Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise

Miyoun Jung; Miyoun Jung

doi:10.3934/math.2024294

AIMS Mathematics

2024, Volume 9, Issue 3: 6013-6040. doi: 10.3934/math.2024294

Previous Article Next Article

Research article

Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise

Miyoun Jung ^,

Department of Mathematics, Hankuk University of Foreign Studies, Yongin, 17035, Korea

Received: 23 November 2023 Revised: 19 January 2024 Accepted: 26 January 2024 Published: 01 February 2024
MSC : 68U10, 65K10, 94A08, 49J40

In this article, we propose a novel group-based sparse representation (GSR) model for restoring color images in the presence of multiplicative noise. This model consists of a convex data-fidelity term, and two regularizations including GSR and saturation-value-based total variation (SVTV). The data-fidelity term is suitable for handling heavy multiplicative noise. GSR enables the retention of textures and details while sufficiently removing noise in smooth regions without producing the staircase artifacts engendered by total variation-based models. Furthermore, we introduce a multi-color channel-based GSR that involves coupling between three color channels. This avoids the generation of color artifacts caused by decoupled color channel-based methods. SVTV further improves the visual quality of restored images by diminishing certain artifacts induced by patch-based methods. To solve the proposed nonconvex model and its subproblem, we exploit the alternating direction method of multipliers, which contributes to an efficient iterative algorithm. Numerical results demonstrate the outstanding performance of the proposed model compared to other existing models regarding visual aspect and image quality evaluation values.

Keywords:

Citation: Miyoun Jung. Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise[J]. AIMS Mathematics, 2024, 9(3): 6013-6040. doi: 10.3934/math.2024294

Related Papers:

[1]	Muhammad Ali Khan, Saleem Abdullah, Alaa O. Almagrabi . Analysis of deep learning technique using a complex spherical fuzzy rough decision support model. AIMS Mathematics, 2023, 8(10): 23372-23402. doi: 10.3934/math.20231188
[2]	Tehreem, Harish Garg, Kinza Ayaz, Walid Emam . Multi attribute decision-making algorithms using Hamacher Choquet-integral operators with complex intuitionistic fuzzy information. AIMS Mathematics, 2024, 9(12): 35860-35884. doi: 10.3934/math.20241700
[3]	Tahir Mahmood, Azam, Ubaid ur Rehman, Jabbar Ahmmad . Prioritization and selection of operating system by employing geometric aggregation operators based on Aczel-Alsina t-norm and t-conorm in the environment of bipolar complex fuzzy set. AIMS Mathematics, 2023, 8(10): 25220-25248. doi: 10.3934/math.20231286
[4]	Shichao Li, Zeeshan Ali, Peide Liu . Prioritized Hamy mean operators based on Dombi t-norm and t-conorm for the complex interval-valued Atanassov-Intuitionistic fuzzy sets and their applications in strategic decision-making problems. AIMS Mathematics, 2025, 10(3): 6589-6635. doi: 10.3934/math.2025302
[5]	Muhammad Qiyas, Neelam Khan, Muhammad Naeem, Saleem Abdullah . Intuitionistic fuzzy credibility Dombi aggregation operators and their application of railway train selection in Pakistan. AIMS Mathematics, 2023, 8(3): 6520-6542. doi: 10.3934/math.2023329
[6]	Chunxiao Lu, Zeeshan Ali, Peide Liu . Selection of artificial neutral networks based on cubic intuitionistic fuzzy Aczel-Alsina aggregation operators. AIMS Mathematics, 2024, 9(10): 27797-27833. doi: 10.3934/math.20241350
[7]	Rana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saeed . Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem. AIMS Mathematics, 2021, 6(3): 2732-2755. doi: 10.3934/math.2021167
[8]	Talha Midrar, Saifullah Khan, Saleem Abdullah, Thongchai Botmart . Entropy based extended TOPOSIS method for MCDM problem with fuzzy credibility numbers. AIMS Mathematics, 2022, 7(9): 17286-17312. doi: 10.3934/math.2022952
[9]	Wajid Azeem, Waqas Mahmood, Tahir Mahmood, Zeeshan Ali, Muhammad Naeem . Analysis of Einstein aggregation operators based on complex intuitionistic fuzzy sets and their applications in multi-attribute decision-making. AIMS Mathematics, 2023, 8(3): 6036-6063. doi: 10.3934/math.2023305
[10]	Jian Qi . Artificial intelligence-based intelligent computing using circular q-rung orthopair fuzzy information aggregation. AIMS Mathematics, 2025, 10(2): 3062-3094. doi: 10.3934/math.2025143

Abstract

1. Introduction

Fibonacci polynomials and Lucas polynomials are important in various fields such as number theory, probability theory, numerical analysis, and physics. In addition, many well-known polynomials, such as Pell polynomials, Pell Lucas polynomials, Tribonacci polynomials, etc., are generalizations of Fibonacci polynomials and Lucas polynomials. In this paper, we extend the linear recursive polynomials to nonlinearity, that is, we discuss some basic properties of the bi-periodic Fibonacci and Lucas polynomials.

The bi-periodic Fibonacci $\left \{ f_{n} \left (t\right) \right \}$ and Lucas $\left \{ l_{n} \left (t \right) \right \}$ polynomials are defined recursively by

$\begin{equation*} f_{0}\left (t \right ) = 0, \quad f_{1}\left (t\right ) = 1, \quad {f_{n}\left ( t\right ) = } \begin{cases} ayf_{n-1}\left ( t \right ) +f_{n-2}\left ( t \right ) \quad & n \equiv 0 \pmod{2}, \\ byf_{n-1}\left (t \right ) +f_{n-2}\left (t \right ) \quad & n \equiv 1 \pmod{2}, \end{cases} \quad n\ge 2, \end{equation*}$

and

$\begin{equation*} l_{0}\left ( t \right ) = 2, \quad l_{1}\left (t \right ) = at, \quad {l_{n}\left ( t \right ) = } \begin{cases} byl_{n-1}\left ( t \right ) +l_{n-2}\left ( t \right ) \quad & n \equiv 0 \pmod{2}, \\ ayl_{n-1}\left ( t \right ) +l_{n-2}\left ( t \right ) \quad & n \equiv 1 \pmod{2}, \end{cases} \quad n\ge 2, \end{equation*}$

where $a$ and $b$ are nonzero real numbers. For $t = 1$ , the bi-periodic Fibonacci and Lucas polynomials are, respectively, well-known bi-periodic Fibonacci $\left \{f_{n} \right \}$ and Lucas $\left \{l_{n} \right \}$ sequences. We let

$\begin{equation*} {\varsigma \left ( n \right ) = } \begin{cases} 0 \quad & n \equiv 0 \pmod{2}, \\ 1 \quad & n \equiv 1 \pmod{2}, \end{cases} \quad n\ge 2. \end{equation*}$

In ^[1], the scholars give the Binet formulas of the bi-periodic Fibonacci and Lucas polynomials as follows:

