The proposed method is based on the concept that the occluded area is filled in with the same type of surface which fits the visible area. To achieve this we first need to estimate the boundaries of the occluded region. Consequently, for each endpoint we estimate the direction of its continuation within the occluded area. The method proposed to estimate the occluded boundary is based on the boundary good continuation constraint. In particular, the endpoint prolongation is performed according the Gestal principle of good continuation and proximity in the order: linearity, co-circularity , closure . In this way we are able to bound the surface which is going to be reconstructed.
We can distinguish three possible cases based on the relation between visible boundaries lying in the proximity of an intersection with the occluded surface.
case A) coincident boundaries;
case B) convergent boundaries;
case C) divergent or parallel boundaries.
This cases are represented in figure 11.
Figure 11: Proposed reconstruction rules
In all the three cases the hidden boundary is the continuation of the visible one. However, in the first case, the boundary extensions across the occluded area are coincident to the line connecting endpoints; in the second case, the boundary extensions over the occluded area intersect within the occluded area (e.g.figure 13); in the third case, the boundary extensions over the occluded area do not intersect (e.g.figure 15). The last case needs an architectural constraint to limit boundary extensions. In the work presented here, we assume that the extension does not pass through walls or the floor. This may overextend some surfaces; the alternative would be more conservative and not reconstruct these cases - awaiting instead additional data.
Hypothetical surfaces can then be created by extending the visible surface regions into the identified bounded area. The method to estimate the occluded surface is based on the surface good continuation constraint. That is, we hypothesise that the surface does not change its shape within the occluded area. The reconstruction is performed in the 3D space in order to achieve a higher accuracy. Given an occluding pixel and an occluded surface we intersect the ray from the sensor through the occluding point with the occluded surface. As this ray overlaps the optical ray of the laser scanning beam, the reconstructed pixel is placed in a position that could actually have been sensed by the sensor. For planes there is usually one intersection. For cylinder and spheres we usually find two intersections. Figure 12 summarizes the recovery method. Figures 13 - 15 show some examples of reconstruction.
Figure 12: Main steps of the reconstruction process (case of convergent boundaries)
Figure 13: Reconstruction of a surface with rules of case B. Occluded surface (left), extended boundaries (center) and reconstructed surface (right).
Figure 14: Reconstruction of a surface with rules of case C. Occluded surface (left), extended boundaries (center) and reconstructed surface (right).
Figure 15: Examples of reconstructed objects: right-side pyramid of scenario 1 (top-left) and right-side pyramid and little house of scenario2 (top-right). Pyramid pyramid before reconstruction (center-left) and pyramid after reconstruction (bottom-left), little house and pyramid before reconstruction (center-right) and little house and pyramid after reconstruction (bottom-right).