Comparison with osculating circle methods

Most previous works on surface reconstruction from occluding contours [Vai 92,Cip 90,Sze 93,Sea 95,Jos 95] make use of osculating circle methods. In such methods, a rim point P is reconstructed by estimating the osculating circle at P and its epipolar correspondents on the previous and the next rim. To this aim, the viewing lines are projected onto a plane and a circle tangent to these three directions is computed. In [Cip 90,Sze 93,Jos 95] one of the epipolar planes is used and in [Vai 92,Sea 95] the radial plane is used. These approaches implicitly suppose that the camera motion is linear [Cip 90,Sze 93,Jos 95] or that the observed surface is locally cylindrical [Vai 92,Sea 95], and assume that the surface remains on the same side of the tangents in the projection plane.

Our approach avoids such constraint, no assumption has to be made on the camera motion, the local surface shape or the side of the tangents on which the surface lies. In addition, it appears that for non-linear camera motions, part of the rim can not be reconstructed by use of an osculating circle method. See for example figure 9 where the three successive camera positions C_-1, C, and C₁ are not aligned.

  

Figure 9: Rims under non-linear camera motion.

In this situation, there is part of the rim to be reconstructed where the point P and its epipolar correspondents are not organised as expected for the estimation of the osculating circle. This corresponds to points P where a₁ and a_-1 have the same sign or, in other words, where the projection of P_-1 and P₁ onto the viewing line at P are on the same side of P. In such cases, the epipolar curves defined at P can not be considered as part of the same curve and methods based on the osculating circle lead to false solutions since they approximate both epipolar curves with a single planar curve: a circle (see for example figure 10).

  

Figure 10: A possible situation for epipolar curves in the case of a non-linear camera motion. The point to be reconstructed is the point P while the position estimated by an osculating circle method is $\tilde{P}$.

This shows that epipolar curves have to be estimated as two different curves and that a method based on the osculating circle allows only a partial reconstruction of the rim in the case of non-linear camera motions.
Our work gives a more general solution to the reconstruction problem. Except for the special cases where the camera motion is in the viewing direction, depth can be computed at any rim point and for any camera motion. Moreover, it gives a unique solution to the reconstruction problem without knowing on which side of the tangent lines the object lies.