$\begin{equation} f_{n}\left (t \right ) = \frac{a^{\varsigma \left (n+1 \right) }}{ \left( ab \right )^{\left \lfloor \frac{n}{2} \right \rfloor } }\left (\frac{\sigma^{n} \left ( t\right ) -\tau ^{n}\left ( t \right ) }{\sigma\left (t \right ) -\tau \left ( t \right ) }\right), \end{equation}$

(1.1)

and

$\begin{equation} l_{n}\left (t \right ) = \frac{a^{\varsigma \left (n \right) }}{ \left( ab \right )^{\left \lfloor \frac{n+1}{2} \right \rfloor } }\left (\sigma ^{n}\left (t\right ) +\tau ^{n}\left ( t\right ) \right), \end{equation}$

(1.2)

where $n\ge 0$ , $\sigma\left (t \right)$ , and $\tau \left (t \right)$ are zeros of $\lambda ^{2}-abt\lambda -ab$ . This is $\sigma \left (t \right) = \frac{abt +\sqrt{a^{2} b^{2} t^{2}+4ab} }{2}$ and $\tau \left (t \right) = \frac{abt -\sqrt{a^{2} b^{2} t^{2}+4ab} }{2}$ . We note the following algebraic properties of $\sigma\left (t \right)$ and $\tau \left (t \right)$ :

$\begin{equation*} \sigma \left ( t \right ) +\tau \left (t \right ) = abt, \quad \sigma \left (t \right ) -\tau \left ( t \right ) = \sqrt{a^{2} b^{2} t^{2} +4ab}, \quad \sigma\left ( t \right ) \tau \left ( t\right ) = -ab. \end{equation*}$

Many scholars studied the properties of bi-periodic Fibonacci and Lucas polynomials; see ^[2,3,4,5,6]. In addition, many scholars studied the power sums problem of second-order linear recurrences and its divisible properties; see ^[7,8,9,10].

Taking $a = b = 1$ and $t = 1$ , we obtain the Fibonacci $\left \{ F_{n} \right \}$ or Lucas $\left \{ L_{n} \right \}$ sequence. Melham ^[11] proposed the following conjectures:

Conjecture 1. Let $m\ge 1$ be an integer, then the sum

$\begin{equation*} L_{1} L_{3} L_{5}\cdots L_{2m+1}\sum\limits_{k = 1}^{n} F_{2k}^{2m+1} \end{equation*}$

can be represented as $\left (F_{2n+1}-1 \right)^{2}R_{2m-1}\left (F_{2n+1} \right)$ , including $R_{2m-1}\left (t \right)$ as a polynomial with integer coefficients of degree $2m-1$ .

Conjecture 2. Let $m\ge 1$ be an integer, then the sum

$\begin{equation*} L_{1} L_{3} L_{5}\cdots L_{2m+1}\sum\limits_{k = 1}^{n} L_{2k}^{2m+1} \end{equation*}$

can be represented as $\left (L_{2n+1}-1 \right)Q_{2m}\left (L_{2n+1} \right)$ , where $Q_{2m}\left (t \right)$ is a polynomial with integer coefficients of degree $2m$ .

In ^[12], the authors completely solved the Conjecture $2$ and discussed the Conjecture $1$ . Using the definition and properties of bi-periodic Fibonacci and Lucas polynomials, the power sums problem and their divisible properties are studied in this paper. The results are as follows:

Theorem 1. We get the identities

$\begin{align} & \sum\limits_{k = 1}^{n}f_{2k}^{2m+1} \left (t\right ) = \frac{a^{2m+1} }{b\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits _{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m+1}{m-j}\left ( \frac{f_{\left ( 2n+1 \right )\left ( 2j+1\right ) }\left ( t \right ) - f_{2j+1}\left ( t \right ) }{l_{2j+1} \left ( t \right ) } \right ), \end{align}$

(1.3)

$\begin{align} & \sum\limits_{k = 1}^{n}f_{2k+1}^{2m+1} \left ( t \right ) = \frac{\left ( ab \right ) ^{m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits_{j = 0}^{m} \binom{2m+1}{m-j} \left ( \frac{f_{\left ( 2n+2\right )\left (2j+1 \right ) }\left (t\right ) -f_{2\left ( 2j+1 \right ) }\left (t\right ) }{l_{2j+1}\left ( t \right ) } \right ), \end{align}$

(1.4)

$\begin{align} & \sum\limits_{k = 1}^{n}l_{2k}^{2m+1} \left (t \right ) = \sum\limits_{j = 0}^{m} \binom{2m+1}{m-j} \left ( \frac{l_{\left ( 2n+1 \right )\left ( 2j+1 \right ) }\left (t \right ) -l_{2j+1}\left ( t\right ) }{l_{2j+1}\left ( t\right ) } \right ), \end{align}$

(1.5)

$\begin{align} &\sum\limits_{k = 1}^{n}l_{2k+1}^{2m+1} \left (t\right ) = \frac{ a ^{m+1} }{b^{m+1} } \sum\limits_{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m+1}{m-j} \left ( \frac{l_{\left ( 2n+2\right )\left ( 2j+1\right ) }\left ( t \right ) -l_{2\left ( 2j+1 \right ) }\left ( t \right ) }{l_{2j+1}\left ( t \right ) } \right ), \end{align}$

(1.6)

where $n$ and $m$ are positive integers.

Theorem 2. We get the identities

$\begin{align} & \sum\limits_{k = 1}^{n}f_{2k}^{2m} \left ( t \right ) = \frac{a^{2m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits_{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m}{m-j} \frac{f_{2j\left ( 2n+1 \right ) }\left (t \right ) }{f_{2j}\left (t\right ) } -\frac{a^{2m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} }\binom{2m}{m}\left ( -1 \right )^{m}\left ( n+ \frac{1}{2 } \right ) , \end{align}$

(1.7)

$\begin{align} & \sum\limits_{k = 1}^{n}f_{2k+1}^{2m} \left ( t \right ) = \frac{\left ( ab \right ) ^{m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits_{j = 0}^{m} \binom{2m}{m-j} \left ( \frac{f_{2j\left ( 2n+2\right ) }\left ( t \right ) -f_{4j }\left ( t \right ) }{f_{2j}\left ( t \right ) } \right )-\frac{\left ( ab \right ) ^{m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} }\binom{2m}{m}n , \end{align}$

(1.8)

$\begin{align} & \sum\limits_{k = 1}^{n}l_{2k}^{2m} \left ( t \right ) = \sum\limits_{j = 0}^{m} \binom{2m}{m-j} \frac{f_{2j\left ( 2n+1 \right ) }\left ( t \right ) }{l_{2j+1}\left (t \right ) }-2^{2m-1}-\binom{2m}{m}\left ( n+\frac{1}{2} \right ) , \end{align}$

(1.9)

$\begin{align} &\sum\limits_{k = 1}^{n}l_{2k+1}^{2m} \left ( t \right ) = \frac{ a ^{m} }{b^{m} } \sum\limits_{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m}{m-j} \left ( \frac{f_{2j\left ( 2n+2\right ) }\left ( t \right ) -f_{4j }\left (t \right ) }{f_{2j}\left ( t \right ) } \right ) -\frac{a^{m} }{b^{m} } \binom{2m}{m} \left ( -1 \right )^{m}n , \end{align}$

(1.10)

where $n$ and $m$ are positive integers.

