Adaptive Fractional-Order Total Variation and Minimax-Concave Based Image Denoising Model

Abstract

Total variation (TV)-based image denoising effectively suppresses noise while preserving edges, but it often introduces staircase artifacts in flat regions. To address this limitation, we propose a novel denoising model that combines adaptive fractional-order total variation with a minimax-concave (MC) penalty in the regularization term. The adaptive fractional-order TV alleviates staircase effects in homogeneous areas while preserving fine details in textured regions. The MC penalty provides a more accurate estimation of image sparsity, improving restoration fidelity compared to traditional L1-based regularization. The resulting model, termed AFTVMC, is efficiently solved using an alternating direction method of multipliers (ADMM). Extensive numerical experiments on synthetic and natural images demonstrate that AFTVMC outperforms classical TV, higher-order LLT, adaptive ATV, and state-of-the-art MCFOTV models in both objective metrics—peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM)—and subjective visual quality, particularly in suppressing staircase artifacts and preserving complex texture details.

1. Introduction

In the field of digital image processing, images are inevitably corrupted by various types of noise, such as Gaussian noise and Poisson noise, during acquisition, transmission, and storage. Noise not only severely degrades the visual quality of images but also significantly hinders subsequent high-level processing tasks, including image segmentation, object recognition, medical diagnosis, and computer vision analysis [1]. Therefore, recovering a clean original image from its noisy observation, known as image denoising, remains a fundamental and critical research topic in image processing and computer vision [2,3].

Image denoising is inherently an ill-posed inverse problem. Regularization methods provide an effective framework for solving such problems, typically formulated as a trade-off between a data fidelity term and a regularization term [4]. The data fidelity term (often using the L2-norm) ensures consistency between the restored image and the observed image, while the regularization term constrains the solution space by incorporating prior knowledge of the image to suppress noise and recover structures. Total Variation (TV) regularization, proposed by Rudin, Osher, and Fatemi, is a landmark work in image denoising [5]. The TV model uses the L1-norm of the image gradient as the regularizer, which tends to produce piecewise constant solutions, effectively removing noise while preserving edge information. However, a well-known drawback of the TV model is the introduction of unnatural “staircase effects” in flat regions, where smooth-intensity transitions are approximated by piecewise constant patches. Furthermore, the TV model suffers from the loss of image contrast [6].

To overcome the staircase effect and contrast loss of the TV model, various improvements have been proposed, which can be broadly categorized as follows:

1.

Higher-order regularization models: Examples include the Lysaker–Lundervold–Tai (LLT) model [7] and the Total Generalized Variation (TGV) model [8,9]. These models incorporate higher-order derivative information to better represent smooth image regions, effectively mitigating staircase effects. However, they often lead to over-smoothed edges or introduce speckle artifacts, compromising detail preservation in texture-rich areas [10].

2.

Fractional-order regularization models: Fractional differential operators with order greater than one possess non-locality and weak singularity [11], enabling them to capture both edge and texture details simultaneously. Models based on fractional-order TV (FOTV) have demonstrated unique advantages in suppressing staircase effects while preserving textures [12,13,14,15]. Moreover, FOTV has been shown to better preserve image contrasts compared to the standard TV model [6]. Nevertheless, as noted by Chen et al. [16], the denoising performance of pure fractional-order TV models on images with high noise levels still requires improvement.

3.

Adaptive and anisotropic models: The core idea of these models is to dynamically adjust the strength or direction of regularization based on local image structure. For instance, the anisotropic TV (ATV) model proposed by Pang et al. [17] introduces an adaptive weighting matrix derived from local gradients, applying weaker smoothing at edges for protection and stronger smoothing in flat regions for noise removal. The works of Wu et al. [18] and Yang et al. [3] also demonstrate the effectiveness of adaptive strategies in image segmentation and denoising. Although the ATV model excels in edge preservation, its performance on images with complex textures remains limited.

4.

Non-convex regularization models: To more accurately estimate signal sparsity (e.g., the sparsity of image gradients), non-convex regularizers are introduced to replace the convex L1-norm. The minimax-concave (MC) penalty is a typical non-convex regularization tool that produces a sharper thresholding function than the L1-norm, leading to more accurate estimation of sparse signals [19]. Chen et al. [16] first combined the MC penalty with fractional-order TV, proposing the MCFOTV model, which achieved superior performance compared to traditional TV and FOTV models. However, this model’s adaptive capability is still insufficient when handling images with high noise levels and complex local structures.

In summary, existing advanced models often excel in only one or a subset of the following goals: suppressing staircase effects, preserving edges and textures, handling high noise levels, or achieving local adaptivity. It remains an open and challenging problem to design a model that can adaptively integrate the texture-preserving capability of fractional-order differentiation with the accurate estimation power of non-convex regularization to simultaneously achieve: (a) effective elimination of staircase effects in flat regions, (b) maximum preservation of details in edge and texture regions, and (c) robust denoising performance across low to high noise levels.

Inspired by the ATV model [17] and the MCFOTV model [16], this paper proposes a novel Adaptive Fractional-order Total Variation and Minimax-Concave-based image denoising model (AFTVMC). The core innovation of our model lies in the organic integration of an adaptive weighting mechanism, a fractional-order differential operator, and the non-convex MC penalty within a unified variational framework. Specifically:

• We employ an adaptive weighting matrix (T) generated from the local gradient information of the noisy image. This allows the model to adaptively adjust the strength of the fractional-order differential operator based on image content (flat areas, edges, textures), enabling finer local control.
• We introduce a fractional-order gradient operator ($D^{\alpha }$) with order $\alpha$ between 1 and 2 as the basis for sparsity measurement. This operator inherently balances edge preservation and staircase effect suppression.
• We apply the minimax-concave (MC) penalty to regularize the weighted fractional-order gradient, aiming for a more accurate estimation of the true image sparsity prior than the traditional L1-norm, thereby enhancing restoration accuracy.

