Pose estimation from an arbitrary number of 2-D to 3D feature correspondences is often done by minimising a nonlinear criterion function using one of the minimal representations for the orientation. However, there are many advantages in using unit quaternions to represent the orientation. Unfortunately, a straightforward formulation of the pose estimation problem based on quaternions results in...

Here Modified Rodrigues Parameters (MRPs) are used for an implementation that avoids making a small angle approximation for the attitude ambiguity. The direct averaging of MRPs is inaccurate because the distance metric between two MRPs is nonlinear to second order. However, the distance metric between two quaternions can be shown to be linear with the addition of a unit norm constraint on the q...

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case ...

Abstract This letter will clarify the fundamental properties of a quaternary neuron whose weights, threshold values, input and output signals are all quaternions, which is an extension of a usual real-valued neuron to quaternions. The main results of this letter are summarized as follows. A quaternary neuron has an orthogonal decision boundary. The 4-bit parity problem which cannot be solved wi...

The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H∗ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyper...

The quaternions were discovered by the Irish mathematician Sir William Rowan Hamilton in 1843. Hamilton was interested in the connection between complex numbers and 2-dimensional geometry. He tried in vain to extend the complex numbers to R, only years later would it be discovered that there is no 3-dimensional normed division algebra. Hamilton’s breakthrough came when he extended the complex n...