As for application of Theorem 1, we get the following:

Corollary 1. We get the congruences:

$\begin{equation} bl_{1} \left ( t \right ) l_{3} \left ( t \right ) \cdots l_{2m+1}\left ( t \right )\sum\limits _{k = 1}^{n}f_{2k}^{2m+1}\left ( t \right ) \equiv 0\pmod{f_{2n+1}\left ( t \right ) -1}, \end{equation}$

(1.11)

and

$\begin{equation} al_{1} \left ( t\right ) l_{3} \left (t \right ) \cdots l_{2m+1}\left ( t \right )\sum\limits _{k = 1}^{n}l_{2k}^{2m+1}\left ( t \right ) \equiv 0\pmod{l_{2n+1}\left (t \right ) -at}, \end{equation}$

(1.12)

where $n$ and $m$ are positive integers.

Taking $t = 1$ in Corollary 1, we have the following conclusions for bi-periodic Fibonacci $\left \{ f_{n} \right \}$ and Lucas $\left \{ l_{n} \right \}$ sequences.

Corollary 2. We get the congruences:

$\begin{equation} bl_{1} l_{3} \cdots l_{2m+1} \sum\limits _{k = 1}^{n}f_{2k}^{2m+1} \equiv 0\pmod{f_{2n+1} -1}, \end{equation}$

(1.13)

and

$\begin{equation} al_{1} l_{3} \cdots l_{2m+1} \sum\limits _{k = 1}^{n}l_{2k}^{2m+1} \equiv 0\pmod{l_{2n+1} -a }, \end{equation}$

(1.14)

where $n$ and $m$ are nonzero real numbers.

Taking $a = b = 1$ and $t = 1$ in Corollary 1, we have the following conclusions for bi-periodic Fibonacci $\left \{ F_{n} \right \}$ and Lucas $\left \{ L_{n} \right \}$ sequences.

Corollary 3. We get the congruences:

$\begin{equation} L_{1} L_{3} \cdots L_{2m+1} \sum\limits _{k = 1}^{n}F_{2k}^{2m+1} \equiv 0\pmod{F_{2n+1} -1}, \end{equation}$

(1.15)

and

$\begin{equation} L_{1} L_{3} \cdots L_{2m+1} \sum\limits _{k = 1}^{n}L_{2k}^{2m+1} \equiv 0\pmod{L_{2n+1} -1 }, \end{equation}$

(1.16)

where $n$ and $m$ are nonzero real numbers.

2. Proofs of theorems

To begin, we will give several lemmas that are necessary in proving theorems.

Lemma 1. We get the congruence

$\begin{equation*} f_{\left ( 2n+1 \right )\left ( 2j+1\right ) } \left ( t \right ) -f_{2j+1 } \left ( t \right ) \equiv 0 \pmod{f_{2n+1 } \left ( t \right )-1}, \end{equation*}$

where $n$ and $m$ are nonzero real numbers.

Proof. We prove it by complete induction for $j\ge 0$ . This clearly holds when $j = 0$ . If $j = 1$ , we note that $abf_{3\left (2n+1 \right) } \left (t \right) = \left (a^{2} b^{2} t^{2} +4ab \right) f_{2n+1}^{3}\left (t \right)-3abf_{2n+1} \left (t\right)$ and we obtain

$\begin{equation*} \begin{split} & f_{3\left ( 2n+1 \right ) }\left ( t \right )-f_{3 }\left ( t \right ) = \left ( abt^{2}+4 \right ) f_{2n+1}^{3}\left ( t\right )-3f_{2n+1} \left (t \right )-\left ( abt^{2}+4 \right )f_{1}^{3} \left (t \right )+3f_{1} \left (t\right ) \\ & = \left ( abt^{2}+4 \right )\left ( f_{2n+1}\left ( t \right )-f_{1}\left ( t \right )\right )\left ( f_{2n+1}^{2}\left (t \right )+ f_{2n+1}\left ( t \right )f_{1}\left ( t \right )+ f_{1}^{2}\left ( t \right ) \right )-3\left ( f_{2n+1} \left ( t\right ) -f_{1}\left ( t \right ) \right ) \\ & = \left ( abt^{2}+4 \right )\left ( f_{2n+1}\left ( t\right )-1\right )\left ( f_{2n+1}^{2}\left (t \right )+ f_{2n+1}\left ( t \right )f_{1}\left ( t \right )+ f_{1}^{2}\left ( t \right ) \right )-3\left ( f_{2n+1} \left ( t \right ) -1 \right ) \\ & \equiv 0 \pmod{f_{2n+1}\left (t \right ) -1 } . \end{split} \end{equation*}$

This is obviously true when $j = 1$ . Assuming that Lemma 1 holds if $j = 1, 2, \dots, k$ , that is,

$\begin{equation*} f_{\left ( 2n+1 \right )\left ( 2j+1\right ) } \left ( t \right ) -f_{2j+1 } \left ( t \right ) \equiv 0 \pmod{f_{2n+1 } \left ( t \right )-1}. \end{equation*}$

If $j = k+1 \ge 2$ , we have

$\begin{equation*} l_{2\left ( 2n+1 \right ) } \left ( t \right ) f_{\left ( 2n+1 \right )\left ( 2j+1 \right ) }\left ( t \right ) = f_{\left ( 2n+1 \right )\left ( 2j+3 \right ) }\left ( t \right ) +abf_{\left ( 2n+1 \right )\left ( 2j-1 \right ) }\left ( t \right ), \end{equation*}$

and

$\begin{equation*} \begin{split} abl_{2\left ( 2n+1 \right ) } \left ( t \right ) = \left ( a^{2} b^{2} t^{2} +4ab \right )f_{2n+1}^{2}\left ( t \right )-2ab \equiv \left ( a^{2} b^{2} t^{2} +4ab \right )f_{1}^{2}\left ( t \right )-2ab \pmod{f_{2n+1}\left ( t \right ) -1 }. \end{split} \end{equation*}$

We have

$\begin{equation*} \begin{split} &f_{\left ( 2n+1 \right )\left ( 2k+3 \right ) } \left (t\right ) -f_{ 2k+3 } \left ( t\right )\\ & = l_{2\left ( 2n+1 \right ) }\left ( t \right ) f_{\left ( 2n+1 \right )\left ( 2k+1 \right ) }\left ( t \right )-abf_{\left (2n+1\right )\left ( 2k-1 \right ) } \left ( t \right ) -l_{2} \left ( t \right )f_{2k+1}\left ( t \right )+abf_{2k-1} \left ( t \right ) \\ & \equiv \left ( \left ( abt^{2}+4 \right )f_{ 1}^{2} \left (t \right ) -2 \right )f_{\left ( 2n+1 \right )\left ( 2k+1 \right ) }\left (t \right )-abf_{\left ( 2n+1 \right )\left ( 2k-1 \right ) } \left (t \right ) \\ &\quad -\left ( \left ( abt^{2}+4 \right )f_{ 1}^{2} \left ( t \right ) -2 \right ) f_{2k+1} \left ( t \right )+abf_{2k-1}\left ( t \right ) \\ & \equiv \left ( \left ( abt^{2}+4 \right )f_{ 1}^{2} \left ( t \right ) -2 \right )\left ( f_{\left ( 2n+1 \right ) \left ( 2k+1 \right ) }\left ( t \right )-f_{2k+1}\left ( t\right ) \right ) -ab\left ( f_{\left ( 2n+1 \right )\left ( 2k-1 \right ) } \left (t \right ) -f_{2k-1}\left ( t \right ) \right ) \\ & \equiv 0 \pmod{f_{2n+1}\left (t \right ) -1 } . \end{split} \end{equation*}$

