In 1981 Cormack studied the Radon transform defined on a family of curves and showed their inversion through their circular harmonic expansions. In this paper we propose to extend this property to a more general family of curves which are defined by a nonlinear-first-order differential equation. In particular we focus on a subclass of this family for which the singular value decomposition shown by Cormack in 1964 can be generalized. Applications based on this subclass clearly appear since the Radon transform defined on three kinds of its curves may serve to model three different Compton scattering tomography modalities. Simulation results using analytical inversions and Chebyshev/Zernike expansions for these three Radon transform show the strength and the validation of proposed inversion methods.
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