Image Processing Reference

In-Depth Information

5.2 NRSFMUNDERWEAK PERSPECTIVE PROJECTION

Recall from Eq.
5.1
that under the weak perspective model, the projection of a 3D point
q
i
can be

written as

u
i

v
i

1

d
(
Rq
i
+

=

t
),

(5.3)

where
R
contains the first two rows of a full rotation matrix,
t
isa2

×

1 translation vector, and
d
is a

scalar.

Given
N
c
such 3D points on a surface, the corresponding equations can be grouped in matrix

form, which yields the system of equations

u
1

···

u
N
c

1

d
(
RQ

=

+

T
),

(5.4)

v
1

···

v
N
c

where
Q
is the 3

N
c
translation matrix

whose columns all contain the same vector
t
. Without loss of generality, and as was proposed in the

factorization algorithm of Tomasi and Kanade
Tomasi and Kanade
[
1992
], the translation
T
can be

eliminated by subtracting the mean of all 2D points, which is equivalent to assuming that the shape

is centered at the origin. This also removes the translation ambiguity mentioned in
Aanaes and Kahl

[
2002
].

A standard assumption of NRSFM methods is that the shape can be approximated with a

linear subspace model, meaning that the shape can be expressed as a linear combination of
N
s
basis

shapes. Under this model, Eq.
5.4
can be re-written as

u
1

×

N
c
matrix of 3D point coordinates, and
T
is the 2

×

N
s

···

u
N
c

=

R

c
k
S
k
,

(5.5)

v
1

···

v
N
c

k
=
1

where each
S
k
is a 3

N
c
matrix containing one basis shape, and
c
k
is its associated coefficient.

Note that since no mesh representation is available here, the basis shapes will depend on the spe-

cific configuration of feature points on the surface. Therefore, they cannot be pre-computed as in

Chapter
4
and must be recovered together with their coefficients. Note also that, without loss of

generality, the scalar
d
has been absorbed in the shape coefficients.

Given outlier-free frame-to-frame correspondences between the 2D surface features in an
N
f

frame video sequence, we can write Eq.
5.5
for each frame, and group all the resulting equations in

a system of the form

⎡

×

⎤

u
1

u
1
N
c

···

⎡

⎣

⎤

⎦

⎡

⎣

⎤

⎦

⎣

v
1

v
N
c

⎦

···

c
1
R
1

c
N
s
R
1

···

S
1

.

S
N
s

.

.

.

.

.

.

=

,

(5.6)

u
N
f

u
N
N
c

c
N
f

c
N
f

···

R
N
f

N
s
R
N
f

···

1

1

v
N
f

1

v
N
f

N
c

···

B

C

W

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