Improper rotation

Abstract: A simple, efficient and general method is described for finding the linear orthogonal transformation to superpose two similar structures given by sets of equivalent points, usually atomic position vectors. Formulae are also given for extracting the independent variables of rotation from the resulting transformation matrix. In addition, general transformations are derived, both in the case of proper rotation and in the case of rotatory inversion, to convert to a molecular frame of reference based on the superposition axis of symmetry.

Abstract: The image transformation due to camera rotation relative to a stationary scene is analyzed, and the associated transformation rules of “features” given by weighted averaging of the image are derived by considering infinitesimal generators on the basis of group representation theory. Three-dimensional vectors and tensors are reduced to two-dimensional invariants on the image plane from the viewpoint of projective geometry. Three-dimensional invariants and camera rotation reconstruction are also discussed. The result is applied to the shape recognition problem when camera rotation is involved.

Abstract: In the rotational inertial navigation system (RINS), the rotation scheme affects the system’s performance directly. Improper rotation scheme may excite devices’ errors instead of reducing the effects caused by them. RINS should choose a suitable rotation scheme based on the error characteristics of its own. The rotation modulation theory for kinds of errors is analyzed, and two rotation schemes, called scheme A and scheme B in this paper, are proposed first. After the merits and demerits of the two schemes are discussed, an improved rotation scheme, called scheme C, is put forward. Scheme C can compensate scale factor errors, biases and installation errors of inertial sensors by rotating inertial measurement unit along the yaw axis and horizontal axis alternately. The simulations based on MATLAB are conducted in specific conditions; the results indicate that the performances of the three schemes are great better than that of general strap-down INS, called scheme D in this paper. Among the three schemes proposed, the performance of scheme C is the best. Experimental results in a real dual-axis rotational inertial navigation system based on fiber optic gyro show that the radical direction position error of the system is about 0.7 n mile in 12 h if scheme C is adopted, which verify the practicability of the rotation scheme proposed in this paper.

Abstract: In an image data rotating system for applying rotation processing to input image data of a large picture plane, the image data is divided into rectangular small blocks and rotation processing is applied to the image data of each block. The rotation processing to the image data of each block is performed by first reducing in the x-axis direction and enlarging in the y-axis direction. Then the first oblique axis transformation is applied. Any error data between the real oblique axis and the image element found in the final raster of the corresponding region is held in a memory. Subsequently, the second oblique axis transformation in the y-axis direction is applied, and error data is held in similar manner. Error data held as the initial value for the rotation processing to the image data of the adjacent region block is used, thereby allowing the rotation of the image data of a large picture plane to an arbitrary angle to be executed at high speed.