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...

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of Principal Component Analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images a...

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...

Pattern recognition techniques have been used to automatically recognize the objects, personal identities, predict the function of protein, the category of the cancer, identify lesion, perform product inspection, and so on. In this paper we propose a novel quaternion-based discriminant method. This method represents and classifies color images in a simple and mathematically tractable way. The p...

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...

Image splicing detection is of fundamental importance in digital forensics and therefore has attracted increasing attention recently. In this paper, a color image splicing detection approach is proposed based on Markov transition probability of quaternion component separation in quaternion discrete cosine transform (QDCT) domain and quaternion wavelet transform (QWT) domain. Firstly, Markov fea...

The performances of three attitude determination algorithms are compared in this paper. The three methods are the Complementary Filter, a Quaternion-based Kalman Filter and a Quaternion-based Gradient Descent Algorithm. An analysis of their performance based on an experimental investigation was undertaken. This paper shows that the Complementary Filter requires the least computational power; Qu...

We use the quaternion Kähler reduction technique to study old and new selfdual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology,...