自适应分数阶总变分迭代恢复消除阶梯伪影
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本文是一篇研究论文，主要关注于利用分裂Bregman迭代（Split Bregman Iteration, SBI）技术解决图像恢复中的两个关键问题：一是减轻变分方法（Variational Method）在处理图像时产生的阶梯状artifact，二是实现对不同区域的自适应参数调整，以优化图像恢复效果。文章的焦点在于提出了一种自适应分数阶总变化（Adaptive Fractional Order Total Variation, AFoTV）正则化模型，该模型引入了分数阶微分（Fractional Calculus），这是一种非整数阶的数学工具，能够在保留更多图像细节的同时，减少阶梯效应。
在传统图像恢复中，总变化（Total Variation, TV）是一种常用的边缘保持技术，但其全局化的平滑性可能导致阶梯状artifact的出现，特别是在处理复杂纹理和细节丰富的图像时。为了克服这个问题，论文提出了一个改进的分数阶微分核掩模（Improved Fractional-Order Differential Kernel Mask, IFODKM）。IFODKM具有更广泛的自由度（Degree of Freedom, DOF），能够根据不同区域的特性设计更精细的滤波，从而在保持边缘的同时减少阶梯状效果。
作者Dazi Lin、Xiangyi Tian、Qibing Jin 和 Kotaro Hirasawa来自北京化工大学信息技术学院，他们在文中展示了如何通过SBI算法有效地解决AFoTV1模型，并演示了IFODKM如何与SBI结合，提高图像恢复的质量，特别是对于那些含有复杂结构和细节的图像。论文还记录了接收日期、修订和接受日期，以及在线发表的时间，关键词包括分数阶微积分、分数阶微分核掩模、分裂Bregman迭代、图像恢复和阶梯状artifact的消除。
这篇文章为图像恢复领域的研究者提供了一种新的策略，即通过自适应分数阶方法和改进的核掩模技术来改善图像恢复质量，这在实际应用中具有重要意义，特别是在处理对细节敏感的图像处理任务时。
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎢
⎤
⎥
⎡
⎢
⎤
⎥
∑
∑
()= (− ) ( − )
()= (− ) ( − )
()
α
α
α
α
α
α
→
=
(− )
→
=
(−)
Dfx
h
Cfx kh
Dfx
h
Cfx kh
lim
1
1
lim
1
1
,
3
x
GL
x
h
k
xx h
k
k
x
GL
x
h
k
xxh
k
k
0
0
/
0
0
/
0
0
1
1
where
()( )
()
=
α
Γα
ΓΓα
+
+−+
C
k
kk
1
11
is the generalized binomial coefficient
and
Γ(⋅
)
is the Gamma function,
α<<
0
1
,
<<xxx
01
.
The proposed method in this paper is based on G-L definition.
Compared with other two definitions, the G-L de fi nition is more
suitable for image processing because it has only one coefficient
for tuning. In the left G-L fractional calculus definition, the tradi-
tional integer order gradient is extended to a fractional gradient
operator, as a result, the typical Laplace mask is improved by a
fractional differential mask. In addition, because the fractional
calculus takes more characteristics of proximity into account, it's
more similar to optical imaging principle and therefore is more
suitable to avoid staircase artifacts.
2.2. Inverse model with fractional-order total variation
As for the characteristics of signals, the additive noise leads to
the quality degradation, and the convolution blur leads to a blur-
ring effect. Then, the visual performance of images comes to low
resolution because of these noises. Therefore, the mathematical
model of image degradation process can be described as Fig. 1.
In Fig. 1,
()∈
×
ux y,
M
N
is the original clear image,
(
)
xy,
denotes
the coordinates of pixel matrix,
(
)
MN,
represents the width and
height of pixel matrix.
(
)
ux y,
is captured by the multiplicative
kernel
()∈
×
h
xy,
MN
, which is similar to the point spread func-
tion.
()∈
×
nx y,
M
N
is the additive white Gaussian noise,
()∈
×
g
xy,
M
N
is the degenerated image).
