## 2 Materials and Methods
### Image Blending  
After the volumes are registered, re-sampling and compounding are performed at each cardiac phase to construct a 3D + time compounded volume with a wide field of view. Before compounding, the volumes are re-sampled to a common grid using cubic b-spline interpolation to ensure that there is complete voxel overlap between volumes so that the blending step (a weighted average of all overlapping volumes at each voxel location) can be performed. The output grid is determined by the extent of all overlapping input images, using isotropic spacing equal to the minimum spacing in any dimension in any input image. We evaluated multiple weighting strategies including the voxel-wise maximum, average and weighted average using two different weighting schemes. The voxel weighting methods we compared were the scaled distance from the image probe and a combination of distance from the probe and a feature detector based on the monogenic signal [14]. The local phase measure from the monogenic signal was previously demonstrated as part of an application-specific loss function for ultrasound compounding [15]. For our purposes, we used the oriented symmetry measure, which returns values from \(-1\) to \(+1\), with positive values being associated with features of interest and negative values being associated with background noise, as visualized in Figure 3. We implemented the 3D extension of the monogenic signal in Python. The distance function assigns higher weights to voxels closer to the image probe, with values ranging from 25 near the probe down to 0 with an inverse-square law dropoff, which we used to approximate the reduction in resolution further away from a phased-array ultrasound probe. For a single voxel position \(p\) in a source volume \(i\), the expression for the distance from the image probe weighting \(d_{i}\) is  
\[d_{i}(p)=25\frac{\left(10-\frac{\frac{10\times\|p-o_{i}\|}{\max \limits_{q\in V_{i}}\left(\|q-o_{i}\|\right)}}{10^{2}}\right.}{}, \tag{1}\]where \(o_{i}\) is the position of the probe origin in volume \(i\). The oriented symmetry weighting \(S_{i}\) is given as  
\[S_{i}(p)= \begin{cases}10\times s_{i}(p)&\text{if }s_{i}(p)<0\\ 25\times s_{i}(p)&\text{if }s_{i}(p)\geq 0\end{cases} \quad, \tag{2}\]  
where \(s_{i}\) is the value of the oriented symmetry measure from the monogenic signal at position \(p\) in volume \(i\). The combined weighting function \(W_{i}\) of distance and oriented symmetry is  
\[W_{i}(p)=\max(0.5,S_{i}(p)+d_{i}(p))\;, \tag{3}\]  
giving the sum of the two weights with a minimum value of 0.5. Both the distance weight \(d_{i}\) and symmetry weight \(S_{i}\) have a maximum value of 25, contributing equally to the overall voxel weighting. Finally, the expression for the final weighted average output intensity \(F(p)\) is  
\[F(p)=\frac{\sum_{V_{i}|V_{i}(p)>0}\big{(}V_{i}(p)\times w_{i}(p)\big{)}}{\sum_{ V_{i}|V_{i}(p)>0}\big{(}w_{i}(p)\big{)}}\;, \tag{4}\]  
where \(V_{i}(p)\) is the intensity value in volume \(i\) at position \(p\), and \(w_{i}\) is either the distance weighting alone or the combined distance and oriented symmetry weighting. A visual comparison of the results of applying the different blending approaches can be seen in Figure 4. The voxel-wise maximum approach produces a volume with very sharp features but passes through any imaging artefacts and highlights registration errors. Simple averaging produces a smoother image but lacks the definition of smaller features and boundary edges appear blurred. Distance weighted averaging further improves image quality, taking advantage of the increased line density nearer to the probe and the corresponding increase in spatial resolution, but small features and edge boundaries are still blurred and lack contrast to the background. The incorporation of the monogenic signal-based feature detector into the weighting function helps to reduce this blurring and makes the structures of interest more distinct without amplifying imaging and registration artefacts.