This completely proves Lemma 1. □

Lemma 2. We get the congruence

$\begin{equation*} al_{\left ( 2n+1 \right )\left ( 2j+1\right ) } \left ( t \right ) -al_{2j+1 } \left ( t \right ) \equiv 0 \pmod{l_{2n+1 } \left ( t \right )-at}, \end{equation*}$

where $n$ and $m$ are nonzero real numbers.

Proof. We prove it by complete induction for $j\ge 0$ . This clearly holds when $j = 0$ . If $j = 1$ , we note that $al_{3\left (2n+1 \right) } \left (t \right) = b l_{2n+1}^{3}\left (t \right)+3al_{2n+1} \left (t \right)$ and we obtain

$\begin{equation*} \begin{split} & a l_{3\left ( 2n+1 \right ) }\left (t \right )-al_{3 }\left ( t \right ) = b l_{2n+1}^{3}\left ( t \right )+3al_{2n+1} \left ( t \right )-bl_{1}^{3} \left (t \right )-3al_{1} \left ( t \right ) \\ & = \left ( l_{2n+1}\left ( t \right )-l_{1}\left ( t \right )\right )\left ( bl_{2n+1}^{2}\left ( t \right )+ bl_{2n+1}\left ( t \right )l_{1}\left ( t \right )+ bl_{1}^{2}\left (t \right ) \right )-3a\left ( l_{2n+1} \left ( t \right ) -l_{1}\left ( t\right ) \right ) \\ & = \left ( l_{2n+1}\left ( t \right )-at\right )\left ( bl_{2n+1}^{2}\left ( t \right )+ bayl_{2n+1}\left ( t \right )+ ba^{2}t^{2} \right )-3a\left ( l_{2n+1} \left ( t \right ) -at \right )\\ & \equiv 0 \pmod{l_{2n+1}\left ( t \right ) -at } . \end{split} \end{equation*}$

This is obviously true when $j = 1$ . Assuming that Lemma 2 holds if $j = 1, 2, \dots, k$ , that is,

$\begin{equation*} al_{\left ( 2n+1 \right )\left ( 2j+1\right ) } \left (t \right ) -al_{2j+1 } \left ( t \right ) \equiv 0 \pmod{l_{2n+1 } \left ( t \right )-at}. \end{equation*}$

If $j = k+1 \ge 2$ , we have

$\begin{equation*} l_{2\left ( 2n+1 \right ) } \left (t \right ) l_{\left ( 2n+1 \right )\left ( 2j+1 \right ) }\left ( t\right ) = l_{\left ( 2n+1 \right )\left ( 2j+3 \right ) }\left ( t\right ) +l_{\left ( 2n+1 \right )\left ( 2j-1 \right ) }\left ( t \right ), \end{equation*}$

and

$\begin{equation*} \begin{split} & al_{2\left ( 2n+1 \right ) } \left ( t \right ) = bl_{2n+1}^{2}\left ( t \right )+2a \equiv bl_{1}^{2}\left ( t \right )+2a \pmod{l_{2n+1}\left ( t \right ) -at }. \end{split} \end{equation*}$

We have

$\begin{equation*} \begin{split} &al_{\left ( 2n+1 \right )\left ( 2k+3 \right ) } \left ( t\right ) -al_{\left ( 2k+3 \right ) } \left ( t \right )\\ & = a\left ( l_{2\left ( 2n+1 \right ) }\left ( t \right ) l_{\left ( 2n+1 \right )\left ( 2k+1 \right ) }\left (t \right )-l_{\left (2n+1\right )\left ( 2k-1 \right ) } \left ( t \right ) \right )-a\left ( l_{2} \left (t \right )l_{2k+1}\left ( t \right )-l_{2k-1} \left (t \right ) \right ) \\ & \equiv \left ( bl_{ 1}^{2} \left ( t \right ) +2a \right )l_{\left ( 2n+1 \right )\left ( 2k+1 \right ) }\left ( t \right )-al_{\left ( 2n+1 \right )\left ( 2k-1 \right ) } \left ( t \right ) -\left ( bl_{ 1}^{2} \left ( t \right ) +2a \right ) l_{2k+1} \left ( t \right )+al_{2k-1}\left ( t \right ) \\ & \equiv \left ( abt^{2} +2 \right )\left ( al_{\left ( 2n+1 \right ) \left ( 2k+1 \right ) }\left ( t\right )-al_{2k+1}\left ( t \right ) \right ) - \left ( al_{\left ( 2n+1 \right )\left ( 2k-1 \right ) } \left ( t\right ) -al_{2k-1}\left (t \right ) \right ) \\ & \equiv 0 \pmod{l_{2n+1}\left ( t \right ) -at } . \end{split} \end{equation*}$

This completely proves Lemma 2. □

Proof of Theorem 1. We only prove (1.3), and the proofs for other identities are similar.

