Search Results (7)

The quaternion discrete Fourier transform (QDFT) is a powerful tool in modern digital signal processing, even though until recently this transformation seemed exotic. In recent years, quite a lot of publications have appeared devoted to effective ways to calculate this transformation. In particular, in one of our previous publications, we presented an economical algorithm for calculating one-dimensional QDFT and showed that this algorithm has the lowest computational complexity among all known algorithms of this type. This generalized algorithm is suitable for computing the QDFT of any sequence in which the number of elements is a power of two. However, as it turned out, there are additional possibilities that make it possible to further reduce the computational complexity of the developed algorithm for each specific N. In this article, we provide some examples of the synthesis of such algorithms for short-length input sequences (samples of signals). In particular, algorithms for N ∊ {2, 3, 4, 5, 6, 7, 8} are presented. A parallel implementation of the proposed algorithm allows for saving more than half of the number of required multipliers in each case compared with the parallel implementation of the naive methods of calculation. Full article

The paper presents the original structure of a processing unit for multiplying quaternions. The idea of organizing the device is based on the use of fast Hadamard transform blocks. The operation principles of such a device are described. Compared to direct quaternion multiplication, the developed algorithm significantly reduces the number of multiplication and addition operations. Hardware implementations of the developed structure, in FPGA and ASIC, are presented. The FPGA blocks were implemented in the Vivado environment. The ASICs were designed using $130\mathrm{nm}$ technology. The developed scripts in VHDL are available in the GitHub repository. Full article

We present an analysis of the Dirac equation when the spin symmetry is changed from SU(2) to the quaternion group, $Q_8$, achieved by multiplying one of the gamma matrices by the imaginary number, i. The reason for doing this is to introduce a bivector into the spin algebra, which complexifies the Dirac field. It then separates into two distinct and complementary spaces: one describing polarization and the other coherence. The former describes a 2D structured spin, and the latter its helicity, generated by a unit quaternion. Full article

Two equations whose variables take values in the Pauli algebra of complex quaternions are shown to be equivalent to the standard Dirac equation and its Hermitian conjugate taken together. They are transformed one into the other by an outer automorphism of the Pauli algebra. Given a solution to the Dirac equation, a new solution is obtained by multiplying it on the right by one of the 16 matrices of the Pauli group. This defines a homomorphism from the Pauli group into the group of discrete symmetries, whose kernel is a cyclic group of order four. The group of discrete symmetries is shown to be the Klein four-group consisting of four elements: the identity Id; the charge conjugation symmetry C; the mass inversion symmetry M; and their composition in either order, CM = MC. The mass inversion symmetry inverts the sign of the mass, leaving the electric charge unchanged. The outer “bar-star” automorphism is identified with the parity operation, resulting in proof of CPT = M or, equivalently, CPTM = Identity. Full article

Advances in micro-electro-mechanical systems technology have led to the emergence of compact attitude measurement sensor products that integrate acceleration, magnetometer, and gyroscope sensors on a single chip, making them important devices in the field of three-dimensional (3D) attitude measurement for unmanned aerial vehicles, smartphones, and other devices. Sensor fusion algorithms for posture measurement have become an indispensable technology in cutting-edge research, such as human posture measurement using wearable sensors, and stabilization problems in robot position and posture measurement. We have also developed wearable sensors and powered suits in our previous research. We needed a technology for the real-time measurement of a 3D human body motion. It is known that quaternions can be used to algebraically handle 3D rotations; however, sensor fusion algorithms for three sensors are presently complex. This is because these algorithms deal with the post-rotation attitude (pure quaternions) rather than rotation information (the rotor) to avoid a double covering problem involving the rotor. If we are dealing with rotation, it may be possible to make the algorithm simpler and faster by dealing directly with the rotor. In this study, to solve the double covering problem involving the rotor, we propose a stateful rotor and develop a technique for uniquely determining the time-varying states of the rotor. The proposed stateful rotor guarantees the continuity of the rotor parameters with respect to angular changes, and this paper confirms its effectiveness by simulating two rotations around an arbitrary axis. In addition, we verify experimentally that a fast sensor fusion method using stateful rotor can be used for attitude calculation. Experiments also confirm that the calculated results converge to the desired rotation angle for two spatial rotations around an arbitrary axis. Since the proposed stateful rotor extends and stabilizes the definition of the rotor, it is applicable to any algorithm that deals with time-varying quaternionic rotors. In this research, an algorithm based on a multiply–add operation is designed to reduce computational complexity as a high-speed calculation for embedded systems. This method is theoretically equivalent to other methods, while contributing to power saving and the cost reduction of products. Full article

Owing to the limitations of imaging principles and system imaging characteristics, infrared images generally have some shortcomings, such as low resolution, insufficient details, and blurred edges. Therefore, it is of practical significance to improve the quality of infrared images. To make full use of the information on adjacent points, preserve the image structure, and avoid staircase artifacts, this paper proposes a super-resolution reconstruction method for infrared images based on quaternion total variation and high-order overlapping group sparse. The method uses a quaternion total variation method to utilize the correlation between adjacent points to improve image anti-noise ability and reconstruction effect. It uses the sparsity of a higher-order gradient to reconstruct a clear image structure and restore smooth changes. In addition, we performed regularization by using the denoising method, alternating direction method of multipliers, and fast Fourier transform theory to improve the efficiency and robustness of our method. Our experimental results show that this method has excellent performance in objective evaluation and subjective visual effects. Full article

Owing to the limitations of the imaging principle as well as the properties of imaging systems, infrared images often have some drawbacks, including low resolution, a lack of detail, and indistinct edges. Therefore, it is essential to improve infrared image quality. Considering the information of neighbors, a description of sparse edges, and by avoiding staircase artifacts, a new super-resolution reconstruction (SRR) method is proposed for infrared images, which is based on fractional order total variation (FTV) with quaternion total variation and the $L_p$ quasinorm. Our proposed method improves the sparsity exploitation of FTV, and efficiently preserves image structures. Furthermore, we adopt the plug-and-play alternating direction method of multipliers (ADMM) and the fast Fourier transform (FFT) theory for the proposed method to improve the efficiency and robustness of our algorithm; in addition, an accelerated step is adopted. Our experimental results show that the proposed method leads to excellent performances in terms of an objective evaluation and the subjective visual effect. Full article