Real Squares Research Articles

A real representation method for special least squares solutions of the quaternion matrix equation $ (AXB, DXE) = (C, F) $

<abstract><p>In this article, our interest is the quaternion matrix equation $ (AXB, DXE) = (C, F) $, and we study its minimal norm centrohermitian least squares solution and skew centrohermitian least squares solution. By applying of the real representation matrices of quaternion matrices and relative properties, we convert the quaternion least squares problems with constrained variables into the corresponding real least squares problems with free variables, and then we obtain the solutions of corresponding problems. The final results can be expressed only by real matrices and vectors, and thus the corresponding algorithms only involves real operations and avoid complex quaternion operations. Therefore, they are portable and convenient. In the end, we give two examples to verify the effectiveness of the purposed algorithms.</p></abstract>

Algebraic techniques for the least squares problems in elliptic complex matrix theory and their applications

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize, and complement some known least squares solutions results from the literature.

Special least squares solutions of the quaternion matrix equation [formula omitted

In this paper, by using the real representations of quaternion matrices, the particular structure of the real representations of quaternion matrices, the Kronecker product of matrices and the Moore–Penrose generalized inverse, we obtain the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation AXB+CXD=E, respectively. Our resulting formulas only involve real matrices, and therefore are simpler than those reported in Yuan (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of quaternion matrices, and therefore are very efficient and portable. Numerical examples are provided to illustrate the efficiency of our algorithms.

Special least squares solutions of the quaternion matrix equation [formula omitted] with applications

In this paper, by applying particular structure of the real representations of quaternion matrices and the Moore–Penrose generalized inverse, we derive the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation AX=B. The resulting formulas only involve real matrices, which are simpler than those reported in (Yuan et al., 2013). The corresponding algorithms only perform real arithmetic which also consider particular structure of the real representations of quaternion matrices, therefore are very efficient and easily understood. Numerical examples are provided to illustrate the efficiency of our algorithms.

Deriving Position of Bending Roll in Roll Bending of Titanium Alloy Wire for Glasses Frame

Titanium alloy is used for a wire for glasses frame due to having high corrosion resistance and human body compatibility. The wire is formed into a three-dimensional design shape by an NC exclusive bending machine. In present, craftsmen adjust the bending roll position by trial and error at every design change. The purpose of this study is suggesting a feedback control method to revise a formed shape into a design shape with a smaller number of trials. It is desirable to introduce a feedforward compensator into a feedback control loop to reduce the trial number. About one of the two-dimensional curves to constitute the three-dimensional curve of the glasses frame, this paper proposes a derivation method of a bending roll position that getting first formed shape near the design shape. This method is based on geometric relations in a steady bending, and equivalent to a feedforward compensator in the control loop. Design shapes are real glasses and squares that have fileted corners, the curvature becomes more than 300 m-1. It is particularly paid attention to the availability of “indentation” as the governing quantity of moment, every moment arm at no load. By the absolute total of the curvature deviation, it is confirmed that the shape formed by the suggested method is closer than that by a conventional method.

On ‘decorrelation’ in solving integer least-squares problems for ambiguity determination

This paper intends to shed light on the decorrelation or reduction process in solving integer least squares (ILS) problems for ambiguity determination. We show what this process should try to achieve to make the widely used discrete search process fast and explain why neither decreasing correlation coefficients of real least squares (RLS) estimates of the ambiguities nor decreasing the condition number of the covariance matrix of the RLS estimate of the ambiguity vector should be an objective of the reduction process. The new understanding leads to a new reduction algorithm, which avoids some unnecessary size reductions in the Lenstra-Lenstra-Lovász (LLL) reduction and still has good numerical stability. Numerical experiments show that the new reduction algorithm is faster than LAMBDA’s reduction algorithm and MLAMBDA’s reduction algorithm (to less extent) and is usually more numerically stable than MLAMBDA’s reduction algorithm and LAMBDA’s reduction algorithm (to less extent).