$\begin{equation*} \begin{split} \sum\limits _{k = 1}^{n}f_{2k}^{2m+1}\left ( t \right )& = \sum\limits_{k = 1}^{n}\left (\frac{a^{\varsigma \left ( 2k+1 \right ) } }{\left (ab \right )^{ \left \lfloor \frac{2k}{2} \right \rfloor} }\cdot \left ( \frac{\sigma ^{2k}\left ( t \right ) - \tau ^{2k} \left (t \right ) }{\sigma\left ( t \right ) - \tau \left (t \right ) } \right ) \right ) ^{2m+1} \\ & = \frac{a^{2m+1} }{\left ( \sigma\left (t \right ) -\tau \left ( t \right ) \right )^{2m+1} }\sum\limits _{k = 1}^{n}\frac{\left ( \sigma ^{2k}\left (t \right ) - \tau ^{2k} \left ( t \right ) \right )^{2m+1} }{\left ( ab \right )^{\left ( 2m+1 \right )k } } \\ & = \frac{a^{2m+1} }{\left ( \sigma\left (t\right ) -\tau \left ( t \right ) \right )^{2m+1} }\sum\limits _{k = 1}^{n}\sum\limits _{j = 0}^{2m+1}\left ( -1 \right )^{j} \binom{2m+1}{j} \frac{ \sigma^{2k\left ( 2m+1-j \right ) } \left ( t \right ) \tau ^{2kj} \left ( t \right ) }{\left ( ab \right )^{\left ( 2m+1 \right )k } } \\ & = \frac{a^{2m+1} }{\left ( \sigma\left ( t \right ) -\tau \left (t \right ) \right )^{2m+1} }\sum\limits _{j = 0}^{2m+1}\left ( -1 \right )^{j} \binom{2m+1}{j} \left ( \frac{1-\frac{\sigma ^{2n\left ( 2m+1-2j \right ) }\left (t \right ) }{\left ( ab \right )^{\left ( 2m+1-2j \right )n } } }{\frac{\left ( ab \right )^{2m+1-2j} }{\sigma ^{2\left ( 2m+1-2j \right ) }\left (t \right ) } -1} \right )\\ & = \frac{a^{2m+1} }{\left (\sigma\left (t \right ) -\tau \left (t\right ) \right )^{2m+1} }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m+1}{j} \left ( \frac{1-\frac{\sigma ^{2n\left ( 2m+1-2j \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( 2m+1-2j \right )n } } }{\frac{\left ( ab \right )^{2m+1-2j} }{\sigma ^{2\left ( 2m+1-2j \right ) }\left (t \right ) } -1} - \frac{1-\frac{\sigma ^{2n\left ( 2j-1-2m \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( 2j-1-2m\right )n } } }{\frac{\left ( ab \right )^{2j-1-2m} }{\sigma ^{2\left ( 2j-1-2m \right ) }\left ( t \right ) } -1}\right )\\ & = \frac{a^{2m+1} }{\left ( \sigma\left ( t \right ) -\tau \left ( t \right ) \right )^{2m+1} }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m+1}{j} \left ( \frac{\frac{\sigma^{2 \left ( 2m+1-2j \right ) }\left ( t\right ) }{\left ( ab \right )^{ 2m+1-2j } }- \frac{\sigma ^{\left ( 2n+2 \right ) \left ( 2m+1-2j \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( n+1 \right ) \left ( 2m+1-2j \right ) } } +1- \frac{\sigma ^{2n \left ( 2j-1-2m \right ) }\left (t \right ) }{\left ( ab \right )^{ \left ( 2j-1-2m \right ) n }} }{1- \frac{\sigma^{2 \left ( 2m+1-2j \right ) }\left ( t \right ) }{\left ( ab \right )^{ \left ( 2m+1-2j \right ) }}} \right )\\ & = \frac{a^{2m+1} }{\left ( \sigma\left ( t \right ) -\tau \left (t \right ) \right )^{2m+1} }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m+1}{j}\\ & \quad \times \left ( \frac{\sigma^{2m+1-2j}\left ( t \right ) - \tau ^{2m+1-2j}\left ( t \right ) -\frac{\sigma^{\left ( 2n+1 \right ) \left ( 2m+1-2j \right ) }\left ( t \right )}{\left ( ab \right )^{\left ( 2m+1-2j \right )n } } +\frac{\tau ^{\left ( 2n+1 \right ) \left ( 2m+1-2j \right ) }\left (t \right )}{\left ( ab \right )^{ \left ( 2m+1-2j \right )n } }}{-\sigma ^{2m+1-2j}\left ( t \right ) -\tau ^{2m+1-2j}\left ( t \right ) } \right ) \\ & = \frac{a^{2m+1} }{b\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits _{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m+1}{m-j}\left ( \frac{f_{\left ( 2n+1 \right )\left ( 2j+1\right ) }\left (t \right ) - f_{2j+1}\left (t \right ) }{l_{2j+1} \left ( t \right ) } \right ) . \end{split} \end{equation*}$

□

Proof of Theorem 2. We only prove (1.7), and the proofs for other identities are similar.

$\begin{equation*} \begin{split} \sum\limits _{k = 1}^{n}f_{2k}^{2m }\left ( t \right )& = \sum\limits_{k = 1}^{n}\left (\frac{a^{\varsigma \left ( 2k +1\right ) } }{\left (ab \right )^{ \left \lfloor \frac{2k}{2} \right \rfloor} }\cdot \left ( \frac{\sigma ^{2k}\left ( t\right ) - \tau ^{2k} \left ( t \right ) }{\sigma\left ( t\right ) - \tau \left (t \right ) } \right ) \right ) ^{2m} \\ & = \frac{a^{2m} }{\left ( \sigma\left (t \right ) -\tau \left (t \right ) \right )^{2m} }\sum\limits _{k = 1}^{n}\frac{\left ( \sigma^{2k}\left ( t\right ) - \tau ^{2k} \left ( t \right ) \right )^{2m} }{\left ( ab \right )^{ 2mk } } \\ & = \frac{a^{2m} }{\left ( \sigma\left ( t \right ) -\tau \left (t \right ) \right )^{2m} }\sum\limits _{k = 1}^{n}\sum\limits _{j = 0}^{2m}\left ( -1 \right )^{j} \binom{2m}{j} \frac{ \sigma^{2k\left ( 2m-j \right ) } \left (t \right ) \tau ^{2kj} \left ( t \right ) }{\left ( ab \right )^{2mk } }\\ & = \frac{a^{2m} }{\left (\sigma\left (t \right ) -\tau \left ( t \right ) \right )^{2m} }\sum\limits _{j = 0}^{2m}\left ( -1 \right )^{j} \binom{2m }{j} \left ( \frac{1-\frac{\sigma ^{2n\left ( 2m-2j \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( 2m-2j \right )n } } }{\frac{\left ( ab \right )^{2m-2j} }{\sigma ^{2\left ( 2m-2j \right ) }\left ( t \right ) } -1} \right ) \end{split} \end{equation*}$

