{"title":"特征函数,特征值,以及四元数和双四元数傅里叶变换的分数化","authors":"S. Pei, Jian-Jiun Ding, Kuo-Wei Chang","doi":"10.5281/ZENODO.42104","DOIUrl":null,"url":null,"abstract":"The discrete quaternion Fourier transform (DQFT) is useful for signal analysis and image processing. In this paper, we derive the eigenfunctions and eigenvalues of the DQFT. We also extend our works to the reduced biquaternion case, i.e., the discrete reduced biquaternion Fourier transform (DRBQFT). We find that an even or odd symmetric eigenvector of the 2-D DFT will also be an eigenvector of the DQFT and the DRBQFT. Moreover, both the DQFT and the DRBQFT have 8 eigenspaces, which correspond to the eigenvalues of 1, -1, i, -i, j, -j, k, and -k. We also use the derived eigenvectors to fractionalize the DQFT and the DRBQFT and define the discrete fractional quaternion transform and the discrete fractional reduced biquaternion Fourier transform.","PeriodicalId":409817,"journal":{"name":"2010 18th European Signal Processing Conference","volume":"136 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2010-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Eigenfunctions, eigenvalues, and fractionalization of the quaternion and biquaternion fourier transforms\",\"authors\":\"S. Pei, Jian-Jiun Ding, Kuo-Wei Chang\",\"doi\":\"10.5281/ZENODO.42104\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The discrete quaternion Fourier transform (DQFT) is useful for signal analysis and image processing. In this paper, we derive the eigenfunctions and eigenvalues of the DQFT. We also extend our works to the reduced biquaternion case, i.e., the discrete reduced biquaternion Fourier transform (DRBQFT). We find that an even or odd symmetric eigenvector of the 2-D DFT will also be an eigenvector of the DQFT and the DRBQFT. Moreover, both the DQFT and the DRBQFT have 8 eigenspaces, which correspond to the eigenvalues of 1, -1, i, -i, j, -j, k, and -k. We also use the derived eigenvectors to fractionalize the DQFT and the DRBQFT and define the discrete fractional quaternion transform and the discrete fractional reduced biquaternion Fourier transform.\",\"PeriodicalId\":409817,\"journal\":{\"name\":\"2010 18th European Signal Processing Conference\",\"volume\":\"136 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2010-08-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"2010 18th European Signal Processing Conference\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5281/ZENODO.42104\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"2010 18th European Signal Processing Conference","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5281/ZENODO.42104","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Eigenfunctions, eigenvalues, and fractionalization of the quaternion and biquaternion fourier transforms
The discrete quaternion Fourier transform (DQFT) is useful for signal analysis and image processing. In this paper, we derive the eigenfunctions and eigenvalues of the DQFT. We also extend our works to the reduced biquaternion case, i.e., the discrete reduced biquaternion Fourier transform (DRBQFT). We find that an even or odd symmetric eigenvector of the 2-D DFT will also be an eigenvector of the DQFT and the DRBQFT. Moreover, both the DQFT and the DRBQFT have 8 eigenspaces, which correspond to the eigenvalues of 1, -1, i, -i, j, -j, k, and -k. We also use the derived eigenvectors to fractionalize the DQFT and the DRBQFT and define the discrete fractional quaternion transform and the discrete fractional reduced biquaternion Fourier transform.