868 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008
into two categories: 1) blind SC grayscale image restoration [4],
[18] and 2) blind MC grayscale image restoration [19], [20].
The amount of works on blind color image restoration is still
very limited [12]. Blind SC grayscale restoration attempts to
estimate the original grayscale image from a single observation
of blurred grayscale image. In contrast, blind MC grayscale
restoration attempts to estimate the original grayscale image
from multiple blurred grayscale images of the same imaging
scene. Efforts to extend these two frameworks to blind color im-
age restoration have achieved only limited success so far. Works
to extend blind SC restoration to blind color image restoration
include [12]. The results are usually unsatisfactory because of
the following: 1) The method ignores interchannel blurring due
to causes such as channel crosstalk, and 2) the information of
interchannel correlation has not been fully utilized. On the other
hand, attempts to extend blind MC restoration to blind color
image restoration encounter the following problems: 1) There
is only a single observation of blurred color image, and 2) the
color image degradation system follows multiinput multioutput
(MIMO) modeling, rather than the single-input multioutput
(SIMO) modeling used by blind MC restoration algorithms.
One of the most challenging issues in addressing blind color
image deconvolution is that the solution to the original MIMO
model is intractable. In other words, we cannot obtain a direct
estimate to the original color image using conventional MIMO
algorithms since the RGB color channels are highly correlated.
Furthermore, the computational cost involved in extending the
1-D MIMO algorithms in communication theory to 2-D images
is significant. In view of this, there is a real need and motivation
to reformulate the original MIMO model into a new framework
so that the solution to the new model will provide a reasonably
good estimate to the original color image, while preserving the
visual quality of the solution.
The contribution of this paper, therefore, is to propose a
new framework that offers a tractable solution to alleviate the
intractable original MIMO deconvolution problem. It also ad-
dresses the issues encountered when extending the SC and the
MC grayscale blind image deconvolution into the color image
domain. In the proposed framework, the blind color image
restoration is performed using a hybrid of single-input single-
output (SISO) and SIMO models. The new method has two
characteristics: 1) In the SISO model, we utilize the local spatial
smoothness of low-frequency component in each channel to
perform image regularization, and 2) in the SIMO model, the
technique exploits the spectral correlation between the high-
frequency subbands of different channels to perform decon-
volution. The method also takes intrachannel and interchannel
blurring into consideration. Experimental results show that the
new method is effective in restoring color images where there
is limited information about the blurring functions.
The organization of the rest of this paper is outlined as
follows. The mathematical model of color image degradation
is introduced in Section II. A wavelet-based deconvolution
scheme for color images and its justification are presented
in Section III. Section IV discusses the approximate subband
deconvolution using SISO modeling. In Section V, the detailed
subband deconvolution based on SIMO modeling is explained.
In Section VI, the optimization procedure is discussed. Simula-
tion results are presented in Section VII. Finally, a conclusion
and further remarks are given in Section VIII.
II. P
ROBLEM FORMULATION
The linear color image degradation processes are commonly
modeled by [6], [12]
y
j
=
i=r,g,b
h
ij
⊗ f
i
+ n
j
,j= r, g, b (1)
where y
j
, f
i
, and n
j
are the jth blurred color channel, ith
original color channel, and jth channel noise, respectively.
h
ii
and h
ij
(i = j) are intrachannel and interchannel PSFs or
blurs. The operator ⊗ denotes the 2-D convolution operation.
The intrachannel blurring usually dominates the interchannel
blurring, or in other words, the coefficients of h
ij
(i = j) are
usually much smaller than that of h
ii
. The objective of blind
color image deconvolution is to estimate f
i
, i = r, g, b given the
three channels of the blurred image y
j
, j = r, g, b. It is obvious
that this is a challenging problem as the information about the
blurring functions h
ij
are not known and each color channel f
i
is highly intercorrelated.
Before outlining the proposed restoration algorithm, we will
first cover some preliminaries on the filter bank used in this
paper. Fig. 1 shows the analysis and synthesis stages of wavelet
decomposition. The low-pass filters a
lp
(x) and a
lp
(y), and
the high-pass filters a
hp
(x) and a
hp
(y) denote the analysis
filter banks performing an undecimated wavelet transform in
the vertical (x) and horizontal (y) directions, respectively.
In contrast, the synthesis filters s
lp
(x), s
lp
(y), s
hp
(x), and
s
hp
(y) reverse the process to obtain the original signal. As the
redundant coefficients in the wavelet transform are useful for
reconstruction in this case, we have chosen not to include the
decimator in wavelet transformation. This operation is known
as redundant discrete wavelet transform. During the analysis
stage, each decomposed channel of the blurred image is given
as follows:
y
j LL
= a
lp
(x) ⊗ [a
lp
(y) ⊗ y
j
(x, y)]
y
j HL
= a
hp
(x) ⊗ [a
lp
(y) ⊗ y
j
(x, y)]
y
j LH
= a
lp
(x) ⊗ [a
hp
(y) ⊗ y
j
(x, y)]
y
j HH
= a
hp
(x) ⊗ [a
hp
(y) ⊗ y
j
(x, y)]
(2)
where y
j LL
is the approximate subband (LL), and y
j HL
, y
j LH
,
y
j HH
are the horizontal (HL), vertical (LH), and diagonal (HH)
detailed subbands of the jth channel of the blurred image,
respectively. Performing wavelet decomposition in similar
fashion to (2) on both sides of (1), and ignoring the additive
noise, we obtain the following:
y
jLL
=
i=r,g,b
h
ij
⊗f
iLL
,y
jHL
=
i=r,g,b
h
ij
⊗f
iHL
y
jLH
=
i=r,g,b
h
ij
⊗f
iLH
,y
jHH
=
i=r,g,b
h
ij
⊗f
iHH
.
(3)
Equation (3) demonstrates that the original system in (1) can
be transformed into an equivalent system consisting of subband
decompositions of the original image f
i
and the observed
blurred image y
j
. The reason for performing this transform is
because it is observed that there exists different characteris-
tics between spectral correlations for different subbands, and
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