Graphics Reference
In-Depth Information
Bruhn et al. [ 76 ] proposed to replace the Horn-Schunck data term with one
inspired by the Lucas-Kanade algorithm, that is,
2
E data (
u , v
) =
w
(
x , y
)(
I 2
(
x
+
u , y
+
v
)
I 1
(
x , y
))
(5.28)
(
x , y
)
where w
(
x , y
)
is a Gaussianwith scale
σ
centered at the point
(
x 0 , y 0
)
at which the flow
is computed. The scale
is usually in the range of one to three pixels. This approach
combines the advantages of Lucas-Kanade's local spatial smoothing, which makes
the data term robust to noise, with Horn-Schunck's global regularization, which
makes the flow fields smooth and dense. Bruhn et al. also extended this approach
to spatiotemporal smoothing when more than two images are available.
Sun et al. [ 476 ] collected ground-truth optical flow fields to learn a more accurate
model of how real images deviate from the brightness constancy assumption. The
learned distribution of I 2
σ
can be used to build a probabilistic
data term (e.g., approximating the distribution by a mixture of Gaussians). The same
approach can be used to learn distributions of filter responses applied to the flow
field (such as the gradient proposed by Brox et al. above).
(
x
+
u , y
+
v
)
I 1
(
x , y
)
5.3.3.2 Changes to the Smoothness Term
Nagel andEnkelmann [ 343 ]were among thefirst tonote theundesirability of applying
the smoothness term uniformly across the entire image. In particular, the smooth-
ness term should be down-weighted perpendicular to image edges, since these often
correspond to depth discontinuities in the scene, where the underlying optical flow
field is not actually smooth. This observation can be encapsulated by modifying the
smoothness term as follows:
u
u
v
v
x
x
x
x
E smoothness
(
u , v
) =
trace
D
(
I 1
(
x , y
))
(5.29)
u
v
u
v
y
y
y
y
2
2
×
2 is the anisotropic diffusion tensor defined by
where D :
R
→ R
2 g g + β
2 I 2 × 2
1
D
(
g
) =
(5.30)
2
2
g
+
2
β
where g
is a constant that controls the
amount of anisotropy. That is, the flow is prevented from being smoothed across
edges in I 1 where the gradient magnitude is significantly larger than
is g rotated clockwise by 90 degrees and
β
.
Figure 5.6 illustrates the shape of the diffusion tensor for an example image, show-
inghowthe smoothingneighborhood is circular (uniform) inflat regions andelliptical
near edges, which prevents the smoothness term from smoothing across edges.
The smoothness term can also be viewed as a prior term on the optical flow
field, since it reflects our assumptions about “good” flows. Similar to Sun et al.,
Roth and Black [ 403 ] collected ground-truth optical flow fields to learn a more accu-
rate model of the distribution of optical flow vectors in real images. As seen earlier,
this leads to a probabilistic smoothness term that Roth and Black modeled as a
product of t-distributions applied to filtered versions of the data (also known as a
Field-of-Experts model).
β
 
Search WWH ::




Custom Search