# Ultrasound Localization Microscopy in Liquid Metal Flows
## 2 Ultrasound Localization Microscopy
### Nonlinear Beamforming  
To apply ULM, particles must be isolated from each other and from other disturbances. Conventionally, delay-and-sum (DAS) beamforming is applied to the raw ultrasound data in order to perform the re-localization of the echo information and, therefore, to image the scattering particles [41]. An example image is presented in Figure 2a. DAS produces too high a concentration of scattering particles and too high a speckle intensity. This is a result of the earlier mentioned scattering particle concentration that is not directly adjustable. The spatial resolution, especially for penetration depths of \(>\)50 mm, is not sufficient to isolate and track particles. To solve this problem, we proposed to apply a nonlinear beamformer. An approach that was based on coherence weighting was introduced by [PERSON] et al. [42]. The so-called filtered delay-multiply-and-sum beamformer (FDMAS) suppresses speckles and enhances the lateral resolution and coherent structures are emphasized by means of nonlinear inter-channel correlation. This method has been further enhanced by other research groups, such as the double-stage FDMAS beamformer [43]. However, these methods lack a dynamic scalability to particle densities and require a high computational effort to process that amount of data.  
[PERSON] et al. proposed the method of \(p\)th root compression delay-and-sum beamforming (\(p\)-DAS) [44]. The ultrasound signals of each tilt are compressed by their \(p\)th root, DAS beamformed and backward raised to the power of \(p\). Afterward, each tilt is added. This method is a simplified version of FDMAS beamforming, but it produces comparable results, has a much higher computational efficiency and, most importantly, the grade of the nonlinear scaling is arbitrarily adjustable by the value \(p\). This allows for the adaptive  
Figure 1: (**a**) The profile of the 1.5 D phased array transducer with inner and outer elements, a center frequency of 4 MHz and 70 % bandwidth; (**b**) the plane wave scanning with seven tilted plane waves of up to \(\pm 30^{\circ}\) at a 1350 Hz _PRF_; (**c**) the receiving of backscattered echoes by the lower 96 inner elements.  
scaling of the beamforming algorithm to the actual particle concentration. Hence, a \(p\)-DAS beamformer was implemented in OpenCL with GPU support in this study. To analyze its properties, the beamforming was conducted with several values of \(p\). From Figure 1(a), it can be seen that the particle profile sharpened as the value of \(p\) increased. This also accounted for the ringing of the acrylic glass window that was visible for up to 20 mm. On the other hand, a loss of information in the corner regions and higher penetration depths could be seen for values of \(p>2\). To find the optimal value, particle tracking for each set of B-Modes was performed, as described in the next section. For \(p=2\), a compromise between the linked trajectories, the average trajectory length and the trajectories in the corner regions and higher penetration depths was found as the optimal value, as shown in Figure 1(b). Furthermore, an apodization method that was based on two steps was applied during the beamforming. First, the sensitivity of each transducer element to each beamformed pixel was derived through the simulation and then normalized. A second apodization was calculated depending on the plane wave tilt angle. Pixels in the direct range were considered as \"1\" and pixels that were out of range were weighted with a Gaussian window that was based on the distance. Clutter filtering was applied using the coherent subtraction of the average beamformed time signals and background removal in the B-Mode data via PtvPy.