Image Denoising Using Partial Differential Equations

Introduction

Typical methods of PDEs

Areas of interest

References

Introduction

In the field of image processing, the noise of an image means the random error introduced to the image during the process of digitisation. These noise is usually unwanted and unpredictable. The images as shown in Figures 1(a) and 1(b) were taken by camera with and without the funciton of denoising respectively.

Fig 1(a) picture without noise	(b) picture with noise

In the past decades, research and development in the use of partial differential equations (PDEs) become an important area in image processing. Comparing with the traditional methods of image denoising PDEs have many advantages, including easy description of local features of an image, employ existing mathematical theory, possible use of many existing numerical algorithms, separate analysis and implementation, preserving most structures and information of an image, deal with the geometric features directly, simulate the dynamic process of image restoration, etc. In the literature, the history of PDE methods dates back to the filter method given by Lee^[1] in 1980. Based on this research, scale space was introduced by Witkin^[2] whereas Koenderink ^[3] made a convolution between image and Gaussian function to implement low-pass filter,which lay a good theoretical foundation of this method. In 1990, Perona and Malik^[4] proposed the anisotropic diffusion based on scale space which paved the use of nonlinear diffusion models in image denoising, image edge detection, image segmentation, image inpainting and so on (See references^[5]-[7], etc.). In 1992,the total variation regularization for image denoising^[8] was proposed (do not use the put forward - I think this is a translation of the Chinese work tichu which is not right but often seen in chinese papers) by Rudin and Osher. The proposed method in fact view image processing from another aspect, i.e., energy functional, making the PDE method more competitive in image denoising.

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Typical Mehtods of PDEs

In this section, a brief review is given of previous work on anisotropic diffusion model and fourth order PDE's model. Several denoising computational results are also shown to illustrate the methodologies.

Anisotropic Diffusion Model

The basic idea proposed by Perona and Malik^[4] is to evolve from an original image I₀(x,y), which is defined in a convex domain Ω⊂R×R, a family of increasingly smooth images I(x,y,t) derived from the solution of the following partial differential equation,that is,anisotropic diffusion model(PM model for short):

From the literature, the choice of coefficent c(x,y,t) is an important isuse because different coefficient would lead to different effect. Perona and Malik choose two different coefficients as follows:

   
Figure 2(a) shows the original Lena image, figure 2(b) has added a 10 db noise, and figure 2(c) shows the result using PM model with the first coefficient showed above.

Fig 2 (a)lena	(b) lena with 10DB noise	(c) result of PM model

Fourth-order PDE's(YK model)

It is easy to see that one of the drawbacks induced by PM model is the "block effect", which means that the grey values are the same in some regions. In order to overcome this shortcoming,You and Kaveh^[7] proposed a fourth-order PDE for image denoising as shown in the equation below.

The result of applying YK model to the noisy image in figure 2(b) is as shown in figure 3.

Fig 3 Result by YK model

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Areas of interest

Currently my main research effort is in image denoising using PDEs, and some important progress has been achieved which will be added here in the near future. Furthermore, I am also intereted in image segmentation and image inpainting.

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References

[1] Lee J S. Digital Image Enhancement and Noise Filtering by Use of Local Statistics [J].IEEE Trans PAMI,1980,2(2):165-168.
[2] J.Babaud, A.Witkin, M.baudin, and R.Duda. Uniqueness of the Gaussian kernel for scale-space filtering. IEEE Trans. Pattern Anal. Machine Intell.Vol, PAMI-8,Jan,1986.
[3] J.Koenderink. The structure of images.Biol.Cybern.1984(50) :363-370.
[4] Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion [J].IEEE Trans PAMI, 1990, 12(7):629-639.
[5] L.Alvarez, P.-L.Lions, and J.-M.Morel. Image selective smoothing and edge detection by nonlinear diffusion II.SIAM J.Numer.Anal.29, 1992:845-866.
[6] J.-L.Morel and S.Solimini. Variational Methods in image Segmentation. Birkhauser, Boston, 1995.
[7] Yu-Li You, Kaveh M. Fourth-Order partial differential equations for noise removal[J].IEEE Transaction Image Processing,2000,9(10):1723-1729.
[8] Leonid I. Rudin, Stanley Osher and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physical D,1992,60(1-4):259-268.

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