4.2.2. Heterogeneous Media

Heterogeneity occurs when the object to be imaged has a complex composition. An example is the transcranial PAT where the sound speed in the skull is 3200 m/s in contrast to 1480 m/s in the water. The speed has jump singularities at the interface of the two constituents. Such singularities can be mingled with those from the initial pressure and appear in the reconstructed image as additional artifacts, casting more challenges for the reconstruction of the initial pressure. We implemented the algorithms in the domain [−1, 1] × [−1, 1] and chose the distribution of the sound speed as

$$\mathcal{L} = 1 - 0.2\sin(2\pi \mathbf{x}) + 0.15\cos(\pi y) + \chi\_{(x-0.5)^2 + (y-0.5)^2 < 0.01\prime} \tag{8}$$

where *χ*(*<sup>x</sup>*−0.5)<sup>2</sup>+(*y*−0.5)<sup>2</sup><0.01 is a function that equals 1 on the disk {(*<sup>x</sup>*, *y*) : (*x* − 0.5)<sup>2</sup> + (*y* − 0.5)<sup>2</sup> < 0.01} and 0 otherwise. The reason for such choice of *c* is as follows. The constant 1 models the speed in soft tissue. The smooth term −0.2 sin(<sup>2</sup>*πx*) + 0.15 cos(*πy*) is added to mimic the slight variation of the sound speed in distinct types of tissue. The non-smooth term *χ*(*<sup>x</sup>*−0.5)<sup>2</sup>+(*y*−0.5)<sup>2</sup><0.01 captures the jumps between different materials such as soft tissue and bones; see Figure 5 for the distribution of the sound speed *c*.

**Figure 5.** Distribution of the sound speed.

With the variable sound speed, we computed the PSNR and SSIM of the reconstructed images with 0%, 10%, 20%, 30%, 40% noise added to the ultrasound signals respectively, as is shown in Figure 6. The DL algorithm still demonstrates the optimal overall performance. Besides being more stable and time-saving, the DL algorithm also yields the least outliers. Here an outlier is a number in the dataset that is less than *Q*1 − 1.5 × (*Q*3 − *Q*1) or greater than *Q*3 + 1.5 × (*Q*3 − *Q*1), where *Q*1 is the lower quartile, and *Q*3 is the upper quartile.

Some reconstructed images are randomly selected from the testing set again to illustrate the difference of the algorithms; see Figure 7. ATR still suffers from the high noise level, but can resolve the jumps in the speed. Landweber iteration, however, introduces additional artifacts on the top right of the reconstructed image where the sound speed jumps as shown in Figure 5. This is partly due to the limited number of iterations, and it is observed that artifact becomes weaker if the number of iterations is increased. The DL reconstruction resolves simultaneously the high noise and jumps in the speed. It remains visually the closest to the true initial source.