Surface regions of illusory images are detected with a slower processing speed than those of luminance-defined images

Research has shown that the processing time for discriminating illusory contours is longer than for real contours. We know, however, little whether the visual processes, associated with detecting regions of illusory surfaces, are also slower as those responsible for detecting luminance-defined images. Using a speed-accuracy trade-off (SAT) procedure, we measured accuracy as a function of processing time for detecting illusory Kanizsa-type and luminance-defined squares embedded in 2D static luminance noise. The data revealed that the illusory images were detected at slower processing speed than the real images, while the points in time, when accuracy departed from chance, were not significantly different for both stimuli. The classification images for detecting illusory and real squares showed that observers employed similar detection strategies using surface regions of the real and illusory squares. The lack of significant differences between the x-intercepts of the SAT functions for illusory and luminance-modulated stimuli suggests that the detection of surface regions of both images could be based on activation of a single mechanism (the dorsal magnocellular visual pathway). The slower speed for detecting illusory images as compared to luminance-defined images could be attributed to slower processes of filling-in of regions of illusory images within the dorsal pathway.

Algorithm Q-LSQR for the least squares problem in quaternionic quantum theory

In this paper, an iterative algorithm for the standard quaternionic least squares problem is proposed without using the real (complex) representation. Our algorithm is implemented in the quaternion field and by means of direct quaternion arithmetic and is a natural generalization of the LSQR algorithm for the real least squares problem.

On the Functional Significance of the P1 and N1 Effects to Illusory Figures in the Notch Mode of Presentation

The processing of Kanizsa figures have classically been studied by flashing the full “pacmen” inducers at stimulus onset. A recent study, however, has shown that it is advantageous to present illusory figures in the “notch” mode of presentation, that is by leaving the round inducers on screen at all times and by removing the inward-oriented notches delineating the illusory figure at stimulus onset. Indeed, using the notch mode of presentation, novel P1and N1 effects have been found when comparing visual potentials (VEPs) evoked by an illusory figure and the VEPs to a control figure whose onset corresponds to the removal of outward-oriented notches, which prevents their integration into one delineated form. In Experiment 1, we replicated these findings, the illusory figure was found to evoke a larger P1 and a smaller N1 than its control. In Experiment 2, real grey squares were placed over the notches so that one condition, that with inward-oriented notches, shows a large central grey square and the other condition, that with outward-oriented notches, shows four unconnected smaller grey squares. In response to these “real” figures, no P1 effect was found but a N1 effect comparable to the one obtained with illusory figures was observed. Taken together, these results suggest that the P1 effect observed with illusory figures is likely specific to the processing of the illusory features of the figures. Conversely, the fact that the N1 effect was also obtained with real figures indicates that this effect may be due to more global processes related to depth segmentation or surface/object perception.

Application of Kalman Filtering with Input Estimation Technique to On-Line Cylindrical Inverse Heat Conduction Problems.

An on-line methodology to solve the inverse heat conduction problems of a hollow cylinder is presented. A new input estimation technique based on the concept of Kalman filtering is developed for estimating unknowns in real time, such as the what of the heat source or the heat flux on the boundary of the hollow cylinder. A recursive relation between the observed value of the residual sequence with unknown heat flux and the theoretical residual sequence of the Kalman filter that assumes known heat flux is formulated. A real time least squares algorithm is derived that uses the residual innovation sequence to compute the magnitude of heat flux. This recursive approach facilitates practical implementation and its capabilities are demonstrated in several typical cases with discontinuous and time-varying heat flux inputs.

Texture segmentation and visual search based on orientation contrast: An infant study with the familiarization-novelty preference method

Texture segmentation and visual search based on differences in line orientation were investigated in 3- to 6-month-old infants with a familiarization-novelty preference procedure. After familiarization to a pair of identical homogenous textures, 5- to 6-month-old infants showed a significant novelty preference for a texture consisting of oriented lines embedded in a background of elements of an orthogonal orientation but not for a single embedded target element. In a second experiment, we tested whether infants base their discrimination on the global perception of the embedded texture as a figure. After familiarization to a pair of identical embedded squarelike textures, infants show a highly significant novelty preference for real diamonds to real squares, although they spontaneously do not prefer diamonds to squares. These results indicate that 5- to 6-month-old infants are not only able to segment textures by differences in line orientation, but that they also have a coherent percept of the embedded texture as a figure of a specific shape.