Deterministic versus stochastic level-set regularization in nonlinear phase contrast tomography

Figures & data

Figure 1. Experimental set-up in propagation-based phase contrast tomography with a single propagation-distance showing the X-ray beam, the rotated coordinate system $(x_{\mathit{\theta }},y_{\mathit{\theta }},z)$ for a rotation angle $\mathit{\theta }$, the sample and the detector.

Figure 2. Ground truth $\mathit{\beta }$ and $\mathit{\delta }$ maps for the object ($O_2$).

Figure 3. Ground truth $\mathit{\delta }$ maps for the object ($O_3$).

Figure 4. Ground truth $\mathit{\delta }$ maps for the object ($O_3$).

Figure 5. Horizontal section of the initial $\mathit{\beta }$ and $\mathit{\delta }$ maps for the object ($0_2$).

Figure 6. Horizontal section of the final error map for $\mathit{\beta }$ and $\mathit{\delta }$ for a PPSNR of 48 dB.

Figure 7. Evolution of the data term with the iterations for the noise levels 48 dB (plain line) and 30 dB (dotted line).

Table 1. Values of the $\mathit{\delta }$ and $\mathit{\mu }$ values for the materials in the object, at 24 keV X-rays from url http://henke.lbl.gov/optical$\_$constants.

Material	$\mathit{\delta }({10}^{-7})$	$\mathit{\mu }(m^{-1})$
PMMA	4.628	41.2
Al	9.396	502.6
Oil	3.60	25.66

Figure 8. Evolution of the RMSE on $\mathit{\delta }$ with the iterations for the noise levels 30 dB (dotted line) and 48 dB (plain line).

Figure 9. Evolution of the RMSE on $\mathit{\beta }$ with the iterations for the noise levels 30 dB (dotted line) and 48 dB (plain line).

Figure 10. Evolution of the RMSE on $\mathit{\beta }$ with the iterations for the noise level 48 dB (dotted line) and $N_{\mathit{\theta }}$=180 (blue line), $N_{\mathit{\theta }}$=125 (black line) and $N_{\mathit{\theta }}$=75 (dotted line) projection angles for the reconstruction with the stochastic level-set algorithm.

Figure 11. Evolution of the RMSE on $\mathit{\beta }$ with the iterations for the noise level 48 dB (dotted line) and $N_{\mathit{\theta }}$=180 (blue line), $N_{\mathit{\theta }}$=125 (black line) and $N_{\mathit{\theta }}$=75 (dotted line) projection angles for the reconstruction with the stochastic level-set algorithm.

Figure 12. Evolution of the data term without topological derivative for the noise level 24 dB (black line) and with the topological derivative for the PPSNR 18 dB (blue line) and 24 dB (dotted line).

Table 2. RMSE for $\mathit{\beta }$ and $\mathit{\delta }$ for the level-set (LS) and Tikhonov regularization approaches.

	RMSE $\mathit{\beta }$, LS	RMSE $\mathit{\delta }$,LS	RMSE $\mathit{\beta }$, Tikhonov	RMSE $\mathit{\delta }$, Tikhonov
PPSNR=30 dB	0.06	0.25	0.14	0.35
PPSNR=48 dB	0.05	0.17	0.1	0.25

Figure 13. Horizontal section of the difference image between the ground truth and the reconstructed $\mathit{\beta }$ maps with the stochastic and the deterministic level-set algorithms for the noise level 24 dB.

Figure 14. Evolution of the data term for the deterministic level-set algorithm for 24 dB (black line), for the intermittent stochastic level-set algorithm for 24 dB (blue line) and for the intermittent stochastic level-set algorithm for 18 dB (dotted line).

Figure 15. Evolution of the RMSE for $\mathit{\beta }$ and $\mathit{\delta }$ for the deterministic level-set algorithm for 24 dB (black line), for the intermittent stochastic level-set algorithm for 24 dB (blue line) and for the intermittent stochastic level-set algorithm for 18 dB (dotted line).

Table 3. RMSE for $\mathit{\beta }$ and $\mathit{\delta }$ for deterministic level-set and stochastic level-set.

	RMSE $\mathit{\beta }$, LS	RMSE $\mathit{\delta }$,LS	RMSE $\mathit{\beta }$, stochastic LS	RMSE $\mathit{\delta }$, stochastic LS
PPSNR=18 dB	0.28	0.85	0.15	0.55
PPSNR=24 dB	0.22	0.80	0.06	0.26

Figure 16. Horizontal section of the difference image between the ground truth and the reconstructed $\mathit{\beta }$ maps with the stochastic and the deterministic level-set algorithms for the noise level 24 dB.

Figure 17. Evolution of the RMSE on $\mathit{\beta }$ with the iterations for the noise level 48 dB (dotted line) and $N_{\mathit{\theta }}$=180 (blue line), $N_{\mathit{\theta }}$=125 (black line) and $N_{\mathit{\theta }}$=75 (dotted line) projection angles for the reconstruction with the stochastic level-set algorithm.

Figure 18. Evolution of the data term without topological derivative for the noise level 24 dB (black line) and with the topological derivative for the PPSNR 18 dB (blue line) and 24 dB (dotted line).

Figure 19. Horizontal section of the reconstructed $\mathit{\beta }$ maps for the intermittent stochastic/deterministic algorithm with and without topological derivative for the noise level 24 dB.

Figure 20. Horizontal section of the reconstructed $\mathit{\delta }$ maps with the topological derivative for the noise level 24 dB and upper view of this section.