Image Processing Reference
Figure 6.15: Using local quadratic models Fayad et al. [ 2010 ]. Top and middle rows: Reconstruction
results on a real sequence. Bottom row: Comparison with the reconstructions of Va ro l et al. [ 2009 ],
depicted by a mesh with green vertices. Note that the Fayad et al. [ 2010 ] reconstruction is more accurate
both because it was obtained from a whole video sequences and because it allows for deformations of the
patches. Courtesy of A. Del Bue.
this homography, the rotation, translation and patch normal can be computed up to a global scale
ambiguity and a twofold normal ambiguity Malis and Vargas [ 2007 ], Zhang and Hanson [ 1995 ].
In other words, each 2D point in a patch can be assigned one of several 3D interpretations. These
ambiguities are resolved by using points that belong to multiple patches, such as those shown in
blue in Fig. 6.14 (a), to enforce consistency. This is done by guaranteeing that they receive the
same interpretation, no matter what patch is used to reconstruct them. First, normal orientations
are chosen consistently across patches. Then, the scale of each patch is optimized such that the
distance between the 3D reconstruction of the same 2D point from different patches is minimized.
One strength of this approach is that each step only involves solving a linear system, which can be
done in closed-form. Furthermore, as opposed to classical NRSFM techniques, this method allows
reconstruction from only two images depicting two different shapes of the same surface.
One drawback of the Va ro l et al. [ 2009 ] method is that it requires sufficiently many corre-
spondences per local patch to reliably estimate the homography. As a consequence, relatively large
portions of the surface are asssumed to be planar, which limits the range of global deformation.