Graphics Reference
In-Depth Information
1
2
→
n
i
→
n
j
m
Patch
i
r
ji
Patch
j
⎡
⎣
⎤
⎦
ρ
1
0
... ...
0
0 ρ
2
... ...
0
... ... ... ... ...
... ... ...
ρ
i
...
00
... ...
ρ
m
1
π
P
=
⎡
⎣
⎤
⎦
0
K
12
...
...
K
(
m
−
1
)
m
K
21
0
...
...
...
K
=
...
...
0
K
ij
...
...
...
K
ji
0
...
K
m
(
m
−
1
)
...
...
...
0
Figure 8.37
Recovering the geometry of a concave Lambertian object from captured images. The sur-
face patches in the geometric model correspond to the pixels in the images. An initial guess
provides the patch normals
n
and reflectances ρ, from which the form factors can be com-
puted. The matrix
P
contains the reflectances;
K
contains the form factors. (After [Nayar
et al. 91].)
where
I
is the identity matrix,
P
is the diagonal “reflectance” matrix of reflectances
the radiance values arranged by facet index
j
.
In Step 2, the calculation to remove the effects of GI is performed. The ra-
diosity solution is computed by solving the system of equations, which amounts
to matrix inversion:
L
L
1
,which
is then subtracted from the pixel values. This is done for each of the three source
images. Then the process starts over at Step 1 using the improved source images,
and is repeated until the geometry converges.
The matrices
K
and
P
do not depend on the light direction. Therefore when
Step 1 is applied to the original images, the recovered shape is the same regardless
of the directions of the light sources as long as the object is concave and fully
illumination. The authors called this the
pseudo shape
, and note that the pseudo
shape appears more shallow than the true shape. Figure 8.38(a) and (b) show a
cross section of an object and its pseudo shape, respectively. Because any concave
Lambertian surface has a unique pseudo shape, it makes the iterative recovery
)
−
1
L
1
. The indirect component is
L
=(
I
−
PK
−
