## 2 Materials and Methods  
### Image Registration  
Following local REB approval, we adapted standard diagnostic TEE acquisition protocols to include multiple transgastric views in addition to the standard mid-esophageal view. Volumes were acquired using ECG gating to match the cardiac phase. Our imaging protocol requires a minimum of one mid-esophageal acquisition and four transgastric acquisitions with approximately 80% spatial overlap or more between adjacent volumes for successful registration of the acquisitions, an example of which can be seen in Figure 1. The transgastric acquisitions should begin at the mitral valve and proceed along the ventricle to the papillary muscles. Compounding is then accomplished by aligning over-sampled data through automated image registration, re-sampling the aligned volumes into a consistent output space, and generating the output image through the voxel-wise blending of the overlapping volumes. The compounding workflow is shown in Figure 2.  
Figure 1: Compounded TEE volume of mitral valve with individual acquisitions outlined.  
Figure 2: Workflow of TEE compounding for the mitral valve.  
Performing image registration of multiple volumes can be achieved using pairwise, fully simultaneous, or semi-simultaneous approaches [13]. We implemented both the semi-simultaneous and fully simultaneous approaches described by [PERSON] et al. [13]. This is done as an extension of our initial work on this method, in which only the semi-simultaneous approach was used [12]. We first perform rigid registration at the end-systole between all volumes using fully simultaneous group-wise registration with the sum of pairwise normalized cross-correlations as the loss function. This gives us a rough global alignment of the volumes and is not dependent on input order. Then, two cycles of semi-simultaneous registration are performed, which we found achieved better agreement between volumes than the fully simultaneous registration alone. Finally, we utilize non-rigid registration in the semi-simultaneous framework at each frame in the acquisitions to account for the slight deviations at different points of the cardiac cycle due to imperfect synchronization. For both semi-simultaneous steps, the loss function used was the sum of normalized cross-correlation between the moving volume and each fixed volume. At each step, optimization was performed using adaptive stochastic gradient descent in a multi-resolution registration framework with four resolution levels, each smoothing the image by a factor of 2 over the previous one. We implemented this approach using the Elastix toolkit ([[http://elastix.isi.uu.nl/](http://elastix.isi.uu.nl/)]([http://elastix.isi.uu.nl/](http://elastix.isi.uu.nl/)), accessed on10 October 2021) on the 3D Slicer platform ([[https://www.slicer.org/](https://www.slicer.org/)]([https://www.slicer.org/](https://www.slicer.org/)), accessed on 10 October 2021). This open-source implementation of our work is available at [[https://github.com/pcarnah/CardiacVolumeStitching](https://github.com/pcarnah/CardiacVolumeStitching)]([https://github.com/pcarnah/CardiacVolumeStitching](https://github.com/pcarnah/CardiacVolumeStitching)), accessed on 10 October 2021.