Generally speaking, the issue of image restoration is an inverse
problem. The classical ROF functional model can be established for
describing this inverse problem
⎡
⎣
⎢
⎤
⎦
⎥
∫
μ
Ω‖∇ ‖ + ‖ − ‖
()
Ω
H
p
u
q
ugdmin
1
.
4
u
p
q
q
q
In this model,
μ
is the regularization parameter, it balances the
noises filter and details protection.
g
and
u
denote the degen-
eration image and the restoration image, respectively.
H
is a linear
sparse operator, while
∇
is the gradient operator or Hamilton
operator.
p
and
q
define the type of functional model. In addition,
[
]=[
]
pq,1,2
defines the
ℓ
1
regularization and
[
]=[
]
pq,2,2
defines
the
ℓ
2
regularization.
Replacing the integer order gradient operator with the frac-
tional gradient operator, the AFOTV model can be represented as
⎡
⎣
⎢
⎤
⎦
⎥
∫
μ
Ω‖∇ ‖ + ‖ − ‖
()
Ω
α
H
p
u
q
ugmin
1
d,
5
u
p
q
q
q
where
α
α∈( ) ∈R0, 1 ,
.
2.3. Brief description of split Bregman iteration
SBI algorithm is an iterative regularization technique, it is an
improvement of Bregman iteration. An unconstrained SBI algo-
rithm is used to deal with the
−ℓAFOTV
1
in this paper.
General image restoration algorithm is to minimize the energy
function described as
()=()+ ( )
()
uJuQug,,
6
where
(
)
u
is the energy function,
(
)J
u
is the error item and
(
)Q
ug,
is the data fidelity item.
(
)J
u
and
(
)Q
ug,
are both the variations of
(
)
Eu g,
.
According to the forms of fractional variation, it is assumed that
⎧
⎨
⎪
⎪
⎩
⎪
⎪
∫
∫
μ
Ω
Ω
()= ‖−‖
()= ‖∇ ‖
()
Ω
Ω
α
HQu g
q
ug
Ju
p
u
,d
1
d
,
7
q
q
p
q
where
(
)Q
u,g
and
(
)J
u
are continuous dimensional and real-va-
lued convex functions, both of which are non-negative and
differentiable.
From the definition of Bregman distance, the Bregman dis-
tances between
u
and
u
k
can be expressed as
{}
()=()−()+< −>
()
u u Ju Ju p u u,,,
8
pk k k
where
p
denotes the sub-gradient of
J
at
u
k
, and
{}
∈∂()= ()≥()+< − >
()
p Ju pJu Ju p u u,.
9
kkk
If
J
is differentiable, then
J
is the sub-gradient of
p
at
u
k
.
2.4. Convergence analysis for split Bregman iteration
Lemma 1.
(
)J
u
and
(
)Q
ug,
are two real-valued convex functions,
(
)Q
ug,
is differentiable,
()<
∞J
u
.
Assuming that
^
=()uQugargmin ,
u
,
()=()+ (
)
uJuQug,
,
where
(
)
u
is solvable. Then, the iterative solutions of
(
)Q
ug,
are monotonically decreasing, which is expressed as
()≤()
()
+
Qu g Qu g,,.
10
kk1
Proof.
^
u
converges to the minimum values of
(
)Q
ug,
, which is
expressed as.
()≤(
^
)+ (
^
)
()
Qu g Ju
k
Qu g,
1
,.
11
k
Lemma 2.
g
is the original clean image,
f
is the degenerated image.
Assuming that.
⎪
⎪
⎧
⎨
⎩
δ(
^
)≤
(
^
)=
()
Qu f
Qu g
,
,0
,
12
2
where
δ
2
denotes the noise levels, and when it satisfies that
δ()>
+
Q
uf,
k 1
2
. Then, the Bregman distances between
u
k
and
^
u
is
monotonically decreasing, which can infer that
(
^
)< (
^
)
()
+
+
uu uu,,,
13
pk pk1
kk1
where
k
is the iteration number.
Fig. 1. The image degradation process model.
D. Li et al. / ISA Transactions 82 (2018) 210–222212
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