conjugate operations. In the (2,2)-model, only two exponential functions with pure quaternion units exit, and therefore, only two types of the quaternion discrete Fourier transform (QDFT) can be defined. One of the crucial benefits of this model is that the unique definition of convolution will help reduce the complexity of quaternion convolution operation calculated using both types of QDFTs.

Screen-Shooting Resilient Watermarking Scheme via Learned Invariant Keypoints and QT.

This paper proposes a screen-shooting resilient watermarking scheme via learned invariant keypoints and QT; that is, if the watermarked image is displayed on the screen and captured by a camera, the watermark can be still extracted from the photo. A screen-shooting resilient watermarking algorithm should meet the following two basic requirements: robust keypoints and a robust watermark algorithm. In our case, we embedded watermarks by combining the feature region filtering model to SuperPoint (FRFS) neural networks, quaternion discrete Fourier transform (QDFT), and tensor decomposition (TD). First we applied FRFS to locate the embedding feature regions which are decided by the keypoints that survive screen-shooting. Second, we structured watermark embedding regions centered at keypoints. Third, the watermarks were embedded by the QDFT and TD (QT) algorithm, which is robust for capturing process attacks. In a partial shooting scenario, the watermark is repeatedly embedded into different regions in an image to enhance robustness. Finally, we extracted the watermarks from at least one region at the extraction stage. The experimental results showed that the proposed scheme is very robust for camera shooting (including partial shooting) different shooting scenarios, and special attacks. Moreover, the efficient mechanism of screen-shooting resilient watermarking could have propietary protection and leak tracing applications.

Geometric resistant polar quaternion discrete Fourier transform and its application in color image zero-hiding

As a typical frequency-domain analysis method, quaternion discrete Fourier transform (QDFT) has been widely used in information hiding in color images. However, due to the sensitivity of QDFT to geometric attacks, existing QDFT-based information hiding schemes have limited ability in resisting geometric attacks. In this study, a kind of novel geometrically resilient polar QDFT (PQDFT) is constructed and the properties of the proposed PQDFT are analyzed. Subsequently, a PQDFT-based color image zero-hiding scheme robust to geometric attacks is proposed for lossless copyright protection of color images, which experimentally shows reasonable resistance against geometric and common attacks, indicating better robustness compared with the existing QDFT-based information hiding schemes and other leading-edge zero-hiding schemes.

A Watermarking Scheme for Color Image Using Quaternion Discrete Fourier Transform and Tensor Decomposition

To protect the copyright of the color image, a color image watermarking scheme based on quaternion discrete Fourier transform (QDFT) and tensor decomposition (TD) is presented. Specifically, the cover image is partitioned into non-overlapping blocks, and then QDFT is performed on each image block. Then, the three imaginary frequency components of QDFT are used to construct a third-order tensor. The third-order tensor is decomposed by Tucker decomposition and generates a core tensor. Finally, an improved odd–even quantization technique is employed to embed a watermark in the core tensor. Moreover, pseudo-Zernike moments and multiple output least squares support vector regression (MLS–SVR) network model are used for geometric distortion correction in the watermark extraction stage. The scheme utilizes the inherent correlations among the three RGB channels of a color image, and spreads the watermark into the three channels. The experimental results indicate that the proposed scheme has better fidelity and stronger robustness for common image-processing and geometric attacks, can effectively resist each color channel exchange attack. Compared with the existing schemes, the presented scheme achieves better performance.

A Fourier transform based audio watermarking algorithm

In this paper; we propose two new substitution schemes for digital audio watermarking based on the Fourier transform. The integration of the watermark will be performed by combining the parity of the successive coefficients values; each variant will represent a different combination. In our experiments several variants of the Fourier transform are used (Discrete Fourier Transform, a Fractional Fourier Transform as well as a Quaternion Discrete Fourier Transform). For each transform, both variants of our watermarking scheme are applied and the results obtained show that our approach offers good imperceptibility and generates watermarked audio sample robust against various attacks with a high-quality watermark. However, using small audio file for our experiments considerably reduces the capacity of our approach in the frequency domain.

A robust blind color image watermarking based on Fourier transform domain

In this work, we propose a new substitution scheme for color images watermarking based on the Fourier transform. This scheme will be declined into two variants in which the image will be divided into three components R, G and B, and then to each component a transform is applied. The parity of the resulting coefficients will then be combined to hide the watermark in the medium frequency band. In our experiments several variants of the Fourier transform are used (Discrete Fourier Transform, a Fractional Fourier Transform as well as a Quaternion Discrete Fourier Transform). For each transform, both variants of our watermarking scheme are applied. The obtained results show that our approach offers good imperceptibility and generates watermarking images robust against various attacks with a high-quality watermark.

