The quaternion representation of color is shown here to be effective in the context of segmenting color images into regions of similar color texture. The advantage of using quaternion arithmetic is that a color can be represented and analyzed as a single entity. A lowdimensional basis for the color textures found in a given image is derived via quaternion principal component analysis (QPCA) of ...

Due to mathematical limitations, conventional phase correlation technique can only be applied to grayscale images or at most complex images. A full colour image must be first converted to a grayscale image before performing the phase correlation, during which the luminance information has been wasted. In this paper, an extension of the phase correlation technique to the quaternion field , based...

An involution is usually defined as a mapping that is its own inverse. In this paper, we study quaternion involutions that have the additional properties of distribution over addition and multiplication. We review formal axioms for such involutions, and we show that the quaternions have an infinite number of involutions. We show that the conjugate of a quaternion may be expressed using three mu...

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...

We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.

The research is Supported by Chongqing University postgraduates’ Science and Innovation Fund (200801A 1 A0070266). Abstract The purpose of this paper is to discuss the inequalities for the trace of self-conjugate quaternion matrix. We present the inequality for eigenvalues and trace of self-conjugate quaternion matrices. Based on the inequality above, we obtain several inequalities for the trac...

This article compares three different algorithms used to compute Euler angles from data obtained by the angular rate sensor (e.g., MEMS gyroscope)-the algorithms based on a rotational matrix, on transforming angular velocity to time derivations of the Euler angles and on unit quaternion expressing rotation. Algorithms are compared by their computational efficiency and accuracy of Euler angles e...

In this paper we present the results from our visual servoing system for a real industrial robot. In contrast to other visual servoing system in the literature, the one presented here, which was derived from [4], uses a quaternion representation of the rotation instead of the more common matrix representation. By doing so, the proposed system avoids potential singularities introduced by the rot...

The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Often, quaternion-based feedback control laws require an additional mechanism that lifts a continuous attitude trajectory to the unit quaternion space. Lifting mechanisms that are memoryless, for example, selecting the quaternion having positive scalar component, h...

This article is an exhaustive revision of concepts and formulas related to quaternions and rotations in 3D space, and their proper use in estimation engines such as the error-state Kalman filter. The paper includes an in-depth study of the rotation group and its Lie structure, with formulations using both quaternions and rotation matrices. It makes special attention in the definition of rotatio...