Foundations of quaternion graph signal processing and related contributions to fractional-order operators
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFPE |
Texto Completo: | https://repositorio.ufpe.br/handle/123456789/50071 |
Resumo: | Signal processing, at its core, is concerned with exploring how different representa- tions of a signal may provide useful ways of manipulating it. These representations may arise from a change in the subjacent algebra over which the signal samples are defined, for example when embedding three- and four-dimensional signals into the quaternion space; or maybe from a different model of the signal domain, as it happened with the development of graph signal processing to deal with network-like data; or even by exploring new linear transforms that map the signal onto a domain in which tasks such as compression, filtering or feature extraction are easier or more efficient. This thesis traverses exactly through these paths, aiming at answering the question of how to extend graph signal processing to the case in which signals and edge weights are quaternion-valued. It proposes a new set of tools which are a basis for what may be called Quaternion Graph Signal Processing (QGSP) and, as byproducts of the research journey, it contributes to the field of fractional transforms in two fronts: by proposing a new approach to the fractionalization of the quaternion discrete Fourier transform (QDFT), alongside the proposition of its multiparametric version, and by proposing a new fractional graph shift operator (GSO). Among the main results, we can mention: (1) the polynomial representation of the fractional GSO for arbitrary graphs was obtained, and it was shown that its use in filter design of finite impulse response and linear and shift-invariant (FIR LSI) filters improve the overall filter quality for a given filter length; (2) the new multiparametric fractional QDFT was used to create a holistic encryption scheme for color images with opacity layer, which was shown to provide satisfactorily large key space and key sensitivity; and (3) the main aspects of spectral analysis, filtering and compression in the context of QGSP were formulated, along with extensive practical examples on real-world data computed through a custom-made and open-source Python package. |
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RIBEIRO, Guilherme Boaviagemhttps://lattes.cnpq.br/3693136022461569http://lattes.cnpq.br/2782095059190056LIMA, Juliano Bandeira2023-05-11T11:37:56Z2023-05-11T11:37:56Z2022-12-23RIBEIRO, Guilherme Boaviagem. Foundations of quaternion graph signal processing and related contributions to fractional-order operators. 2022. Tese (Doutorado em Engenharia Elétrica) - Universidade Federal de Pernambuco, Recife, 2022.https://repositorio.ufpe.br/handle/123456789/50071Signal processing, at its core, is concerned with exploring how different representa- tions of a signal may provide useful ways of manipulating it. These representations may arise from a change in the subjacent algebra over which the signal samples are defined, for example when embedding three- and four-dimensional signals into the quaternion space; or maybe from a different model of the signal domain, as it happened with the development of graph signal processing to deal with network-like data; or even by exploring new linear transforms that map the signal onto a domain in which tasks such as compression, filtering or feature extraction are easier or more efficient. This thesis traverses exactly through these paths, aiming at answering the question of how to extend graph signal processing to the case in which signals and edge weights are quaternion-valued. It proposes a new set of tools which are a basis for what may be called Quaternion Graph Signal Processing (QGSP) and, as byproducts of the research journey, it contributes to the field of fractional transforms in two fronts: by proposing a new approach to the fractionalization of the quaternion discrete Fourier transform (QDFT), alongside the proposition of its multiparametric version, and by proposing a new fractional graph shift operator (GSO). Among the main results, we can mention: (1) the polynomial representation of the fractional GSO for arbitrary graphs was obtained, and it was shown that its use in filter design of finite impulse response and linear and shift-invariant (FIR LSI) filters improve the overall filter quality for a given filter length; (2) the new multiparametric fractional QDFT was