Artifacts in the inversion of the broken ray transform in the plane

Figure 1.  Left: a general broken ray, where $ l_1 $ and $ l_2 $ are related by a diffeomorphism. Right: a broken ray in the reflection case

Figure 2.  The small neighborhood $ U_k $ and $ (x_k, \xi^k) $, for $ k = 1, 2 $

Figure 3.  A sketch of a broken ray reflected on a smooth boundary and the notation

Figure 4.  Two broken rays intersect when $ \alpha_2 $ increases as $ \alpha_1 $ increases

Figure 5.  In (a) and (b), the bold part is the intersection region where the incoming rays hit there and reflect with conjugate points

Figure 6.  Artifacts and caustics. Form left to right: $ f $, $ {{B}}^*\Lambda {{B}} f $, and caustics caused by reflected light

Figure 7.  Local reconstruction by Landweber iteration

Figure 8.  Inside a circular mirror, a sequence of broken rays and conjugate points on them

Figure 9.  Reconstruction of $ f_1 $ and $ f_2 $ from global data, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error

Figure 10.  Reconstruction from global data for Modified Shepp-Logan phantom $ f_3 $, where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $ is the relative error

Figure 11.  The error plot for the reconstruction of $ f_1, f_2, f_3 $ in order. The first two has the same range of color bar

Figure 12.  Reconstruction of two coherent states. Left to right: true $ f $, the envelopes (caused by trajectories that carry singularities and are reflected only once), $ f^{(100)} $ (where $ e = \frac{\|f - f^{(100)}\|_2}{\|f\|_2} $), the error

Figure 13.  Another case of radial singularities. Left to right: true $ f $, reconstruction $ f^{(100)} $, error for $ f $ with radial singularities after 100 iterations. The relative error $ e $ is defined as before

Figure 14.  Left to right: true $ f $, backprojection $ f^{(1)} $, $ f^{(100)} $