- Research
- Open Access

# Hybrid regularizers-based adaptive anisotropic diffusion for image denoising

- Kui Liu
^{1, 2}Email author, - Jieqing Tan
^{1}and - Liefu Ai
^{2}

**Received:**5 January 2016**Accepted:**14 March 2016**Published:**2 April 2016

## Abstract

To eliminate the staircasing effect for total variation filter and synchronously avoid the edges blurring for fourth-order PDE filter, a hybrid regularizers-based adaptive anisotropic diffusion is proposed for image denoising. In the proposed model, the \(H^{-1}\)-norm is considered as the fidelity term and the regularization term is composed of a total variation regularization and a fourth-order filter. The two filters can be adaptively selected according to the diffusion function. When the pixels locate at the edges, the total variation filter is selected to filter the image, which can preserve the edges. When the pixels belong to the flat regions, the fourth-order filter is adopted to smooth the image, which can eliminate the staircase artifacts. In addition, the split Bregman and relaxation approach are employed in our numerical algorithm to speed up the computation. Experimental results demonstrate that our proposed model outperforms the state-of-the-art models cited in the paper in both the qualitative and quantitative evaluations.

## Keywords

- Image denoising
- Total variation
- Fourth-order filter
- Split Bregman method
- Relaxation method

## Introduction

With the popularity of image sensor, digital images play a key role in people’s daily life. Unfortunately, images are ineluctably contaminated by noise during acquisition, transmission, and storage. Therefore, image denoising is still an open and complex problem in image processing and computer vision (Chatterjee and Milanfar 2010). Image denoising aims to recovering the original image *u* from the observed noisy image \(u_0\), where \(u_0=u+n\), and *n* is the zero-mean Gaussian white noise with standard deviation \(\sigma\).

*g*(

*x*) is an adaptive edge-stopping function, which is defined in Strong (1997) as follow,

*g*(

*x*) is smaller near the edges and larger away from the boundaries, so the model of (2) has the capability of preserving the edges while removing noise because the diffusion is stopped across edges.

In addition, Nikolova replaced the \(\ell ^2\)-norm with the \(\ell ^1\)-norm in the fidelity term of TV model in Nikolova (2002). Osher et al. (2005) proposed an iterative regularization method of TV model. Chen et al. (2010) presented an adaptive total variation method based on the difference curvature. Wang et al. (2011) put forward a modified TV model.

*g*(

*x*) also denotes the edge-stopping function defined as in (3). The results of experiments indicate that the model of (5) performs better than the pure second-order or hight-order models.

In recent years, efficient computational algorithms for solving the denoising models have emerged in large numbers, for instance, fixed point iteration, gradient descent methods, primal-dual methods, relaxation methods, Bregman iteration and split Bregman method, and so on. These methods are efficient for image denoising while preserving the edges.

Inspired by Li et al. (2007) and Liu (2015), we propose a novel adaptive anisotropic diffusion model, incorporating the advantages of the total variation filter and the fourth-order filter, and develop an efficient computational algorithm. The main contributions of our paper can be generalized as follows. First of all, the hybrid regularization term of the novel model is composed of total variation regularization and a fourth-order filter. The fidelity term uses the \(H^{-1}\)-norm as opposed to the more commonly used \(\ell ^1\)-norm or \(\ell ^2\)-norm. The two above-mentioned filters can be adaptively selected according to the diffusion function. When the pixels locate at the edges, the total variation filter is selected to filter the image, which can preserve the edges. When the pixels belong to the flat regions, the fourth-order filter is adopted to smooth the image, which can eliminate the staircase artifacts. Another main contribution is that the split Bregman and relaxation approach are successively employed in our numerical algorithm to speed up the computation. Experimental results demonstrate that our proposed model achieves higher quality in both the qualitative and quantitative aspects than that of the state-of-the-art models cited in the paper.

The remainder of this paper is organized as follows. In “Preliminaries” section, we give some definitions. In “The new model and algorithms” section, we give the proposed model and numerical implementation in detail. The experimental results are given in “Experiments” section. Finally, this paper is concluded in the fifth section.

## Preliminaries

In this section, we give a brief overview of some necessary notations and definitions for the proposed model, which will be used in the subsequent sections.

