Appendix The Line - Surface Intersection Algorithm

pointing towards the sensor and a i is a binary number depending on the intersection of l and Q i , and The reason behind taking a weighted sum of the intersection points is that the sensor measurement of the position of a surface point is less reliable if the local surface is facing away from the sensor. The weight w i is a reflection of this heuristic. In general, the sensor direction information is available from the range image sensor setup and calibration. For a Cartesian range image (depth map) without sensor information, we can simply take. If there are more than one intersections between l and the object surface, we can perform a clustering to separate the intersections into groups corresponding to each real intersection, and choose the closest cluster to compute the above weighted sum. In practice, we have not had to use such a clustering algorithm. This is because the result of the intersection algorithm depends on an initial point p (which is the vertex point in this paper). When the vertices are far from the object surface, can be from any clusters. But we are less concerned with the actual location of the intersection at the time. As the mesh grows and the vertex get closer to the surface, almost all the intersections computed are from the closest cluster, since it is the closest local minimum when considering the intersection process as a minimization. For robustness purpose, we have implemented a simple filtering scheme to eliminate gross outliers in the intersections based on the distribution of the intersections found. 1 if the intersection exists , 0 if there is no intersection ,    = n s n s 0 0 1 , , () τ = q i 28 When we have more than one range images, the intersection of the line with all the range images are computed. Let , be the set of range images and l be the line in consideration. The intersection of l and the surface can be defined as the weighted sum of all the intersections: (A.1) Figure 10. Intersecting a line with a digital surface illustrated in a 2-D case. Directed Line l Tangents Q P Start point True intersection Approximate Intersection Projections p q 0 q 1 0 Y X n p