Blind watermarking for color images using EMMQ based on QDFT

In this study, we developed a novel scheme for the blind watermarking of color images. The proposed scheme incorporates extreme pixel adjustment (EPA), multi-bit partly sign-altered mean modulation (MPSAM), mixed modulation (MM), and particle swarm optimization (PSO) within a scheme based on crisscross inter-block quaternion discrete Fourier transform (QDFT). Accordingly, the proposed scheme employing EPA, MPSAM, MM, and QDFT is referred to as EMMQ. The image is separated into non-overlapping 8 × 8 pixel blocks, whereupon MPSAM is used to map multiple bits within a single block using multiple coefficients in one of the four transformed components of QDFT. The use of MM to embed the watermark allows for superior image quality and strong resistance to image processing attacks. Our use of PSO also makes it possible to optimize the EMMQ parameters, thereby enabling outstanding robustness without compromising imperceptibility. Experiment results demonstrate the efficacy of the proposed scheme in resisting a variety of image-processing attacks, with performance superior to that of existing watermarking schemes in terms of imperceptibility and robustness for a given payload capacity.

Quaternion Discrete Fourier Transform-Based Color Image Watermarking Method Using Quaternion QR Decomposition

In this paper, a new Quaternion Discrete Fourier Transform (QDFT)-based digital color image watermarking method is presented. In addition, the Quaternion QR (QQR) decomposition is applied in digital watermarking technology for the first time. First of all, the QDFT and QQR decomposition are performed on the host image, respectively, to acquire the scalar part of the quaternion matrix for watermark information embedding. After that, we divide the scalar part of the quaternion matrix generated by the QQR decomposition into blocks and calculate the entropy. The block with high entropy is selected to embed the watermark information. Then the watermark information is embedded into the extracted block using the quantization index modulation method. We conducted a large number of tests and experimental results indicate that the presented approach obtains excellent robustness against Scaling, Rotation, Median filtering, `Salt & Pepper' noise, and JPEG Compression. Compared with the existing methods, the presented method achieves better performance.

Alpha-Rooting Color Image Enhancement Method by Two-Side 2-D Quaternion Discrete Fourier Transform Followed by Spatial Transformation

In this paper a quaternion approach of enhancement method is proposed in which color in the image is considered as a single entity. This new method is referred as the alpha-rooting method of color image enhancement by the two-dimensional quaternion discrete Fourier transform (2-D QDFT) followed by a spatial transformation. The results of the proposed color image enhancement method are compared with its counterpart channel-by-channel enhancement algorithm by the 2-D DFT. The image enhancements are quantified to the enhancement measure that is based on visual perception referred as the color enhancement measure estimation (CEME). The preliminary experiment results show that the quaternion approach of image enhancement is an effective color image enhancement technique.

Enhancement of Underwater Color Images by Two-side 2-D Quaternion Discrete Fourier Transform

The research problem is to find an effective enhancement method for enhancing raw underwater color images. In underwater, as depth increases the high wavelength regions of the light spectrum are absorbed by the water and the light spectrum consists only of low wavelength regions such as green and blue and therefore, the image captured underwater looks green or greenish blue. This paper proposes an enhancement algorithm for improving the quality of raw underwater images by the method of alpha-rooting by two-side 2-D quaternion discrete Fourier transform (QDFT) with color correction done by multiscale retinex (MSR). The results of proposed enhancement are compared with the alpha-rooting method, by transforming color images to 2-D grayscale images. The enhancement are measured with reference to the metric color enhancement measure estimation.

Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Quaternion Discrete Fourier Transform and the 2-D Discrete Fourier Transform

Color in an image is resolved into 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by the genetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.

Geometric correction based color image watermarking using fuzzy least squares support vector machine and Bessel K form distribution

This paper presents a robust color image watermarking algorithm based on fuzzy least squares support vector machine (FLS-SVM) and Bessel K form (BKF) distribution, which is a recently developed geometric correction algorithm. We firstly compute the quaternion discrete Fourier transform (QDFT) of the maximum central region of the original color image. Then the watermark is embedded into the magnitudes of low-frequency information of QDFT. In watermark decoding process, the synchronous correction based on FLS-SVM model is used. When training the FLS-SVM model, we firstly perform the quaternion wavelet transform (QWT) of the grayscale images that correspond to the color training images, and then use BKF distribution to fit the empirical histogram of coefficients of the QWT, and finally use the shape parameters and scale parameters of BKF distribution to construct image feature vector. Experimental results show that the proposed algorithm is not only invisible, but also has outstanding robustness against common image processing attacks and geometric attacks.

