{"title":"正交元素在先验重构中的应用:傅里叶和多项式技术","authors":"R. Andrews, A. G. Law, A.D. Strilaeff, R. Sloboda","doi":"10.1109/PACRIM.1989.48311","DOIUrl":null,"url":null,"abstract":"The mathematical setting assumed is the Hilbert space. Two image reconstruction problems are summarized. In one (from emission tomography), an unknown member, f, of the space is sought as a linear combination of linearly independent elements g/sub 1/, g/sub 2/, . . ., g/sub n/, under the hypothesis that the inner products are known for 1<or=j<or=n. The other situation, from signal sampling, has an analogous mathematical structure, but it involves a transform, T, and the Fourier transform is used. The Fourier techniques can be carried out efficiently by taking advantage of the FFT, but ill-conditioning can appear in the linear system which arises in the a priori reconstruction process. With, instead, reconstruction in the original space, a polynomial choice for the basis function provides some interesting machinery through properties inherited from orthogonality.<<ETX>>","PeriodicalId":256287,"journal":{"name":"Conference Proceeding IEEE Pacific Rim Conference on Communications, Computers and Signal Processing","volume":"63 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1989-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Application of orthogonal elements in a-priori reconstruction: Fourier and polynomial techniques\",\"authors\":\"R. Andrews, A. G. Law, A.D. Strilaeff, R. Sloboda\",\"doi\":\"10.1109/PACRIM.1989.48311\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The mathematical setting assumed is the Hilbert space. Two image reconstruction problems are summarized. In one (from emission tomography), an unknown member, f, of the space is sought as a linear combination of linearly independent elements g/sub 1/, g/sub 2/, . . ., g/sub n/, under the hypothesis that the inner products are known for 1<or=j<or=n. The other situation, from signal sampling, has an analogous mathematical structure, but it involves a transform, T, and the Fourier transform is used. The Fourier techniques can be carried out efficiently by taking advantage of the FFT, but ill-conditioning can appear in the linear system which arises in the a priori reconstruction process. With, instead, reconstruction in the original space, a polynomial choice for the basis function provides some interesting machinery through properties inherited from orthogonality.<<ETX>>\",\"PeriodicalId\":256287,\"journal\":{\"name\":\"Conference Proceeding IEEE Pacific Rim Conference on Communications, Computers and Signal Processing\",\"volume\":\"63 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1989-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Conference Proceeding IEEE Pacific Rim Conference on Communications, Computers and Signal Processing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/PACRIM.1989.48311\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Conference Proceeding IEEE Pacific Rim Conference on Communications, Computers and Signal Processing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/PACRIM.1989.48311","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Application of orthogonal elements in a-priori reconstruction: Fourier and polynomial techniques
The mathematical setting assumed is the Hilbert space. Two image reconstruction problems are summarized. In one (from emission tomography), an unknown member, f, of the space is sought as a linear combination of linearly independent elements g/sub 1/, g/sub 2/, . . ., g/sub n/, under the hypothesis that the inner products are known for 1>
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