Image Processing Reference
In-Depth Information
As discussed in Chapter 3 , given N c point correspondences between a reference image in which the
3D shape is known and an input image, recovering the new shape in that image amounts to solving
the linear system of Eq. 3.16 . We write it here again as
v 1
v N v
0 , where x
v i contains the 3D coordinates of the i th vertex of the N v -vertex triangulated mesh representing the
surface, and M is a matrix that depends on the coordinates of correspondences in the input image
and on the camera internal parameters. A solution of this system defines a surface such that 3D
feature points that project at specific locations in the reference image reproject at matching locations
in the input image. Solving this system in the least-squares sense therefore yields surfaces for which
the overall reprojection error is small.
Note, however, that this is not strictly equivalent to minimizing the reprojection error because
computing the actual reprojection of a 3D point on the image plane would involve a division by the
depth factors d i of Eq. 3.15 , thus yielding nonlinear terms. In essence, solving this linear system
is equivalent to performing a Direct Linear Transformation (DLT) Hartley and Zisserman [ 2000 ],
which gives a different weight to each correspondence according to its distance to the camera and
therefore potentially reduces accuracy. Even more problematically, M is a 2 N c × 3 N v matrix with
at least N v singular values that are very small, as shown in Fig. 3.3 . Because the system is so ill-
conditioned, many different shapes can produce very similar projections, and even small imprecisions
in the point coordinates, and consequently in the coefficients of M , can lead to large reconstruction
In this chapter, we will review some of the approaches that have been proposed to overcome
these ambiguities and increase accuracy either by enforcing temporal consistency across images in
video sequences, or by enforcing additional geometric constraints, such as smoothness and preser-
vation of geodesic distances across the surface.
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