Graphics Reference
In-Depth Information
Ultimately, robust matting is equivalent to a random walk problem in which we
minimize
1
0
1
0
M
(2.71)
with
L
+
diag
(
w
F
)
+
diag
(
w
B
)
−
w
F
−
w
B
w
F
w
F
M
=
−
0
(2.72)
w
B
w
B
−
0
where
L
is the standardmatting Laplacian and
w
F
and
w
B
are
N
1 vectors of terminal
weights. Expanding Equation (
2.71
) results in the equivalent objective function
×
α
L
)
W
F
(
α
−
)
+
α
W
B
α
α
+
(
α
−
1
N
×
1
1
N
×
1
(2.73)
where
W
F
and
W
B
are
N
N
diagonal matrices with the vectors
w
F
and
w
B
on the
diagonals, respectively. This objective function is quadratic in
×
and thus results in a
slightly modified linear system from the one used in closed-formmatting.
Rhemann et al. [
389
] suggested a modification to the objective function that more
explicitly involves the estimates
α
α
i
and confidences
c
i
:
α
L
α
+
λ(
α
−
α
)
D
min
(
α
−
α
)
(2.74)
where
α
is an
N
×
1 vector that acts as a prior estimate of
α
at every pixel in the matte,
λ
is a tunable parameter, and
D
is a diagonal matrix with the confidences
c
i
on the
diagonal. In this way, it's clear that when the confidence in the
α
i
estimate is high,
α
i
=
α
i
, and when the
confidence is low, the usual neighborhood constraints from the matting Laplacian
Robust matting was later refined into an algorithm called
soft scissors
[
529
] that
solves the matting problem incrementally based on real-time user input. That is, a
local trimap is generated on the fly as a user paints a wide stroke near the boundary
of a foreground object. The pixels on either edge of the stroke are used to build local
foreground and background models, and the stroke automatically adjusts its width
based on the local image properties. The pixels interior to the stroke are treated as
unknown and their
the objective function puts a higher weight on the prior that
's are estimated with the robust matting algorithm. Since this
region is relatively small, the drawing andmatte estimation canproceed at interactive
rates.
Rhemann et al. [
391
] also extended robust matting by incorporating a sparsity
prior on
α
is created from an underlying sharp-edged
(nearly binary) matte with a constant point-spread function (PSF) induced by the
camera. The underlying matte and PSF are iteratively estimated and used to bias the
matting result to be less blurry. They later extended their technique to allow the PSF
to spatially vary [
390
].
α
that presumes the observed
α
2
Rhemann et al. [
389
] also defined the
α
i
estimates and confidence terms slightly differently from
robust matting, and generated the foreground samples based on a geodesic-distance approach
instead of a Euclidean-distance one.
