Graphics Reference
In-Depth Information
Table 5.1: Robust penalty functions typically used for optical flow.
β
is an
adjustable parameter, and
ε
is a very small number (e.g., 0.001) so that the
Charbonnier penalty function is nearly equal to
|
z
|
.
Name
Definition
ρ(
z
)
6
log
1
2
4
2
z
1
Lorentzian
ρ(
z
)
=
+
β
2
0
−2
0
2
z
3
2
√
z
2
2
Charbonnier
ρ(
z
)
=
+
ε
1
0
−2
0
2
z
2
Generalized
Charbonnier
z
2
2
)
β
ρ(
z
)
=
(
+
ε
1
0
−2
0
2
z
In the same way, we can robustify the smoothness term in optical flow. In this
case, we need to robustly compare a vector at each pixel (i.e., the four optical flow
gradients) to that of its neighbors. A typical robust smoothness term has the form
∂
2
u
x
,
∂
u
y
,
∂
v
x
,
∂
v
E
smoothness
(
u
,
v
)
=
ρ
(5.32)
∂
∂
∂
∂
y
where
is again one of the robust functions in Figure
5.1
(it need not be the same
penalty function used for the data term). In particular, when
ρ
is the (approximate)
L
1
penalty function, theoptical flowmethod is said touse a
total variation
regularization
(e.g., [
74
,
538
]). An alternative approach is to apply a different
ρ
ρ
to each of the four
gradient terms and sum the results.
5.3.3.4 Occlusions
The techniques discussed so far implicitly assume that every pixel in
I
2
corresponds
to some pixel in
I
1
. However, this assumption is violated at
occlusions
— regions
visible in one image but not the other. Occlusions are generally caused by objects
close to a stationary camera that move in the interval between taking the images,
or by changes in perspective due to a moving camera. In either case, background
pixels formerly visible in
I
1
will be hidden behind objects in
I
2
, and background pixels