###
**Definition 1**

*u*in \(\Omega\) is defined as,

*div*is the divergence operator, \(C_c^1(\Omega ,{\mathbb {R}}^n)\) is the subset of continuously differentiable vector functions of compact support contained in \(\Omega\), and \(L^\infty (\Omega )\) is the essential supremum norm.

###
*Remark 1*

Let the Sobolev space be \(W ^{1,1}(\Omega ):=\{u\in L^1(\Omega )|\nabla u\in L^1(\Omega )\}\). If \(\Vert u\Vert _{BV^2(\Omega )}=\int _\Omega |\nabla u|+\Vert u\Vert _{ W ^{1,1}(\Omega )}\), the space \(BV^2(\Omega )\) is a Banach space.

###
**Definition 2**

*u*is defined as,

###
**Definition 3**

*u*in \(\Omega\) is defined by,

###
**Definition 4**

*u*in \(\Omega\) is defined as,

###
**Definition 5**

## The new model and algorithms

### The proposed model

*u*and \(u_0\) are the recovered image and the noisy image, respectively. Seen from Eq. (12), the \(H^{-1}\)-norm is considered as the fidelity term and the regularization term is composed of a total variation regularization and a fourth-order filter in the proposed model. \(\Vert u_0-u\Vert ^2_{H^{-1}}=\int _\Omega |\nabla (\Delta ^{-1}(u_0-u))|^2d\Omega\), and \(\Delta ^{-1}\) is the inverse Laplace operator. The diffusivity function

*g*(

*x*) is defined as,

*C*lies between 0 to 1. When \(g(x)\rightarrow 0\), it means that the pixels locate at the edges. Then total variation filter is selected to filter the image, which can preserve the edges. When \(g(x)\rightarrow 1\), it means that the pixels belong to the flat regions. Then the fourth-order filter is adopted to smooth the image, which can eliminate the staircase artifacts. We replace

*g*(

*x*) with

*g*in the next part of this article. Figure 1 shows the results of image denoising by our proposed model and model from Li et al. (2007), which demonstrates that the model whose fidelity term uses the \(H^{-1}\)-norm yields better results in image denoising since \(H^{-1}\)-norm is appropriate to represent textured or oscillatory patterns.

### The numerical algorithm for the proposed model

We apply a split Bregman method (Cai et al. 2009) to solve Eq. (12). The idea of split Bregman method is to use splitting operator and Bregman iteration to solve various inverse problems (Goldstein and Osher 2009).

*z*,

*b*,

*k*is the number of iterations.

#### Solve the first subproblem in Eq. (16)

At present, the Euler–Lagrange equation method is usually used to solve the problem similarity to the first subproblem in Eq. (16). However, it works slowly. To accelerate the computation speed, the split Bregman algorithm and relaxation algorithm are adopted to solve the first subproblem in Eq. (16).

#### Solve the second subproblem in Eq. (16)

*z*, which is as follows,

## Experiments

*u*and \({\bar{u}}\) are respectively the recovered image and the original image. Generally, the larger the value of the PSNR, the better the performance. However, PSNR is inconsistent with human visual judgments. SSIM, MS-SSIM, and FSIM are close to the human vision system, so we also use them to assess the noise removal quality. SSIM is defined by,

*u*, respectively, \(\sigma _{u{\bar{u}}}\) is the covariance of

*u*and \({\bar{u}}\), and \(c_1\) and \(c_2\) are two constants to avoid instability. MS-SSIM is defined by,

*i*between images

*u*and \({\bar{u}}\) are defined as follows,

*u*and \({\bar{u}}\) at scale

*i*; \(\sigma _u\) (resp. \(\sigma _{{\bar{u}}}\)) is the standard deviation of

*u*(and \({\bar{u}}\)) at scale

*i*, and \(\sigma _{u{\bar{u}}}\) is the covariance between

*u*and \({\bar{u}}\) at scale

*i*. \(c_1\), \(c_2\) and \(c_3\) are three small constants to avoid instability. In this paper, the values of the exponents \(\alpha _M\), \(\beta _i\) and \(\gamma _i\) are set as the same as those in Wang et al. (2003). FSIM is defined by,

*x*is the similarity measure, which is defined as product of the similarity function \(S_{PC}(x)\) on Phase Congruency (PC) and similarity function \(S_{G}(x)\) on Gradient Magnitude (GM). \(S_{PC}(x)\) and \(S_{G}(x)\) are defined as follows,

*u*and \({{\bar{u}}}\), respectively, and \(G_u\) and \(G_{{{\bar{u}}}}\) denote the GM maps extracted from

*u*and \({{\bar{u}}}\), respectively; \(T_1\) and \(T_2\) are two small positive constants to avoid instability.