Through this combination, our model aims to inherit the local adaptivity of the ATV model, the texture-preserving capability of FOTV models, and the accurate estimation power of the MC penalty, thereby achieving higher-quality restoration with fewer artifacts for various images, especially those containing rich textures and different noise levels.

For numerical solution, we adopt the efficient Alternating Direction Method of Multipliers (ADMM) to solve the proposed optimization problem. Through theoretical analysis, we select appropriate parameters (e.g., a small $\mathsf{\beta }$ value) to ensure that the overall objective function remains convex under specific conditions despite the non-convex term, guaranteeing stable convergence of the ADMM algorithm. Extensive experiments on synthetic and natural images demonstrate that compared to the classical TV model, the LLT model, the ATV model, and the state-of-the-art MCFOTV model, our proposed AFTVMC model shows significant advantages in both objective metrics (Peak Signal-to-Noise Ratio—PSNR, Structural Similarity Index—SSIM) and subjective visual quality, particularly in suppressing staircase effects and preserving complex texture details.

The remainder of this paper is organized as follows: Section 2 details the proposed AFTVMC model and its mathematical formulation. Section 3 describes the ADMM-based numerical algorithm. Section 4 presents extensive numerical experiments and comparative analysis with existing advanced methods. Section 5 concludes the paper and outlines future research directions.

2. The Proposed Model

Let the image domain $\mathrm{\Omega }\subset R^2$, the process of image contamination by noise can be described as $f=u+w,$ where $f:\mathrm{\Omega }\to R$ is the observed noisy image, $u:\mathrm{\Omega }\to R$ is the clear image, $w:\mathrm{\Omega }\to R$ represents the Gaussian noise. To recover u from the degraded image f, we propose the following AFTVMC denoising model:

$$\underset{u}{\min }\left\{{\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{∥T{D}^{\alpha }u∥}_1-\underset{v}{\min }\left\{{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u-v∥}_2^2+{∥v∥}_1\right\}\right\},$$

(1)

where $\lambda$ and $\beta$ are nonnegative parameters, $D^{\alpha }$ denotes the derivative of a real order $\alpha$ and T represents the adaptive weighted matrix.

There are various definitions of fractional-order derivative, among which the most popular are Grünwald–Letnikov(G-L), Riemann–Liouville (R-L), and frequency domain definitions [20]. We have adopted an easily implementable G-L fractional derivative. The G-L fractional derivative of the function $f\left(t\right)$ is as follows [11]

$${}_{a}D_t^{\alpha }f\left(t\right)=\underset{{\displaystyle \genfrac{}{}{0pt}{}{h\to 0\hfill }{nh=t-a\hfill }}}{lim}{h}^{-\alpha }\sum _{r=0}^n{(-1)}^r{C}_{\alpha }^{r}f(t-rh),$$

where $C_k^{\alpha }={\displaystyle \frac{\mathrm{\Gamma }\left(\alpha +1\right)}{\mathrm{\Gamma }\left(k+1\right)\mathrm{\Gamma }\left(\alpha -k+1\right)}}$ is the generalized binomial coefficient, $\mathrm{\Gamma }(·)$ denotes the Gamma function and $a,t$ are terminals of fractional derivative. The G-L fractional $\alpha$-order gradient of a two-dimensional image u at a point $(x,y)\in \mathrm{\Omega }$ is defined as

$$D^{\alpha }u(x,y)=[D_x^{\alpha }u(x,y);D_y^{\alpha }u(x,y)],$$

where $D_x^{\alpha }u(x,y),D_y^{\alpha }u(x,y)$ represent the G-L fractional derivative of $u(x,y)$ along the x and y directions, respectively. In the paper, we set $1<\alpha \le 2$.

Many adaptive regularization denoising models have been proposed [17,18,21,22,23,24]. They reflect the local structure of u and lead to a good estimation of the edges. Inspired by Pang et al. [17], Bollt et al. [23], Yang et al. [3], Wu et al. [18], we use an adaptive weighted matrix operator T for model (1), as defined by

$$T\left(\begin{array}{c}{\widehat{v}}_1(x,y)\\ {\widehat{v}}_2(x,y)\end{array}\right)=\left[\begin{array}{cc}{t}_1(x,y)& 0\\ 0& t_2(x,y)\end{array}\right]\left(\begin{array}{c}{\widehat{v}}_1(x,y)\\ {\widehat{v}}_2(x,y)\end{array}\right)$$

$$\begin{array}{c}=\left[\begin{array}{cc}{\displaystyle \frac{1}{1+\kappa {\left|G_{\widehat{\sigma }}(x,y)\ast {\nabla }_{x}f(x,y)\right|}^2}}& 0\\ 0& {\displaystyle \frac{1}{1+\kappa {\left|G_{\widehat{\sigma }}(x,y)\ast {\nabla }_{y}f(x,y)\right|}^2}}\end{array}\right]\times \left(\begin{array}{c}{\widehat{v}}_1(x,y)\\ {\widehat{v}}_2(x,y)\end{array}\right),\hfill \end{array}$$

where $G_{\widehat{\sigma }}(x,y)$ is the Gaussian kernel, $\kappa$ and $\widehat{\sigma }$ are nonnegative parameters. ${\nabla }_{x}f(x,y)$ and ${\nabla }_{y}f(x,y)$ are first-order derivative of noisy image f in the x-axis direction and in the y-axis direction, respectively. The adaptive weighting matrix T utilizes the gradient information of the noisy image, enabling model (1) to effectively describe local structures. If $\kappa =0$, then T becomes the identity matrix I and model (1) can be transformed into the MCFOTV model in [16].

