## 2 Materials and Methods
### Simulation Study
CNR was calculated based on the formula provided in [50], as the absolute value of the difference between the average object and background pixels is divided by the square root of the sum of the variances of the object and background pixels. Areas of greater signal with visually well-defined boundaries of the imaging target were chosen to be object pixels. Background area was chosen near the boundary of the reconstruction where the object was known to not exist. From least-dense to most-dense grid-size, the CNR values were found to be: 1.07, 1.38, 1.44, and 1.45. From the result of negligible difference in CNR with further increasing grid-size and simplicity in our simulated environment (no scatterers) as well as maximal use of our computational power, our use of \(750\times 750\times 750\) voxels grid-size is justified. For the simulation study, an imaging target consisting of three thin orthogonal rectangular legs with lengths of 1 cm and widths of 2 mm each, was defined in k-Wave simulation [47] as shown in Figure 1a. The simulated object is similar to the imaging target that we used in the experiments. The initial pressure was set to 10 Pa and sampling frequency was set to 50 MHz, same as what was used in the experiment. Image reconstruction is a process through which transducer signals are transformed into an image. Our reconstruction volume for both simulation and experimental data was \(3\times 3\times 3\) cm\({}^{3}\) and made up of \(400\times 400\times 400\) voxels with 75 \(\upmu\)m width in every direction. Different beamformers treat transducer signals differently to generate an image [51]; some of them are simple to implement such as delay-and-sum (DAS), some are very accurate and exact such as universal back projection (UBP), and some are more robust to noise and artifacts such as delay-multiply-and-sum (DMAS). DAS is the most commonly used beamforming method. It is popular due to its simplicity and the fact that it can be used to reconstruct both US and PA images which is ideal for commercial products [52; 53; 54]. DAS is a blind beamforming method which treats all the detected pressure waves the same way. Mathematically, DAS is the simplified version of the UBP algorithm at which only the first term is used to backproject (see equation 20 in [55]); UBP algorithm is the exact time-domain back-projection method, as illustrated in [55]. The different frequencies accompanying UBP in Figure 1c are the cut-off frequencies of the filter used in UBP. This filter is utilized to mitigate the effects of the amplified noise by the ramp filter existing in UBP formula. In DAS, sidelobes of the radiation pattern of the US transducers affect the image quality. To suppress the effects of sidelobes and off-axis signals, coherence factor (CF) (defined as the ratio of the coherent and incoherent summation of the recorded signals [56]) can be used along with DAS, we call that DAS-CF. CF is an index to show focusing quality and works based on the ratio of mainlobe energy to the total energy. While DAS, UBP and filtered back projection follow a general delay and sum procedure, delay-multiply-and-sum (DMAS) is another beamforming method outperforming DAS in terms of resolution and contrast that use a correlation process between the received signals [56]. It should be noticed that all these algorithms are based on an ideal point transducer element with an infinite angular view. Figure 1c demonstrates the performance of different reconstruction algorithms including DAS, DAS-CF, DMAS, and UBP with multiple cutoff frequencies in terms of CNR of the simulated imaging target consisting of three thin orthogonal rectangular legs with 50 transducer elements. DMAS [56] algorithm was chosen in this study for image reconstruction because it provided images with the highest CNR. DMAS algorithm has several variations, we implemented the fast DMAS following the method described in [57].