  Quaternion wavelets are redundant wavelet transforms generalizing
complex-valued non-decimated wavelet transforms. In this paper we propose a
matrix-formulation for non-decimated quaternion wavelet transforms and define
spectral tools for use in machine learning tasks. Since quaternionic algebra is
an extension of complex algebra, quaternion wavelets bring redundancy in the
components that proves beneficial in wavelet based tasks. Specifically, the
wavelet coefficients in the decomposition are quaternion-valued numbers that
define the modulus and three phases.
  The novelty of this paper is definition of non-decimated quaternion wavelet
spectra based on the modulus and phase-dependent statistics as low-dimensional
summaries for 1-D signals or 2-D images. A structural redundancy in
non-decimated wavelets and a componential redundancy in quaternion wavelets are
linked to extract more informative features. In particular, we suggest an
improved way of classifying signals and images based on their scaling indices
in terms of spectral slopes and information contained in the three quaternionic
phases. We show that performance of the proposed method significantly improves
when compared to the standard versions of wavelets including the complex-valued
wavelets.
  To illustrate performance of the proposed spectral tools we provide two
examples of application on real-data problems: classification of sounds using
scaling in high-frequency recordings over time and monitoring of steel rolling
process using the fractality of captured digitized images. The proposed tools
are compared with the counterparts based on standard wavelet transforms.