The objective function of model (1) is as follows

$${\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{∥T{D}^{\alpha }u∥}_1-\underset{v}{\min }\left\{{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u-v∥}_2^2+{∥v∥}_1\right\}.$$

It can be converted into

$$\underset{v}{\max }\left\{{\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{∥T{D}^{\alpha }u∥}_1-{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u-v∥}_2^2-{∥v∥}_1\right\},$$

i.e.,

$$\begin{array}{c}\underset{v}{\max }\left\{{\displaystyle \frac{\lambda }{2}}{∥f∥}_2^2-\lambda 〈f,u〉+{\displaystyle \frac{\beta }{2}}〈T{D}^{\alpha }u,v〉-{\displaystyle \frac{\beta }{2}}{∥v∥}_2^2-{∥v∥}_1\right\}\hfill \\ +{∥T{D}^{\alpha }u∥}_1+{\displaystyle \frac{\lambda }{2}}{∥u∥}_2^2-{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u∥}_2^2.\hfill \end{array}$$

(2)

Because the first two terms in (2) are convex, (2) tends to be convex when ${\displaystyle \frac{\lambda }{2}}{∥u∥}_2^2-{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u∥}_2^2$ is convex. The value of the parameter $\beta$ affects the convexity of the objective function. A higher $\beta$-value will result in the function being non-convex, which makes it challenging to find the optimal solution for the model (1). Based on these analyses, we set $\beta$ to 0.001 in this paper.

3. Algorithm

3.1. Preliminaries

A digital image is usually defined on the grid points of a rectangular area. Then, images $u,f$ will be matrices of size $M\times N$ and $u,f\in R^{M\times N}$. Each pixel in the images u and f are denoted as $u_{i,j}$ and $f_{i,j}$, where $i\in \{1,2,...,M\},j\in \{1,2,...,N\}$. The first-order derivative in the weighted matrix T is discretized using first-order difference with the periodic boundary condition, with the following format

$${\left({\nabla }_{x}f\right)}_{i,j}=\left\{\begin{array}{c}{f}_{i,j}-f_{i-1,j},if1<i\le M,1\le j\le N,\\ f_{i,j}-f_{M,j},ifi=1,1\le j\le N,\end{array}\right.$$

$${\left({\nabla }_{y}f\right)}_{i,j}=\left\{\begin{array}{c}{f}_{i,j}-f_{i,j-1},if1\le i\le M,1<j\le N,\\ f_{i,j}-f_{i,N},if1\le i\le M,j=1.\end{array}\right.$$

G-L discrete fractional $\alpha$-order gradient with the periodic boundary condition is defined as [25]

$${\left(D^{\alpha }u\right)}_{i,j}=\left[{\left(D_x^{\alpha }u\right)}_{i,j};{\left(D_y^{\alpha }u\right)}_{i,j}\right],$$

(3)

with

$$\begin{array}{c}{\left(D_x^{\alpha }u\right)}_{i,j}={\displaystyle \sum _{k=0}^{K-1}}{(-1)}^k{C}_k^{\alpha }{u}_{i-k,j},{\left(D_y^{\alpha }u\right)}_{i,j}={\displaystyle \sum _{k=0}^{K-1}}{(-1)}^k{C}_k^{\alpha }{u}_{i,j-k}.\hfill \end{array}$$

(4)

where the parameter K denotes the number of neighboring pixels at each pixel and $u_{-\nu ,j}=u_{M-\nu ,j},v\in \{0,1,2,\dots ,K-1\}$, $u_{i,-\mu }=u_{i,N-\mu },\mu \in \{0,1,2,\dots ,K-1\}$.

When $\alpha =1$ in (4), then $C_0^1=1$, $C_1^1=1$, and $C_k^1=0$, for $2\le k\le K-1$. Thus, (3) deduces to first-order differences along the x-axis and the y-axis. When $\alpha =2$ in (4), then $C_0^2=1$, $C_1^2=2$, $C_2^2=1$, and $C_k^2=0$ for $3\le k\le K-1$. Therefore, (3) deduces to second-order differences along the coordinate axes. In this paper, we take $K=20$.

The conjugation of fractional-order derivative satisfies the following relationship [10]

$${\left(D^{\alpha }\right)}^{*}=\overline{{\left(-1\right)}^{\alpha }}di{v}^{\alpha }.$$

For $p=\left(p^1;p^2\right),p^1,p^2\in R^{M\times N}$, the discrete fractional-order divergence is defined as [10,25]

$${\left(di{v}^{\alpha }p\right)}_{i,j}={(-1)}^{\alpha }\sum _{k=0}^{K-1}{(-1)}^k{C}_k^{\alpha }(p_{i+k,j}^1+p_{i,j+k}^2),$$

where $p_{M+\nu ,j}^1=p_{\nu ,j}^1,v\in \{0,1,2,\dots ,K-1\}$, $p_{i,N+\mu }^2=p_{i,\mu }^2,\mu \in \{0,1,2,\dots ,K-1\}$.

