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Image de-noising using Markov Random Field in Wavelet Domain

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Abstract

Removing noise from original image is still a challenging problem for researchers. There have been several published algorithm and each approach has its assumptions, advantages and disadvantages. Markov Random Field is n-dimensional random process defined on a on a discrete lattice. Markov Random Field is a new branch of probability theory that promises to be important both in theory and application of probability. This paper is an attempt to present the basic idea of the subject and its application in image denoising to the wider audience. In this paper, a novel approach for image denoising is introduced using ICM (Iterated Conditional Modes) approach of Markov Random Fields model.

Figures (3)
Where H \h[i,j] is the entire set of random variables H without the element h[ij] and N[i, j] represents the set of hfi,j]’s neighboring vertices. It is well known which consequences this setup has on the joint distribution of the random variables in H. Before we can elaborate on these however, we need to introduce the concept of a cilque [4][5][6]. A subset of H is called a clique if it is a singleton or if every pair of elements h[i,j] in that subset is neighbors in the corresponding graph. The lattice shaped MRF considered in this paper is depicted in Figure 1. We easily identify a single clique for each vertex h[i,j] and also identify cliques of the form {hfi,j], h{i,j-1]} or {hlij], hli- 1,j]} for each pair of adjacent vertices. According to the Hammersley-Clifford theorem [3] an MRF has equivalent Gibbs distribution is given by
Where H \h[i,j] is the entire set of random variables H without the element h[ij] and N[i, j] represents the set of hfi,j]’s neighboring vertices. It is well known which consequences this setup has on the joint distribution of the random variables in H. Before we can elaborate on these however, we need to introduce the concept of a cilque [4][5][6]. A subset of H is called a clique if it is a singleton or if every pair of elements h[i,j] in that subset is neighbors in the corresponding graph. The lattice shaped MRF considered in this paper is depicted in Figure 1. We easily identify a single clique for each vertex h[i,j] and also identify cliques of the form {hfi,j], h{i,j-1]} or {hlij], hli- 1,j]} for each pair of adjacent vertices. According to the Hammersley-Clifford theorem [3] an MRF has equivalent Gibbs distribution is given by
The performance of RMF method is illustrated with a quantitative and qualitative performance measure. The qualitative measure is the visual quality of the resulting image. The Peak-signal-to-noise (PSNR) is used as the quantitative measure. Different images, like natural, medical, aerial and under water, are used with four different noise, Gaussian, salt and pepper, speckle and Poisson. Fig (1), (4) and (7) show aerial, medical and underwater images respectively. Fig(2), (5) and (8 show original images with Gaussian , Poisson and speckle noise with PSNR values 30.9688 dB, 27.8729 dB and 27.247dB . And fig. (3), (6) and (9) are the denoised images with PSNR 38.5907dB, 35.324dB and 32.2061dB. (4) and (7) show aerial, medical and underwater images respectively. Fig(2), (5) and (8 show original images with Gaussian , Poisson and speckle noise with PSNR values 30.9688 dB, 27.8729 dB and 27.247dB . And fig. (3), (6) and (9) are the denoised images with PSNR 38.5907dB, 35.324dB and 32.2061dB.
The performance of RMF method is illustrated with a quantitative and qualitative performance measure. The qualitative measure is the visual quality of the resulting image. The Peak-signal-to-noise (PSNR) is used as the quantitative measure. Different images, like natural, medical, aerial and under water, are used with four different noise, Gaussian, salt and pepper, speckle and Poisson. Fig (1), (4) and (7) show aerial, medical and underwater images respectively. Fig(2), (5) and (8 show original images with Gaussian , Poisson and speckle noise with PSNR values 30.9688 dB, 27.8729 dB and 27.247dB . And fig. (3), (6) and (9) are the denoised images with PSNR 38.5907dB, 35.324dB and 32.2061dB. (4) and (7) show aerial, medical and underwater images respectively. Fig(2), (5) and (8 show original images with Gaussian , Poisson and speckle noise with PSNR values 30.9688 dB, 27.8729 dB and 27.247dB . And fig. (3), (6) and (9) are the denoised images with PSNR 38.5907dB, 35.324dB and 32.2061dB.

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References (13)

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  11. Kim Marie Mueller, "Construction and behavior of Multinomial Markov random field models", Iowa State University,Ames, Iowa ,2010(Theses and Dissertations)
  12. Chaohui Wang,, Nikos Paragios,"Markov Random Fields in Vision Perception: A Survey" ISSN 0249-6399, September 2012(Research Report) AUTHORS
  13. First Author -Shweta Chaudhary, B.E in Electronics and Telecommunication, M.E in VLSI and Embedded System, G.H. Raisoni Institute of Engineering and Technology, Pune, India Email: c.shweta.02@gmail. ISSN 2250-3153 www.ijsrp.org Second Author -Prof. A. L. Wanare, B.E. in Electronics, M.E in Electronics, PhD in Image Processing, D. Y. Patil College of Engineering, Pune, India. E-mail: anilwanare15@gmail.com. Correspondence Author -Shweta Chaudhary, +91- 9811864309, +91-9811864093, +91-11-25611221, E-mail: c.shweta.02@gmail.com

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