Quaternion Technique Research Articles

Quaternion Capsule Neural Network With Region Attention for Facial Expression Recognition in Color Images

Facial expression recognition (FER) is an essential subject of computer vision and human-computer interaction. It has been reported that many factors are closely related to the FER performance such as the pose, facial muscle variations, and the ignored color information in facial images. In this study, we propose a quaternion capsule neural network (Q-CapsNet) with region attention mechanism for FER in color images. The proposed Q-CapsNet is an end-to-end deep learning framework, which adopts the concept of quaternion theory to the Capsule Neural Network and further uses an attention mechanism to focus on the facial region of interests (ROIs). The Q-CapsNet addresses the FER problems in the following aspects. First, the internal dependencies between color channels are captured by the quaternion technique, which is of great importance in color image processing. Second, the Q-CapsNet could disentangle latent geometry information from facial images by quaternion capsules and quaternion routing algorithm. Third, inspired by the fact that facial expressions are mainly determined by several vital facial regions, the region attention mechanism is introduced to extract emotional features from the ROIs of the face. The proposed Q-CapsNet is evaluated on four public color FER datasets including MMI, Oulu-CASIA, RAF-DB, and SFEW. The comprehensive experiments and visualization results demonstrate that the proposed network outperforms comparative models and the state-of-the-art FER methods.

Color Facial Expression Recognition by Quaternion Convolutional Neural Network With Gabor Attention

Facial expression recognition (FER) plays an important role in cognitive psychology research. In FER studies, deep convolutional neural networks (CNNs) and attention mechanisms have shown great advantages over traditional techniques. Although many attention-based CNN models have been developed, most of them are proposed to deal with grayscale images. For color images, these network models usually take their grayscale versions as the input or convert them to grayscale data by the first hidden layer. Furthermore, their attention modules focus on only grayscale information. This mechanism does not fully consider the correlation between color channels, which will degrade the performance of color FER. To alleviate these shortcomings, we introduce quaternion techniques to CNNs and propose a quaternion CNN integrated with an attention mechanism (QA-CNN) for color FER. The proposed QA-CNN takes into account not only the correlation between color channels but also expressional attention. The new attention mechanism is based on a multidirectional quaternion Gabor filter. Besides, the proposed QA-CNN reduces the number of network parameters by 75% compared with the real-valued CNN with the same structure. Experimental results on three widely used data sets demonstrate the effectiveness of the proposed QA-CNN by showing clear performance improvements over other state-of-the-art FER methods.

The quaternion-based spatial-coordinate and orientation-frame alignment problems.

The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.

Disturbance rejection for interconnected positive systems and its application to formation control of quadrotors

This study deals with disturbance rejection of interconnected positive (IP) systems by output feedback laws that are installed as an outer loop of the IP system. Since the IP system without disturbances has a first integral, which is a linear function of the state and takes a constant value along the trajectories, this study utilises this function as an index of detecting the disturbances. Then, dynamic and static disturbance rejection output feedback laws with the first integral are proposed, respectively. Although the pole assignment problem of the IP system with the static output feedback law is not solvable in general, this study reveals that the system matrix of the closed-loop system has a structure called rank-one updated matrices. This special structure implies that the simple eigenvalue zero of the original IP system moves to a stable one by the feedback law while the other stable eigenvalues never move. This study applies these feedback laws to formation control of quadrotors based on IP systems where the dynamics of a quadrotor are linearised to be positive and stable second-order subsystems by dual quaternion techniques. Numerical examples illustrate the quadrotors achieve a designed shape of formation against the disturbances.

From the Kinematics of Precession Motion to Generalized Rabi Cycles

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.

A canonical form-based approach to affine registration of DTI

Due to the orientation feature of diffusion tensor images (DTI), tensors need to be reoriented during an affine registration. There exists two active reorientation schemes: finite strain (FS) and preserving principal direction (PPD). However, FS scheme limits its application on rigid deformation and PPD scheme suffers from computation load caused by the iteration. In order to overcome these shortcomings, we propose a canonical form-based affine registration of DTI, named as CFARD. We transform voxel sets into canonical forms where an affine registration is simplified as a rigid registration, while still preserves the effects of non-rigid components. This transforming thus extends the application of FS scheme to affine deformation. Furthermore, to reduce computation load, the quaternion technique is skillfully employed to seek a closed-form solution of the optimal rotation where no iteration is required. Extensive experiments are conducted on synthetic and real DTI data from the human brain. In contrast to four existing algorithms, the proposed CFARD improves the consistency between tensor orientation and the anatomical structures after deformation, and performs a better balance between accuracy and computational complexity.

