Compton scattering tomography (CST) is an alternative imaging process which reconstructs, in a two-dimensional slice, the electron density of an object by collecting radiation emitted from an external source and scattered throughout this object. The collected data at specific scattering energies appears essentially as the integral of the electron density on definite families of arcs of circles. Reconstruction of the unknown electron density is achieved by the inversion of the corresponding circular-arcs Radon transforms (CART). We review two existing CST modalities, their corresponding CART and establish their numerical inversion algorithms in the formalism of the so-called circular harmonic decomposition (CHD) for a function. The quality of the reconstructed images is illustrated by numerical simulations on test phantoms. Comparison with standard tomography performances demonstrates the efficiency and interest of this inversion method via CHD in imaging science such as biomedical imaging and non-destructive industrial testing.