Image Processing Reference
In-Depth Information
When the surface can be assumed to be lit by a distant light source, these additional con-
straints can be obtained from shading information around corresponding points to constrain the
intensities of surrounding surface patches in the input and reference images to be related through
a Lambertian reflectance model
Moreno-Noguer
et al.
[
2009
]. The shading information yields a
system of cubic equations on the weights
β
i
. Since there are many such cubic constraints, they
are solved by extended linearization
Courtois
et al.
[
2000
]. While extended linearization does not
guarantee an exact solution of the constraints, it is more practical than other techniques such as
Groebner bases, which cannot handle that many equations. In
Moreno-Noguer
et al.
[
2010
], the
approach of
Moreno-Noguer
et al.
[
2009
] was extended to allow the use of more generic shading
models. Instead of writing the vector
c
as a weighted sum of singular vectors, the fact that solving the
system of Eq.
4.7
is ill-conditioned was addressed as follows. The least-squares solution of Eq.
4.7
can be expressed as
=
(
B
B
)
−
1
B
b
,
c
(4.10)
=
=−
where
B
Mx
0
. Recall from Chapter
3
that the coefficients of the matrix
M
of Eq.
4.1
are ultimately derived from point correspondences, which contain some amount of uncertainty.
Assuming the image coordinates of these point correspondences to be normally distributed around
their true values, the covariance matrix for the distribution of
c
can be expressed as
MS
and
b
J
β
u
J
β
,
c
=
(4.11)
where
∂(
B
B
)
−
1
∂
u
+
(
B
B
)
−
1
∂
B
b
∂
u
B
b
J
β
=
(4.12)
is the Jacobian of
(
B
B
)
−
1
B
b
with respect to the 2D correspondence coordinates, which can
be computed analytically.
u
is a diagonal covariance matrix representing the distribution of the
2D point coordinates around their true locations. The algorithm then samples the possible shapes
around the mean shape given by the least-squares solution of Eq.
4.10
according to the covariance
matrix
c
of Eq.
4.11
. An additional source of information, such as motion or shading, is then used
to evaluate the quality of the samples and resample the solution space more finely around the most
promising ones. When using shading, a single light source whose position is unknown and may be
either distant or nearby is assumed. For each sample
c
, the light source position is estimated so that
the image synthesized by shading the corresponding surface is as similar as possible to the original
one. To speedup convergence, the samples for which this optimization yields the smallest residuals
are favored in the resampling step. The fact that the algorithm provides a reliable way to generate 3D
shape hypotheses makes the use of nearby light-sources practical. Without these hypotheses, such
illumination conditions are difficult to handle, since they involve solving a non-convex minimization
problem. This is all the more true since the lighting parameters are initially unknown and must be
estimated from the images. In other words, while the lighting model used in
Moreno-Noguer
et al.
[
2010
] is still too simple to be truly general, the approach could, in theory at least, handle much
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