The paper gives a new representation of conformal groups in n dimensions terms hyperquaternions defined as tensor products quaternion algebras (or subalgebra thereof). Being Clifford algebras, provide good pseudo-orthogonal such $$O(p+1,q+1)$$ isomorphic to the nD group with $$n=p+q.$$ yields simple expressions generators, independently matrices or operators. canonical decomposition and invaria...

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several ...

Quaternion-valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a σ-bandli...

Quaternion models have been developed for both direction of arrival estimation and beamforming based on crossed-dipole arrays in the past. However, for almost all the models, especially for adaptive beamforming, the desired signal is still complex-valued and one example is the quaternion-Capon beamformer. However, since the complex-valued desired signal only has two components, while there are ...

The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hype...

The quaternion representation of colour is shown to be effective in the context of colour texture region segmentation in digital colour images. The advantage of using quaternion arithmetic is that a colour can be represented and analyzed as a single entity. A basis for the colour textures occurring in a given image is derived via quaternion principal component analysis of a training set of colo...

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditiona...