Journal of Scientific Computing

, Volume 67, Issue 1, pp 103–129 | Cite as

Removing Mixture of Gaussian and Impulse Noise by Patch-Based Weighted Means

  • Haijuan Hu
  • Bing Li
  • Quansheng LiuEmail author


We first establish a law of large numbers and a convergence theorem in distribution to show the rate of convergence of the non-local means filter for removing Gaussian noise. Based on the convergence theorems, we propose a patch-based weighted means filter for removing an impulse noise and its mixture with a Gaussian noise by combining the essential idea of the trilateral filter and that of the non-local means filter. Experiments show that our filter is competitive compared to recently proposed methods. We also introduce the notion of degree of similarity to measure the impact of the similarity among patches on the non-local means filter for removing a Gaussian noise, as well as on our new filter for removing an impulse noise or a mixed noise. Using again the convergence theorem in distribution, together with the notion of degree of similarity, we obtain an estimation for the PSNR value of the denoised image by the non-local means filter or by the new proposed filter, which is close to the real PSNR value.


Gaussian noise Impulse noise Mixed noise Trilateral filter Non-local means filter Convergence theorems Degree of similarity Estimation of PSNR  



The work has been supported by the Fundamental Research Funds for the Central Universities in China (No.N130323016), the Research Funds of Northeastern University at Qinhuangdao (No. XNB201312, China), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry(48-2), the National Natural Science Foundation of China (Grant Nos. 11171044 and 11401590), and the Science and Technology Research Program of Zhongshan (Grant No. 20123A351, China). The authors are very grateful to the reviewers for their valuable remarks and comments which led to a significant improvement of the manuscript. They are also grateful to Prof. Raymond H. Chan and Dr. Yiqiu Dong for kindly providing the code of ROLD-EPR.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNortheastern University at QinhuangdaoQinhuangdaoChina
  2. 2.College of Information Science and Engineering, Northeastern UniversityShenyangChina
  3. 3.CNRS UMR 6205, LMBAUniversité de Bretagne-SudVannesFrance
  4. 4.Department of MathematicsZhongshan PolytechnicZhongshanChina

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