*nth*and \((n+1)th\) iteration, and \(\varepsilon\) is a given positive number. We set \(\varepsilon =10^{-3}\) in the experiments.

Performance comparison of the recovered results with different methods in Fig. 2

Method | PSNR | SSIM | MS-SSIM | FSIM | Iterations | Time(s) |
---|---|---|---|---|---|---|

TV model | 31.97 | 0.91 | 0.97 | 0.94 | 246 | 11.20 |

LLT model | 30.62 | 0.86 | 0.95 | 0.75 | 216 | 87.21 |

NLM model | 32.15 | 0.89 | 0.97 | 0.89 | 1 | 275.42 |

BLS-GSM | 31.94 | 0.92 | 0.96 | 0.91 | 1 | 4.68 |

Hybrid model | 32.67 | 0.92 | 0.96 | 0.90 | 197 | 93.72 |

Proposed model | 33.49 | 0.93 | 0.97 | 0.93 | 147 | 81.72 |

Comparison results in PSNR(dB) for different levels of Gaussian noise

Method | Lena | Peppers | Barbara | Cameraman | House |
---|---|---|---|---|---|

\(\sigma =5\) | |||||

Noise | 35.76 | 34.18 | 33.15 | 33.97 | 35.17 |

TV model | 37.46 | 36.86 | 35.96 | 35.95 | 37.12 |

LLT model | 36.92 | 35.94 | 35.67 | 35.67 | 36.34 |

NLM model | 37.41 | 36.48 | 36.72 | 36.76 | 37.81 |

BLS-GSM | 37.16 | 36.37 | 36.18 | 36.63 | 37.53 |

Hybrid model | 37.07 | 36.19 | 36.11 | 36.07 | 37.25 |

Proposed model | 37.44 | 36.52 | 36.82 | 36.81 | 37.89 |

\(\sigma =10\) | |||||

Noise | 27.72 | 28.18 | 28.15 | 28.12 | 28.12 |

TV model | 33.72 | 33.19 | 30.07 | 32.39 | 34.27 |

LLT model | 32.61 | 31.98 | 30.12 | 31.87 | 32.14 |

NLM model | 34.79 | 33.35 | 33.81 | 33.41 | 35.41 |

BLS-GSM | 34.64 | 33.25 | 33.57 | 33.26 | 35.32 |

Hybrid model | 34.56 | 33.13 | 33.41 | 33.22 | 35.25 |

Proposed model | 35.07 | 33.86 | 34.15 | 34.27 | 35.64 |

\(\sigma =20\) | |||||

Noise | 22.17 | 22.16 | 22.15 | 22.20 | 22.18 |

TV model | 31.51 | 30.54 | 28.78 | 29.47 | 31.21 |

LLT model | 30.67 | 29.37 | 27.79 | 29.32 | 30.29 |

NLM model | 31.69 | 31.02 | 29.92 | 30.82 | 31.82 |

BLS-GSM | 31.54 | 30.76 | 29.64 | 30.59 | 31.54 |

Hybrid model | 31.45 | 30.71 | 29.19 | 30.38 | 31.43 |

Proposed model | 31.97 | 31.17 | 30.32 | 30.98 | 32.29 |

\(\sigma =40\) | |||||

Noise | 16.11 | 16.09 | 16.12 | 16.08 | 16.14 |

TV model | 29.22 | 27.95 | 26.93 | 27.62 | 28.85 |

LLT model | 28.46 | 26.07 | 25.92 | 27.19 | 27.95 |

NLM model | 29.57 | 28.19 | 27.89 | 28.59 | 29.42 |

BLS-GSM | 29.36 | 28.13 | 27.67 | 28.35 | 29.17 |

Hybrid model | 29.49 | 28.12 | 27.86 | 28.47 | 29.23 |

Proposed model | 30.13 | 28.94 | 28.42 | 29.15 | 30.27 |

\(\sigma =50\) | |||||

Noise | 14.17 | 14.23 | 14.21 | 14.16 | 14.19 |