$\begin{equation*} \begin{split} & = \frac{a^{2m} }{\left (\sigma\left (t \right ) -\tau \left ( t \right ) \right )^{2m} }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m}{j} \left ( \frac{1-\frac{\sigma ^{2n\left ( 2m-2j \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( 2m-2j \right )n } } }{\frac{\left ( ab \right )^{2m-2j} }{\sigma ^{2\left ( 2m-2j \right ) }\left ( t \right ) } -1} +\frac{1-\frac{\sigma ^{2n\left ( 2j-2m \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( 2j-2m\right )n } } }{\frac{\left ( ab \right )^{2j-2m} }{\sigma ^{2\left ( 2j-2m \right ) }\left (t\right ) } -1}\right )\\ & \quad +\frac{a^{2m} }{\left ( \sigma \left ( t \right ) -\tau \left (t \right ) \right )^{2m} } \left ( -1 \right ) ^{m+1}\binom{2m}{m}n\\ & = \frac{a^{2m} }{\left (\sigma\left ( t \right ) -\tau \left (t \right ) \right )^{2m} }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m}{j} \left ( \frac{\frac{\sigma ^{2 \left ( 2m -2j \right ) }\left ( t \right ) }{\left ( ab \right )^{ 2m -2j } }- \frac{\sigma ^{\left ( 2n+2 \right ) \left ( 2m -2j \right ) }\left ( t \right ) }{\left ( ab \right )^{\left ( n+1 \right ) \left ( 2m -2j \right ) } } -1+\frac{\sigma^{2n \left ( 2j -2m \right ) }\left ( t \right ) }{\left ( ab \right )^{ \left ( 2j -2m \right ) n }} }{1- \frac{\sigma^{2 \left ( 2m -2j \right ) }\left ( t \right ) }{\left ( ab \right )^{ 2m-2j }}} \right )\\ & \quad +\frac{a^{2m} }{\left (\sigma \left ( t \right ) -\tau \left ( t \right ) \right )^{2m} } \left ( -1 \right ) ^{m+1}\binom{2m}{m}n \\ & = \frac{a^{2m } }{\left ( \sigma\left ( t \right ) -\tau \left ( t \right ) \right )^{2m } }\sum\limits _{j = 0}^{m}\left ( -1 \right )^{j} \binom{2m }{j} \left ( \frac{\sigma^{2m -2j}\left ( t \right ) - \tau ^{2m -2j}\left (t\right ) -\frac{\sigma^{\left ( 2n+1 \right ) \left ( 2m -2j \right ) }\left (t \right )}{\left ( ab \right )^{n\left ( 2m -2j \right ) } } +\frac{\tau ^{\left ( 2n+1 \right ) \left ( 2m-2j \right ) }\left (t \right )}{\left ( ab \right )^{n\left ( 2m-2j \right ) } }}{ \tau ^{2m -2j}\left ( t \right )-\sigma ^{2m -2j}\left ( t \right ) } \right ) \\ & \quad +\frac{a^{2m} }{\left ( \sigma \left (t \right ) -\tau \left ( t \right ) \right )^{2m} } \left ( -1 \right ) ^{m+1}\binom{2m}{m}n\\ & = \frac{a^{2m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \sum\limits _{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m}{m-j}\left ( \frac{f_{2j\left ( 2n+1 \right ) }\left ( t \right ) - f_{ 2j}\left (t \right ) }{f_{ 2j} \left (t \right ) } \right ) +\frac{a^{2m} }{\left ( a^{2} b^{2} t^{2} +4ab \right )^{m} } \left ( -1 \right ) ^{m+1}\binom{2m}{m}n . \end{split} \end{equation*}$

□

Proof of Corollary 1. First, from the definition of $f_{n}\left (t \right)$ and binomial expansion, we easily prove $\left (f_{2n+1}\left (t \right)-1, a^{2} b^{2} t^{2} +4ab \right) = 1$ . Therefore, $\left (f_{2n+1}\left (t \right)-1, \left (a^{2} b^{2} t^{2} +4ab \right)^{m} \right) = 1$ . Now, we prove (1.11) by Lemma 1 and (1.3):

$\begin{equation*} \begin{split} &b l_{1} \left ( t\right ) l_{3} \left ( t \right ) \cdots l_{2m+1}\left ( t \right )\sum\limits _{k = 1}^{n}f_{2k}^{2m+1}\left ( t\right ) \\ & = l_{1} \left ( t\right ) l_{3} \left (t \right ) \cdots l_{2m+1}\left ( t \right ) \left ( \frac{a^{2m+1} }{\left ( \sigma\left (t \right ) -\tau \left (t \right ) \right )^{2m} } \sum\limits_{j = 0}^{m}\left ( -1 \right )^{m-j} \binom{2m+1}{m-j} \left ( \frac{f_{\left ( 2n+1 \right )\left ( 2j+1 \right ) }\left ( t \right ) -f_{2j+1}\left ( t \right ) }{l_{2j+1}\left (t\right ) } \right ) \right )\\ & \equiv 0\pmod{f_{2n+1}\left ( t \right ) -1}. \end{split} \end{equation*}$

Now, we use Lemma 2 and (1.5) to prove (1.12):

$\begin{equation*} \begin{split} &al_{1} \left ( t\right ) l_{3} \left ( t \right ) \cdots l_{2m+1}\left ( t\right )\sum\limits _{k = 1}^{n}l_{2k}^{2m+1}\left ( t \right ) \\ & = l_{1} \left ( t \right ) l_{3} \left (t \right ) \cdots l_{2m+1}\left ( t \right ) \left ( \sum\limits_{j = 0}^{m} \binom{2m+1}{m-j} \left ( \frac{al_{\left ( 2n+1 \right )\left ( 2j+1 \right ) }\left ( t\right ) -al_{2j+1}\left (t \right ) }{l_{2j+1}\left ( t \right ) } \right ) \right )\\ & \equiv 0\pmod{l_{2n+1}\left ( t \right ) -at}. \end{split} \end{equation*}$

□

3. Conclusions

In this paper, we discuss the power sums of bi-periodic Fibonacci and Lucas polynomials by Binet formulas. As corollaries of the theorems, we extend the divisible properties of the sum of power of linear Fibonacci and Lucas sequences to nonlinear Fibonacci and Lucas polynomials. An open problem is whether we extend the Melham conjecture to nonlinear Fibonacci and Lucas polynomials.

Use of AI tools declaration

The authors declare that they did not use Artificial Intelligence (AI) tools in the creation of this paper.

