# Image Human Thorax Using Ultrasound Traveltime Tomography with Supervised Descent Method
## 4 Numerical Experiments
### Numerical Experiments
#### 4.3.1 Comparison with Traditional Gradient-Based Method  
The performance of SDM inversion is compared with traditional Gauss-Newton inversion. The ground truth is the thorax model shown in Figure 2. The results of the two methods are shown in Figure 9. We conduct inversion based on SPR method traveltime data and k-wave simulation traveltime data. Figure 8: The normalized model misfit versus the No. of iterations in the training stage. Figure 7: Example of the training model. Red Block: high-velocity block (2200 m/s). Blue region: low-velocity (1500 m/s) background. For inversion based on SPR method data, see Figure 9a,b. SDM finishes the inversion in 49.9 s. Although the Gauss-Newton method takes 102.9 s. SDM takes only half of the time the traditional method takes. In addition, from the results we find that the acoustic velocity distribution computed by SDM has fewer artifacts around the thorax and more precise tissue boundaries for ribs. The edges of lungs, ribs, and spine are clear. Heart and lungs are hard to distinguish because their acoustic velocity is close. However, both methods cannot correctly reconstruct the acoustic velocity of marrow because few paths traverse this region. According to Fermat's Principle, the wave will propagate along the path that takes the shortest time. The acoustic velocity of the marrow is significantly smaller than the spine around it, which means path would like to propagate in the spine rather than the marrow and result in low sensitivity of the marrow region. This example verifies the feasibility of imaging human thorax using ultrasound traveltime tomography with SDM. It also shows the strong generalization ability of SDM because little prior information on the shape of the thorax is added to the training set. The training model contains no structure information of thorax but only a rectangular high-velocity block and a homogeneous background. For inversion based on k-Wave simulation traveltime data, see Figure 9c,d, the results are similar to the inversion based on SPR method data, which further verifies the modeling and inversion algorithm. The traveltime data used in the following numerical experiment is all computed by SRP method with \(n_{nodes}=5\). A better initial model would help the inversion. The new initial model is obtained by smoothing the true model with a Gauss filter, see Figure 10a. The result of SDM inversion is shown in Figure 10b. Comparing the new result with the result shown in Figure 9a, we found that the area of the heart, marrow, and bones is better reconstructed. In the future, the initial model can be derived from CT or other imaging modalities. Figure 9: The inversion results of SDM inversion and Gauss–Newton inversion based on different traveltime data. (**a**) SDM inversion based on SPR method traveltime data, (**b**) Gauss–Newton inversion based on SPR method traveltime data, (**c**) SDM inversion based on k-Wave simulation traveltime data, (**d**) Gauss–Newton inversion based on k-Wave simulation traveltime data. Colorbar: 1200–2200 m/s. As a result (**a**), SDM takes 49.9 s to finish the inversion. As a result (**b**), the Gauss–Newton method takes 102.9 s.  
The inversion results cannot be improved by adding more transducers nor using a higher-resolution grid. We increase the number of transducers and the new array with 40 transducers is shown in Figure 11. We also prepare a new grid whose size is 57 \(\times\) 86. We conduct reconstruction on the same normal thorax model with previous array and new grid, new array and previous grid, and new array and new grid. New training sets are constructed for each task using the previous method. The inversion results are shown in Figure 11. Compared with the inversion results shown in Figure 9, no significant improvement can be seen. The reason the results have not improved might be that this traveltime tomography problem is too ill-posed. The current method cannot obtain the model out of the local minima. In addition, more transducers and higher-resolution grid will increase the time of inversion.