The Broken Ray Transform on the Square

We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for (formula presented) perturbations of a vanishing absorption parameter (formula presented). Given an open subset (formula presented) of the boundary, we measure the attenuation of all broken rays starting and ending at with the standard optical reflection rule applied to (formula presented). Using the analytic microlocal approach of Frigyik et al. for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays. Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order (formula presented), which leads to the desired result.