## Shunfeng Wang* , Jiacen Xie* , Yuhui Zheng** , Jin Wang*** and Tao Jiang*## |

| Corrupted image Y and X^{(0)}=Y,penalty parameter β,regularization parameters λ and ε, the radius r. |

Step 1. | Choose the most likely Gaussian mixing weights k for each patch _{max}P; _{i}X |

Calculate z using (10); _{i}^{(n+1)} | |

Pre-estimate image X using (11); ^{(n+1)} | |

Let I, calculate ^{(n+1)}=X^{(n+1)}μ and _{k}σ; _{k}^{2} | |

LetP,calculate ^{(n+1)}=I^{(n+1)}a using (3) and (4); _{k},b_{k} | |

Calculate q using (5) and let ^{(n+1)}q; ^{(n+1)}=X^{(n+1)} | |

Repeat Steps 1-6 until the stopping criterion is satisfied. | |

Output. | De-noised image X. ^{(n+1)} |

In the iteration process, the input image is the same as the guided image, and Eqs. (3) and (4) can be simplified as:

The parameter ε determines whether a pixel is in a boundary region (Table 2).

Table 2.

| The pixel value varies greatly, [TeX:] $$\sigma _ { k } ^ { 2 } >> \varepsilon , \text { so } a _ { k } \rightarrow \mathbf { 1 } , b _ { k } \rightarrow \mathbf { 0 } , q \rightarrow I$$ and the edge information of the image is better preserved. |

| The pixel values are almost unchanged, [TeX:] $$\sigma _ { k } ^ { 2 } \ll \varepsilon , \text { so } a _ { k } \rightarrow 0 , b _ { k } \rightarrow \mu _ { k } , q \rightarrow \overline { \mu } _ { k }$$, and the flat regions are smoother. |

It can been seen from the above analysis that the edge-preserving effect of the guided filter depends on the guided image, while the EPLL model can present a better guiding image for guided filtering. In contrast, the second smoothing of flat regions obtained by the guided filtering can improve the noise suppression effect of the EPLL and avoid the ladder effect. These two methods complement and reinforce each other perfectly. The next sections prove the validity of the proposed method via experiments.

In this section, we discuss the performance of the proposed method. In this study’s experiments, the GMM with 200 mixture components was learned from a set of 2 × 10^{6} images patches, which were sampled from the Berkeley Segmentation Database Benchmark (BSDS300). In order to verify the effectiveness of the proposed method, it was compared with guided filtering and EPLL in terms of visual effects and numerical results. To the images used in this study’s experiments were added Gaussian noise with zero mean and standard variance, σ = 15 or σ = 30. The parameters for EPLL in the experiments were as follows: image patch size *√L* = 8, regularization parameter *λ = L/σ ^{2}*, and penalty parameter β = 1⁄σ

Fig. 1 displays the de-noised results of the three methods on Couple image with dimensions of 512×512. Here, Fig. 1(a) is the original clean image; Fig. 1(b) is the noisy image generated by adding Gaussian white noise with zero mean and standard variance σ = 30 to the original image; Fig. 1(c) shows the result of the guider filtering, but the edges, details, and other information have not been preserved well; Fig. 1(d) shows the de-noising results of the EPLL model, where it can be seen that mottling occurs in certain areas; and Fig. 1(e) shows the de-noising result of the proposed method. This shows a better visual effect at the boundary of the wall. The boundary is preserved and the transition is more natural in the smooth region. As shown in Fig. 2, the local enlargement of the results of the three algorithms confirms the above analysis. Figs. 3 and 4 show the same result. The proposed method makes the restoration result smoother and preserves more details. Thus, it is reasonable to conclude that the proposed method is superior to EPLL in terms of numerical results, as shown in Table 3.

Table 3.

Image | Noise standard variance | Guided filtering | EPLL | Our method |

Boat | σ=15 | 29.99 | 31.75 | 31.99 |

σ=30 | 27.34 | 28.29 | 28.61 | |

Plane | σ=15 | 28.75 | 28.97 | 29.54 |

σ=30 | 25.34 | 26.10 | 26.60 | |

Barbara | σ=15 | 30.11 | 30.41 | 30.62 |

σ=30 | 27.56 | 27.98 | 28.23 | |

Hill | σ=15 | 30.18 | 31.51 | 31.74 |

σ=30 | 27.61 | 28.57 | 28.86 | |

Couple | σ=15 | 29.60 | 31.70 | 31.96 |

σ=30 | 27.12 | 28.07 | 28.35 |

Based on the complementarity of EPLL and guided filtering, this paper proposes a method of coupling the expected patch log likelihood and guided filtering for image de-noising. It uses the EPLL model to construct the guided image for guided filtering, which can provide better structural information for guided filtering. Meanwhile, by the secondary smoothing of guided image filtering in the image homogenization areas, we can improve the noise suppression effect in those areas, and reduce the ladder effect brought about by EPLL.

The experimental results show that the proposed method is better than the previous two methods, in terms of both the visual effect and numerical performance. This combination makes full use of the advantages of the two methods while making up for their shortcomings, which makes the two complement and progress each other. Of course, there are still some shortcomings in this method. For example, the selection of the parameters ε and iteration times are all artificially set, and their values directly determine whether the algorithm will be overly smooth or not. As such, any future research will focus on this aspect.

She received the M.S. degree in the Nanjing University of Information Science and Technology, China in 2002. Now she is a Professor at the College of Mathematics and Statistics, Nanjing University of Information Science and Technology. Her research interests cover pattern recognition, and information processing.

He received the B.S. and M.S. degrees from Nanjing University of Posts and Telecommunications, China in 2002 and 2005, respectively. He received Ph.D. degree from Kyung Hee University Korea in 2010. Now, he is a professor in the School of Computer Communication Engineering, Changsha University of Science Technology. His research interests mainly include routing protocol and algorithm design, performance evaluation and optimization for wireless ad hoc and sensor networks. He is a member of the IEEE and ACM.

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