Image Processing Reference

In-Depth Information

perspective case, it can easily be checked that the last column of the matrix can be expressed as a

linear combination of the others, which then are linearly independent. Thus, matrix
M
m
of Eq.
3.15

has still full rank minus 1. This reflects the well-known scale ambiguity in monocular vision.

Representing the problem as in Eq.
3.15
was convenient to discuss the rank of the matrix.

However, in practice, we want to recover the vertex coordinates but are not interested in having the

d
i
as unknowns. We therefore eliminate them by rewriting Eq.
3.15
as

⎡

⎤

v
1

...

v
N
v

⎣

⎦
=

M

0
,

(3.16)

with

⎡

⎤

a
1
T
1

b
1
T
1

c
1
T
1

0

...

...

⎣

⎦

...

...

...

...

...

...

u
i
A
3

v
i
A
3

,

0

b
j
T
j

c
j
T
j

d
j
T
j

0

...

M

=

,
and
T
i

=

A
2
×
3
−

...

...

...

...

...

...

a
l
T
l

0

c
l
T
l

0

e
l
T
l

...

...

...

...

...

...

...

where
A
3
represents the last row of matrix
A
and
A
2
×
3
its first two rows. Since
M
has the same rank

as matrix
M
m
by construction, the previous and following results are valid for both representations

of the problem.

3.3.2.4 Effective Rank

In the previous paragraph, we showed that
M
has at most full rank minus one. However, this does

not tell the whole story: In general, it is ill-conditioned and many of its singular values are so small

that, in practice, it should be treated as a matrix of even lower rank. To illustrate this point, we

projected randomly sampled points on the facets of the synthetic 88-vertices mesh of Fig.
3.3
(a)

using a known camera model. We then computed the singular values of
M
, which we plot in Fig.
3.3

(b). Even though only one of these values is exactly zero, we can see that they drop down drastically

after the first 2
N
v
=
176. This shows that, even though the matrix may have full rank minus 1, the

solution of the linear system would be very sensitive to noise. Therefore, in a real situation, we would

actually be closer to having
N
v
ambiguities. In Fig.
3.4
, we show the effect of adding two of the

corresponding singular vectors—one associated to the zero singular value and the other to a small

one—to the mesh in its reference position.

Intuitively, the 3D-to-2D correspondences constrain the mesh vertices to move along lines of

sight but their exact distance to the camera is poorly constrained because changing it only results in

minor reprojection errors for points lying inside the facets. As a consequence, the number of degrees

of freedom corresponds to the one derived for the weak perspective case in Section
3.3.1.3
, except

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