Graphics Reference
In-Depth Information
features that come directly from an automated method like those discussed in the
previous chapter will probably not be sufficient. We now turn to optical flow algo-
rithms, which have the natural interpretation of moving pixels from the first image
toward the second. Every pixel intensity in the image plays a role in the deformation.
5.3
OPTICAL FLOW
The general problem of estimating the dense correspondence field corresponding to
the apparent motion of brightness patterns is called optical flow , one of the oldest
concepts in computer vision [ 167 ]. We are given two images of the same scene at
different times, possibly from different perspectives, and the goal is to estimate a
motion vector
corre-
spond. We first introduce the classical optical flow methods of Horn-Schunck and
Lucas-Kanade, and then describe more modern refinements and extensions that
produce high-quality optical flow fields. Algorithms for both optical flow and stereo
are typically defined over grayscale images, but can be naturally extended to color
images by adding terms for each channel. As discussed in Section 5.9 , optical flow is
commonly used in visual effects for slowing down or speeding up shots, adding new
textures onto a moving object, or creating interesting transitions between objects.
(
u , v
)
at every point
(
x , y
)
such that I 1
(
x , y
)
and I 2
(
x
+
u , y
+
v
)
5.3.1
The Horn-Schunck Method
Horn and Schunck [ 203 ] were among the first to formalize a computational method
for determining the optical flow between an image pair. First, we assume that the two
images are samples of a function I of both space and time, so that I 1
(
x , y
) =
I
(
x , y , t
)
and I 2
for some time t . The key assumption of optical flow is that
image intensities of corresponding points are preserved over time; that is,
(
x , y
) =
I
(
x , y , t
+
1
)
I
(
x
+
u , y
+
v , t
+
1
) =
I
(
x , y , t
)
(5.16)
This is known as the brightness constancy assumption . Equation ( 5.16 ) actually
combines several assumptions, including that the surfaces in the scene have the
same brightness regardless of viewing direction (the Lambertian assumption ), that
the illumination of the scene doesn't change, and that the image formation process
is ideal (e.g., there is no vignetting or darkening of the image toward its edges).
If we perform a Taylor expansion of Equation ( 5.16 ) around
(
x , y , t
)
assuming that
u and v are small, we obtain
+
+
I
I
I
x u
y v
=
0
(5.17)
t
where all the quantities are functions evaluated at
(
x , y , t
)
. Put another way,
Equation ( 5.17 ) tells us that
u
v
=−
I
·
I
(5.18)
t
Thus, the component of the flow vector in the direction of the gradient
I is given by
I
t
2
I
(5.19)
2
I
Search WWH ::




Custom Search