used to create a holistic encryption scheme for color images with opacity layer, which was shown to provide satisfactorily large key space and key sensitivity; and (3) the main aspects of spectral analysis, filtering and compression in the context of QGSP were formulated, along with extensive practical examples on real-world data computed through a custom-made and open-source Python package.CAPESA área de Processamento de Sinais, em sua essência, preocupa-se em explorar como diferentes representações de um sinal podem fornecer maneiras úteis de manipulá-lo. Essas representações podem surgir, por exemplo, a partir de uma mudança na álgebra subjacente sobre a qual as amostras do sinal são definidas, como ocorre ao representar sinais de três ou quatro dimensões como vetores de um espaço vetorial quaterniônico. Outras representações podem advir de mudanças no domínio do sinal, como ocorreu com o desenvolvimento do processamento de sinais de grafos, para lidar com sinais estruturados em rede; ou ainda podem vir da exploração de novas transformadas lineares que mapeiam o sinal em um domínio em que tarefas como compressão, filtragem ou extração de recursos sejam mais fáceis ou eficientes. Esta tese desenvolve-se exatamente por esses caminhos, visando res- ponder à pergunta de como estender o processamento de sinais sobre grafos para o caso em que os sinais e pesos das arestas são quaterniônicos. Propõe-se um novo conjunto de ferramentas que são a base do que pode ser chamado de Processamento de Sinais Quater- niônicos sobre Grafos (QGSP, quaternion graph signal processing) e, como subprodutos da jornada de pesquisa, contribui-se para o campo das transformadas fracionárias em duas frentes: propondo uma nova abordagem para a fracionalização da transformada discreta de Fourier quaterniônica (QDFT, quaternion discrete Fourier transform), juntamente com a proposta de sua versão multiparamétrica, e propondo um novo operador de deslocamento sobre grafo fracionário (GSO, graph shift operator). Entre os principais resultados, podemos mencionar: (1) a representação polinomial do GSO fracionário para grafos arbitrários foi obtida, e foi demonstrado que seu uso no projeto de filtros de resposta finita de impulso e lineares e invariantes a deslocamento (FIR LSI, finite impulse response and linear and shift-invariant) melhora a qualidade geral do filtro para um determinado comprimento de filtro; (2) a nova QDFT fracionária multiparamétrica foi usada para criar um esquema de criptografia holística para imagens coloridas com camada de opacidade, que fornece um espaço de chave satisfatoriamente grande e com grande sensibilidade à mudança de chave; e (3) os principais aspectos da análise espectral, filtragem e compressão no contexto do QGSP foram formulados, juntamente com extensos exemplos práticos em dados do mundo real calculados por meio de um pacote Python personalizado e de código aberto.engUniversidade Federal de PernambucoPrograma de Pos Graduacao em Engenharia EletricaUFPEBrasilAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessEngenharia elétricaQuatérniosTransformada discreta de Fourier quaterniônica fracionáriaOperador de deslocamento de grafo fracionárioProcessamento de sinal de grafo quaterniônicoFoundations of quaternion graph signal processing and related contributions to fractional-order operatorsinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisdoutoradoreponame:Repositório Institucional da UFPEinstname:Universidade Federal de Pernambuco (UFPE)instacron:UFPEORIGINALTESE Guilherme Boaviagem Ribeiro.pdfTESE Guilherme Boaviagem Ribeiro.pdfapplication/pdf7634459https://repositorio.ufpe.br/bitstream/123456789/50071/1/TESE%20Guilherme%20Boaviagem%20Ribeiro.pdf16756a02a5c807d42f262186841750beMD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; 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dc.title.pt_BR.fl_str_mv |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
title |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
spellingShingle |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators RIBEIRO, Guilherme Boaviagem Engenharia elétrica Quatérnios Transformada discreta de Fourier quaterniônica fracionária Operador de deslocamento de grafo fracionário Processamento de sinal de grafo quaterniônico |
title_short |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
title_full |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
title_fullStr |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