Acknowledgments

The authors would like to thank the editor and referees for their helpful suggestions and comments, which greatly improved the presentation of this work. All authors contributed equally to the work, and they have read and approved this final manuscript. This work is supported by Natural Science Foundation of China (12126357).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1]	C. J. Oliver, S. Quegan, Understanding synthetic aperture radar images, SciTech Publishing, Inc., Raleigh, NC, 2004.
[2]	R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, Statistics of speckle in ultrasound B-scans, IEEE Trans. Sonics Ultrason., 30 (1983), 156–163. https://doi.org/10.1109/T-SU.1983.31404 doi: 10.1109/T-SU.1983.31404
[3]	J. M. Schmitt, S. Xiang, K. M. Yung, Speckle in optical coherence tomography, J. Biomed. Opt., 4 (1999), 95–105. https://doi.org/10.1117/1.429925 doi: 10.1117/1.429925
[4]	J. W. Goodman, Some fundamental properties of speckle, J. Opt. Soc. Amer., 66 (1976), 1145–1150. https://doi.org/10.1364/JOSA.66.001145 doi: 10.1364/JOSA.66.001145
[5]	J. S. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980), 165–168. https://doi.org/10.1109/TPAMI.1980.4766994 doi: 10.1109/TPAMI.1980.4766994
[6]	V. S. Frost, J. A. Stiles, K. S. Shanmugan, J. C. Holtzman, A model for radar images and its application to adaptive digital filtering of multiplicative noise, IEEE Trans. Pattern Anal. Mach. Intell., PAMI-4 (1982), 157–166. https://doi.org/10.1109/TPAMI.1982.4767223 doi: 10.1109/TPAMI.1982.4767223
[7]	Y. Yu, S. T. Acton, Speckle reducing anisotropic diffusion, IEEE Trans. Image Process., 11 (2002), 1260–1270. https://doi.org/10.1109/TIP.2002.804276 doi: 10.1109/TIP.2002.804276
[8]	K. Krissian, C. F. Westin, R. Kikinis, K. G. Vosburgh, Oriented speckle reducing anisotropic diffusion, IEEE Trans. Image Process., 16 (2007), 1412–1424. https://doi.org/10.1109/TIP.2007.891803 doi: 10.1109/TIP.2007.891803
[9]	S. Parrilli, M. Poderico, C. V. Angelino, L. Verdoliva, A nonlocal SAR image denoising algorithm based on LLMMSE wavelet shrinkage, IEEE Trans. Geosci. Remote Sens., 50 (2012), 606–616. https://doi.org/10.1109/TGRS.2011.2161586 doi: 10.1109/TGRS.2011.2161586
[10]	G. Aubert, J. F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), 925–946. https://doi.org/10.1137/060671814 doi: 10.1137/060671814
[11]	J. Shi, S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci., 1 (2008), 294–321. https://doi.org/10.1137/070689954 doi: 10.1137/070689954
[12]	Y. M. Huang, M. K. Ng, Y. W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20–40. https://doi.org/10.1137/080712593 doi: 10.1137/080712593
[13]	G. Steidl, T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting methods, J. Math. Imaging Vis., 36 (2010), 168–184. https://doi.org/10.1007/s10851-009-0179-5 doi: 10.1007/s10851-009-0179-5
[14]	S. Yun, H. Woo, A new multiplicative denoising variational model based on m-th root transformation, IEEE Trans. Image Process., 21 (2012), 2523–2533. https://doi.org/10.1109/TIP.2012.2185942 doi: 10.1109/TIP.2012.2185942
[15]	Y. M. Huang, L. Moisan, M. K. Ng, T. Zeng, Multiplicative noise removal via a learned dictionary, IEEE Trans. Image Process., 21 (2012), 4534–4543. https://doi.org/10.1016/j.dsp.2015.12.012 doi: 10.1016/j.dsp.2015.12.012
[16]	Y. Dong, T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), 1598–1625. https://doi.org/10.1137/120870621 doi: 10.1137/120870621
[17]	M. Kang, S. Yun, H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM J. Imaging Sci., 6 (2013), 875–903. https://doi.org/10.1137/11086077X doi: 10.1137/11086077X
[18]	Z. Li, Y. Lou, T. Zeng, Variational multiplicative noise removal by DC programming, J. Sci. Comput., 68 (2016), 1200–1216. https://doi.org/10.1007/s10915-016-0175-z doi: 10.1007/s10915-016-0175-z
[19]	J. Lu, L. Shen, C. Xu, Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Appl. Comput. Harmon. Anal., 41 (2016), 518–539. https://doi.org/10.1016/j.acha.2015.10.003 doi: 10.1016/j.acha.2015.10.003
[20]	H. Na, M. Kang, M. Jung, M. Kang, An exp model with spatially adaptive regularization parameters for multiplicative noise removal, J. Sci. Comput., 75 (2018), 478–509. https://doi.org/10.1007/s10915-017-0550-4 doi: 10.1007/s10915-017-0550-4
[21]	H. Na, M. Kang, M. Jung, M. Kang, Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters, Inverse Probl. Imaging, 13 (2019), 117–147. https://doi.org/10.3934/ipi.2019007 doi: 10.3934/ipi.2019007
[22]	X. Liu, J. Lu, L. Shen, C. Xu, Y. Xu, Multiplicative noise removal: Nonlocal low-rank model and its proximal alternating reweighted minimization algorithm, SIAM J. Imaging Sci., 13 (2020), 1595–1629. https://doi.org/10.1137/20M1313167 doi: 10.1137/20M1313167
[23]	W. Wang, M. Yao, M. K. Ng, Color image multiplicative noise and blur removal by saturation-value total variation, Appl. Math. Model., 90 (2021), 240–264. https://doi.org/10.1016/j.apm.2020.08.052 doi: 10.1016/j.apm.2020.08.052
[24]	M. Jung, Saturation-value based higher-order regularization for color image restoration, Multidim. Syst. Sign. P., 34 (2023), 365–394. https://doi.org/10.1007/s11045-023-00867-x doi: 10.1007/s11045-023-00867-x
[25]	L. I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259–268. https://doi.org/10.1016/0167-2789(92)90242-F doi: 10.1016/0167-2789(92)90242-F
[26]	G. Sapiro, Vector-valued active contours, Proceedings of CVPR, 680–685, San Francisco, CA, USA, 1996. https://doi.org/10.1109/CVPR.1996.517146
[27]	P. Blomgren, T. F. Chan, Total variation methods for restoration of vector valued images, IEEE T. Image Process., 7 (1998), 304–309. https://doi.org/10.1109/83.661180 doi: 10.1109/83.661180
[28]	R. Kimmel, R. Malladi, N. Sochen, Images as embedded maps and minimal surfaces: Movies, color, texture, and volumetric medical images, Int. J. Comput. Vis., 39 (2000), 111–129. https://doi.org/10.1023/A:1008171026419 doi: 10.1023/A:1008171026419
[29]	T. Chan, S. Kang, J. Shen, Total variation denoising and enhancement of color images based on the CB and HSV color models, J. Vis. Commun. Image R., 12 (2001), 422–435. https://doi.org/10.1006/jvci.2001.0491 doi: 10.1006/jvci.2001.0491
[30]	H. Attouch, G. Buttazzo, G. Michaille, Variational analysis in Sobolev and BV spaces: Applications to pdes and optimization, MPS-SIAM Ser. Optim., 6, SIAM, Philadelphia, 2006.
[31]	X. Bresson, T. F. Chan, Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse Probl. Imag., 2 (2008), 255–284. https://doi.org/10.3934/ipi.2008.2.455 doi: 10.3934/ipi.2008.2.455
[32]	Y. Wen, M. Ng, Y. Huang, Efficient total variation minimization methods for color image restoration, IEEE T. Image Process., 17 (2008), 2081–2088. https://doi.org/10.1109/TIP.2008.2003406 doi: 10.1109/TIP.2008.2003406
