The Radon transform (RT) on straight lines deals as mathematical foundation for many tomographic modalities (e.g. Xray scanner, Positron Emission Tomography), using only primary radiation. In this paper, we consider a new RT defined on a pair of half-lines forming a letter V, arising from the modeling a two-dimensional emission imaging process by Compton scattered gamma rays. We establish its analytic inverse, which is shown to support the feasibility of the reconstruction of a two-dimensional image from scattered radiation collected on a one-dimensional collimated camera. Moreover, a filtered back-projection inversion method is also constructed. Its main advantages are algorithmic efficiency and computational rapidity. We present numerical simulations to illustrate the working. To sum up, the V-line RT leads not only to a new imaging principle, but also to a new concept of detector with high energetic resolution capable to collect the scattered radiation.