Zonal-Alpha-Rooting Color Image Enhancement by the Two-Side 2D Quaternion Discrete Fourier Transform

Zonal-Alpha-Rooting Color Image Enhancement by the Two-Side 2D Quaternion Discrete Fourier Transform

Tensor transform-based quaternion fourier transform algorithm

The traditional Fourier transform is a fundamental tool in signal and image processing, which in color imaging transforms images either in monochrome or as separate color channels, thereby ignoring interactions between the color channels. The quaternion discrete Fourier transform (QDFT), a generalization of the Fourier transform, represents the three color channels of an image in the RGB color space as a vector field in the quaternion arithmetic allowing for simultaneous analysis of all color data. It is expected that this new color imaging model which captures the interactions between the different color channels will improve upon existing image processing and recognition systems. Efficient QDFT algorithms are derived by combining classical DFT transforms, enabling standard fast Fourier transform (FFT) software to provide a fast QDFT numerical implementation. We developed an open-source quaternion Fourier transform tool that calculates the 2-D QDFT by the column-row method which is performed by calculating multiple 1-D QDFTs where each 1-D transform is calculated by two complex 1-D DFTs. The matrices involved in these calculations necessitate the inclusion of a discussion on symplectic decomposition. Our main contribution is the presentation a new algorithm for the tensor representation of the QDFT that is structurally simpler and uses less multiplications than its predecessors. For instance, for the images of size N×N, where N=2r,r>2, the tensor 2-D QDFT saves about 2N2r real multiplications.

Color image watermarking based on quaternion Fourier transform and improved uniform log-polar mapping

In this paper, we propose a blind color image watermarking scheme based on quaternion discrete Fourier transform (QDFT) and on an improved uniform log-polar mapping (IULPM). The proposed watermarking scheme embeds dual watermarks: one is a meaningful binary image watermark and the other is a bipolar watermark. The former is embedded in the real part of mid-frequency QDFT coefficients using quantization index modulation. The latter is used to resynchronize the watermark after the watermarked image has been attacked, making the scheme resistant to geometric attacks. In particular, the IULPM allows for greater accuracy when estimating the rotation angle of a geometric attack. At the same time, the watermark embedding employs the image holistically, rather than in a block pattern. Experimental results demonstrate that the proposed scheme achieves better performance of robustness against both common signal operations and geometric attacks compared to other existing schemes.

Robust hashing for image authentication using quaternion discrete Fourier transform and log-polar transform

In this work, a novel robust image hashing scheme for image authentication is proposed based on the combination of the quaternion discrete Fourier transform (QDFT) with the log-polar transform. QDFT offers a sound way to jointly deal with the three channels of color images. The key features of the present method rely on (i) the computation of a secondary image using a log-polar transform; and (ii) the extraction from this image of low frequency QDFT coefficients' magnitude. The final image hash is generated according to the correlation of these magnitude coefficients and is scrambled by a secret key to enhance the system security. Experiments were conducted in order to analyze and identify the most appropriate parameter values of the proposed method and also to compare its performance to some reference methods in terms of receiver operating characteristics curves. The results show that the proposed scheme offers a good sensitivity to image content alterations and is robust to the common content-preserving operations, and especially to large angle rotation operations.

Full 4-D quaternion discrete Fourier transform based watermarking for color images

Among the few existing color watermarking schemes, some use quaternion discrete Fourier transform (QDFT). By modulating at least one component of QDFT coefficients, they spread the watermark over two or three of the RGB color channels. However, these schemes do not fully utilize the four-dimensional (4-D) QDFT frequency domain and some also suffer from a watermark energy loss directly at the embedding stage. In this paper, we first establish the links that exist between the DFT of the three RGB color channels and the components of QDFT coefficients while considering a general unit pure quaternion. Then, for different unit pure quaternions i, j, k or their linear combinations, we discuss the symmetry constraints one should follow when modifying QDFT coefficients in order to overcome the previous drawbacks. We also provide a general watermarking framework to illustrate the overall performance gain in terms of imperceptibility, capacity and robustness we can achieve compared to other QDFT based algorithms, i.e. when fully considering the 4-D QDFT domain. From this framework we derive three schemes, depending on whether i, j or k is used. Provided theoretical analysis and experimental results show that these algorithms offer better performance in terms of capacity and robustness to most common attacks, including JPEG compression, noise, cropping and filtering and so on, than other QDFT based algorithms for the same watermarked image quality.