# Ultrasound Localization Microscopy in Liquid Metal Flows
## 3 Characterization of the Spatial Resolution  
The spatial resolution of ULM is characterized by how well the center position of a single scatterer can be isolated. This also equals the localization uncertainty \(\sigma_{l}\). In general, the localization uncertainty at a specific position \(P(x,y)\) consists of a systematic uncertainty due to the finite size and ununiform shape of the particle profiles and image distortions, as well as a random uncertainty due to noise:  
\[\sigma_{l}=\sqrt{\sigma_{l,\text{syst.}}^{2}+\sigma_{l,\text{rand.}}^{2}}. \tag{1}\]  
In accordance with the guide for the expression of uncertainty in measurement (GUM), the measurement uncertainty can be derived from a calibration measurement, in which the random uncertainty can be estimated using the standard deviation. The systematic uncertainty can be estimated using the average deviation from a reference. For ULM, this consists of two components in the \(x\) and \(y\) directions, through which the absolute localization deviation can be calculated using the root mean square:  
\[\Delta l=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}. \tag{2}\]  
This is an unknown systematic deviation, which has to be considered with a uniform probability distribution as a worst case scenario. Therefore, this is translated into a Gaussian uncertainty by:  
\[\sigma_{l,\text{syst.}}=\frac{\Delta l}{\sqrt{3}}. \tag{3}\]  
As a reference, a set-up with a single scatterer was built [38]. It consisted of a fiber with a 50 \(\upmu\)m diameter, which acted as scatterer. It was placed alongside the elevation direction of the ultrasound array inside a container. The container was designed to provide the same acoustic environment as was used in the demonstration experiment in Section 4 with an acrylic wall and GaInSn as the working fluid, as shown in Figure 3a. The fiber was moved through the container by two linear stages (LTM 80 and OWIS GmbH). The stages had a position deviation of 2.5 \(\upmu\)m and a yaw angle deviation of 10 \(\upmu\)m, which was negligible compared to the deviations of the ultrasound images. With this set-up, the point spread function was measured for several points on a defined grid inside the container. The center position was estimated using the same algorithm as ULM. The uncertainty \(\sigma_{l}\) was calculated according to Equation (1) and is depicted in Figure 3b. The overall uncertainty was derived using the local average of all \(N\) calibration points \(P_{N}\) using:  
\[\overline{\sigma}_{l}=\frac{1}{N}\sum_{n}^{N}\sigma_{l}(P_{n})=188\,\upmu\text {m}=0.28\;\lambda. \tag{4}\]  
For comparison, the derived uncertainty was expressed relative to the wavelength. It can be seen in Figure 3b that the uncertainty was not dependent on the penetration depth, as long as a sufficient signal-to-noise ratio, or rather peak-to-background ratio, was present and the diverging spatial resolution was sufficient to separate particles. This was also an issue in shadowed regions or away from the directivity of the phased array. We were able to extract calibration points at a distance of 100 \(\upmu\)m from the side wall, which was supposedly within the region of the viscous boundary layer in the experiment (\(<\)0.5 cm).