686 CHINESE OPTICS LETTERS / Vol. 7, No. 8 / August 10, 2009
Hopfield neural network-based image restoration with
adaptive mixed-norm regularization
Yuannan Xu (
NNN
III
), Liping Liu (
444
www
±±±
), Yuan Zhao (
ëëë
),
Chenfei Jin (
), and Xiudong Sun (
DDD
ÁÁÁ
)
∗
Department of Physics, Harbin Institute of Technology, Harbin 150001, China
∗
E-mail: xdsun@hit.edu.cn
Received December 3, 2008
To overcome the shortcomings of traditional image restoration model and total variation image restora-
tion model, we propose a novel Hopfield neural network-based image restoration algorithm with adaptive
mixed-norm regularization. The new error function of image restoration combines the L
2
-norm and L
1
-
norm regularization types. A method of calculating the adaptive scale control parameter is introduced.
Experimental results demonstrate that the proposed algorithm is better than other algorithms with single
norm regularization in the improvement of signal-to-noise ratio (ISNR) and vision effect.
OCIS codes: 100.3020, 100.3190, 200.4260.
doi: 10.3788/COL20090708.0686.
Image restoratio n deals with the recovery of an origi-
nal scene from its degraded image
[1−3]
. Since neural
network-based image restoration does not generate ring-
ing effect which relates to matrix inversion for solving
Euler-Lagrange equation, it can restor e a higher qual-
ity image. Most neural netwo rk-based image restor a-
tion algorithms use L
2
-norm as regularization item
and an isotropic regulariz ing operator such as Laplace
operator
[4−10]
. We call it traditiona l image restoration
model here. Although this type of regularization based
on L
2
-norm and Laplace ope rator has a good ability to
smooth and remove noise, it blurs the edges of resto red
image more or less and the vision effect is not very good.
To preserve edges, a total va riation-based image restora -
tion model is proposed
[11−15]
. However, the model is
mainly suitable for the images with smooth patch and
evident edge. By experiments, we find that the restored
image appears ladder effect when image dissatisfy this
condition, for e xample, the change of slo pe edge is slow.
Sometimes, noise can be wrongly c onsidered a s false edge.
To overcome these shortcomings of the two image restora-
tion models, we propose a novel Hopfield neural network-
based image restoration algorithm using adaptive mixed-
norm regulariza tion.
The image degradation model in a matrix operation
can be expressed as
g = Hf + n, (1)
where g, f, and n are vectors corresponding to the lexi-
cographically orga nized degraded image, original image,
and additive noise, respectively, H is the blur matrix
corresponding to a point spread function (PSF). For the
case of a convolution system, H usua lly takes the form
of a block Toeplitz matrix. Commonly, image restora-
tion is to minimize an error mea surement such as the
constrained squared error function:
E =
1
2
kg − Hfk
2
2
+
1
2
λ
D
ˆ
f
2
2
, (2)
where k·k
2
is L
2
norm,
ˆ
f is the restored image estimate
vector, λ is the re gularizing parameter, D is a block
Toeplitz matrix generated by the regularizing operator.
In this image res toration model, the regularizing operator
is a Laplace operator, which can be written as
[16]
d
Laplace
=
1
8
"
0 1 0
1 −4 1
0 1 0
#
. (3)
The error function base d on total variation is given by
E =
1
2
kg − Hfk
2
2
+
1
2
λ
∇
ˆ
f
1
, (4)
where k·k
1
is L
1
norm, ∇ is the gradient operator,
|∇f| =
p
(∂f/∂x)
2
+ (∂f/∂y)
2
. In this image restoration
model, the action of smoothing filter is not imposed so
that the edge information can be preserved.
For obtaining a b e tter solution to the image restora-
tion, we propose to combine the be nefits of the tradi-
tional model and total variation model. To fit Hopfield
neural network based processing, a novel error function
with mixed-norm regularizatio n is proposed:
E =
1
2
kg − Hfk
2
2
+
1
2
λη
D
ˆ
f
2
2
+
1
2
λ(1 − η)
∇
ˆ
f
1
, (5)
where η is a regularizing scale control parameter and its
span is [0, 1]. The image restoration model with mixed-
norm regula rization is aimed to overcome the shortcom-
ings and keep the advantages of the two above-mentioned
models. Blurring of image deta il, false edge, and ladder
effect can be removed to a certain extent, and a better
image restoration effect can be achieved.
Hopfield neural network method is designed to min-
imize a quadratic programming problem. The energy
function of Hopfield neural network has the following
form:
E
HNN
= −
1
2
ˆ
f
T
W
ˆ
f − b
T
f + c , (6)
where W is the interconnection weight matrix and the
(i, j)th element of W corresponds to the interconnection
1671-7694/2009/080686-04
c
2009 Chinese Optics Letters
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