## 6 Discussion  
The main goal of this paper is to evaluate the quality of surface reconstruction from point clouds using four point-based methods, especially when the inputs are extremely sparse point clouds from freehand 3D ultrasound. The BPA method is intuitive and robust. If given sufficiently dense input data, it is capable of reconstructing a satisfactory surface from the original manifold. However, the BPA method failed to produce a closed surface from sparse point clouds, as observed in Experiment 2 and Experiment 3. In Experiment 2, the BPA method was the worst method when more than six contours were used. The Power Crust method produced watertight surfaces when the input sample was sufficiently dense. However, when the input data were extremely sparse, as in Experiment 2 and Experiment 3 (_i.e._ two contours), it failed completely. The Poisson method also created satisfactory surfaces when given sufficiently dense input data. However, the Poisson method required oriented normal vectors at the input points to define the vector field. When the input data were extremely sparse, the vector field could not be correctly defined, and the Poisson method failed to reconstruct the correct surfaces. Hence, the Poisson method was the worst of the four methods in all experiments when reconstructed from extremely sparse contours. In all cases, the surface reconstructed by the VSR method showed the greatest visual resemblance to the original surface, compared with the other three methods. In most cases, the metrics also indicated that the quantitative differences between the VSR reconstructed surface and the original surface were the smallest. Even given extremely sparse input data, the VSR method could reconstruct a close approximation to the original surface. Therefore, the strengths of the VSR method were as follows: (1) the VSR method could reconstruct satisfactory surfaces from sparse data; (2) the surfaces reconstructed by the VSR method were smooth; (3) although the VSR and Poisson methods were both implicit-function-based methods, the VSR method did not need to compute the oriented normal vectors at the input point clouds to define a vector field. However, the second strength of VSR was also a weakness. Because the reconstructed surfaces by VSR method were smooth, the reconstruction of objects with high curvature might fail if the surface was sampled too sparsely, _i.e._, with not enough contours. The VSR method has another limitation concerning contradictory inputs as shown by [PERSON] _et al._[PERSON] _et al._ (2016), _i.e._, contours that define different surfaces, although they should be located on the same surface. This issue was demonstrated in Figure 14. In Figure 14, we used three same contours of the kidney as in the first row of Figure 6, except that the horizontal contour was edited manually to form contradictory contours. In row (a) of Figure 14, we could see that the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the edited version of the horizontal contour (red) did not intersect with the vertical contour (blue). The quantitative assessments of the quality of surface reconstruction from the contradictory contours are shown in Table 5.  
_Appl. Sci._**2016**, \(6\), 114  
Figure 14: Surface reconstruction of contradictory contours of the kidney. (**a**) Contradictory contour: the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the edited version of the horizontal contour (red) did not intersect with the vertical contour (blue); (**b**) surface reconstruction from the original contours; (**c**) surface reconstruction from the contradictory contours. For row b and c, from left to right: the original surface of the kidney, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. (**d**) Detail of the surface reconstruction near the intersection of the contradictory contours.