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AdvancesinAppliedMathematics
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,2023,12(4),1732-1743
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.124180
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AnOptimizationAlgorithmonSignal
ReconstructionandItsApplicationin
ImageRestoration
ZunyangWang
1
,ChaoGuo
1
,HongchunSun
2
∗
1
Department of Electronic and Communication Engineering, Beijing Institute of Electronic Science
andTechnology,Beijing
2
SchoolofMathematicsandStatistics,LinyiUniversity,LinyiShandong
Received:Mar.24
th
,2023;accepted:Apr.18
th
,2023;published:Apr.27
th
,2023
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[J].
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1732-1743.DOI:10.12677/aam.2023.124180
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Abstract
In thispaper, we furtherconsider anoptimizationmethodfor solvingthesignal recon-
struction and imagedenoising problem.To this end, a new algorithmwith Armijo-like
line search is proposed.Global convergence results of the new algorithmis established
in detail.Furthermore, we also show that the method is sublinearly convergent rate of
O
(1
/k
2
)
.Finally,theefficiencyoftheproposedalgorithmisillustratedthroughsome
numericalexamplesonsparsesignalrecoveryandimagedenoising.
Keywords
TheSignalReconstructionandImageDenoisingProblem,Algorithm,Global
Convergence,SublinearlyConvergentRate
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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=
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k
+
∇
f
(
y
k
)+
L
k
(
x
k
−
y
k
) = 0
.
(2.11)
Ù
¥
ξ
k
∈
∂ϕ
(
x
k
)
.
g
•
e
¡
à
`
z
¯
K
min
(
ρϕ
(
x
)+
L
k
2
x
−
y
k
−
1
L
k
∇
f
(
y
k
)
2
)
,
(2.12)
Ä
u
ƒ
'
•
`
5
n
Ø
([10]),
•
T
¯
K
)
8
†
§
-
½:
8
´
˜
—
.
(
Ü
(2.11),
Œ
•
x
k
´
(2.12)
)
,
=
x
k
:=
argmin
(
ρϕ
(
x
)+
L
k
2
x
−
y
k
−
1
L
k
∇
f
(
y
k
)
2
)
,
(2.13)
5
2.2
d
L
k
=
η
m
k
β
Ú
η>
1
,
k
L
k
≥
β
.
,
,
Ï
L
k
/η
Ø
÷
v
(2.7)
½
(2.8),
(
Ü
Ú
n
2.1,
k
L
k
<η
k
A
>
A
k
.
Ï
d
,
k
β
≤
L
k
<η
k
A
>
A
k
.
(2.14)
e
5
§
?
Ø
Ž
{
2.1
Â
ñ
5
±
9
Â
ñ
„
Ý
.
•
d
,
‰
Ñ
±
e
Ú
n
.
Ú
n
2.2
é
u
?
¿
x
∈R
n
,
k
F
(
x
)
−
F
(
x
k
)
≥
L
k
2
x
k
−
y
k
2
+
L
k
x
−
y
k
,y
k
−
x
k
,
∀
k
≥
1
.
(2.15)
DOI:10.12677/aam.2023.1241801736
A^
ê
Æ
?
Ð
ƒ
y
²
.
d
(2.5),
F
(
p
L
k
(
y
k
))
≤
Q
L
k
(
p
L
k
(
y
k
)
,y
k
)
.
(2.16)
d
(2.13),
x
k
=
p
L
k
(
y
k
).
(
Ü
(2.16),
Œ
F
(
x
)
−
F
(
x
k
)
≥
F
(
x
)
−
Q
L
k
(
x
k
,y
k
)
.
(2.17)
Ï
f
Ú
ϕ
Ñ
´
à
,
Œ
F
(
x
)=
f
(
x
)+
ρϕ
(
x
)
≥
[
f
(
y
k
)+
h
x
−
y
k
,
∇
f
(
y
k
)
i
]+[
ρϕ
(
x
k
)+
ρ
h
x
−
x
k
,ξ
k
)
i
]
,
(2.18)
Ù
¥
ξ
k
∈
∂ϕ
(
x
k
).
