Graphics Reference
In-Depth Information
green
green
red
red
(a)
(b)
Figure 2.9.
(a) A tougher example of a scatterplot of the colors in labeled foreground and back-
ground regions. Black dots represent background and white dots represent foreground. In this
case, the foreground and background densities are neither well separated nor well represented
by a single Gaussian. (b) Gaussian mixture models fit to the foreground and background samples
do a better job of separating the distributions.
F
U
Figure 2.10.
The local foreground and back-
ground samples in a window around each
pixel can be used to compute the distribu-
tions for Bayesian matting.
distributions remains, but the Gaussian mixture components are better separated
and model the data more tightly.
In the multiple-Gaussian case, solving Equation (
2.10
) directly is no longer
straightforward, but Chuang et al. [
99
] suggested a simple approach. We consider
each possible pair of (foreground, background) Gaussians independently, and solve
for the best
F
,
B
, and
by alternating Equations (
2.16
)-(
2.17
). Then we compute the
log likelihood given by the argument of Equation (
2.10
) for each result. We need to
include the determinants of
α
for each
pair, since they are not all the same — these factors were ignored in Equation (
2.15
).
Finally, we choose the estimates for
F
,
B
, and
F
and
B
when evaluating log
P
(
F
)
and log
P
(
B
)
α
that produce the largest value of
Equation (
2.10
).
For complicated foregrounds and backgrounds, it makes sense to determine the
foreground and background distributions in Equation (
2.15
) locally at a pixel, rather
than globally across the whole image. This can be accomplished by creating a small
(relative to the image size) window around the pixel of interest and using the colors of
F
for pixels
inside both the window and the unknown region are estimated, they can supplement
the samples. Generally, the estimation begins at the edges of the unknown area and
and
B
inside thewindow to build the local pdfs (Figure
2.10
). As
F
,
B
, and
α