3.2. ADMM Strategy

In the following, we solve model (1) using the ADMM algorithm, which decomposes an optimization problem into several subproblems. The convergence analysis of the ADMM algorithm is available in [26]. The model (1) can be reformulated as follows

$$\underset{u}{\min }\underset{v}{\max }\left\{{\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{∥T{D}^{\alpha }u∥}_1-{\displaystyle \frac{\beta }{2}}{∥T{D}^{\alpha }u-v∥}_2^2-{∥v∥}_1\right\}.$$

By introducing auxiliary variables x and w, the above optimization problem can be transformed into an equivalent form

$$\underset{u}{\min }\underset{v}{\max }\left\{{\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{∥x∥}_1-{\displaystyle \frac{\beta }{2}}{∥x-v∥}_2^2-{∥v∥}_1\right\}s.t.x=Tw,w=D^{\alpha }u.$$

(5)

Based on the augmented Lagrangian method, we rewrite the problem (5) as follows

$$\begin{array}{c}L(u,v,x,w,\xi ,\lambda )={\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2-{\displaystyle \frac{\beta }{2}}{∥x-v∥}_2^2+{∥x∥}_1-{∥v∥}_1+{\displaystyle \frac{{\gamma }_1}{2}}{∥x-Tw∥}_2^2\hfill \end{array}$$

$$\begin{array}{c}-{\mu }^T\left(x-Tw\right)+{\displaystyle \frac{{\gamma }_2}{2}}{∥w-D^{\alpha }w∥}_2^2-{\xi }^T\left(w-D^{\alpha }w\right),\hfill \end{array}$$

where ${\gamma }_1$ and ${\gamma }_2$ are the penalty parameters, $\mu$ and $\xi$ are the Lagrange multipliers. Next, we elaborate on how to solve subproblems on $x,w,v,$ and u, respectively.

3.2.1. X-Subproblem

X-subproblem can be written as

$$\begin{array}{c}{x}^{k+1}=\underset{x}{\arg \min }\left\{-{\displaystyle \frac{\beta }{2}}{∥x-v^k∥}_2^2+{∥x∥}_1+{\displaystyle \frac{{\gamma }_1}{2}}{∥x-T{w}^k∥}_2^2\right.\left.-{\left({\mu }^k\right)}^T\left(x-T{w}^k\right)\right\}.\hfill \end{array}$$

The above equation can be equivalently transformed into

$$\begin{array}{c}{x}^{k+1}=\underset{x}{\arg \min }\left\{{∥x∥}_1+{\displaystyle \frac{{\gamma }_1-\beta }{2}}{∥x-{\displaystyle \frac{{\gamma }_{1}T{w}^k+{\mu }^k-\beta v^k}{{\gamma }_1-\beta }}∥}_2^2\right\}.\hfill \end{array}$$

Using the soft-thresholding operator in [27], we can obtain the closed-form solution as

$$x_{i,j}^{k+1}=\max \left\{{∥t_{i,j}∥}_2-{\displaystyle \frac{1}{{\gamma }_1-\beta }},0\right\}{\displaystyle \frac{t_{i,j}}{{∥t_{i,j}∥}_2}},$$

(6)

where $t_{i,j}={\displaystyle \frac{{\gamma }_{1}T{w}^k+{\mu }^k-\beta v^k}{{\gamma }_1-\beta }}$.

3.2.2. W-Subproblem

W-subproblem can be expressed as

$$\begin{array}{c}{w}^{k+1}=\underset{w}{\arg \min }\{{\displaystyle \frac{{\gamma }_1}{2}}{∥x^{k+1}-Tw∥}_2^2-{\left({\mu }^k\right)}^T\left(x^{k+1}-Tw\right)+{\displaystyle \frac{{\gamma }_1}{2}}{∥w-D^{\alpha }{u}^k∥}_2^2\hfill \end{array}$$

(7)

$$\begin{array}{c}\left.-{\left({\xi }^k\right)}^T\left(w-D^{\alpha }{u}^k\right)\right\}.\hfill \end{array}$$

(8)

By directly solving, we obtain

$$w_{i,j}^{k+1}={\displaystyle \frac{{\gamma }_2{D}_x^{\alpha }{u}_{i,j}^k-T_{i,j}{u}_{i,j}^k+{\gamma }_1{T}_{i,j}{x}_{i,j}^{k+1}+{\xi }_{i,j}^k}{{\gamma }_2+{\gamma }_1{T}_{i,j}^2}}.$$

(9)

3.2.3. V-Subproblem

V-subproblem corresponding to the following optimization problem

$$v^{k+1}=\arg \underset{v}{\max }\left\{-{\displaystyle \frac{\beta }{2}}{∥x^{k+1}-v∥}_2^2-{∥v∥}_1\right\}=\arg \underset{v}{\min }\left\{{\displaystyle \frac{\beta }{2}}{∥x^{k+1}-v∥}_2^2+{∥v∥}_1\right\}.$$

Similar to the x-subproblem, we also use the softthresholding operator to obtain the solution as

$$v_{i,j}^{k+1}=\max \left\{{∥x_{i,j}^{k+1}∥}_2-{\displaystyle \frac{1}{\beta }},0\right\}{\displaystyle \frac{x_{i,j}^{k+1}}{{∥x_{i,j}^{k+1}∥}_2}}.$$

(10)

3.2.4. U-Subproblem

U-subproblem can be written as

$$\begin{array}{c}{u}^{k+1}=\underset{u}{\arg \min }\left\{{\displaystyle \frac{\lambda }{2}}{∥f-u∥}_2^2+{\displaystyle \frac{{\gamma }_2}{2}}{∥w^{k+1}-D^{\alpha }u∥}_2^2-{\left({\xi }^k\right)}^T\left(x^{k+1}-T{w}^{k+1}\right)\right\},\hfill \end{array}$$

