{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T18:11:27Z","timestamp":1787595087130,"version":"build-2736575974"},"reference-count":19,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in $\\mathbb{R}^{2}$ for which the transform is rapidly computable and invertible. We describe a fast algorithm using $O(N\\log{N})$ operations, where $N =n^{2}$ is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764\u2013784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.<\/jats:p>","DOI":"10.1137\/060650301","type":"journal-article","created":{"date-parts":[[2008,2,14]],"date-time":"2008-02-14T11:55:06Z","timestamp":1202990106000},"page":"785-803","source":"Crossref","is-referenced-by-count":57,"title":["A Framework for Discrete Integral Transformations II\u2014The 2D Discrete Radon Transform"],"prefix":"10.1137","volume":"30","author":[{"given":"A.","family":"Averbuch","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R. R.","family":"Coifman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D. L.","family":"Donoho","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Israeli","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Y.","family":"Shkolnisky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I.","family":"Sedelnikov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,14]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/060650283"},{"key":"R2","unstructured":"A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, J. Wald\u00e8n, and Y. Shkolnisky,\n                      Fast Slant Stack: A Notion of Radon Transform for Data in a Cartesian Grid Which Is Rapidly Computible, Algebraically Exact, Geometrically Faithful and Invertible\n                      , manuscript, 2001."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1109\/TASSP.1987.1165108"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539793256673"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595285718"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S003613999732425X"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/05064182X"},{"key":"R8","unstructured":"S. R. Deans,\n                      The Radon Transform and Some of Its Applications\n                      , revised ed., Krieger Publishing Company, Melbourne, FL, 1993."},{"key":"R9","doi-asserted-by":"crossref","unstructured":"D. L. Donoho and A. G. Flesia,\n                      Digital ridgelet transform based on true ridge functions\n                      , in Beyond Wavelets, vol. 10, J. Stoecker and G. V. Welland, eds., Academic Press, New York, 2003, pp. 1\u201330.","DOI":"10.1016\/S1570-579X(03)80029-0"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.1987.4307847"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1109\/42.7788"},{"key":"R12","unstructured":"K. Fourmont,\n                      Schnelle Fourier-tranformation bei nicht\u00e4quidistanten Gittern und tomographische anwendungen\n                      , Ph.D. thesis, Universit\u00e4t M\u00fcnster, M\u00fcnster, Germany, 1999."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0031-3203(95)00057-7"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1109\/78.539055"},{"key":"R15","unstructured":"A. K. Jain,\n                      Fundamentals of Digital Image Processing\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1989."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1109\/83.236530"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1109\/34.254058"},{"key":"R18","unstructured":"I. Sedelnikov,\n                      Discrete Diffraction Transform\n                      , Master's thesis, School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel, 2004."},{"key":"R19","unstructured":"A. Zygmund,\n                      Trigonometric Series\n                      , 2nd ed., Cambridge University Press, Cambridge, UK, 1993."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060650301","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:56:06Z","timestamp":1787334966000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060650301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":19,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060650301"],"URL":"https:\/\/doi.org\/10.1137\/060650301","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}