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This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

The security of information is necessary for the success of any system. So, there is a need to have a robust mechanism to ensure the verification of any person before allowing him to access the stored data. So, for purposes of increasing the security level and privacy of users against attacks, cancelable biometrics can be utilized. The principal objective of cancelable biometrics is to generate new distorted biometric templates to be stored in biometric databases instead of the original ones. This paper presents effective methods based on different discrete transforms, such as Discrete Fourier Transform (DFT), Fractional Fourier Transform (FrFT), Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT), in addition to matrix rotation to generate cancelable biometric templates, in order to meet revocability and prevent the restoration of the original templates from the generated cancelable ones. Rotated versions of the images are generated in either spatial or transform domains and added together to eliminate the ability to recover the original biometric templates. The cancelability performance is evaluated and tested through extensive simulation results for all proposed methods on a different face and fingerprint datasets. Low Equal Error Rate (EER) values with high AROC values reflect the efficiency of the proposed methods, especially those dependent on DCT and DFrFT. Moreover, a comparative study is performed to evaluate the proposed method with all transformations to select the best one from the security perspective. Furthermore, a comparative analysis is carried out to test the performance of the proposed schemes with the existing schemes. The obtained outcomes reveal the efficiency of the proposed cancelable biometric schemes by introducing an average AROC of 0.998, EER of 0.0023, FAR of 0.008, and FRR of 0.003.

Performing some special tasks using electrooculography (EOG) in daily activities is being developed in various areas. In this paper, simple rotation matrixes were introduced to help the operator move a 2-DoF planar robot manipulator. The EOG sensor, NF 5201, has two output channels (Ch1 and Ch2), as well as one ground channel and one reference channel. The robot movement was the indicator that this system could follow gaze motion based on EOG. Operators gazed into five training target points each in the horizontal and vertical line as the preliminary experiments, which were based on directions, distances and the areas of gaze motions. This was done to get the relationships between EOG and gaze motion distance for four directions, which were up, down, right and left. The maximum angle for the horizontal was 46°, while it was 38° for the vertical. Rotation matrixes for the horizontal and vertical signals were combined, so as to diagonally track objects. To verify, the errors between actual and desired target positions were calculated using the Euclidian distance. This test section had 20 random target points. The result indicated that this system could track an object with average angle errors of 3.31° in the x-axis and 3.58° in the y-axis.

The first step in this paper is to standardize each index based on factor analysis, then extract factors based on eigenvalues, and then use a rotation matrix to rotate and name the factor variables. The soccer player development system was created by determining the principal components based on the eigenvectors of the eigenvalues. The relationship between exogenous and endogenous latent variables was investigated by establishing structural equations to investigate the linear combination of variables. The model was used to conduct correlation and path analysis on the influencing factors of youth soccer player development. The study showed family factors strongly correlated with youth soccer player socialization r=0.656, p<0.01. Coaching factors moderately correlated with youth soccer player socialization r=0.456, p<0.01. The study of this paper provides some reference value for the development of youth soccer players.

The type of throw of a spin bowler can be analysed in the laboratory using a motion analysis system. However, there is still no method to determine the type of throw using other means and less effort. To solve this problem, we revised the traditional classification of spin bowling throws and analysed whether spin bowling throws are separate entities or continuous concepts. We used an advanced smart cricket ball with high-speed gyroscopes to record the bowling actions and mathematically transformed the spin axis from the ball coordinate system (BCS) to the global coordinate system (GCS). We developed a visualisation method to map spin bowling throws from the yaw and pitch angles of the ball’s spin axis in the GCS. We compared the data from the smart ball with the data from the motion analysis system and profiled seven spin bowlers using the new method. The results of this study have shown that spin bowling throws are continuous concepts and that all differences between the two spin axis measurement methods were within 95% limits of agreement. The Smart Ball is sufficiently accurate to measure the direction of the ball’s spin axis in the GCS and is therefore well suited for profiling spin bowlers. Hybrid deliveries between sidespin, top/backspin, and swerve maximise the deviations of the ball in flight from the straight flight path in all three planes of the GCS. Hybrid throws between sidespin, top/backspin, and spin maximise the ball’s deviation from the straight trajectory in all three planes of the GCS.

