# A High-Efficiency Super-Resolution Reconstruction Method for Ultrasound Microvascular Imaging
## 2 Materials and Methods
### Trajectory Tracking  
Visualization (rendering) of the processed sequence data involves the creation of a new super-resolution image based on the coordinates of the localized microbubbles. The original scatter plot method [14] is a typical all-or-none tracking method. When a microbubble appears at a certain place, it is simply recorded and reflected with the constant gray level in the reconstruction result. It cannot display the density of microbubbles and the exact shape of the blood flow vessels. Thus, it has a negative impact on the imaging resolution. We solved this problem based on the histograms.  
Histograms are often used to estimate the density of data by counting the number of observations that fall into each bin [14]. In our case, a two-dimensional histogram of microbubble positions can be created with the bin size corresponding to the pixel size of the final super-resolution image. Thus, for every localized microbubble (group), the bin value (i.e., the image brightness) at the corresponding microbubble positions is incremented by one. For a random sample of size \(h\), the classic histogram takes the value  
\[\hat{f}(x)=\frac{\text{bin count}}{h}. \tag{8}\]  
The histogram visualization optionally supports \"jittering\" of the microbubbles between frames. When enabled, a random number drawn from the normal distribution, with a standard deviation equal to the computed (or user-specified) localization uncertainty, is added to the coordinates of every microbubble position before creating the histogram. This step is applied several times and all generated histograms are averaged together. As the number of jitters increases, the final image approaches the result of the Gaussian rendering, whose result is often considered as the golden standard in the SMLM. It can reflect the density of microbubbles and depict an accurate vascular boundary. For a small number of jitters, the histogram visualization is much faster than the Gaussian rendering but the resulting images may appear noisy. To solve this problem, we introduced an improved visualization algorithm.  
This visualization algorithm uses a density estimation approach based on ASH [16]. The averaged shifted histogram or ASH is a nonparametric probability density estimator derived from a collection of histograms [28], as shown in Figure 2. In the one-dimensional case, this method works by averaging \(n\) histograms with the same bin width \(w\), but with the origin of each histogram shifted by \(\frac{w}{n}\) from the previous histogram. In the multidimensional case, there are \(n^{d}\) multidimensional histograms averaged in total, i.e., for \(n\) shifts in each of \(d\) dimensions. In our implementation, the width of the histogram bin is determined as \(w=na\), where \(a\) is the pixel size of the super-resolution image. The number of shiftsin the lateral and axial directions can be specified independently. A simple calculation shows that the ordinary ASH is given by  
\[\hat{f}(x)=\frac{1}{n}\sum_{i=1-n}^{n-1}\frac{(n-|i|)\text{bin count}_{i}}{h}. \tag{9}\]  
Theoretically, the time complexity of the proposed method is \(O(N)\), where \(N\) is the number of microbubbles to visualize. This complexity level is close to the conventional scatter plot approach. However, the real speed of our visualization method is also influenced by the number of histograms to average, which makes it a bit slower than the scatter plot. From the perspective of the reconstruction performance, the proposed method based on the ASH approach provides similar results as Gaussian rendering with a constant localization uncertainty. However, the ASH approach is orders of magnitude faster than Gaussian rendering.