A^
(2.1)
…
-
x
=
x
k
,y
=
y
k
,
Q
L
k
(
x
k
,y
k
) = [
f
(
y
k
)+
x
k
−
y
k
,
∇
f
(
y
k
)
+
L
k
2
x
k
−
y
k
2
]+
ρϕ
(
x
k
)
.
(2.19)
A^
(2.18),(2.19)
Ú
(2.17),
Œ
F
(
x
)
−
F
(
x
k
)
≥
F
(
x
)
−
Q
L
k
(
x
k
,y
k
)
≥
x
−
x
k
,
∇
f
(
y
k
)+
ρξ
k
−
L
k
2
x
k
−
y
k
2
=
L
k
x
−
x
k
,y
k
−
x
k
−
L
k
2
x
k
−
y
k
2
=
L
k
(
x
−
y
k
)+(
y
k
−
x
k
)
,y
k
−
x
k
−
L
k
2
x
k
−
y
k
2
=
L
k
x
−
y
k
,y
k
−
x
k
+
L
k
2
x
k
−
y
k
2
.
(2.20)
Ù
¥
1
˜
‡
ª
¤
á
´
Ä
u
(2.11).
Ú
n
2.3
b
x
∗
´
(1.1)
?
¿˜
)
,
{
x
k
}
Ú
{
y
k
}
´
Ž
{
2.1
)
S
.
K
%η
k
A
>
A
k
β
2
L
k
τ
2
k
α
k
1
−
2
L
k
+1
τ
2
k
+1
α
k
+1
1
≥
α
k
+1
2
2
−
α
k
2
2
,
(2.21)
Ù
¥
α
k
1
:=
F
(
x
k
)
−
F
(
x
∗
)
,α
k
2
:=
τ
k
x
k
−
(
τ
k
−
σ
)
x
k
−
1
−
σx
∗
.
y
²
.
é
u
?
¿
x
∈R
n
,
d
(2.15)
k
e
¡
Ø
ª
¤
á
F
(
x
)
−
F
(
x
k
+1
)
≥
L
k
+1
2
x
k
+1
−
y
k
+1
2
+
L
k
+1
x
−
y
k
+1
,y
k
+1
−
x
k
+1
.
(2.22)
3
(2.22)
¥
,
©
O
-
x
:=
x
k
Ú
x
:=
x
∗
,
Œ
F
(
x
k
)
−
F
(
x
k
+1
)
≥
L
k
+1
2
x
k
+1
−
y
k
+1
2
+
L
k
+1
x
k
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.23)
DOI:10.12677/aam.2023.1241801737
A^
ê
Æ
?
Ð
ƒ
Ú
F
(
x
∗
)
−
F
(
x
k
+1
)
≥
L
k
+1
2
x
k
+1
−
y
k
+1
2
+
L
k
+1
x
∗
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.24)
d
α
k
1
½
Â
,(2.23)
Ú
(2.24)
U
•
2
L
k
+1
(
α
k
1
−
α
k
+1
1
)
≥
x
k
+1
−
y
k
+1
2
+2
x
k
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.25)
Ú
−
2
L
k
+1
α
k
+1
1
≥
x
k
+1
−
y
k
+1
2
+2
x
∗
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.26)
3
(2.25)
ü
>
¦
±
τ
k
+1
−
σ
(
τ
k
+1
−
σ
)
2
L
k
+1
(
α
k
1
−
α
k
+1
1
)
≥
(
τ
k
+1
−
σ
)
x
k
+1
−
y
k
+1
2
+2(
τ
k
+1
−
σ
)
x
k
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.27)
3
(2.26)
ü
>
Ó
¦
±
σ
−
σ
2
L
k
+1
α
k
+1
1
≥
σ
x
k
+1
−
y
k
+1
2
+2
σ
x
∗
−
y
k
+1
,y
k
+1
−
x
k
+1
(2.28)
(2.27)
\
(2.28)
2
L
k
+1
((
τ
k
+1
−
σ
)
α
k
1
−
τ
k
+1
α
k
+1
1
)
≥
τ
k
+1
x
k
+1
−
y
k
+1
2
+2
τ
k
+1
y
k
+1
−
(
τ
k
+1
−
σ
)
x
k
−
σx
∗
,x
k
+1
−
y
k
+1
(2.29)
3
(2.29)
ü
>
Ó
¦
±
τ
k
+1
,
…
A^
(2.9),
Œ
2
L
k
+1
(
%τ
2
k
α
k
1
−
τ
2
k
+1
α
k
+1
1
)
≥
τ
2
k
+1
x
k
+1
−
y
k
+1
2
+2
τ
k
+1
τ
k
+1
y
k
+1
−
(
τ
k
+1
−
σ
)
x
k
−
σx
∗
,x
k
+1
−
y
k
+1
(2.30)
-
a
:=
τ
k
+1
y
k
+1
,b
:=
τ
k
+1
x
k
+1
9
c
:= (
τ
k
+1
−
σ
)
x
k
+
σx
∗
.
