Graphics Reference
In-Depth Information
I 1 ,
, I j W }
j
1 ,
j
W
where W is the number of pixels in the window and
{
...
and
{ α
...
,
α
}
rep-
resent the ordered list of image colors and
α
values inside window j . More compactly,
we can write Equation ( 2.27 )as
G j a j
b j
2
N
J
( α
, a , b
) =
α
(2.28)
j
j
=
1
where
's in window j followed by three
0's. If we suppose that the matte is known, then this vector is constant and we can
minimize Equation ( 2.27 ) for the individual
α
j is a
(
W
+
3
) ×
1 vector containing the
α
{
a j , b j
}
as a standard linear system:
a j
b j
G j G j ) 1 G j
= (
α j
(2.29)
α
That is, the optimal a and b in eachwindow for a givenmatte
are linear functions
α
of the
values. This means we can substitute Equation ( 2.29 ) into Equation ( 2.26 )
to get
J
( α ) =
min
a , b
J
( α
, a , b
)
(2.30)
G j a j
b j
2
N
=
min
a , b
α
(2.31)
j
j
=
1
G j a j
b j
α j
2
N
=
(2.32)
j
=
1
j
N
2
G j G j
) 1 G j
=
G j
(
α
α
(2.33)
j
j
=
1
I ( W + 3 ) × ( W + 3 )
N
1 α j
G j G j ) 1 G j
=
G j (
α j
(2.34)
j
=
= α L
α
(2.35)
In the last equation, we've collected all of the equations for the windows into a
singlematrix equation for the N
N matrix L is called the matting
Laplacian . It is symmetric, positive semidefinite, and quite sparse if the window size
is small. This matrix plays a key role in the rest of the chapter.
Working out the algebra in Equation ( 2.34 ), one can compute the elements of the
matting Laplacian as:
×
1 vector
α
. The N
×
1
I i µ k )
W I 3 × 3 1
1
W
L
(
i , j
) =
δ ij
+ (
k +
(
I j µ k )
(2.36)
k
| (
i , j
)
w k
where
k are the mean and covariance matrix of the colors in window k and
δ ij is the Kronecker delta. Frequently, the windows are taken to be 3
µ k and
×
3, so W
=
9. The
notation k
w k in Equation ( 2.36 ) means that we only sum over the windows k
that contain both pixels i and j ; depending on the configuration of the pixels, there
could be from 0 to 6 windows in the sum (see Problem 2.11 ).
| (
i , j
)
 
Search WWH ::




Custom Search