(11)

which is a linear system as follows

$$\begin{array}{c}\left\{\lambda +{\gamma }_2{\left(D^{\alpha }\right)}^{*}{D}^{\alpha }\right\}{u}^{k+1}=\lambda f+{\gamma }_2{\left(D^{\alpha }\right)}^{*}\left({\gamma }_2{w}^{k+1}-{\xi }^k\right).\hfill \end{array}$$

(12)

Under the assumption of the periodic boundary of u, ${\left(D^{\alpha }\right)}^{*}{D}^{\alpha }$ is block circulant with circulant blocks (BCCB). So the above equation can be solved efficiently by the fast Fourier transform (FFT). Then its solution can be obtained by

$$u^{k+1}={\mathcal{F}}^{-1}\left({\displaystyle \frac{\lambda \mathcal{F}\left(f\right)+{\gamma }_2\mathcal{F}\left({\left(D^{\alpha }\right)}^{*}\left({\gamma }_2{w}^{k+1}-{\xi }^k\right)\right)}{\lambda \mathcal{F}\left(\mathcal{I}\right)+{\gamma }_2\mathcal{F}\left({\left(D^{\alpha }\right)}^{*}{D}^{\alpha }\right)}}\right).$$

(13)

The algorithm with ADMM is summarized in Algorithm 1.

Algorithm 1: ADMM for AFTVMC Denoising Model

1. Input: $f,\lambda ,{\gamma }_1,{\gamma }_2,\alpha ,\beta$;

2. Initialize: $u^1=f,{\xi }^1=0,{\eta }^1=0,v^1=0,w^1=\left[D_x^{\alpha }f;D_y^{\alpha }f\right]$;

3. For $k=1,2,...$

Update $x^{k+1}$ by (6),

Update $w^{k+1}$ by (9),

Update $v^{k+1}$ by (10),

Update $u^{k+1}$ by (13),

${\xi }^{k+1}={\xi }^k-{\gamma }_2(w^{k+1}-D^{\alpha }{u}^{k+1})$,

${\mu }^{k+1}={\mu }^k-{\gamma }_1(x^{k+1}-T{w}^{k+1})$;

4. End for until stopping rule (${\displaystyle \frac{\parallel u^k-u^{k-1}{\parallel }_2}{\parallel u^{k-1}{\parallel }_2}}\le {10}^{-5}$ or iterations $>500$) is met;

5. Output $u=u^{k+1}$ as the denoised image.

4. Numerical Experiments

In this section, we conduct comprehensive denoising experiments to validate the effectiveness and superiority of the proposed Adaptive Fractional-order Total Variation and Minimax-Concave (AFTVMC) model. We compare our method with several state-of-the-art TV-based models, including the classical TV model [5], the high-order LLT model [7], the adaptive ATV model [17], and the recent MCFOTV model [16]. All experiments are performed using MATLAB (R2018a) on a PC with a 2.50 GHz 12th Gen Intel(R) Core(TM) i5-12500H CPU and 16.0 GB RAM. The iterative process for each model is terminated when the relative difference between successive solutions satisfies ${\displaystyle \frac{\parallel u^k-u^{k-1}{\parallel }_2}{\parallel u^{k-1}{\parallel }_2}}\le {10}^{-5}$ or the number of iterations exceeds 500.

4.1. Evaluation Criteria

To evaluate denoising performance, we consider three aspects: quantitative metrics, convergence behavior, and visual quality. As quantitative measures, we employ the structural similarity index measure (SSIM) and the peak signal-to-noise ratio (PSNR) [28]. SSIM is defined by

$$SSIM(x,y)={\displaystyle \frac{\left(2{\mu }_x{\mu }_y+C_1\right)\left(2{\sigma }_{xy}+C_2\right)}{\left({\mu }_x^2+{\mu }_y^2+C_1\right)\left({\sigma }_x^2+{\sigma }_y^2+C_2\right)}},$$

where ${\mu }_x$, ${\mu }_y$ are local means, ${\sigma }_x$, ${\sigma }_y$ are standard deviations, ${\sigma }_{xy}$ is cross-covariance, and $c_1$, $c_2$ are constants. PSNR is defined as

$$PSNR(u,I)=10{log}_{10}{\displaystyle \frac{MA{X}^2}{\frac{1}{MN}{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(u_{i,j}-I_{i,j}\right)}^2}},$$

where u and I denote the denoised and reference images, respectively. In our experiments, images are normalized to $[0,1]$, hence $MAX=1$. Larger PSNR and SSIM indicate better denoised performance.

4.2. Benchmark Images for Evaluation

Six benchmark images are used: “lena” ($512\times 512$), “boat” ($512\times 512$), “pepper” ($256\times 256$), “barbara” ($512\times 512$), “man” ($1024\times 1024$), and “bird” ($128\times 128$), as illustrated in Figure 1. These images include both smooth regions and textured structures, making them suitable for evaluating denoising models. Each image is normalized to $[0,1]$, and additive white Gaussian noise is generated using MATLAB’s imnoise, and the noise level is set to $\sigma \in \{15,20,25,30,35,40\}$, covering both low- and high-noise scenarios. Here, $\sigma$ denotes the noise level parameter used in the noise generation procedure.

4.3. Parameter Settings

The regularization parameter $\lambda$, which balances noise suppression and detail preservation, is empirically chosen from the range $[1,60]$. The fractional order $\alpha$ is searched over $[1.1,2.0]$ with step $0.1$ (see

Table 1. The PSNR of the AFTVMC model with various $\alpha$ and noise levels.