Polarimetric detection for vector-sensor processing in quaternion proper Gaussian noises

The polarimetric detection problem for 2-D vector sensor is reformulated based on quaternion technique. The quaternion-formed detectors are deduced and analyzed in different quaternion proper Gaussian noises, mainly for Q-proper and C-proper cases. The entropy method is applied to verify the equivalence of quaternion and complex second order statistics (SOS). The similar behaviors of quaternion and complex detectors are thus obtained when SOS is known a priori. Specially, in the case of unknown Q-proper SOS, quaternion and complex detection statistics follow the same F-distribution except for the distinctive second degrees of freedom. This means the improved behavior of quaternion adaptive detector can be achieved. Further, the estimate accuracy of SOS is evaluated fairly by Kullback---Leibler divergence. Consequently, the improved performance derives from the prior structure involved in the quaternion estimate. Numerical simulations also verify the analysis.

Quaternion polar complex exponential transform for invariant color image description

Moments and moment invariants have been widely used as a basic feature descriptors in image analysis, pattern recognition, and image retrieval. However, they are mainly used to deal with the binary or gray-scale images, which lose some significant color information. Recently, quaternion techniques were introduced to conventional image moments (including Fourier–Mellin moments, Zernike/Pseudo Zernike moments, and Bessel–Fourier moments, etc.) for describing color images, and some quaternion moment and moment invariants were developed. But, the conventional image moments usually cannot effectively capture the image information, especially the edges. Besides, the kernel computation of them involves computation of a number of factorial terms, which inevitably cause the numerical stability of these moments. Based on effective polar complex exponential transform (PCET) and algebra of quaternions, we introduced the quaternion polar complex exponential transform (QPCET) for describing color images in this paper, which can be seen as the generalization of PCET for gray-level images. It is shown that the QPCETs can be obtained from the PCET of each color channel. We derived and analyzed the rotation, scaling, and translation (RST) invariant property of QPCET. We also discussed the problem of color image retrieval using QPCET. Experimental results are provided to illustrate the efficiency of the proposed color image descriptors.

A new grid deformation technology with high quality and robustness based on quaternion

Quality and robustness of grid deformation is of the most importance in the field of aircraft design, and grid in high quality is essential for improving the precision of numerical simulation. In order to maintain the orthogonality of deformed grid, the displacement of grid points is divided into rotational and translational parts in this paper, and inverse distance weighted interpolation is used to transfer the changing location from boundary grid to the spatial grid. Moreover, the deformation of rotational part is implemented in combination with the exponential space mapping that improves the certainty and stability of quaternion interpolation. Furthermore, the new grid deformation technique named “layering blend deformation” is built based on the basic quaternion technique, which combines the layering arithmetic with transfinite interpolation (TFI) technique. Then the proposed technique is applied in the movement of airfoil, parametric modeling, and the deformation of complex configuration, in which the robustness of grid quality is tested. The results show that the new method has the capacity to deal with the problems with large deformation, and the “layering blend deformation” improves the efficiency and quality of the basic quaternion deformation method significantly.

Lattice Boltzmann implementation of the three-dimensional Ben-Naim potential for water-like fluids

We develop a three-dimensional lattice Boltzmann (LB) model accounting for directional interactions between water-like molecules, based on the so-called Ben-Naim (BN) potential [A. Ben-Naim, Molecular Theory of Water and Aqueous Solutions: Part I: Understanding Water (World Scientific Publishing Company, 2010); "Statistical mechanics of 'waterlike' particles in two dimensions. I. Physical model and application of the Percus-Yevick equation," J. Chem. Phys. 54, 3682 (1971)]. The water-like molecules are represented by rigid tetrahedra, with two donors and two acceptors at the corners and interacting with neighboring tetrahedra, sitting on the nodes of a regular lattice. The tetrahedra are free to rotate about their centers under the drive of the torque arising from the interparticle potential. The orientations of the water molecules are evolved in time via an overdamped Langevin dynamics for the torque, which is solved by means of a quaternion technique. The resulting advection-diffusion-reaction equation for the quaternion components is solved by a LB method, acting as a dynamic minimizer for the global energy of the fluid. By adding thermal fluctuations to the torque equation, the model is shown to reproduce some microscopic features of real water, such as an average number of hydrogen bonds per molecules (HBs) between 3 and 4, in a qualitative agreement with microscopic water models. Albeit slower than a standard LB solver for ordinary fluids, the present scheme opens up potentially far-reaching scenarios for multiscale applications based on a coarse-grained representation of the water solvent.