[33]	P. Rodriguez, B. Wohlberg, A generalized vector-valued total variation algorithm, In: Image Processing (ICIP 2009), Cairo, Egypt, 2009, 1309–1312. https://doi.org/0.1109/ICIP.2009.5413587
[34]	C. Wu, X. C. Tai, Augmented lagrangian method, dual methods, and split bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300–339. https://doi.org/10.1137/090767558 doi: 10.1137/090767558
[35]	S. Ono, I. Yamada, Decorrelated vectorial total variation, Proceedings of CVPR, Columbus, OH, USA, 2014, 4090–4097. https://doi.org/10.1109/CVPR.2014.521
[36]	R. Bergmann, R. H. Chan, R. Hielscher, J. Persch, G. Steidl, Restoration of manifold-valued images by half-quadratic minimization, Inverse Probl. Imag., 10 (2016), 281–304. https://doi.org/10.3934/ipi.2016001 doi: 10.3934/ipi.2016001
[37]	J. Duran, M. Moeller, C. Sbert, D. Cremers, Collaborative total variation: A general framework for vectorial TV models, SIAM J. Imaging Sci., 9 (2016), 116–151. https://doi.org/10.1137/15M102873X doi: 10.1137/15M102873X
[38]	Z. Jia, M. K. Ng, W. Wang, Color image restoration by saturation-value total variation, SIAM J. Imaging Sci., 12 (2019), 972–1000. https://doi.org/10.1137/18M1230451 doi: 10.1137/18M1230451
[39]	A. Buades, B. Coll, J. M. Morel, A non-local algorithm for image denoising, Proceedings of CVPR, San Diego, CA, USA, 2005, 60–65. https://doi.org/10.1109/CVPR.2005.38
[40]	K. Dabov, A. Foi, V. Katkovnik, K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE T. Image Process., 16 (2007), 2080–2095. https://doi.org/10.1109/TIP.2007.901238 doi: 10.1109/TIP.2007.901238
[41]	S. Kindermann, S. Osher, P. W. Jones, Deblurring and denoising of images by nonlocal functionals, Multiscale Model. Sim., 4 (2005), 1091–1115. https://doi.org/10.1137/050622249 doi: 10.1137/050622249
[42]	A. Elmoataz, O. Lezoray, S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing, IEEE T. Image Process., 17 (2008), 1047–1060. https://doi.org/10.1109/TIP.2008.924284 doi: 10.1109/TIP.2008.924284
[43]	G. Peyré, Image processing with nonlocal spectral bases, Multiscale Model. Sim., 7 (2008), 703–730. https://doi.org/10.1137/07068881X doi: 10.1137/07068881X
[44]	X. Zhang, M. Burger, X. Bresson, S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3 (2010), 253–276. https://doi.org/10.1137/090746379 doi: 10.1137/090746379
[45]	M. Jung, X. Bresson, T. F. Chan, L. A. Vese, Nonlocal Mumford-Shah regularizers for color image restoration, IEEE T. Image Process., 20 (2011), 1583–1598. https://doi.org/10.1109/TIP.2010.2092433 doi: 10.1109/TIP.2010.2092433
[46]	M. Elad, M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE T. Image Process., 15 (2006), 3736–3745. https://doi.org/10.1109/TIP.2008.2008065 doi: 10.1109/TIP.2008.2008065
[47]	J. Mairal, F. Bach, J. Ponce, G. Sapiro, A. Zisserman, Non-local sparse models for image restoration, Proceedings of ICCV, Tokyo, Japan, 2009, 2272–2279. https://doi.org/10.1109/ICCV.2009.5459452
[48]	W. Dong, L. Zhang, G. Shi, X. Wu, Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization, IEEE T. Image Process., 20 (2011), 1838–1857. https://doi.org/10.1109/TIP.2011.2108306 doi: 10.1109/TIP.2011.2108306
[49]	W. Dong, L. Zhang, G. Shi, X. Li, Nonlocally centralized sparse representation for image restoration, IEEE T. Image Process., 22 (2013), 1620–1630. https://doi.org/10.1109/TIP.2012.2235847 doi: 10.1109/TIP.2012.2235847
[50]	W. Dong, G. Shi, X. Li, Nonlocal image restoration with bilateral variance estimation: A low-rank approach, IEEE T. Image Process., 22 (2013), 700–711. https://doi.org/10.1109/TIP.2012.2221729 doi: 10.1109/TIP.2012.2221729
[51]	S. Gu, L. Zhang, W. Zuo, X. Feng, Weighted nuclear norm minimization with application to image denoising, Proceedings of CVPR, Columbus, OH, USA, 2014, 2862–2869. https://doi.org/10.1109/CVPR.2014.366
[52]	T. Huang, W. Dong, X. Xie, G. Shi, X. Bai, Mixed noise removal via Laplacian scale mixture modeling and nonlocal low-rank approximation, IEEE T. Image Process., 26 (2017), 3171–3186. https://doi.org/10.1109/TIP.2017.2676466 doi: 10.1109/TIP.2017.2676466
[53]	J. Zhang, D. Zhao, W. Gao, Group-based sparse representation for image restoration, IEEE T. Image Process., 23 (2014), 3336–3351. https://doi.org/10.1109/TIP.2014.2323127 doi: 10.1109/TIP.2014.2323127
[54]	J. Zhang, S. Ma, Y. Zhang, W. Gao, Image deblocking using group-based sparse representation and quantization constraint prior, Proceedings of ICIP, Quebec City, QC, Canada, 2015,306–310. https://doi.org/10.1109/ICIP.2015.7350809
[55]	W. Shi, C. Chen, F. Jiang, D. Zhao, W. Shen, Group-based sparse representation for low lighting image enhancement, Proceedings of ICIP, Phoenix, AZ, USA, 2016, 4082–4086. https://doi.org/10.1109/ICIP.2016.7533127
[56]	S. Liu, G. Zhang, Y. T. Soon, An over-complete dictionary design based on GSR for sar image despeckling, IEEE Geosci. Remote Sens. Lett., 14 (2017), 2230–2234. https://doi.org/10.1109/LGRS.2017.2758397 doi: 10.1109/LGRS.2017.2758397
[57]	S. Lee, M. Kang, Group sparse representation for restoring blurred images with Cauchy noise, J. Sci. Comput., 83 (2020), 41. https://doi.org/10.1007/s10915-020-01227-8 doi: 10.1007/s10915-020-01227-8
[58]	Z. Zha, X. Yuan, B. Wen, J. Zhou, J. Zhang, C. Zhu, A Benchmark for sparse coding: When group sparsity meets rank minimization, IEEE T. Image Process., 29 (2020), 5094–5109. https://doi.org/10.1109/TIP.2020.2972109 doi: 10.1109/TIP.2020.2972109
[59]	Z. Zha, X. Yuan, B. Wen, J. Zhang, J. Zhou, C. Zhu, Image restoration using joint patch-group based sparse representation, IEEE T. Image Process., 29 (2020), 7735–7750. https://doi.org/10.1109/TIP.2020.3005515 doi: 10.1109/TIP.2020.3005515
[60]	Z. Zha, X. Yuan, B. Wen, J. Zhou, C. Zhu, Group sparsity residual constraint with non-local priors for image restoration, IEEE T. Image Process., 29 (2020), 8960–8975. https://doi.org/10.1109/TIP.2020.3021291 doi: 10.1109/TIP.2020.3021291
[61]	Y. Kong, C. Zhou, C. Zhang, L. Sun, C. Zhou, Multi-color channels based group sparse model for image restoration, Algorithms, 15 (2022), 176. https://doi.org/10.3390/a15060176 doi: 10.3390/a15060176
[62]	S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1–122. https://doi.org/10.1561/2200000016 doi: 10.1561/2200000016
[63]	Y. Wang, W. Yin, J. Zeng, Global convergence of ADMM in nonconvex nonsmooth optimization, J. Sci. Comput., 78 (2019), 1–2. https://doi.org/10.1007/s10915-018-0757-z doi: 10.1007/s10915-018-0757-z
[64]	J. Mota, J. Xavier, P. Aguiar, M. Puschel, A proof of convergence for the alternating direction method of multipliers applied to polyhedral-constrained functions, arXiv.1112.2295, 2011.
[65]	Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600–612. https://doi.org/10.1109/TIP.2003.819861 doi: 10.1109/TIP.2003.819861

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)