Heading estimation is a central problem for indoor pedestrian navigation using the pervasively available smartphone. For smartphones placed in a pocket, one of the most popular device positions, the essential challenges in heading estimation are the changing device coordinate system and the severe indoor magnetic perturbations. To address these challenges, we propose a novel heading estimation approach based on a rotation matrix and principal component analysis (PCA). Firstly, through a related rotation matrix, we project the acceleration signals into a reference coordinate system (RCS), where a more accurate estimation of the horizontal plane of the acceleration signal is obtained. Then, we utilize PCA over the horizontal plane of acceleration signals for local walking direction extraction. Finally, in order to translate the local walking direction into the global one, we develop a calibration process without requiring noisy compass readings. Besides, a turn detection algorithm is proposed to improve the heading estimation accuracy. Experimental results show that our approach outperforms the traditional uDirect and PCA-based approaches in terms of accuracy and feasibility.

This paper investigates a stochastic-contraction-stability-based convex optimization control (SCOC) scheme for the trajectory tracking problem of stochastic autonomous underwater vehicles, where the rotation matrix is employed to represent the attitude and achieve arbitrary 6-DOF rotation with a large angle maneuver. Firstly, in contrast to the complexity introduced by directly using a rotation matrix in controller design, a novel error dynamics is developed for AUVs with an alternative attitude error vector. It has the potential to avoid singularities and unwinding phenomena under the influence of stochastic disturbance. A linear sliding manifold, together with the convex optimization, is utilized to construct an optimal contraction performance metric, which has the capability to minimize the upper bound of mean squared tracking errors greedily. The innovation of the SCOC lies in the transformation of the non-convex upper-bound minimization problem into an equivalent convex optimization framework with Riccati inequality constraints. The stochastic contraction theory demonstrates that SCOC can guarantee all-time exponential boundedness of the error for any initial condition with L 2 -robustness properties. To implement the sampling-based SCOC, the discrete-time stochastic contraction stability and its explicit relationship with the continuous-time counterpart will be introduced. Eventually, the theoretical analysis and comprehensive simulations are carried out to validate the feasibility and effectiveness of the proposed control scheme.

With the development of aerospace technology, rigid-flexible coupling spacecraft with strong nonlinearity dominates, bringing huge challenges for numerical tools. This paper aims to accurately and quickly analyze the dynamic behavior of rigid-flexible coupling spacecraft. In our work, rigid body and flexible body are both described by Euler parameters for avoiding singular angles. Then, for this type of strong nonlinear dynamic system, the BN-stable method that is unconditionally stable for nonlinear initial value problems proposed by the first author is introduced to solve the transient responses. To keep the accuracy of the numerical rotation matrix, the relation between the angular velocity and Euler parameters, and the constraint in the level of velocity, are introduced to reformulate the original BN-stable method. Through the dynamic analysis for rigid satellite bodies and flexible solar wings, we find that the proposed strategy can effectively simulate the dynamic responses of spacecraft, and compared to the currently popular strategy, our strategy enjoys considerable advantages in accuracy, stability, and dissipation.

Land cover classification is an important application for polarimetric synthetic aperture radar (PolSAR). Target polarimetric response is strongly dependent on its orientation. Backscattering responses of the same target with different orientations to the SAR flight path may be quite different. This target orientation diversity effect hinders PolSAR image understanding and interpretation. Roll-invariant polarimetric features such as entropy, anisotropy, mean alpha angle, and total scattering power are independent of the target orientation and are commonly adopted for PolSAR image classification. On the other aspect, target orientation diversity also contains rich information which may not be sensed by roll-invariant polarimetric features. In this vein, only using the roll-invariant polarimetric features may limit the final classification accuracy. To address this problem, this work uses the recently reported uniform polarimetric matrix rotation theory and a visualization and characterization tool of polarimetric coherence pattern to investigate hidden polarimetric features in the rotation domain along the radar line of sight. Then, a feature selection scheme is established and a set of hidden polarimetric features are selected in the rotation domain. Finally, a classification method is developed using the complementary information between roll-invariant and selected hidden polarimetric features with a support vector machine (SVM)/decision tree (DT) classifier. Comparison experiments are carried out with NASA/JPL AIRSAR and multi-temporal UAVSAR data. For AIRSAR data, the overall classification accuracy of the proposed classification method is 95.37% (with SVM)/96.38% (with DT), while that of the conventional classification method is 93.87% (with SVM)/94.12% (with DT), respectively. Meanwhile, for multi-temporal UAVSAR data, the mean overall classification accuracy of the proposed method is up to 97.47% (with SVM)/99.39% (with DT), which is also higher than the mean accuracy of 89.59% (with SVM)/97.55% (with DT) from the conventional method. The comparison studies clearly demonstrate the efficiency and advantage of the proposed classification methodology. In addition, the proposed classification method achieves better robustness for the multi-temporal PolSAR data. This work also further validates that added benefits can be gained for PolSAR data investigation by mining and utilization of hidden polarimetric information in the rotation domain.

In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed.