Graphics Reference
In-Depth Information
difficult to obtain exact knowledge of each background image, to ensure that these
don't change, and to ensure that
F
is exactly the same (both in terms of intensity and
position) in front of both backgrounds. Therefore, triangulation is typically limited
to extremely controlled circumstances (for example, a static object in a lab setting).
If Equation (
2.5
) does not hold exactly due to differences in
F
and
between back-
grounds or incorrect values of
B
, the results will be poor. For example, we can see
slight errors in the toy example in Figure
2.7
due to “spill” from the background onto
the foreground, and slight ghosting in the nest example due to tiny registration errors.
Blue-screen, green-screen, and difference matting are pervasive in film and TV
production. A huge part of creating a compelling visual effects shot is the creation
of a matte for each element, which is often a manual process that involves heuristic
combinations and manipulations of color channels, as described in Section
2.11
.
These heuristics vary from shot to shot and even vary for different regions of the
same element. For more discussion on these issues, a good place to start is the topic
by Wright [
553
]. The topic by Foster [
151
] gives a thorough discussion of practical
considerations for setting up a green-screen environment.
α
2.3
BAYESIAN MATTING
In the rest of this chapter, we'll focus on methods where only one image is obtained
and no knowledge of the clean plate is assumed. This problem is called
natural
imagematting
. The earliest natural image matting algorithms assumed that the user
supplied a trimap along with the image to be matted. This means we have two large
collections of pixels known to be background and foreground. The key idea of the
algorithms in this section is to build probability density functions (pdfs) from these
labeled sets, which are used to estimate the
α
,
F
, and
B
values of the set of unknown
pixels in the region
U
.
2.3.1
The Basic Idea
Chuang et al. [
99
] were the first to pose the matting problem in a probabilistic frame-
work called
Bayesian matting
. At each pixel, we want to find the foreground color,
background color, and alpha value that maximize the probability of observing the
given image color. That is, we compute
argmax
F
,
B
,
P
(
F
,
B
,
α
|
I
)
(2.7)
α
We'll show how to solve this problem using a simple iterative method that results
from making some assumptions about the form of this probability. First, by Bayes'
rule, Equation (
2.7
) is equal to
1
argmax
F
,
B
,
P
(
I
|
F
,
B
,
α)
P
(
F
,
B
,
α)
(2.8)
P
(
I
)
α
since it doesn't depend on the parameters to be estimated,
and we can assume that
F
,
B
, and
We can disregard
P
(
I
)
α
are independent of each other. This reduces
Equation (
2.8
) to:
argmax
F
,
B
,
P
(
I
|
F
,
B
,
α)
P
(
F
)
P
(
B
)
P
(α)
(2.9)
α
