This paper discusses quaternion L geometric weighting averaging working on the multiplicative Lie group of nonzero quaternionsH∗, endowed with its natural bi-invariant Riemannian metric. Algorithms for computing the Riemannian L center of mass of a set of points, with 1 ≤ p ≤ ∞ (i.e., median, mean, L barycenter and minimax center), are particularized to the case of H∗. Two different approaches ...

Recently, solutions to inverse problems have been required in various engineering fields. The neural network inversion method has been studied as one of the neural network-based solutions. On the other hand, the extension of the neural network to a higher-dimensional domain, e.g., complex-value or quaternion, has been proposed, and a number of higher-dimensional neural network models have been ...

This paper presents an algorithm to average a set of quaternion observations. The average quaternion is determined by minimizing the weighted sum of the squared Frobenius norms of the corresponding attitude matrix differences, subject to the unit-norm constraint in the determined solution. Two cases are presented: one that incorporates scalar weights and one that incorporates general weights on...

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...

Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordinates, such as Euler angles. Due to the coordinate singularities associated with local coordinate charts, it is common in engineering applications to adopt the unit quaternion representation, and the numerical simulations typically impose the unit length 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.

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...