Fourier Inversion of the Attenuated X-Ray Transform

A variably attenuated x-ray transform is shown to be invertible via an integral formula for the inversion of the exponential x-ray transform. The attenuation must be known and constant in a convex set containing the unknown emitter. However the attenuation can be otherwise arbitrary. If $\mu $ denotes the attenuation constant of the exponential x-ray transform then the integral formula computes the Fourier transform of the emitter on all of $R^n $ from the values of the Fourier transform on the set $A^\mu = \{ {\sigma + i\mu \omega \in C^n |\omega \in S^{n - 1} ,\sigma \bot \omega } \}$. Of course F. Natterer [Numer. Math., 32 (1979), pp. 431–438] showed that the values of the Fourier transform of the emitter can be obtained from the Fourier transform of the exponential x-ray transform. In essence however the basic method is analytic continuation from the set $A^\mu $. A consequence of the integral formula is a uniqueness theorem for attenuated x-ray transforms of the type considered here: if the transforms of two objects agree at infinitely many directions, then the objects are the same.