Image	$\mathit{\sigma }$	$\mathit{\alpha }$
		1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9	2
lena	15	32.0150	32.1642	32.2931	32.4086	32.5048	32.5804	32.6341	32.6671	32.6831	32.6845
	20	31.2245	31.2181	31.2186	31.2308	31.2501	31.2684	31.2826	31.2882	31.2837	31.2685
	25	29.9022	30.0252	30.1219	30.2083	30.2804	30.3387	30.3795	30.4044	30.4155	30.4118
	30	29.5136	29.5232	29.5307	29.5531	29.5792	29.6019	29.6181	29.6252	29.6221	29.6067
	35	28.9406	28.9432	28.9485	28.9656	28.9909	29.0187	29.0430	29.0564	29.0567	29.0419
	40	28.2809	28.2903	28.2959	28.3109	28.3344	28.3549	28.3694	28.3739	28.3687	28.3495
boat	15	30.6809	30.7453	30.7794	30.7900	30.7816	30.7570	30.7170	30.6649	30.6016	30.5318
	20	29.4218	29.4636	29.4760	29.4682	29.4441	29.4069	29.3610	29.3062	29.2452	29.1790
	25	28.3858	28.4466	28.4741	28.4815	28.4717	28.4477	28.4129	28.3705	28.3227	28.2682
	30	27.4005	27.5019	27.5670	27.6073	27.6268	27.6287	27.6151	27.5901	27.5581	27.5198
	35	26.8944	26.9548	26.9838	26.9926	26.9856	26.9684	26.9414	26.9064	26.8692	26.8279
	40	26.4076	26.4378	26.4430	26.4317	26.4100	26.3820	26.3478	26.3095	26.2708	26.2300
pepper	15	31.8172	31.8304	31.8256	31.8282	31.8431	31.8566	31.8562	31.8407	31.8101	31.7573
	20	30.2425	30.3138	30.3628	30.4130	30.4597	30.4994	30.5273	30.5438	30.5451	30.5230
	25	29.0004	29.0650	29.0982	29.1279	29.1569	29.1824	29.2072	29.2304	29.2475	29.2526
	30	28.0025	28.0659	28.0971	28.1219	28.1504	28.1857	28.2230	28.2538	28.2698	28.2696
	35	27.2813	27.3449	27.3744	27.3999	27.4244	27.4524	27.4835	27.5097	27.5326	27.5466
	40	26.6106	26.6678	26.6777	26.6802	26.6873	26.6980	26.7126	26.7312	26.7475	26.7598
barbara	15	28.1651	28.3136	28.4536	28.5868	28.7131	28.8309	28.9384	29.0317	29.1079	29.1640
	20	26.7034	26.8447	26.9725	27.0890	27.1933	27.2829	27.3559	27.4099	27.4429	27.4545
	25	25.5283	25.6545	25.7649	25.8628	25.9495	26.0226	26.0792	26.1174	26.1361	26.1347
	30	24.6202	24.7366	24.8396	24.9327	25.0140	25.0820	25.1342	25.1694	25.1882	25.1910
	35	24.0048	24.1009	24.1857	24.2605	24.3250	24.3767	24.4147	24.4379	24.4480	24.4450
	40	23.6436	23.7134	23.7717	23.8222	23.8641	23.8967	23.9199	23.9310	23.9302	23.9182
man	15	30.7079	30.8562	30.9838	31.0928	31.1833	31.2554	31.3105	31.3489	31.3721	31.3809
	20	29.4806	29.6058	29.7081	29.7912	29.8554	29.9031	29.9361	29.9547	29.9619	29.9590
	25	28.4486	28.5547	28.6377	28.7037	28.7554	28.7945	28.8221	28.8384	28.8460	28.8452
	30	27.6053	27.6857	27.7465	27.7962	27.8344	27.8635	27.8845	27.8968	27.9030	27.9025
	35	26.7526	26.8277	26.8844	26.9305	26.9684	27.0006	27.0254	27.0403	27.0500	27.0537
	40	26.0632	26.1182	26.1596	26.1944	26.2244	26.2504	26.2708	26.2836	26.2921	26.2956
bird	15	32.9145	32.9772	33.0805	33.2178	33.3593	33.4964	33.6159	33.7186	33.8028	33.8453
	20	31.8634	31.8124	31.8500	31.9392	32.0521	32.1652	32.2760	32.3801	32.4687	32.5038
	25	30.8911	30.8039	30.7820	30.8171	30.9071	31.0114	31.1165	31.2108	31.2788	31.3121
	30	29.9077	29.8697	29.9073	29.9796	30.0720	30.1900	30.3067	30.4125	30.4997	30.5480
	35	29.1492	28.9984	28.9337	28.9606	29.0477	29.1588	29.2799	29.3874	29.4696	29.5246
	40	28.3207	28.3018	28.3132	28.3550	28.4065	28.4934	28.5983	28.7033	28.7878	28.8426

α is the fractional order in model (1), and σ is the noise level. The bold entries denote the best PSNR for each image at each noise level, and its α is adopted.

), indicating that the denoising performance is sensitive to $\alpha$. Therefore, $\alpha$ is tuned for each test image at each noise level to maximize PSNR. The resulting PSNR values are reported in Table 1, where the best PSNR for each image and noise level (and its corresponding $\alpha$) is highlighted in bold.

We also evaluate the effect of the adaptive weighting matrix T by comparing the cases $T\ne I$ and $T=I$. The results in Figure 2 indicate that incorporating the adaptive matrix improves performance, and the parameters $\kappa$ and $\widehat{\sigma }$ in T should be selected according to a proper choice as suggested in related work [17]. Moreover, the penalty parameters ${\gamma }_1$ and ${\gamma }_2$ mainly influence the convergence speed rather than the final denoising quality; we set ${\gamma }_1={\gamma }_2={10}^{\alpha +1}$ in all experiments.

