The Broken Ray Transform On The Square

Abstract. We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for C20 perturbations of the con-stant unit weight. Given an open subset E of the boundary, we measure the attenuation of all broken rays starting and ending at E with the standard optical reflection rule. Using the analytic microlocal approach of Frigyik, Ste-fanov, and Uhlmann 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 −1 plus a smoothing term with C∞0 Schwartz kernel, which leads to the desired result. 1