Furthermore, it can be observed that the correlation
is nearly linear.
Therefore, the correlation can be expressed by a
line estimated by the linear mean square error
(LMSE) estimator. Then, the chrominance
components can be estimated from the luminance
components by the following linear equation:
1yu
ˆ
ba
,
(14)
where
yu
y
yu
μaμba ,
)var(
),cov(
,
(15)
and
1
denotes the vector having the same size as
and with all the entries having the value 1.
Here,
u
μ
is the mean of the chrominance
components, and
y
μ
is the mean of the luminance
components. The object function to be minimized
for each segmented region is
2
mlmlml
e
,,.
u
ˆ
u
,
(16)
where the subscript
ml,
denotes the region
corresponding to the
m
th basis vector of the scale -l
sub matrix. The local approximated chrominance
function based on the linear model in Eq. (14) can be
expressed as
mlmlmlmlml
ba
,,,.,
1yu
ˆ
.
(17)
Now we can observe the fact that the vector
ml ,
y
corresponds to the vector
il 2,
c
in Eq. (13) and
the vector
ml ,
1
to the vector
12, il
c
in Eq. (12).
Therefore, by solving the problem in Eq. (9) with the
matrix containing the colorization vectors in Eq. (12)
and Eq. (13), we are actually computing the linear
coefficients
mlml
ba
,.
,
of the local linear regression
model. However, since the minimization problem of
Eq. (9) produces the coefficients that minimize the
total error in the chrominance differences between
the original and the reconstructed color image, the
linear coefficients are computed to minimize the
following total error in the linear regression model:
lm
ml
eE
.
.
(18)
We also use an initial color reconstruction
method to further reduce the compressed file size.
For the sub-matrix corresponding to scale 1, all the
coefficients corresponding to all the basis vectors in
the sub-matrix are extracted in the encoder and sent
to the decoder. Then, in the decoder, an initial
reconstruction of the color components is done using
these coefficients. This initial reconstruction of the
color components covers the whole domain of the
reconstructed color image, and thus prevents that
some regions become uncoloured. Furthermore,
since all the coefficients are sent for the sub-matrix
of scale-1, the coefficients can be sent in a pre-
defined order, and therefore, there is no need to send
the position information of the coefficients. This
reduces the size of data to be sent and thus gives a
higher compression rate.
Then, for the residual image, i.e., the image
obtained from the subtraction of the original color
image and the initially reconstructed color image,
the coefficients are extracted by the
L
0
minimization.
The whole system flow of the proposed
algorithm is described in Figure 5. In the encoder,
the original color image is divided into the
luminance channel and the chrominance channels.
The chrominance channels are sub-sampled
according to the 4:2:0 format, since the resolution of
chrominance channels are lower than that of
luminance channel. The luminance channel is
encoded with conventional methods such as JPEG or
JPEG2000. The encoded bit stream is sent to the
decoder, and the decompressed luminance channel is
also sub-sampled to the size of the chrominance
channel. The decompressed and sub-sampled
luminance channel is segmented with the multi-
meanshift algorithm. Then, the type 1 and type 2
colorization basis vectors are constructed for each
segmented region. After that, the initial
reconstruction of the chrominance channels is
performed using the sub matrix of scale-1. Then,
further colorization coefficients are extracted from
the residual image.
The bit stream of the luminance channel and the
colorization coefficients are sent to the decoder. The
bit stream of the luminance channel is decoded and
sub-sampled. Using the decompressed luminance
channel, the colorization matrix is constructed in the
same manner as in the encoder. The colorization
process is performed by multiplying the colorization
matrix and the colorization coefficients sent from the
encoder. Then, the colorized chrominance channels
are up-sampled to the size of the luminance channel.
An inverse YUV conversion of the decompressed
luminance channel and the reconstructed
chrominance channels reconstructs the color image.
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