4.4. Image Denoising

4.4.1. Metric Comparison (PSNR, SSIM)

We first present a quantitative comparison among TV, LLT, ATV, MCFOTV, and the proposed AFTVMC in terms of PSNR and SSIM. The denoising results for all test images at all noise levels are summarized in

Table 2. SSIM and PSNR obtained by five denoising models under all noise levels.

Noise Image	Models	$\mathit{\sigma }=15$		$\mathit{\sigma }=20$		$\mathit{\sigma }=25$		$\mathit{\sigma }=30$		$\mathit{\sigma }=35$		$\mathit{\sigma }=40$
		PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM	PSNR	SSIM
lena	TV	32.2737	0.8573	30.9751	0.8296	30.0408	0.8104	29.2534	0.7894	28.6340	0.7674	27.9789	0.7640
	LLT	32.4521	0.8567	31.1233	0.8319	30.1334	0.8113	29.2943	0.7898	28.7330	0.7755	28.0409	0.7628
	ATV	32.4256	0.8597	31.1337	0.8349	30.2252	0.8154	29.4507	0.7964	28.8399	0.7782	28.1688	0.7584
	MCFOTV	32.4717	0.8581	31.1446	0.8308	30.1737	0.8116	29.3398	0.7901	28.7714	0.7764	28.0836	0.7647
	AFTVMC	32.6845	0.8618	31.2882	0.8426	30.4155	0.8132	29.6252	0.8024	29.0564	0.7917	28.3739	0.7738
boat	TV	30.5012	0.8140	29.1690	0.7754	28.1442	0.7437	27.3797	0.7158	26.7040	0.6931	26.1434	0.6715
	LLT	30.5139	0.8126	29.1363	0.7730	28.1021	0.7394	27.2752	0.7127	26.5839	0.6882	25.9928	0.6660
	ATV	30.6820	0.8172	29.3788	0.7801	28.4014	0.7486	27.6172	0.7227	26.9460	0.7000	26.3848	0.6799
	MCFOTV	30.6242	0.8158	29.2868	0.7779	28.2854	0.7453	27.4293	0.7161	26.7982	0.6910	26.2378	0.6704
	AFTVMC	30.7900	0.8194	29.4760	0.7817	28.4815	0.7506	27.6287	0.7225	26.9926	0.6981	26.4430	0.6779
pepper	TV	31.3150	0.8820	29.8609	0.8530	28.7007	0.8322	27.6987	0.8049	26.9277	0.7862	26.1661	0.7620
	LLT	31.2961	0.8805	29.8454	0.8521	28.5695	0.8187	27.5487	0.7925	26.8589	0.7675	26.1027	0.7554
	ATV	31.7219	0.8911	30.3094	0.8700	29.1729	0.8450	28.1438	0.8207	27.3618	0.8019	26.5427	0.7796
	MCFOTV	31.3298	0.8834	29.8771	0.8544	28.5696	0.8187	27.5679	0.7914	26.8582	0.7772	26.1028	0.7554
	AFTVMC	31.8566	0.8999	30.5451	0.8783	29.2526	0.8496	28.2698	0.8244	27.5466	0.8054	26.7598	0.7878
barbara	TV	28.5862	0.8100	26.9629	0.7603	25.8187	0.7169	25.0373	0.6831	24.3833	0.6592	23.9037	0.6431
	LLT	28.9141	0.8133	27.1623	0.7549	25.8664	0.6987	24.9474	0.6673	24.1913	0.6285	23.6374	0.6101
	ATV	28.8497	0.8275	27.1908	0.7770	25.9690	0.7359	25.0591	0.6912	24.3919	0.6673	23.9083	0.6361
	MCFOTV	28.9091	0.8150	27.1626	0.7552	25.8695	0.7034	24.9476	0.6673	24.1914	0.6285	23.6375	0.6101
	AFTVMC	29.1640	0.8196	27.4545	0.7776	26.1361	0.7267	25.1910	0.6947	24.4480	0.6617	23.9310	0.6447
man	TV	30.8834	0.7877	29.4651	0.7401	28.3854	0.7011	27.4545	0.6691	26.6395	0.6367	25.9203	0.6167
	LLT	31.1876	0.8004	29.7420	0.7554	28.6161	0.7183	27.6650	0.6853	26.8249	0.6571	26.0854	0.6322
	ATV	31.0585	0.7913	29.6495	0.7452	28.5643	0.7073	27.6194	0.6746	26.7790	0.6458	26.0311	0.6205
	MCFOTV	31.1877	0.8004	29.7480	0.7549	28.6246	0.7179	27.6736	0.6850	26.8249	0.6571	26.0855	0.6322
	AFTVMC	31.3809	0.8041	29.9619	0.7608	28.8460	0.7246	27.9030	0.6934	27.0537	0.6651	26.2956	0.6409
bird	TV	32.9522	0.9006	31.5693	0.8777	30.4361	0.8520	29.4807	0.8323	28.6358	0.8235	27.9817	0.8055
	LLT	32.7531	0.8740	31.2190	0.8472	29.9444	0.8164	29.3278	0.8071	28.2246	0.7881	27.7324	0.7808
	ATV	33.3542	0.9109	32.0309	0.8890	30.9539	0.8753	29.9178	0.8485	29.2357	0.8443	28.2889	0.8159
	MCFOTV	32.7533	0.8740	31.2230	0.8433	29.9446	0.8164	29.3279	0.8071	28.2246	0.7881	27.7325	0.7808
	AFTVMC	33.8453	0.9070	32.5038	0.8915	31.3121	0.8716	30.5480	0.8582	29.5246	0.8457	28.8426	0.8287

Bold values represent the maximum PSNR or SSIM values among different denoising models for the corresponding noisy image.

