EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1624-1633
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Image Impulse Noise Reduction Using a Conjugate
Gradient of Alternative Parameter

Hawraz N. Jabbar1, Yeldez J. Subhi2, Basim A. Hassan3,∗

1 Department of Mathematics, College of Science, University of Kirkuk, Iraq
2 Department of Petroleum, College of Engineering, University of Al-Kitab, Iraq
3 Department of Mathematics, College of Computers Sciences and Mathematics,
University of Mosul, Iraq

Abstract. Conjugate gradient approaches emphasise the conjugate formula. This study creates
a new conjugate coefficient for the conjugate gradient approach to restore pictures using Perry’s
conjugacy condition and a quadratic model. Algorithms have global convergence and descent.
The new technique performed better in numerical testing. The new conjugate gradient technique
outperforms the FR method. The new technique performed better in numerical testing. The new
conjugate gradient technique outperforms the FR method.

2020 Mathematics Subject Classifications: 65K10, 49M37, 90C06

Key Words and Phrases: Alternative Parameter gradient, Global convergence, Image restora-
tion problems

1. Introduction

Gradient methods are a type of first-order approach that have been proven to be
effective in solving nonlinear optimization problems, which are often very large in size.
These methods are commonly used in image processing applications.

Adaptive median filter and variational technique benefits are combined in a two-phase
strategy in [1]. The first step for salt-and-pepper noise uses an adaptive median filter [15].
Let X represent the genuine picture and A = {1, 2, 3, · · ·M}×{1, 2, 3, · · ·N} represent X’s
index set. The collection of noise pixel indices discovered in the first phase is denoted by
the symbol N ⊂ A. The current challenge is to reduce the functional as much as feasible:

fα(u) =
∑

(i,j)∈N

[
|ui,j − yi,j |+

β

2
(2× S1

i,j + S2
i,j)

]
(1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4849

Email addresses: hawrazmath@uokirkuk.edu.iq (H. N. Jabbar),
mailto.yeldez.subhi@uoalkitab.edu.iq (Y. J. Subhi), basimah@uomosul.edu.iq (B. A. Hassan)

https://www.ejpam.com 1624 © 2023 EJPAM All rights reserved.



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a regularization parameter β, S1
i,j = 2

∑
(m,n)∈Pi,j∩Ncφα(ui,j−ym,n

, S2
i,j =

∑
(m,n)∈Pi,j∩N ϕα(ui,j − ym,n)

and an edge-preserving potential function φα =
√
α+ x2, α > 0 to improve the accuracy

of a system. Let yi,j represent the observed pixel value of the image at position (i, j), and
let Pi,j represent the set of the pixel’s four closest neighbours at location (i, j) ∈ A and
ui,j = ⌊ui,j⌋(i,j)∈N represent a lexicographically arranged column vector of length c. The
number of items in N is given by c.

The term |ui,j − yi,j | in equation (1) enables noisy pixels to be recognised but provides
a modest bias on damaged pixel restoration [1],[15]. It advises removing the term from
the equation and considering the functional of the following form:

fα(u) =
∑

(i,j)∈N

⌊(2× S1
i,j + S2

i,j)⌋ (2)

One of the important iterative approaches for conjugate gradient (CG) removes impulsive
noise:

f(u∗) = min
x∈RN

f(u) (3)

The conjugate gradient approach is a kind of iterative algorithm that generates a sequence
using the following format:

uk+1 = uk + αkdk (4)

where dk denotes the direction of the search and αk denotes the average step size uncovered
by a reliable exact line search, as in:

αk = −
gTk dk

dTkQdk
(5)

Formula [13] adds details. Under the Wolfe scenario, step length is defined as αk:

f(uk + αkdk) ≤ f(uk) + δαkg
T
k dk

dTk g(uk + αkdk) ≥ σ dTk gk
(6)

where 0 < δ < σ < 1. You may find further information in [11]. The conjugate gradient
method’s formula for choosing the search direction is as follows:

dk+1 = −gk+1 + βkdk (7)

where βk is a scalar.
Two examples of formulas for are the Dai-Yuan (DY) technique [2] and the Fletcher-

Reeves (FR) method [4]. They take the shape of the following:

