Cormack and Hounsfield received the 1979 Nobel prize in medicine for their work on CT scans. Cormack, a physicist, published his mathematical work on this in 1963, to essentially no response. Hounsfield, an engineer, built the first CT scanner in 1971 unaware of Cormack's work. Cormark included the following in his Nobel prize speech: "If a fine beam of gamma-rays of intensity $I_0$ is incident on the body and the emerging intensity is $I$, then the
measurable quantity is $g = \ln(I_0/I) = \int_L f ds$, where $f$ is
the variable absorption coefficient along the line $L$.
Hence if $f$ is a function in two dimensions, and $g$ is
known for all lines [...], the question is: Can $f$ be
determined if $g$ is known? This seemed like a problem
which would have been solved before, probably in the
19th century, but a literature search and enquiries of
mathematicians provided no information about it.
Fourteen years would elapse before I learned that Radon
had solved this problem in 1917."
Fourteen years after Cormack's work means 1977, so Radon's work was rediscovered by the people involved with creating CT scan technology only after CT scan's had been around for several years. (Search on "Radon transform" for more information.)
Radon's work was rediscovered multiple times:
1) Cramer and Wold (1936) in probability theory,
2) Ambartsumian (1936) in astronomy,
3) Bracewell (1956) in astronomy,
4) De Rosier and Klug (1968) in chemistry.
In fact, Radon's basic idea was worked out before Radon, by Funk (1916) and Lorentz (1905). This work of Lorentz was unpublished, but a formula he found is mentioned in a paper by Bockwinel in 1906. More on this history is in Cormack's survey paper
Computed tomography: some history and recent developments, pp. 35--42 in "Computed tomography: Proceedings of Symposia in Applied Mathematics" 27, AMS, 1983.
Shortly before the work of Cormack, Olendorf (a medical doctor in LA) published a paper in 1961 describing a crude CT scanner he had built out of household parts, such as model railroad tracks (!) but it went unnoticed. Hounsfield acknowledged it, but Olendorf was not included in the Nobel prize list with Cormack and Hounsfield. He once said in an interview "I think Professor Cormack was selected [for the Nobel prize] because he worked out all the line integrals mathematically. [...] I didn't provide any mathematical treatment of it, and that apparently carried a lot of weight with the Nobel committee. See http://en.wikipedia.org/wiki/William_H._Oldendorf for more on his story.
The mathematical and engineering concepts in CT scan technology, with applications
to medical imaging, were worked out in an obscure journal in Kiev by S. T. Tetelbaum
in 1957-58, before Olendorf!