d
Pythagoras
'
X
k
b
−
a
k
2
+2
h
b
−
a,a
−
c
i
=
k
b
−
c
k
2
−k
a
−
c
k
2
,
Ú
(2.30),
Œ
2
L
k
+1
(
%τ
2
k
α
k
1
−
τ
2
k
+1
α
k
+1
1
)
≥
τ
k
+1
x
k
+1
−
(
τ
k
+1
−
σ
)
x
k
−
σx
∗
2
−
τ
k
+1
y
k
+1
−
(
τ
k
+1
−
σ
)
x
k
−
σx
∗
2
(2.31)
DOI:10.12677/aam.2023.1241801738
A^
ê
Æ
?
Ð
ƒ
d
(2.10),
w
,
Œ
τ
k
+1
y
k
+1
=
τ
k
+1
x
k
+(
τ
k
−
σ
)(
x
k
−
x
k
−
1
).
Š
â
(2.31)
Ú
α
k
2
½
Â
,
Œ
2
L
k
+1
(
%τ
2
k
α
k
1
−
τ
2
k
+1
α
k
+1
1
)
≥
α
k
+1
2
2
−
α
k
2
2
.
(2.32)
2
(
Ü
(2.14),
Œ
%η
k
A
>
A
k
β
2
L
k
τ
2
k
α
k
1
−
2
L
k
+1
τ
2
k
+1
α
k
+1
1
≥
α
k
+1
2
2
−
α
k
2
2
.
(2.33)
Ú
n
2.4
d
(2.9)
ê
{
τ
k
}
÷
v
τ
k
≥
k
2
(
k
≥
1)
,
Ù
¥
τ
1
= 1
.
y
²
.
|
^
ê
Æ
8
B
{
y
²
.
k
= 1
ž
,
d
τ
1
= 1,
w
,
k
τ
1
≥
1
2
.
b
k
ž
·
K
¤
á
,
=
τ
k
≥
k
2
.
d
(2.9),
¿
A^
σ
≥
1
,%
≥
1,
Œ
τ
k
+1
=
σ
+
p
σ
2
+
%τ
2
k
2
≥
σ
+
√
%k
2
≥
1+
k
2
.
Ï
d
U
ý
Ï
(
J
.
y
3
,
y
²
Ž
{
2.1
Â
ñ
(
J
.
½
n
2.1
b
x
∗
´
(1.1)
?
¿˜
)
,
{
x
k
}
Ú
{
y
k
}
d
Ž
{
2.1
)
S
.
K
k
F
(
x
k
+1
)
−
F
(
x
∗
)
≤
c
(
k
+1)
2
.
(2.34)
Ù
¥
c
´
~
ê
.
y
²
.
-
a
k
=
2
L
k
τ
2
k
α
k
1
,
b
k
=
α
k
2
2
,
A^
(2.21),
a
k
+1
+
b
k
+1
≤
%η
k
A
>
A
k
β
a
k
+
b
k
.
