Let the two-dimensional cylinder function be defined by
(1)
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Then the Radon transform is given by
(2)
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where
(3)
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is the delta function. Rewriting in polar coordinates then gives
(4)
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Now use the harmonic addition theorem to write
(5)
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with a phase shift. Then
(6)
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(7)
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(8)
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Then use
(9)
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which, with , becomes
(10)
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Define
(11)
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(12)
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(13)
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so the inner integral is
(14)
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(15)
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and the Radon transform becomes
(16)
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(17)
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(18)
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Converting to using ,
(19)
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(20)
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(21)
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which could have been derived more simply by
(22)
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