{"id":{"repo_id":"gatech","oai_identifier":"oai:repository.gatech.edu:1853/61244"},"canonical_url":"https://search.dev.ndltd.org/etd/gatech/oai:repository.gatech.edu:1853/61244","repository":{"repo_id":"gatech","name":"Georgia Tech","base_url":"https://repository.gatech.edu/server/oai/request"},"display":{"title":"Assessing self-similarity in redundant complex and quaternion wavelet domains: Theory and applications","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Kong, Tae Woon"],"institution":"Georgia Institute of Technology","degree_name":null,"degree_level":"Doctoral","degree_discipline":null,"degree_department":"Industrial and Systems Engineering","school":null,"contributors":[],"advisors":["Vidakovic, Brani"],"committee_chairs":[],"committee_members":["Mei, Yajun","Paynabar, Kamran","Kang, Sung Ha","Lee, Kichun"],"year":2019,"date_issued":"2019-03-25","date_published":"2019-03-25","updated_at":"2026-07-27T19:49:22Z","subjects":["Wavelets","Classification","Feature extraction"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1853/61244","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Vidakovic, Brani"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Mei, Yajun","Paynabar, Kamran","Kang, Sung Ha","Lee, Kichun"]},{"key":"dc:contributor.department","label":"Department","values":["Industrial and Systems Engineering"]},{"key":"dc:creator","label":"Author","values":["Kong, Tae Woon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-05-29T14:02:49Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-05-29T14:02:49Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-03-25"]},{"key":"dc:publisher","label":"Institution","values":["Georgia Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Wavelets","Classification","Feature extraction"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1853/61244"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Assessing self-similarity in redundant complex and quaternion wavelet domains: Theory and applications"]}]}],"canonical_facts":{"dc:contributor.advisor":["Vidakovic, Brani"],"dc:contributor.committeemember":["Mei, Yajun","Paynabar, Kamran","Kang, Sung Ha","Lee, Kichun"],"dc:contributor.department":["Industrial and Systems Engineering"],"dc:creator":["Kong, Tae Woon"],"dc:date.accessioned":["2019-05-29T14:02:49Z"],"dc:date.available":["2019-05-29T14:02:49Z"],"dc:date.issued":["2019-03-25"],"dc:description.abstract":["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. 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