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represents the attenuation data of a tomographic scan through an inhomogeneous medium whose density is represented by the function
51:, and coincides with it in two dimensions. In higher dimensions, the X-ray transform of a function is defined by integrating over
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of the X-ray transform is therefore of practical importance because it allows one to reconstruct an unknown density
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370:, Translations of Mathematical Monographs, vol. 220, Providence, R.I.:
310:"The ultrahyperbolic differential equation with four independent variables"
59:
as in the Radon transform. The X-ray transform derives its name from X-ray
209:{\displaystyle Xf(L)=\int _{L}f=\int _{\mathbf {R} }f(x_{0}+t\theta )dt}
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435:, Progress in Mathematics (2nd ed.), Boston, M.A.:
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Gelfand, I. M.; Gindikin, S. G.; Graev, M. I. (2003) ,
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43:in 1938 that is one of the cornerstones of modern
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287:Mathematical Methods in Image Reconstruction
67:) because the X-ray transform of a function
285:Natterer, Frank; Wübbeling, Frank (2001).
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398:Geometric analysis on symmetric spaces
264:can be written as an X-ray transform.(
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367:Selected topics in integral geometry
226:is an initial point on the line and
47:. It is very closely related to the
112:defined on the set of all lines in
266:Gelfand, Gindikin & Graev 2003
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249:The X-ray transform satisfies an
234:giving the direction of the line
83:from its known attenuation data.
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345:Berenstein, Carlos A. (2001) ,
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104:, then the X-ray transform of
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402:American Mathematical Society
372:American Mathematical Society
326:10.1215/S0012-7094-38-00423-5
251:ultrahyperbolic wave equation
426:Helgason, Sigurdur (1999),
396:Helgason, Sigurdur (2008),
352:Encyclopedia of Mathematics
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295:10.1137/1.9780898718324.fm
463:X-ray computed tomography
314:Duke Mathematical Journal
260:The Gaussian or ordinary
262:hypergeometric function
289:. Philadelphia: SIAM.
242:on the Euclidean line
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308:Fritz, John (1938).
230:is a unit vector in
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458:Integral transforms
429:The Radon Transform
95:continuous function
92:compactly supported
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37:integral transform
16:Integral transform
453:Integral geometry
411:978-0-8218-4530-1
381:978-0-8218-2932-5
347:"X-ray transform"
55:rather than over
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29:ray transform
27:(also called
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329:. Retrieved
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320:: 300–322.
57:hyperplanes
21:mathematics
447:Categories
437:Birkhauser
331:23 January
272:References
268:, 2.1.2).
61:tomography
41:Fritz John
357:EMS Press
195:θ
162:∫
146:∫
77:Inversion
63:(used in
65:CT scans
35:) is an
420:2463854
390:2000133
253:called
110:Xƒ
97:on the
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219:where
106:ƒ
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81:ƒ
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69:ƒ
23:, the
433:(PDF)
90:is a
53:lines
406:ISBN
376:ISBN
333:2013
322:doi
291:doi
116:by
31:or
19:In
449::
416:MR
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386:MR
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316:.
312:.
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360:.
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324::
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297:.
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244:L
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228:θ
224:0
221:x
204:t
201:d
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192:t
189:+
184:0
180:x
176:(
173:f
167:R
158:=
155:f
150:L
142:=
139:)
136:L
133:(
130:f
127:X
114:R
102:R
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