. Overall, AFTVMC achieves the best PSNR and/or SSIM in most cases, demonstrating its strong denoising capability over a wide range of image contents and noise levels.

More specifically, AFTVMC attains the highest PSNR for nearly all images at all noise levels. In terms of SSIM, AFTVMC also performs competitively and achieves the best results in most cases. Nevertheless, there are a few cases in which ATV yields slightly higher SSIM than AFTVMC. For example, ATV performs better on “lena” at noise level $\sigma =25$, on “barbara” at $\sigma =15$ and 35, on “boat” at $\sigma =30,35,$ and 40, and on “bird” at $\sigma =15$ and 25. These results indicate that ATV may have a slight advantage in preserving structural similarity under some specific texture and noise conditions. In contrast, AFTVMC shows more stable overall performance by combining adaptive fractional-order regularization with the minimax-concave penalty, which leads to a better balance between noise removal and detail preservation.

4.4.2. Convergence Rate Comparison (PSNR vs. Iterations; Relative Error vs. Iterations)

To compare convergence behavior and denoising efficiency, we record the relative error and PSNR at each iteration for all five methods. Figure 3 illustrates the curves of (i) relative error versus iteration number and (ii) PSNR versus iteration number at $\sigma =30$. The relative error curves show that all methods converge rapidly, while AFTVMC and MCFOTV converge the fastest. Furthermore, the PSNR-versus-iterations curves indicate that AFTVMC reaches its peak PSNR with fewer iterations, suggesting higher denoising efficiency under the same computational budget.

4.4.3. Visual Comparison (Profile Plots First, Then Global and Local Views)

We evaluate visual performance in three aspects: intensity profile (cross-section) plots, global denoised images, and local zoom-ins.

(1)

Profile comparison.

To highlight structural fidelity along a line, we first examine the intensity profiles of the “bird” image at noise level $\sigma =30$ along the 39th column. The profile plots are shown in Figure 4. The noisy profile exhibits large fluctuations. In certain regions, such as the second peak, the TV and ATV models deviate more from the reference profile compared with the other methods, resulting in greater contrast loss. While the reference profile rises smoothly, the denoised curves from the TV and ATV models exhibit oscillations. In flat regions, the denoised curves from the LLT and MCFOTV models also exhibit more pronounced oscillations. In comparison, the proposed model provides a closer fit to the reference profile.

(2)

Global and local visual comparisons.

Following the profile analysis, the global denoised results and corresponding local patches are shown in Figure 5 and Figure 6. and for “lena”, “boat”, “pepper”, “barbara”, “man”, and “bird” at $\sigma =30$. TV shows staircase artifacts in flat regions. TV and ATV oversmooth fine textures (e.g., brim details in “lena”, structural lines in “boat”, and fine features in “pepper”). LLT and MCFOTV suppress noise but slightly attenuate textures. In contrast, AFTVMC effectively preserves both edges and textures while reducing speckle artifacts due to the adaptive fractional-order regularization. Overall, AFTVMC provides the most balanced visual quality across global structures and local details.

Figure 4. Intensity profiles along column 39 of the “bird” image at noise level $\sigma =30$. Panel (a) compares the clean image (green) with the noisy image (red). Panels (b–f) compare the clean image (green) with the profiles of the denoised images (red) obtained by TV, LLT, ATV, MCFOTV, and AFTVMC, respectively. These plots illustrate how closely each denoising method preserves the original structure and details along the selected column.

Figure 4. Intensity profiles along column 39 of the “bird” image at noise level $\sigma =30$. Panel (a) compares the clean image (green) with the noisy image (red). Panels (b–f) compare the clean image (green) with the profiles of the denoised images (red) obtained by TV, LLT, ATV, MCFOTV, and AFTVMC, respectively. These plots illustrate how closely each denoising method preserves the original structure and details along the selected column.

Figure 5. Denoising effects of five models for “lena”, “boat” and “pepper” with $\sigma =30$, the denoising images (lines 1, 3, 5) and the locally enlarged images (lines 2, 4, 6). Images from left to right are restored results using the models as TV, LLT, ATV, MOFCTV, and AFTVMC.

Figure 5. Denoising effects of five models for “lena”, “boat” and “pepper” with $\sigma =30$, the denoising images (lines 1, 3, 5) and the locally enlarged images (lines 2, 4, 6). Images from left to right are restored results using the models as TV, LLT, ATV, MOFCTV, and AFTVMC.

Figure 6. Denoising effects of five models for “barbara”, “man” and “bird” with $\sigma =30$, the denoising images (lines 1, 3, 5) and the locally enlarged images (lines 2, 4, 6). Images from left to right are restored results using the models as TV, LLT, ATV, MOFCTV, and AFTVMC.

Figure 6. Denoising effects of five models for “barbara”, “man” and “bird” with $\sigma =30$, the denoising images (lines 1, 3, 5) and the locally enlarged images (lines 2, 4, 6). Images from left to right are restored results using the models as TV, LLT, ATV, MOFCTV, and AFTVMC.

5. Conclusions

In this paper, we propose a new denoising model that combines adaptive fractional-order TV and minimax-concave. We employ the ADMM iterative method to solve the new model. In numerical experiments, we compare our results with other state-of-the-art TV-based models from three perspectives: visual effects, quantitative indicators, and convergence rate. The numerical results confirm the effectiveness and superiority of the AFTVMC model. In future work, we will expand the method to address the removal of Poisson noise and Cauchy noise.