The algorithm described in the previous section is designed to reconstruct the observed portions of the surface. Unseen portions of the surface will appear as holes in the reconstruction. While this result is an accurate representation of the known surface, the holes are esthetically unsatisfying and can present a stumbling block to follow-on algorithms that expect continuous meshes. In , for example, the authors describe a method for parameterizing patches that entails generating evenly spaced grid lines by walking across the edges of a mesh. Gaps in the mesh prevent the algorithm from creating a fair parameterization. As another example, rapid prototyping technologies such as stereolithography typically require a ``watertight'' model in order to construct a solid replica .
One option for filling holes is to operate on the reconstructed mesh. If the regions of the mesh near each hole are very nearly planar, then this approach works well. However, holes in the meshes can be (and frequently are) highly non-planar and may even require connections between unconnected components. Instead, we offer a hole filling approach that operates on our volume, which contains more information than the reconstructed mesh.
The key to our algorithm lies in classifying all points in the volume as being in one of three states: unseen, empty, or near the surface. Holes in the surface are indicated by frontiers between unseen regions and empty regions (see Figure 6). Surfaces placed at these frontiers offer a plausible way to plug these holes (dotted in Figure 6). Obtaining this classification and generating these hole fillers leads to a straightforward extension of the algorithm described in the previous section:
Initialize the voxel space to the ``unseen'' state.
Update the voxels near the surface as described in the previous section. As before, these voxels take on continuous signed distance and weight values.
Follow the lines of sight back from the observed surface and mark the corresponding voxels as ``empty''. We refer to this step as space carving.
Perform an isosurface extraction at the zero-crossing of the signed distance function. Additionally, extract a surface between regions seen to be empty and regions that remain unseen.
Figure 6: Volumetric grid with space carving and hole filling. (a) The regions in front of the surface are seen as empty, regions in the vicinity of the surface ramp through the zero-crossing, while regions behind remain unseen. The green (dashed) segments are the isosurfaces generated near the observed surface, while the red (dotted) segments are hole fillers, generated by tessellating over the transition from empty to unseen. In (b), we identify the three extremal voxel states with their corresponding function values.
In practice, we represent the unseen and empty states using the function and weight fields stored on the voxel lattice. We represent the unseen state with the function values , and the empty state with the function values , , as shown in Figure 6b. The key advantage of this representation is that we can use the same isosurface extraction algorithm we used in the previous section without the restriction on interpolating voxels of zero weight. This extraction finds both the signed distance and hole fill isosurfaces and connects them naturally where they meet, i.e., at the corners in Figure 6a where the dotted red line meets the dashed green line. Note that the triangles that arise from interpolations across voxels of zero weight are distinct from the others: they are hole fillers. We take advantage of this distinction when smoothing surfaces as described below.
Figure 6 illustrates the method for a single range image, and provides a diagram for the three-state classification scheme. The hole filler isosurfaces are ``false'' in that they are not representative of the observed surface, but they do derive from observed data. In particular, they correspond to a boundary that confines where the surface could plausibly exist. In practice, we find that many of these hole filler surfaces are generated in crevices that are hard for the sensor to reach.
Because the transition between unseen and empty is discontinuous and hole fill triangles are generated as an isosurface between these binary states, with no smooth transition, we generally observe aliasing artifacts in these areas. These artifacts can be eliminated by prefiltering the transition region before sampling on the voxel lattice using straightforward methods such as analytic filtering or super-sampling and averaging down. In practice, we have obtained satisfactory results by applying another technique: post-filtering the mesh after reconstruction using weighted averages of nearest vertex neighbors as described in . The effect of this filtering step is to blur the hole fill surface. Since we know which triangles correspond to hole fillers, we need only concentrate the surface filtering on the these portions of the mesh. This localized filtering preserves the detail in the observed surface reconstruction. To achieve a smooth blend between filtered hole fill vertices and the neighboring ``real'' surface, we allow the filter weights to extend beyond and taper off into the vicinity of the hole fill boundaries.
We have just seen how ``space carving'' is a useful operation: it tells us much about the structure of free space, allowing us to fill holes in an intelligent way. However, our algorithm only carves back from observed surfaces. There are numerous situations where more carving would be useful. For example, the interior walls of a hollow cylinder may elude digitization, but by seeing through the hollow portion of the cylinder to a surface placed behind it, we can better approximate its geometry. We can extend the carving paradigm to cover these situations by placing such a backdrop behind the surfaces being scanned. By placing the backdrop outside of the voxel grid, we utilize it purely for carving space without introducing its geometry into the model.