βFR
k =

∥gk+1∥2

∥gk∥2
, βDY

k+1 =
∥gk+1∥2

dTk yk
(8)

Many studies have been done on the characteristics of convergence shown by conjugate
gradient methods. These investigations got underway with Zoutendijk [16], who showed



1626

how the FR approach converges globally when precise line searching is carried out. Up to
now, a number of researchers have created brand-new conjugate gradient coefficient equa-
tions that perform superbly numerically and lead to the global solution. While this is only
a prototype, the conjugate gradient technique has known more advanced adaptations [3],
[5] and [12]. Accurate descent conjugate gradient methods have been developed developed
in a number of different ways. Examples of CG approaches given by Wu and Chen [14]
include the following:

βWC
k =

yTk+1gk+1

dTk yk
+

2(fk − fk+1) + gTk sk

dTk yk
(9)

Both in terms of their theoretical use and their practical usefulness, these strategies excel.
The calculation of the search direction is the primary point of differentiation between the
conjugate gradient technique and the Wu and Chen algorithm. Please see [6], [9] and [7]
for further information on the optimization techniques and references.

We create a new class of formulas by basing them on the quadratic model, and then
we examine and report on the theoretical properties as well as the numerical performance
of these formulas.

2. A New Parameter For βk

The new parameter will be derived by using Taylor series as defined by:

f(u) = f(uk+1) + gTk+1(u− uk+1) +
1

2
(u− uk+1)

TQ(uk)(u− uk+1) (10)

Now, choosing u = uk, and we can calculate the derivative as:

gk+1 = gk +Q(uk)sk (11)

From (10) and (11), yield:

sTkQ(uk)sk = 2(fk − fk+1) + 2yTk sk + 2gTk sk (12)

Based on (11) and (12), yield: dk+1

−sTk gk+1 = 2(fk+1 − fk)− 2yTk sk − 3gTk sk. (13)

Perry’s conjugecy condition is defined by:

dTk+1yk = −sTk gk+1 (14)

Based on (12) and (14), yield:

βkd
T
k yk = gTk+1yk + 2(fk+1 − fk)− 2yTk sk − 3gTk sk. (15)

As resulted:

βk =
gTk+1yk + 2(fk+1 − fk)− 2yTk sk − 3gTk sk

dTk yk
. (16)

The ways that the aforementioned parameter produced are known as the New.



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3. Global convergence

The aim of this part is to study the global convergence aspects of the method. We
start by doing the following:

Hypotheses

(i) In the level set Ω = {u : u ∈ Rn, f(u) ≤ f(u1)} is bounded.

(ii) In some neighborhood Λ in Ω, the gradient g(u) of the function f(u) is satisfying
Lipchitz condition as:

∥g(t1)− g(t2)∥ ≤ L∥t1 − t2∥,∀t1, t2 ∈ Λ (17)

where L is Lipchitz constant. The Assumption 1 above implies that there exists
µ > 0 such that:

(∇f(r1)−∇f(r2))
T ≥ µ∥r1 − r2∥2, ∀r1, r2 ∈ Rn (18)

See [8] and [10].

Theorem 1. Let our assumption hold. Then:

dTk+1gk+1 ≤ −c∥gk+1∥2. (19)

Proof. If k = 0 after that gT0 d0 = −∥g0∥2. Let dTk gk < 0 for all k. Multiply (7) by
gk+1, we obtain:

dTk+1gk+1 = −gTk+1gk+1 + βks
T
k gk+1 (20)

By substituting (16) into (20) and using (11), we obtain:

dTk+1gk+1 = −∥gk+1∥2 + (
gTk+1yk

sTk yk
−

sTk gk+1

sTk yk
)sTk gk+1 (21)

It’s implies:

dTk+1gk+1 = −∥gk+1∥2 +
gTk+1yks

T
k gk+1

sTk yk
−

(sTk gk+1)
2

sTk yk
(22)

Apply Cauchy-Schwartz inequality wT v ≤ 1
2(∥w∥

2 + ∥v∥2, where w = (yTk sk)gk+1 and
v = (sTk gk+1)yk we get:

gTk+1yks
T
k gk+1

sTk yk
≤

1
2 [∥gk+1∥2(yTk sk)2 + (sTk gk+1)

