On the visibility of point clouds

Is it possible to determine the visible subset of points directly from a given point cloud? Interestingly, in [7] it was shown that this is indeed the case - despite the fact that points cannot occlude each other, this task can be performed without surface reconstruction or normal estimation. The operator is very simple - it first transforms the points to a new domain and then constructs the convex hull in that domain. Points that lie on the convex hull of the transformed set of points are the images of the visible points. This operator found numerous applications in computer vision, including face reconstruction, keypoint detection, finding the best viewpoints, reduction of points, and many more. The current paper addresses a fundamental question: What properties should a transformation function satisfy, in order to be utilized in this operator? We show that three such properties are sufficient: the sign of the function, monotonicity, and a condition regarding the function's parameter. The correctness of an algorithm that satisfies these three properties is proved. Finally, we show an interesting application of the operator - assignment of visibility-confidence score. This feature is missing from previous approaches, where a binary yes/no visibility is determined. This score can be utilized in various applications, we illustrate its use in view-dependent curvature estimation.