PhD Defense by Taewoon Kong

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  • Date/Time:
    • Friday March 15, 2019
      9:00 am - 11:00 am
  • Location: Groseclose 226A (SCL Resources room)
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Summary Sentence: Assessing Self-Similarity in Redundant Complex and Quaternion Wavelet Domains: Theory and Applications

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Title:  Assessing Self-Similarity in Redundant Complex and Quaternion Wavelet Domains: Theory and Applications

Advisors: Dr. Brani Vidakovic (ISyE)


Committee Members

Dr. Yajun Mei (ISyE)

Dr. Kamran Paynabar (ISyE)

Dr. Sung Ha Kang (School of Mathematics)

Dr. Kichun Sky Lee (Industrial Engineering in Hanyang University)


Date and Time: Friday, March 15th, 9:00 am

Location: Groseclose 226A (SCL Resources room)



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.


In Chapter 2, we propose spectral tools based on non-decimated complex wavelet transforms implemented by their matrix formulation. This non-decimated complex wavelet spectra utilizes both real and imaginary parts of complex-valued wavelet coefficients via their modulus and phase. A structural redundancy in non-decimated wavelets and a componential redundancy in complex wavelets act in a synergy when extracting wavelet-based informative descriptors. In particular, we suggest an improved way of separating signals and images based on their scaling indices in terms of spectral slopes and information contained in the phase in order to improve performance of classification. It is also worth mentioning that the proposed method can handle signals of an arbitrary size and in 2-D case, rectangular images of possibly different and non-dyadic dimensions because of the matrix formulation of non-decimated wavelet transform. This is in contrast to the standard wavelet transforms where algorithms for handling objects of non-dyadic dimensions requires either data preprocessing or customized algorithm adjustments.


Quaternion wavelets are another redundant wavelet transforms generalizing complex-valued wavelet transforms. In Chapter 3, 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. Since quaternionic algebra is an extension of complex algebra, quaternion wavelets bring more 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. 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. A structural redundancy in non-decimated wavelets and a componential redundancy in quaternion wavelets are linked to extract more informative features.


Dual relation that is an alternative representation to analyze the same fields has been studied in various fields including optimization, physics, engineering, and mathematics, etc. However, there is no dual approach for wavelet spectra method as far as we know, thus, the duality concept is worth considering to measure a self-similarity index from a different perspective. In Chapter 4, we suggest a dual wavelet spectra based on non-decimated wavelet transform in real, complex, and quaternion domains. Specifically, as dual representations of original wavelet spectra based on modulus, distributions of levels (scales) with regard to each energy (squared wavelet coefficients) bracket are established as opposed to distributions of energies with regard to each level on original wavelet spectra. Moreover, the energy is replaced with phases of complex- and quaternion-valued wavelet coefficients to obtain dual phase-dependent statistics. They can improve classification performance when used together with their original counterparts suggested in previous chapters because of complementary interactions.


In Chapter 5, to demonstrate the use of three defined spectral methodologies, we provide four examples of application on real-data problems: classification of visual acuity using scaling in pupil diameter dynamic in time and diagnostic, classification of sounds using scaling in high-frequency recordings over time, screening digital mammogram images using the fractality of digitized images of the background tissue, 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 in terms of computing time and evaluation metrics for classification problems.

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Phd Defense
  • Created By: Tatianna Richardson
  • Workflow Status: Published
  • Created On: Mar 4, 2019 - 12:04pm
  • Last Updated: Mar 4, 2019 - 12:04pm