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

  • * Corresponding author: Yang Zhang

    * Corresponding author: Yang Zhang

The first author is supported by NSF grant DMS-1600327

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  • We study the integral transform over a general family of broken rays in $ \mathbb{R}^2 $. One example of the broken rays is the family of rays reflected from a curved boundary once. There is a natural notion of conjugate points for broken rays. If there are conjugate points, we show that the singularities conormal to the broken rays cannot be recovered from local data and therefore artifacts arise in the reconstruction. As for global data, more singularities might be recoverable. We apply these conclusions to two examples, the V-line transform and the parallel ray transform. In each example, a detailed discussion of the local and global recovery of singularities is given and we perform numerical experiments to illustrate the results.

    Mathematics Subject Classification: Primary: 35R30, 44A12; Secondary: 65R32.

    Citation:

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  • 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)} $

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