## 1 Introduction  
Three-dimensional (3D) ultrasound imaging is used extensively as a diagnostic and guidance tool for cardiac procedures. Three-dimensional echocardiography allows for the acquisition of volumetric data of the heart, which can be analysed in any plane. The current standard of care for mitral valve procedures includes diagnostic imaging with a 3D transesophageal echocardiography (TEE) probe [1, 2]. This method of imaging provides a clear view of the mitral valve, and including color Doppler allows the cardiologist/cardiac surgeon to identify the mitral valve pathology. While echocardiography is a powerful imaging technique, it nevertheless has some major limitations. The field of view is limited when using 3D transducers, which can limit the range of anatomy that can be easily viewed, and structures further away from the image probe may suffer from poor spatial resolution, while thin structures parallel to the ultrasound beam suffer from signal dropout artefacts. Ultrasound compounding, or mosaicing, has been proposed by several groups to address limitations of 3D ultrasound and improve imaging capability. By registering and blending together adjacent acquisitions from different poses, we can expand the field of view and address the issue of signal dropout, producing higher quality images with greater information for the clinician. Several image compounding techniques have been proposed to register a set of ultrasound volumes, all of which demonstrate improved image quality [3] and provide an avenue for combining common cardiac ultrasound views into a single volume with reduced noise, reduced speckle, and fewer signal dropout artefacts. Researchers have demonstrated 3D ultrasound compounding techniques with applications in cardiac, fetal and breast imaging [4; 5; 6]. Common across all compounding methods are two critical steps: global registration of all volumes and blending the overlapping regions of the registered volumes to generate the resulting image [3]. Evaluation of registration frameworks has identified three main approaches, consisting of sequential alignment, semi-simultaneous and fully simultaneous registration [7]. Using a sequential alignment approach, each acquisition is registered to the next; however, this technique suffers from drift and error accumulation. The semi-simultaneous approach uses each volume as the moving volume in turn and every other volume as fixed for multiple cycles until convergence is met. This approach balances computational complexity, as only the parameters of a single transform need to be considered in optimization; furthermore, it has global alignment because every volume is considered at every step. The final approach, fully simultaneous group-wise registration, optimizes the transformation parameters of all volumes simultaneously, applying a loss function as the sum of pairwise losses. This approach is optimal for registration quality; however, it is limited by computational complexity due to the number of parameters that need to be optimized. For mitral valve imaging using standard en-face views, the limitations of 3D ultrasound result in the structures beyond the valve, including the chordae tendineae, papillary muscles, and left-ventricular outflow tract being difficult to identify. While imaging from a different position (e.g., transgastric view) can capture these structures, the field of view limits the utility of these images as at these positions the entire mitral valve apparatus cannot be captured in a single volume. Currently, choosing the treatment plan that provides the greatest benefit to the patient is one of the biggest clinical challenges for cardiologists and cardiac surgeons [8]. Determining optimal neochord length is one of the main issues that cardiac surgeons must address in MV repair procedures [9], and it is particularly challenging to define this length due to a general lack of accurate anatomical information from standard diagnostic imaging, which includes only the en-face view of the mitral valve.