2∥yk∥2]
(sTk yk)

2
(23)

Putting (23) in (22) we have:

dTk+1gk+1 ≤ −∥gk+1∥2 +
1/2[∥gk+1∥2(yTk sk)2 + (sTk gk+1)

2∥yk∥2]
(sTk yk)

2
−

(sTk gk+1)
2

sTk yk
(24)



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Using (17) in (24), it’s ensures:

dTk+1gk+1 ≤ −1

2
∥gk+1∥2 + [

1

2
L− 1]

(sTk gk+1)
2

sTk yk
(25)

As results:
dTk+1gk+1 ≤ −c∥gk+1∥2 (26)

so it has been proved.

Any conjugate gradient technique with Wolfe line search converges. A weak Zoutendijk
condition [16] is sufficient.

Lemma 1. In the event that the hypotheses are true, let {uk} produced by (3), dk be the
direction of descent, αk meet Wolfe conditions, and if:∑

k≥0

1

∥dk+1∥2
= ∞ (27)

Then
lim
k→∞

inf∥gk∥ = 0. (28)

Theorem 2. New Algorithm converges globally whenever our assumption hold, i.e.:

lim
k→∞

inf∥gk∥ = 0 (29)

Proof. But it holds from (9), that:

∥dk+1∥ = ∥ − gk+1 + βNew
k sk∥ (30)

Putting (16) in (30) by using (11), implies:

∥dk+1∥ = ∥ − gk+1 +
gTk+1yk

dTk yk
sk −

sTk gk+1

dTk yk
sk∥ (31)

Finally, using (17) and (18), it gets as:

c∥dk+1∥ ≤ ∥gk+1∥+
∥gk+1∥L∥sk∥2

µ∥sk∥2
+

∥gk+1∥∥sk∥2

µ∥sk∥2
≤ (1 +

L

µ
+

1

µ
)∥gk+1∥ ≤ [

µ+ L+ 1

µ
]∥gk+1∥

(32)
Therefore, ∑

k≥1

1

∥dk∥2
≥ (

µ

µ+ L+ 1
)
1

Γ

∑
k≥1

1 = ∞ (33)

Applying Lemma, this research concludes that limk→∞ inf∥gk∥ = 0.



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4. Numerical Results

In order to demonstrate how well New reduces salt-and-pepper impulse noise, we pro-
vide some numerical data. We evaluate the FR approach as well as the New method. We
function in this manner. With MATLAB r2017a, each code is present. Then they are
executed by a computer. These are the circumstances that will cause both techniques to
stop:

∥f(uk)∥ ≤ 10−4(1 + |f(uk)|) and
|f(uk)− f(uk−1)|

|f(uk)|
≤ 10−4 (34)

In addition to the test text, the test images include Lena, House, the Cameraman, and
Elaine. Also featured is the test photograph. In order to provide a qualitative assessment
of the performance of the restoration, we use the PSNR (peak signal to noise ratio) in a
manner that is analogous to [1], [15]. The following definition applies to the restoration
performance:

PSNR = 10log10
2552

1
MN

∑
i,j (u

r
i,j − u∗i,j)

2
(35)

Although the pixel values of the original image are represented by uri,j , those of the image
that was restored to its original condition are represented by u∗i,j . In this research, we show
how many function evaluations and iterations are needed to finish the whole denoising
procedure as well as the image’s PSNR results from this method. The new approach
outperforms the FR methodology, which takes a very long time to finish, in terms of
speed. Table (1) may be accessed here, and it contains the supporting data. The PSNR
values produced with the new technique and those acquired with the FR method are quite
comparable to one another.

Table 1: Numerical results of FR, New algorithms.



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Figure 1: Lena image.

Figure 2: House image.



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Figure 3: Elaine image.

Figure 4: Cameraman image.



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5. Conclusions

We also discussed the conjugate gradient approach new, in addition to a newly dis-
covered formula for a conjugate gradient. We were able to find the Wolfe line’s global
convergence using search criteria. It has been shown that new may significantly reduce
the quantity of simulation evaluations of functions and iterations while keeping the visual
quality constant.

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