TV model | 27.51 | 26.05 | 24.77 | 25.89 | 27.73 |

LLT model | 26.89 | 25.07 | 23.91 | 25.11 | 26.84 |

NLM model | 27.95 | 26.21 | 25.69 | 26.32 | 27.06 |

BLS-GSM | 27.67 | 26.03 | 25.43 | 26.13 | 26.74 |

Hybrid model | 27.76 | 26.14 | 25.17 | 26.27 | 26.97 |

Proposed model | 28.21 | 26.97 | 25.88 | 26.62 | 27.37 |

Comparison results in SSIM for different levels of Gaussian noise

Method | Lena | Peppers | Barbara | Cameraman | House |
---|---|---|---|---|---|

\(\sigma =5\) | |||||

Noise | 0.85 | 0.85 | 0.89 | 0.84 | 0.80 |

TV model | 0.93 | 0.91 | 0.92 | 0.93 | 0.97 |

LLT model | 0.91 | 0.89 | 0.93 | 0.92 | 0.94 |

NLM model | 0.94 | 0.92 | 0.96 | 0.94 | 0.97 |

BLS-GSM | 0.94 | 0.92 | 0.96 | 0.96 | 0.98 |

Hybrid model | 0.94 | 0.93 | 0.96 | 0.96 | 0.98 |

Proposed model | 0.94 | 0.93 | 0.97 | 0.97 | 0.97 |

\(\sigma =10\) | |||||

Noise | 0.61 | 0.61 | 0.71 | 0.63 | 0.53 |

TV model | 0.89 | 0.88 | 0.87 | 0.86 | 0.90 |

LLT model | 0.87 | 0.86 | 0.85 | 0.86 | 0.89 |

NLM model | 0.90 | 0.89 | 0.90 | 0.90 | 0.92 |

BLS-GSM | 0.91 | 0.88 | 0.91 | 0.92 | 0.92 |

Hybrid model | 0.91 | 0.88 | 0.90 | 0.90 | 0.91 |

Proposed model | 0.91 | 0.89 | 0.91 | 0.91 | 0.90 |

\(\sigma =20\) | |||||

Noise | 0.34 | 0.43 | 0.48 | 0.41 | 0.35 |

TV model | 0.86 | 0.85 | 0.82 | 0.83 | 0.84 |

LLT model | 0.84 | 0.83 | 0.80 | 0.82 | 0.83 |

NLM model | 0.87 | 0.86 | 0.83 | 0.85 | 0.85 |

BLS-GSM | 0.86 | 0.84 | 0.83 | 0.84 | 0.87 |

Hybrid model | 0.87 | 0.85 | 0.82 | 0.83 | 0.85 |

Proposed model | 0.89 | 0.87 | 0.84 | 0.86 | 0.87 |

\(\sigma =40\) | |||||

Noise | 0.15 | 0.21 | 0.26 | 0.22 | 0.16 |

TV model | 0.79 | 0.77 | 0.75 | 0.77 | 0.80 |

LLT model | 0.75 | 0.74 | 0.73 | 0.74 | 0.76 |

NLM model | 0.82 | 0.80 | 0.76 | 0.76 | 0.82 |

BLS-GSM | 0.79 | 0.77 | 0.74 | 0.76 | 0.83 |

Hybrid model | 0.81 | 0.79 | 0.76 | 0.78 | 0.81 |

Proposed model | 0.82 | 0.79 | 0.75 | 0.79 | 0.83 |

\(\sigma =50\) | |||||

Noise | 0.11 | 0.17 | 0.15 | 0.18 | 0.13 |

TV model | 0.73 | 0.72 | 0.69 | 0.71 | 0.73 |

LLT model | 0.70 | 0.69 | 0.65 | 0.69 | 0.70 |

NLM model | 0.73 | 0.74 | 0.71 | 0.70 | 0.75 |

BLS-GSM | 0.74 | 0.73 | 0.69 | 0.72 | 0.75 |

Hybrid model | 0.73 | 0.73 | 0.70 | 0.71 | 0.74 |

Proposed model | 0.74 | 0.74 | 0.72 | 0.73 | 0.76 |

Comparison results in MS-SSIM for different levels of Gaussian noise

Method | Lena | Peppers | Barbara | Cameraman | House |
---|---|---|---|---|---|

\(\sigma =5\) | |||||

Noise | 0.97 | 0.98 | 0.98 | 0.97 | 0.97 |

