Theoretical self-similar processes have been an essential tool for modeling a wide range of real-world signals or images that describe phenomena in engineering, physics, medicine, biology, economics, geology, chemistry, and so on. However, it is often difficult for general modeling methods to quantify a self-similarity due to irregularities in the signals or images. Wavelet-based spectral tools have become standard solutions for such problems in signal and image processing and achieved outstanding performances in real applications. This thesis proposes three novel wavelet-based spectral tools to improve the assessment of self-similarity. First, we propose spectral tools based on non-decimated complex wavelet transforms implemented by their matrix formulation. A structural redundancy in non-decimated wavelets and a componential redundancy in complex wavelets act in a synergy when extracting wavelet-based informative descriptors. Next, we step into the quaternion domain and propose a matrix-formulation for non-decimated quaternion wavelet transforms and define spectral tools for use in machine learning tasks. We define non-decimated quaternion wavelet spectra based on the modulus and three phase-dependent statistics as low-dimensional summaries for 1-D signals or 2-D images. Finally, we suggest a dual wavelet spectra based on non-decimated wavelet transform in real, complex, and quaternion domains. This spectra is derived from a new perspective that draws on the link of energies of the signal with the temporal or spatial scales in the multiscale representations.
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Assessing self-similarity in redundant complex and quaternion wavelet domains: Theory and applications
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