Reconstruction from Range Data: Volumetric Regularization

The surface reconstruction technique that we present is an extension of the variational implicit surfaces of [Turk 99] (Variational Implicit Surfaces main page). This approach is based on the calculus of variation and is similar to surface regularization in that it defines an energy functional to be minimized. Unlike surface regularization, however, the energy functional is defined in R³ rather than R². Hence, the functional does not act on the space of surfaces, but rather, on the space of 3D functions. This distinction from surface regularization is critical in that the reconstructed surface is a level- set of the energy-minimizing 3D implicit function, but the surface does not, itself, minimize an energy functional. In [Turk 99], the functional that was minimized is the thin-plate energy in 3D. Turk and O'Brien argue that the level-set of a function which minimizes such an energy is also a smoothly varying function. The following figure shows a side view of the reconstruction of a toy dinosaur, generated by minimizing the functional used by Turk and O'Brien. Notice that the limbs of the dinosaur are fused, and that the tail has become a disconnected component.

The space carved data and surface constraint points used to generate the implicit function are discussed in Constraint Specification and Results. Next, we discuss how the reconstruction can be improved by data approximation using volumetric regularization.