# Implementation of Constrained Swept Synthetic Aperture Using a Mechanical Fixture
## 2 Materials and Methods
### Image Reconstruction and Analysis  
Reference images (i.e., no sweep) were formed using the plane wave imaging data with standard plane wave focusing methods [40]. Swept synthetic aperture images were formed using standard diverging wave focusing methods with varying transmit and receive aperture positions [12] according to either the translation stage (or simulated) position or the estimated motion (from plane wave transmissions). In both cases, diverging wave data were resampled to retain frames at roughly 0.5 mm spatial intervals, reducing the computation required for image formation and avoiding artifacts due to the varying temporal profiles of the manual sweeps. It was assumed that all motion occurred along the lateral axis within the imaging plane and that there was no rotation of the transducer. For the interleaved acquisition sequence, the motion estimates from the plane wave frames were interpolated to provide estimates of positions for both the left and right diverging wave transmissions. The focused images from the left and right transmissions were coherently combined, and frames were weighted with a Tukey window to reduce high spatial frequency components of the synthesized transmit aperture present in the SSA image compared to a fully sampled large array [12]. Image analysis was performed using standard methods for resolution and lesion detectability measurement. The lateral point spread function was displayed using a slice through the peak of the point target with the highest value normalized to 0 dB. Lesion detectability was quantified using the generalized contrast-to-noise ratio (gCNR):  
\[\text{gCNR}=1-\int_{-\infty}^{\infty}\min\{f(x)g(x)\}\,dx, \tag{4}\]  
where \(f\) and \(g\) are the normalized histograms of two image regions [41]. Histogram matching was applied to images across different conditions to visually match the speckle background appearance (the mean and variance) despite variations in the peak value across images [42]. speed of sound used in estimation (including image formation for speckle tracking) was introduced, varying it from 1240 to 1840 m/s compared to the correct 1540 m/s. This error led to slightly varying depths of estimation and effective kernel sizes but had largely no effect on the channel correlation approach. The speckle tracking approach, due to the distortion of the images produced, showed average positive and negative estimation errors at larger distances up to approximately 30 \(\upmu\)m. This bias is likely the result of the stretched speckle pattern used in correlation. Finally, the impact of kernel choice on the estimates was studied. The depth of estimation and the axial kernel extent were varied for both methods and the root mean square error of the estimates were calculated (all showed roughly linear profiles as in previous cases). Some increase in error at larger depths and reduction of error with a larger axial kernel were observed. Greater sensitivity and larger errors were found for speckle tracking, but both methods had average errors under 5 \(\upmu\)m. While better performance was observed  
Figure 5: Demonstration of speckle tracking and channel correlation for displacement estimation using individual estimates on pairs of frames. Estimates performed with a 10 \(\times\) 10 mm kernel. (**Left**) Displacement estimates and error over a short distance, less than one element pitch. Error bars represent 10 speckle realizations. A \(\pm\)2.5 mm search region was used. (**Right**) Estimates and error over larger distances, up to 5 mm. A \(\pm\)7.5 mm search region was used. Errors depend on the choice of sampling, kernels and subsample estimation method. Figure 6: Challenges to and optimization of estimates in simulation. (**Left**) Additive channel noise reduces accuracy compared to Figure 5. (**Middle**) Errors in assumed speed of sound affects speckle tracking accuracy but not channel correlation (true speed of sound 1540 m/s). (**Right**) Speckle tracking is more sensitive than channel correlation to choice of kernel location and depth.