title_full_unstemmed |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
title_sort |
Foundations of quaternion graph signal processing and related contributions to fractional-order operators |
author |
RIBEIRO, Guilherme Boaviagem |
author_facet |
RIBEIRO, Guilherme Boaviagem |
author_role |
author |
dc.contributor.authorLattes.pt_BR.fl_str_mv |
https://lattes.cnpq.br/3693136022461569 |
dc.contributor.advisorLattes.pt_BR.fl_str_mv |
http://lattes.cnpq.br/2782095059190056 |
dc.contributor.author.fl_str_mv |
RIBEIRO, Guilherme Boaviagem |
dc.contributor.advisor1.fl_str_mv |
LIMA, Juliano Bandeira |
contributor_str_mv |
LIMA, Juliano Bandeira |
dc.subject.por.fl_str_mv |
Engenharia elétrica Quatérnios Transformada discreta de Fourier quaterniônica fracionária Operador de deslocamento de grafo fracionário Processamento de sinal de grafo quaterniônico |
topic |
Engenharia elétrica Quatérnios Transformada discreta de Fourier quaterniônica fracionária Operador de deslocamento de grafo fracionário Processamento de sinal de grafo quaterniônico |
description |
Signal processing, at its core, is concerned with exploring how different representa- tions of a signal may provide useful ways of manipulating it. These representations may arise from a change in the subjacent algebra over which the signal samples are defined, for example when embedding three- and four-dimensional signals into the quaternion space; or maybe from a different model of the signal domain, as it happened with the development of graph signal processing to deal with network-like data; or even by exploring new linear transforms that map the signal onto a domain in which tasks such as compression, filtering or feature extraction are easier or more efficient. This thesis traverses exactly through these paths, aiming at answering the question of how to extend graph signal processing to the case in which signals and edge weights are quaternion-valued. It proposes a new set of tools which are a basis for what may be called Quaternion Graph Signal Processing (QGSP) and, as byproducts of the research journey, it contributes to the field of fractional transforms in two fronts: by proposing a new approach to the fractionalization of the quaternion discrete Fourier transform (QDFT), alongside the proposition of its multiparametric version, and by proposing a new fractional graph shift operator (GSO). Among the main results, we can mention: (1) the polynomial representation of the fractional GSO for arbitrary graphs was obtained, and it was shown that its use in filter design of finite impulse response and linear and shift-invariant (FIR LSI) filters improve the overall filter quality for a given filter length; (2) the new multiparametric fractional QDFT was used to create a holistic encryption scheme for color images with opacity layer, which was shown to provide satisfactorily large key space and key sensitivity; and (3) the main aspects of spectral analysis, filtering and compression in the context of QGSP were formulated, along with extensive practical examples on real-world data computed through a custom-made and open-source Python package. |
publishDate |
2022 |
dc.date.issued.fl_str_mv |
2022-12-23 |
dc.date.accessioned.fl_str_mv |
2023-05-11T11:37:56Z |
dc.date.available.fl_str_mv |
2023-05-11T11:37:56Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
RIBEIRO, Guilherme Boaviagem. Foundations of quaternion graph signal processing and related contributions to fractional-order operators. 2022. Tese (Doutorado em Engenharia Elétrica) - Universidade Federal de Pernambuco, Recife, 2022. |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufpe.br/handle/123456789/50071 |
identifier_str_mv |
RIBEIRO, Guilherme Boaviagem. Foundations of quaternion graph signal processing and related contributions to fractional-order operators. 2022. Tese (Doutorado em Engenharia Elétrica) - Universidade Federal de Pernambuco, Recife, 2022. |
url |
https://repositorio.ufpe.br/handle/123456789/50071 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.publisher.program.fl_str_mv |
Programa de Pos Graduacao em Engenharia Eletrica |
dc.publisher.initials.fl_str_mv |
UFPE |
dc.publisher.country.fl_str_mv |
Brasil |
publisher.none.fl_str_mv |
Universidade Federal de Pernambuco |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFPE instname:Universidade Federal de Pernambuco (UFPE) instacron:UFPE |
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Repositório Institucional da UFPE |
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