Summary: Compton scattering tomography (CST) is an emerging two-dimensional imaging concept exploiting the scattered radiation while a specimen is illuminated by a gamma source. In the last decade, the study of first-order scattered photons has led to model the measured flux by Radon transforms over circles. Such transforms were shown to be invertible due to their strong relation with the classical Radon transform, i.e., the line integrals. In this paper, we study the smoothness properties and the regularization issues for such transforms and then build suitable reconstruction methods based on the approximate inverse, which facilitates the extraction of features (for instance the contours). However, these previously derived models neglect physical factors such as the attenuation of the beam. This leads us to incorporate a weight in the forward transforms similarly to the attenuated Radon transform in single-photon emission computed tomography (SPECT). In this case, no analytical inversion of the corresponding weighted transforms is known. Therefore, we develop a strategy to treat the weighted case via the contour extraction. Finally we establish a new modality on CST in which the \(\gamma\)-ray source is fixed and the scattered radiation is measured by high-energy sensitive detectors located on an annulus. The resulting system has the advantage of being rotation free and yet circumventing theoretically limited angle artifacts. All of these results are testified by simulation results.