Reconstruction from Range Data: Interpolation vs. Approximation

Scattered data interpolation is the process of estimating previously unknown data values using neighboring data values that are known. In the case of surface reconstruction, the surface passes exactly through the known data points. In between the data points, the location of the surface is estimated by interpolation from the known points. Data interpolation is best used when the data values are precise. In the case of vision-based data, however, there is some uncertainty in the validity of the data points. Using data interpolation to construct the surface is no longer ideal because the surface may not actually pass exactly through the given data points. If the uncertainty of the data points is known, a surface that better represents the data would pass close to the data points rather than through them. Constructing such a surface is known as data approximation. Many vision-based techniques for capturing 3D surface points have an associated error distribution or confidence range for the data points.

We can allow the surface to pass close to, but not necessarily through, the known data points by relaxing the constraints of the linear system. We use the formulation discussed in [Giro 93]. A simple derivation is presented therein which shows that a summation of weighted radial basis functions as given in equation above is the solution to minimizing a cost functional of the following form:

The surface is much smoother, and the limbs are not as fused. For example, the head and arms are well separated. However, the feet and tail remain fused, and the tail is not well attached to the body. The reconstruction may be described as blobby. This characteristic is evident in many implicit surface reconstructions, such as the blobby Gaussian reconstructions. In the next section, we describe how we reduce blobbiness and enhance detail in our reconstructions by using a radial basis function that achieves multiple orders of smoothness.