TV model | 0.99 | 0.99 | 0.99 | 0.98 | 0.99 |

LLT model | 0.96 | 0.99 | 0.95 | 0.96 | 0.97 |

NLM model | 0.98 | 0.99 | 0.99 | 0.98 | 0.98 |

BLS-GSM | 0.99 | 0.99 | 0.98 | 0.98 | 0.98 |

Hybrid model | 0.98 | 0.98 | 0.98 | 0.97 | 0.97 |

Proposed model | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

\(\sigma =10\) | |||||

Noise | 0.93 | 0.95 | 0.95 | 0.93 | 0.92 |

TV model | 0.97 | 0.98 | 0.97 | 0.97 | 0.97 |

LLT model | 0.96 | 0.97 | 0.95 | 0.96 | 0.96 |

NLM model | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 |

BLS-GSM | 0.98 | 0.98 | 0.97 | 0.98 | 0.98 |

Hybrid model | 0.97 | 0.97 | 0.98 | 0.99 | 0.98 |

Proposed model | 0.98 | 0.98 | 0.98 | 0.99 | 0.98 |

\(\sigma =20\) | |||||

Noise | 0.83 | 0.89 | 0.88 | 0.83 | 0.82 |

TV model | 0.95 | 0.97 | 0.92 | 0.93 | 0.96 |

LLT model | 0.94 | 0.95 | 0.92 | 0.91 | 0.93 |

NLM model | 0.97 | 0.97 | 0.96 | 0.96 | 0.97 |

BLS-GSM | 0.96 | 0.96 | 0.96 | 0.95 | 0.97 |

Hybrid model | 0.96 | 0.95 | 0.96 | 0.95 | 0.96 |

Proposed model | 0.97 | 0.96 | 0.97 | 0.96 | 0.96 |

\(\sigma =40\) | |||||

Noise | 0.66 | 0.76 | 0.73 | 0.68 | 0.65 |

TV model | 0.91 | 0.93 | 0.87 | 0.89 | 0.92 |

LLT model | 0.88 | 0.89 | 0.85 | 0.82 | 0.85 |

NLM model | 0.93 | 0.94 | 0.92 | 0.93 | 0.94 |

BLS-GSM | 0.93 | 0.93 | 0.91 | 0.93 | 0.94 |

Hybrid model | 0.92 | 0.93 | 0.90 | 0.92 | 0.93 |

Proposed model | 0.93 | 0.94 | 0.92 | 0.94 | 0.94 |

\(\sigma =50\) | |||||

Noise | 0.60 | 0.70 | 0.67 | 0.63 | 0.60 |

TV model | 0.89 | 0.91 | 0.85 | 0.89 | 0.90 |

LLT model | 0.84 | 0.86 | 0.82 | 0.80 | 0.82 |

NLM model | 0.91 | 0.92 | 0.89 | 0.91 | 0.92 |

BLS-GSM | 0.91 | 0.92 | 0.88 | 0.90 | 0.92 |

Hybrid model | 0.90 | 0.91 | 0.88 | 0.89 | 0.91 |

Proposed model | 0.92 | 0.92 | 0.90 | 0.91 | 0.92 |

Comparison results in FSIM for different levels of Gaussian noise

Method | Lena | Peppers | Barbara | Cameraman | House |
---|---|---|---|---|---|

\(\sigma =5\) | |||||

Noise | 0.99 | 0.96 | 0.99 | 0.95 | 0.95 |

TV model | 0.99 | 0.97 | 0.99 | 0.97 | 0.97 |

LLT model | 0.99 | 0.97 | 0.99 | 0.97 | 0.97 |

NLM model | 0.99 | 0.98 | 0.99 | 0.98 | 0.97 |

BLS-GSM | 0.99 | 0.97 | 0.99 | 0.98 | 0.97 |

Hybrid model | 0.99 | 0.97 | 0.98 | 0.97 | 0.96 |

Proposed model | 0.99 | 0.99 | 0.99 | 0.98 | 0.98 |

\(\sigma =10\) | |||||

Noise | 0.95 | 0.87 | 0.96 | 0.86 | 0.86 |

TV model | 0.98 | 0.95 | 0.96 | 0.93 | 0.93 |

LLT model | 0.97 | 0.94 | 0.97 | 0.93 | 0.93 |

NLM model | 0.98 | 0.97 | 0.98 | 0.95 | 0.95 |

