Cormack-type inversion of exponential Radon transform (Q2709878)

The exponential Radon transform of an imaginary parameter \(\mu=i\beta\) is defined as NEWLINE\[NEWLINE F(s,\theta,\beta) = \int f(s\omega +t\overline{\omega}) e^{i\beta t} dt, \qquad \omega=(\cos \theta,\sin \theta). NEWLINE\]NEWLINE The author uses a modified form of an explicit inversion formula of the attenuated Radon transform given by \textit{O. Tretiak} and \textit{C. Metz} [SIAM J. Appl. Math. 39, 341--354 (1980; Zbl 0459.44003)] to solve a generalized Cormack equation. The latter is a Volterra type equation which can be deduced from incorporating the Fourier expansions of the function \(f\) and its transform \(F\) into the definition of the Radon transform. The author then uses methods from complex analysis and some facts for Chebyshev polynomials of the second kind to solve the integral equation. The resulting solutions are known to be numerically unstable, but the reconstruction algorithm can be modified to circumvent this problem.