Digital Signal Processing Reference
In-Depth Information
We relax the lower bound by three deviations to include the possible shadow
colors and add the upper bound with three deviations to exclude the unwanted
brighter colors. Thus, we can eliminate the incorrect luminance pixels as possible.
The pixels, which meet the signal subspace criterion stated in (
2.25
), could be very
possible mixed color pixels.
In order to further exclude the mixed color pixels, we should use the noise space
criterion to remove the pixel color pixels from the signal subspace pixels, which
satisfy the criterion stated in (
2.25
). The noise subspace criterion can be discussed
in two cases:
p
=
1and
p
=
2. For
p
=
1, the noise subspace now becomes the span
{
,
}
of
v
1
v
2
. We should perform the noise subspace criterion as
ˆ
v
i
|
y
i
,
k
|
=
|
·
g
k
| >
λ
i
+
k
in
σ
λ
i
,
for
i
=
2
,
3
,
(2.27)
to remove the unwanted pixels, where
k
in
is a constant to specify the confidence
interval of the noise. We know
th
at the pixels, whose projections to the noise sub-
space should be as small as
ˆ
3, are matched with the desired color
modal. For any other pixels with m
ix
ed colors, their color vectors project onto the
λ
i
for
i
=
2
,
noise subspace will be larger than
ˆ
3. Similarly, we can keep the desired
pixels
once we find
their projections to the noise subspace are beyond the limits of
k
in
·
λ
i
for
i
=
2
,
ˆ
λ
i
±
k
2
n
·
σ
λ
i
for
i
=
2
,
3. For
p
=
1, we perform the detection of
ˆ
ˆ
v
2
g
k
≥
k
1
n
·
λ
i
+
k
2
n
·
σ
λ
2
≥|
y
2
,
k
|
=
·
k
1
n
·
λ
i
−
k
2
n
·
σ
λ
2
(2.28)
and
ˆ
ˆ
y
3
,
k
|
=
v
3
g
k
≥
k
1
n
·
λ
i
+
k
2
n
·
σ
λ
3
≥|
·
k
1
n
·
λ
i
−
k
2
n
·
σ
λ
3
(2.29)
to remove the unwanted pixels. We set the constants
k
1
n
=
3and
k
2
n
=
3inour
experiment to achieve the best results. In (
2.28
)and(
2.29
),
σ
λ
2
and
σ
λ
3
denote the
deviation of the second and third eigenvalues respectively given by
2
2
[
ξ
2
ξ
2
]
≈
λ
2
λ
2
=
σ
E
N
,
(2.30)
and
2
3
N
[
ξ
3
ξ
3
]
≈
λ
2
λ
3
=
σ
E
.
(2.31)
In summary, we utilize the complete projections of both signal and noise sub-
spaces to detect the desired color pixels. The signal-subspace projection helps
to classify the desired pixels while the noise-subspace projection provides the
information to eliminate the unmatched pixels. Theoretically, the noise plane re-
lated to the smallest eigenvalue can highlight the most wanted object after removing
