Graphics Reference
In-Depth Information
0.35
0.7
0.5
0.25
0.3
0.15
0.1
0.05
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
α
α
(a)
(b)
Figure 2.11.
(a) The normalized histogram of
α
values for the ground-truth matte for the middle
example in Figure
2.4
. (b) The normalized histogram of
α
values just over the trimap's unknown
region, superimposed by a beta distribution with
η
=
τ
=
1
4
.
proceeds toward its center. We'll say more about the issue of local pixel sampling in
Section
2.6.1
.
While the original Bayesian matting algorithm treated the prior term
P
(α)
as a
constant, later researchers observed that
P
is definitely not a uniformdistribution.
This stands to reason, since there are a relatively large number of pixels that are
conclusively foreground (
(α)
0) compared to mixed pixels,
which typically occur along object boundaries. Figure
2.11
illustrates the distributions
of
α
=
1) or background (
α
=
for a real image; the left panel shows that over the whole image the distribution is
highly nonuniform, and the right panel shows that even over the trimap's uncertain
region, the distribution is biased toward
α
values close to 0 and 1. Wexler et al. [
544
]
and Apostoloff and Fitzgibbon [
16
] suggested modeling this behavior with a beta
distribution of the form
α
(η
+
τ)
(η)(τ)
α
η
−
1
−
α)
τ
−
1
P
(α)
=
(
1
(2.18)
1
4
is superimposed on Figure
2.11
bto
give a sense of the fit. Unfortunately, incorporating amore complex prior for
A sketch of a beta distribution with
η
=
τ
=
α
makes
Equation (
2.10
) harder to solve.
It's also important to remember that the pixels in the
image are highly correlated,
sowe should be able to do amuch better job by enforcing that the
α
values of adjacent
pixels be similar (the same type of correlation holds, though more weakly, for the
background and foreground images). We will discuss algorithms that exploit this
coherence in the rest of this chapter.
α
CLOSED-FORM MATTING
2.4
In Bayesian Matting, we assumed that the foreground and background distributions
were Gaussians (i.e., that the samples formed ellipsoidal clusters in color space).
However, it turns out that in many natural images, the foreground and background
distributions look more like lines or skinny cigar shapes [
355
]. In fact, this is visible
