{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T13:11:22Z","timestamp":1740143482001,"version":"3.37.3"},"reference-count":15,"publisher":"Wiley","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-0923297","EB011785"],"award-info":[{"award-number":["CMMI-0923297","EB011785"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000070","name":"National Institute of Biomedical Imaging and Bioengineering","doi-asserted-by":"publisher","award":["CMMI-0923297","EB011785"],"award-info":[{"award-number":["CMMI-0923297","EB011785"]}],"id":[{"id":"10.13039\/100000070","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Biomedical Imaging"],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:p>This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems leads to different reconstruction formulas and explain why the Radon formula cannot directly work with truncated projection data. Also, we introduce a gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry.<\/jats:p>","DOI":"10.1155\/2011\/285130","type":"journal-article","created":{"date-parts":[[2010,10,26]],"date-time":"2010-10-26T15:32:42Z","timestamp":1288107162000},"page":"1-16","source":"Crossref","is-referenced-by-count":0,"title":["Inverse Fourier Transform in the Gamma Coordinate System"],"prefix":"10.1155","volume":"2011","author":[{"given":"Yuchuan","family":"Wei","sequence":"first","affiliation":[{"name":"Department of Radiation Oncology, Wake Forest University School of Medicine, Winston-Salem, NC 27157, USA"},{"name":"Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hengyong","family":"Yu","sequence":"additional","affiliation":[{"name":"Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USA"},{"name":"Department of Radiology, Division of Radiologic Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USA"},{"name":"Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Virginia Tech., Blacksburg, VA 24061, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2656-7705","authenticated-orcid":true,"given":"Ge","family":"Wang","sequence":"additional","affiliation":[{"name":"Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Wake Forest University Health Sciences, Winston-Salem, NC 27157, USA"},{"name":"Biomedical Imaging Division, VT-WFU School of Biomedical Engineering and Sciences, Virginia Tech., Blacksburg, VA 24061, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"issue":"4","key":"1","first-page":"235","volume":"15","year":"2007","journal-title":"Journal of X-Ray Science and Technology"},{"issue":"1","key":"2","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1088\/0266-5611\/20\/1\/014","volume":"20","year":"2004","journal-title":"Inverse Problems"},{"year":"2004","key":"3"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/22\/2\/004"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/22\/4\/N02"},{"year":"1996","key":"6"},{"volume":"1","year":"1977","key":"7"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1118\/1.1828673"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171203209315"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139901387186"},{"issue":"4","key":"11","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1016\/S0196-8858(03)00099-X","volume":"32","year":"2004","journal-title":"Advances in Applied Mathematics"},{"key":"12a","first-page":"267","volume":"69","year":"1917","journal-title":"Berichte \u00fcber die Verhandlungen der S\u00e4chsische Akademie der Wissenschaften"},{"issue":"4","key":"12b","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1109\/TMI.1986.4307775","volume":"5","year":"1986","journal-title":"IEEE Transactions on Medical Imaging"},{"issue":"4","key":"13","first-page":"429","volume":"20","year":"1975","journal-title":"Soviet Physics\u2014Crystallography"},{"year":"2001","series-title":"Springer Undergraduate Mathematics Series","key":"14"}],"container-title":["International Journal of Biomedical Imaging"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijbi\/2011\/285130.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijbi\/2011\/285130.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijbi\/2011\/285130.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T11:30:27Z","timestamp":1497871827000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijbi\/2011\/285130\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":15,"alternative-id":["285130","285130"],"URL":"https:\/\/doi.org\/10.1155\/2011\/285130","relation":{},"ISSN":["1687-4188","1687-4196"],"issn-type":[{"type":"print","value":"1687-4188"},{"type":"electronic","value":"1687-4196"}],"subject":[],"published":{"date-parts":[[2011]]}}}