## 1 Introduction
### Surface Reconstruction for Freehand 3D Ultrasound Imaging  
To avoid a voxel-based intermediate step, _i.e._, volume reconstruction, we can reconstruct the 3D surface of the VOI directly from contours (cross-sections) segmented from the original 2D B-scans in a prerequisite step. Because the contours are non-parallel, a method that does not assume parallel contours is required. Such reconstruction methods are scarce. Moreover, because manual segmentation is still the only universally reliable method for ultrasound data in clinical applications [9], it is one of the most time-consuming processes during a tumor ablation surgery. It would be preferable to reconstruct surfaces from only a few ultrasound contours, which would be valuable in a clinical setting as it reduces manual work and reduces time spent on manual segmentation. Therefore, only a small number of B-scans are recorded and manually segmented to guarantee short segmentation time in clinical applications. However, a small number of contours provide only sparse input data to surface reconstruction. Thus, a surface reconstruction method that can handle both non-parallel and sparse contours is necessary. These two requirements (_i.e._, non-parallel and sparse) make the surface reconstruction of freehand 3D US a challenging task. Candidate surface reconstruction methods for freehand 3D US usually fall into two groups: contour-based methods, and point-based methods. Contour-Based methods try to construct a triangle mesh directly from the segmented ultrasound contours and are usually ineffective when applied on non-parallel contours. Point-based methods first convert all contours into a point cloud and then construct triangle meshes from this point cloud. Hence, point-based methods can address non-parallel contours, but they are still ineffective when applied on sparse input data. [PERSON] _et al._[PERSON] _et al._ (2016) proposed an algorithm for volume estimation based on polyhedral approximation, and several researchers ([PERSON] _et al._, 2016) made use of this method for surface reconstruction. Two contours were sampled as a set of lines joining boundary points, and then the volume was calculated from these points by forming tetrahedra with a common central point on each plane. The surface was formed by triangulating each boundary point in turn between the contours. This would only be accurate for fairly similar contours; the contours are not allowed to intersect each other. [PERSON] _et al._[PERSON] _et al._ (2016) used a CAD package to triangulate between cross sections of saphenous vein phantoms. The cross sections were nearly parallel or nearly circular, and consequently the resulting surface was conical or cylindrical. [PERSON] _et al._[PERSON] _et al._ (2016) proposed a method to reconstruct non-parallel contours. It was performed directly in the triangle-mesh domain and required dense contours for input. It resulted in disconnected geometry when the projections of the curves were not overlapping between planes; thus, it is incapable of reconstructing sparse data in our case. Most of the above surface reconstruction methods could not address arbitrarily oriented contours. The orientation of each contour was typically restricted so that the contours were nearly parallel or did not intersect mutually. In fact, methods that can handle mutually intersected contours are uncommon. Although the method by [PERSON] _et al._[PERSON] _et al._ (2016) could handle non-parallel contours, it could not tackle sparse data. After the conversion of contours to a 3D point cloud, the orientation of the contours becomes irrelevant and we can disregard the non-parallel contours. Currently, there are many surface reconstruction methods that can be used to reconstruct a surface from point clouds ([PERSON] _et al._, 2012; [PERSON] _et al._, 2013; [PERSON] _et al._, 2014; [PERSON] _et al._, 2015; [PERSON] _et al._, 2016; [PERSON] _et al._, 2017; [PERSON] _et al._, 2018; [PERSON] _et al._, 2019; [PERSON] _et al._, 2018).