Image Reconstruction Through Regularization by Envelope Guided Conjugate Gradients

29 Figure 12: Tail strategy for 10 million registered photon pairs. all images in enhanced display. 28 Figure 11: Tail strategy for 10 million registered photon pairs. The result of the quadratic case was used as start for the other cases; all images in enhanced display. top row: quadratic, log for = 2 13 , logcosh for = 2 5 ; bottom row: Huber for, = 2 8 , multiquadric for = 2 8 , rational for = 2 17. Again the convex, asymptotically linear penalty terms removed most high frequency noise and were less sensitive to the precise choice of. The most insensitive formula was the multiquadric penalty (8), which produced excellent results over a wide range of 's; see Figure 12. A hitherto unexplored semirational formula, (27) behaved very similar, but avoids the calculation of square roots. 27 Figure 10: Same as previous gure, but with corners deened by the diierence quotient bringing the penalty close to the absolute value penalty slow down the pcg method, resulting in a degradation of the image when, as in our experiments, the number of iterations is xed. In Figure 8, we show, for the Huber penalty (9), the dependence of the results of the corner strategy on. The same is done for the tail strategy in Figure 9. Moreover, to show the innuence of the formula used to compute the corner, we did these calculations also using version 2, i.e., equation (22) as a measure of bentness, giving the results displayed in Figure 10. Here (but not in general) version 2 produced the most pleasing results of all our methods. Case 2: 10 million photon pairs The high photon case is, in comparison, much easier, and produces reconstructed images of good quality. Best results were obtained using 32 iterations of the tail strategy with the quadratic penalty function (4) to compute a good starting point, used with the tail strategy for the nonlinear penalties (another 32 iterations). Figure 11 shows the results for optimal choices of the scaling parameter , found by trying all powers of 2 in a reasonable range. 26 Figure 9: Sensitivity for Huber penalty (enhanced display): tail strategy for Because of the noise in the phantom, the randomness in choosing the direction of the annihilating photons and the inaccuracy resulting from the discretization used to set up the matrix, the noise level only barely allows the reconstruction …