市販３ＤＶＯＧデータのＦｉｃｋ座標系から主軸解析への変換とその信頼性

The output from the recently developed infrared video-oculography is in terms of Euler's angle coordinates, which do not express the eye movements precisely. More precise representation of the eye movements requires transcription on a rotational axis reference frame. Description of the rotation of the eyeballs expressed in terms of the Euler's angle can be transformed to a rotational axis reference frame through quaternion conversion. Transcription on a rotational axis reference frame was accomplished by consecutively combining the rotational quaternions describing the 3 components of Euler's angle. The directions of the primary position and Listing's plane were also calculated using the quaternion technique. The algorithm developed by us was provided here, and practical data were obtained to evaluate the adequacy of these transcriptions. Further, the accuracy of the commercialized video-oculography product was discussed. Using our algorithm to manage the video-oculography data, we could accurately determine the positional (Listing's) plane and primary position. False torsion of the raw data from the video-oculography could be successfully managed. However, further reduction of the measuring noise would be desirable, especially on a roll plane. Conversion through a rotational quaternion represents an excellent method for transcribing eye movements in an adequately realistic form. Video-oculography could serve as a basis for convenient and precise description of the eye movements. With further improvement, this instrument could substantially facilitate research into the dynamics of the eye movements.

Deriving Linear Transformations in 3D Using Quaternion Algebra

We propose a method for delivering linear transformation in the two parallel projections that works well with quaternion techniques. The article also presents a completed vector method to derive a linear transformation of the two kinds of parallel projections. Here, we propose a novel method that can achieve a unified framework based on quaternion techniques.

Development of computer-based environment for simulating the voluntary upper-limb movements of persons with disability.

Upper-limb orthotic systems have been designed for restoring the upper-limb functions of individuals with disabilities resulting from spinal cord injury (SCI), stroke and muscular dystrophy. These systems employ either functional electrical stimulation or external power. It is proposed that, instead of time-consuming and complicated monitoring using sensors and motion analysis, a software simulator with both angular displacement and acceleration parameters can facilitate the design of a control strategy for an orthosis. Reaching movements of three cervical SCI subjects are used to verify the simulator. A motion analysis system is used to measure the range of motion and joint angles during hand reaching. Results indicate that quaternion and spline curve techniques are suitable for interpolation of the hand reaching movements. The information needed for good simulation only compress the shoulder and elbow joint angles in a few key postures. Stimulated acceleration signals on the upper-arm segment have a high correlation coefficient (> 0.9) and a small root mean squared error (< 0.11 g) with a real bi-axial accelerometer.

Error analysis of direction cosines and quaternion parameters techniques for aircraft attitude determination

Matrix differential equations for the determination of aircraft attitude orientation and the associated errors equations are developed in a four-dimensional space for both the direction cosines and quaternion parameter approaches. In this four-dimensional space, the corresponding algorithms, using accumulated gyro outputs, are developed and evaluated for the case of classical coning motion with time-varying angular velocity direction vector. Computation using symbolic algebra is considered in finding a closed-form solution for error propagation for a time-varying angular velocity direction vector. Examining the coning motion case shows that the true (analytic) scale and skew errors generated by the gyro drift are negligibly small in the direction cosines approach. Further, the true (analytic) scale errors in the quaternion technique are smaller than the computed (numerical) errors. Graphically illustrated are the true and computed errors in both the direction cosines and quaternion techniques for comparison.

The quaternion method of regularizing integral equations of the theory of elasticity

Using quaternions [1], a new method of regularizing integral equations of the theory of elasticity is proposed which is identical for plane and spatial problems. The regularizors and the regularized equations are presented. The method does not use the theory of a symbol, as was done earlier [2–4], or the transition to the complex form [5]. The quaternion technique has not been applied so far to integral equations of the theory of elasticity but has been used to obtain the general solutions of Lamé's equations. ‡ ‡ GRIGOR'YEV Yu. M., Solving spatial statics problems of the theory of elasticity by the methods of the theory of quaternion functions. Candidate disseration, Novosibirsk, 1985.

Model-based recognition and positioning of polyhedra using intensity-guided range sensing and interpretation in 3-D space

This paper presents a design of 3-D multi-sensory (visual and ranging) system capable of identifying and locating polyhedral objects in six degrees of spatial freedom. Due to differences in sensor characteristics, e.g. the fast data acquisition in intensity domain and the abundance of information inherent in range domain, sensors should operate in a cooperative manner to increase system efficiency and to yield information otherwise unavailable, or hard to obtain, from any single sensor. The adopted strategy of intensity-guided range sensing firstly extracts features (e.g. the line drawing and the “control points”) from intensity images and then uses them as a guide to selectively sense ranges. Through the interfusion of both types of information, 3-D object description in terms of a relational graph structure is thus constructed for recognition and positioning purposes. To identify the object, interpretation tree search in 3-D space is applied to the matching between the constructed relational graph and those stored in database. Since high level primitives (such as faces) derived from intensity and range informations are utilized for interpretation (or, matching), the search space can be significantly reduced. Finally, a non-iterative method based on the quaternion technique is adopted to solve the transformation between sensor and model coordinates for further robotic applications.