Image Processing Reference

In-Depth Information

CHAPTER

4

PerformingTemplate-Based

Reconstruction

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

⎣

⎦
,

Mx

=

0
,
where
x

=

(4.1)

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

errors.

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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