BLS-GSM | 0.98 | 0.96 | 0.97 | 0.95 | 0.94 |

Hybrid model | 0.97 | 0.95 | 0.96 | 0.94 | 0.94 |

Proposed model | 0.98 | 0.96 | 0.97 | 0.96 | 0.95 |

\(\sigma =20\) | |||||

Noise | 0.86 | 0.73 | 0.88 | 0.72 | 0.71 |

TV model | 0.95 | 0.91 | 0.92 | 0.88 | 0.88 |

LLT model | 0.94 | 0.90 | 0.92 | 0.86 | 0.88 |

NLM model | 0.96 | 0.94 | 0.94 | 0.91 | 0.92 |

BLS-GSM | 0.95 | 0.93 | 0.94 | 0.90 | 0.91 |

Hybrid model | 0.94 | 0.92 | 0.93 | 0.89 | 0.91 |

Proposed model | 0.96 | 0.93 | 0.94 | 0.91 | 0.92 |

\(\sigma =40\) | |||||

Noise | 0.72 | 0.54 | 0.74 | 0.56 | 0.53 |

TV model | 0.93 | 0.86 | 0.89 | 0.81 | 0.84 |

LLT model | 0.91 | 0.81 | 0.86 | 0.76 | 0.82 |

NLM model | 0.93 | 0.89 | 0.92 | 0.85 | 0.88 |

BLS-GSM | 0.92 | 0.88 | 0.92 | 0.84 | 0.87 |

Hybrid model | 0.92 | 0.87 | 0.91 | 0.84 | 0.87 |

Proposed model | 0.94 | 0.89 | 0.92 | 0.85 | 0.88 |

\(\sigma =50\) | |||||

Noise | 0.65 | 0.48 | 0.67 | 0.51 | 0.48 |

TV model | 0.89 | 0.83 | 0.87 | 0.79 | 0.82 |

LLT model | 0.86 | 0.78 | 0.84 | 0.73 | 0.79 |

NLM model | 0.91 | 0.88 | 0.90 | 0.83 | 0.85 |

BLS-GSM | 0.90 | 0.87 | 0.89 | 0.82 | 0.85 |

Hybrid model | 0.89 | 0.86 | 0.88 | 0.81 | 0.84 |

Proposed model | 0.91 | 0.88 | 0.90 | 0.82 | 0.85 |

## Conclusions

To eliminate the so-called staircase effect in total variation filter and avoid the edges blurring for fourth-order PDE filter, we propose an adaptive anisotropic diffusion model for image denoising, which is composed of a hybrid regularization term combining a total variation filter and a fourth-order filter and the fidelity term using the \(H^{-1}\)-norm. We also develop an efficient algorithm to solve our proposed model. Numerical experiments show that our proposed model has the highest PSNR, SSIM, MS-SSIM, and FSIM values among the six methods, and can preserve important structures, such as edges and corners.

## Declarations

### Authors’ contributions

Dr. KL carried out the study design and drafted the manuscript. Prof. JT analyzed the theory and revised the manuscript. Dr. KL and Dr. LA participated in software programming and analysis. All authors read and approved the final manuscript.

### Acknowledgements

This work is supported by the National Science Foundation of China (No. 61472466) and NASF-Guangdong Joint Foundation (Key Project) (No. U1135003).

**Competing interests**

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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