4.2.1. Homogeneous Media

Soft biological tissue is made up mostly of water and therefore can be viewed approximately as a homogeneous medium. The sound speed in water is 1480 m/s, which can be taken as the speed in tissue. The specific value of the constant is irrelevant to the performance of the algorithms, as the value can always be adjusted by choosing different units. What matters is that the speed must be uniform everywhere. Therefore, we simply take the speed as 1 in our numerical test, i.e., we set *c*(*r*) = 1. Ultrasound signals are generated using the forward model (1). The signals are then exploited in TR, ATR, Landweber iteration, and DL algorithms, respectively, to reconstruct the original image. The input of the neural network is the output of the first iteration of ATR, which is denoted as ATR-.

Based on an independent testing set of 200 CT image patches, we investigated these reconstruction algorithms with 0%, 10%, 20%, 30%, 40% noise added to the ultrasound signals respectively, and reported the box plots of the image quality that was evaluated by PSNR and SSIM in reference to the ground-truth simulated initial source in Figure 3. The experimental results show the DL algorithm has at least two advantages over the others. On the one hand, it is relatively more stable as the noise level increases. Its PSNR outperforms the others at all noise levels, with the only exception of the ideal zero-noise case where ATR provides the best reconstruction. On the other hand, the DL algorithm is time-saving. Although training the neural network takes large amount of time, its output is almost immediate once the training process is accomplished. In contrast, the ATR and Landweber iteration takes several minutes to achieve an image of satisfactory quality.

A few reconstructed images randomly selected from the testing set are displayed in Figure 4. Each column corresponds to the reconstruction using the algorithm labeled at the bottom. The ATR- column consists of images from the first iteration of ATR, which are the inputs to the neural network. The IS column consists of the ground-truth initial source images used to generate the ultrasound signals by the forward model. Among all the algorithms, TR gives the worst result, which can be

expected as it does not resolve reflected artifacts. ATR tends to lose detailed information at a high noise level (see the last row). Landweber performs better in resolving noise as it has regularization effect, which helps remove the high-frequency content of the image. The DL algorithm exhibits superior reconstruction in general.

**Figure 3.** The box plots of the image quality evaluated by PSNR and SSIM on the testing set in a homogeneous medium. From top row to bottom row: noise increases from 0 to 0.4. Diamonds () indicate outliers.

**Figure 4.** Reconstruction in a homogeneous medium.
