Reconstruction from Range Data: Constraint Specification

In the previous sections we have discussed our implicit formulation and the radial basis used to construct the implicit function. Each radial basis that make up the function is centered at a constraint point. We now discuss how we obtain constraint points from real and synthetic range data. When range data is acquired using cameras, the camera position and direction provide additional information that can be used for surface reconstruction. Together, this information allows us to define the free space as shown below. The free space is where the object does not exist in 3D space [Curl 96]. Each camera can see a subset of surface locations which are recorded on the image plane of the camera. The rays from the camera to these surface locations do not intersect the object anywhere else except on the surface. Free space occurs all along the rays that produce an image. We can use this a priori knowledge about the object surface locations and the free space to define constraints which lie on or outside of the object. In the figure below, blue stars indicate points on the object, and red dashes indicate points outside. The exterior constraints are those locations where we want our implicit function to be negative, and the surface constraints are where the implicit function should evaluate to zero. Note that we do not have any knowledge about the behaviour of the object behind the surface locations with respect to one camera position and direction. However, if we have images and surface locations from viewpoints surrounding the object, we can completely define the existence space - surface, exterior, and interior - of the object. Suppose we have range images and camera positions from viewpoints on a sphere around the object. The interior of the object is known by virtue of surface enclosure.

We uniformly sample the range data in the manner described above. Note that we do not use all the surface points in our reconstruction algorithm. If all surface points were used as constraints, the system matrix used to solve for the weights of the radial basis function would become too large. In addition, overfitting (or overshoots) may occur if all the data were used.