(2.35)
(
Ü
0
<
%η
k
A
>
A
k
β
<
1,
Ï
L
O
Ž
Œ
a
k
+1
≤
a
k
+1
+
b
k
+1
≤
a
k
+
b
k
≤
a
k
−
1
+
b
k
−
1
≤···≤
a
1
+
b
1
,
(2.36)
…
a
1
=
2
L
1
(
F
(
x
1
)
−
F
(
x
∗
)),
b
1
=
k
(
x
1
−
x
0
)+
σ
(
x
0
−
x
∗
)
k
2
.
2
(
Ü
(2.14),
¿
A^
Ú
n
2.4,
Œ
F
(
x
k
+1
)
−
F
(
x
∗
)
≤
2
η
k
A
>
A
k
[
2
L
1
(
F
(
x
1
)
−
F
(
x
∗
))+
k
(
x
1
−
x
0
)+
σ
(
x
0
−
x
∗
)
k
2
]
(
k
+1)
2
.
-
c
= 2
η
k
A
>
A
k
[
2
L
1
(
F
(
x
1
)
−
F
(
x
∗
))+
k
(
x
1
−
x
0
)+
σ
(
x
0
−
x
∗
)
k
2
],
K
k
(
J
¤
á
.
DOI:10.12677/aam.2023.1241801739
A^
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3.
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Š
¢
!
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ø
˜
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u
¯
K
(1.1)
ê
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,
±
y
²
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{
2.1
k
5
.
¤
k
“
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MATLAB
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ρ
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001,
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5
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.
25
,%
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.
15,
…
n
= 2
N
(
N
∈
Z
+
),
m
= floor(
n/
4),
k
= floor(
m/
8),
d
MATLAB
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)
ÿ
þ
Ý
A
.
d
MATLAB
©
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p=randperm(n);x(p(1:k))=randn(k,1)
)
©
&
Ò
¯
x
.
Ž
{
ªŽ
I
O
•
k
F
k
−
F
k
−
1
k
k
F
k
−
1
k
<
10
−
10
,
Ù
¥
F
k
L
«
¯
K
(1.1)
3
S
“
:
x
k
8
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¼
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3
z
g
ÿ
Á
¥
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Ï
L
e
¡
ú
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é
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RelErr =
k
ˆ
x
−
¯
x
k
k
¯
x
k
,
Ù
¥
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x
L
«
£
Â
&
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ä
k
\
5
p
d
x
D
(
D
Õ
&
Ò
¡
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!
A^
Ž
{
2.1
5
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E
\
5
p
d
x
D
(
‚
¸
[
D
Õ
&
Ò
.
-
n
=2
13
,m
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11
,k
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8
.
ã
1
‰
Ñ
©
&
Ò
!
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þ
&
ÒÚ
d
2.1
Ž
{
-
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Ò
(
ù
:
I
P
).
w
,
,
l
ã
1
1
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Ú
1
n
‡
ã
¥
,
©
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k
ƒ
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:
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L
²
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{
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¡
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©
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Figure1.
SignalrecoveryonadditiveGaussianwhitenoise(
n
= 2
13
)
ã
1.
ä
k
\
5
p
d
x
D
(
D
Õ
&
Ò
¡
E
(
n
= 2
13
)
DOI:10.12677/aam.2023.1241801740
A^
ê
Æ
?
Ð
ƒ
,
,
é
u
Ø
Ó
‘
ê
!
Ø
Ó
&
Ò
¯
K
(1.1),
A^
Ž
{
2.1
?
1
\
5
p
d
x
D
(
‚
¸
[
D
Õ
&
Ò
¡
E
.CPU
ž
m
(CPUTime)
!
S
“g
ê
(Iter)
Ú
ƒ
é
Ø
(RelErr)
3
L
1.
l
L
¥
Œ
±
w
Ñ
,
Ž
{
2.1
U
Ð
¡
E
Ø
Ó
‘
ê
D
Õ
&
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,
…
CPU
ž
m
,
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O
(
,
é
‘
ê
O
\
Ø
¯
a
.
L
²
Ž
{
2.1
k
Ð
-
½
5
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