_{1}

^{*}

In this paper, we present a noise removal technique by combining the P-M model with the LLT model. The combined technique takes full use of the advantage of both filters which is able to preserve edges and simultaneously overcomes the staircase effect. We use a weighting function in our model, and compare this model with the P-M model as well as other fourth-order functional both in theory and numerical experiment.

In image processing, the methods based on variation and PDEs play an important role. In recent years, many scholars have done a lot of research in this respect, and many good results have been achieved on image such as enhancing, sharping, and denoising images. See for instance [

By using the functional variation principle to figure out the corresponding Euler-Lagrange equation, we can find the solution of Equation (1) to approach the original image. These second order partial differential equations (PDEs) models [

The P-M model has a good performance in many details and it is the origin of the other models. It is well known that the model represented by Perona and Malik [

where

creasing function.

minimized the energy functional (2), thus and obtained the following equation

here g is the diffusion coefficient, and g(・) is a non negative monotonically decreasing function with g(0) = 1 as

well as

sity of the level image is the global minima of the energy functional from (2) and (4). One should notice that there are many fast methods to minimize (2) such as [

The P-M model can remove the noise, at the same time it will produce the staircase effect, while the fourth-order model can remove the noise and will not generate the blocky effect. Since the Laplacian of the image at a pixel is zero only if the image is planar in its neighborhood, there are many fourth-order PDEs models attempt to remove noise by approaching piecewise planar image. Piecewise planar image looks more natural than step images which anisotropic diffusion, in other word, the fourth-order model will not produce the staircasing effect. It also is a strength of the LLT model. This model is put forward by Lysaker and Tai et al. In [

where the parameter λ_{2} > 0 which balances the relative weights of the two terms. This model is called the LLT

model. For simplicity we introduce the notation

Through the calculation of this variation, the gradient descent PDE of the minimization is

To avoid singularities in the above systems, we replace

Since the problem is convex, the steady state solution of the gradient descent PDE (9) is the minimizer of the energy functional. Numerical experiments show that the LLT model is able to greatly supress the staircase effect, but blur image and produce the speckle effect.

In order to find an appropriate scale parameter in our approach, firstly, we will give the specific calculation method [

where

These equations give us some dynamic values, which appear to converge as

In Section 2, the P-M model and the LLT model are presented in front of us, and in this section, we will combine these models with a weighting function. The image enhancement techniques for image deboising is a good approach to combine the advantages of the second-order PDEs method and the fourth-order PDEs method, thus has been popular, such as [

And the LLT model as follows

Set Ω is a rectangular domain,

where

And the initial condition is:

To some extent, these two models are able to suppress the noise, but it is well known that the first model is prone to massive effect, while the second model is tend to blur images and produce the speckles. Both methods have their pros and cons depending on the character of the image interest. We should find out a way for demonstrating the merits of these two models, at the same time, the shortcoming is suppressed by adjusting the parameter. Considering (14) and (15), we try to create a new model by a convex combination

of different ways. one can find that when

paper, we find out a α = 0.625 by repeated experiments, it works fine. As for more detailed algorithms, we will give them in the following.

As in [

where

Note that N, S, E, W represent the North, the South, the East and the West, respectively. And the symbol σ indicates 4-nearest-neighbors differences:

And the diffusion coefficient

Similarly, we can obtain the LLT discrete form: firstly, see [

These are discrete format of the P-M model and the LLT, but our focus points are the following:

Here, the value of α is 0.625, it is a preferred value of α. When α is different, the effect of image denoising is distinct. In the following numerical experiments, it will prove that a combination model is effective and accurate.

In this section, we will present numerical results obtained by applying our proposed model to image denoising. For comparison, we also present some results from application of the P-M model (14) and the LLT model (15) to the same images. From the result of experiment, we will utilize the pictures to verify these models, especially demonstrate the advantages of the convex combining model: the new model can reduce the blocky effects and avoid leaving the speckle artifacts. The following two classical single PDE schemes are:

(a) The model proposed by Perona and Malik [

(b) Lysaker and Tai et al. raised such model [

Note that, to avoid singular in the LLT model, we utilize the above expression. The contrast parameter k > 0 can be chosen in a variety of ways. Further, we proposed our model to compare with the above models.

(c) Convex combination of the above two models.

From the expression, we know that our model is built on the basis of two classical models. Our model is more natural, and the effect is good in comparison with the model which is obtained by an energy functional that contains gradient functions and Laplace functions. Our experimental pictures are 256 × 256 sized gray-scale image Lena. We consider the noisy image f = u + n where u is the original image, and the n is a white Gaussian noise with the expectation zero and standard deviation σ. In

where u is the original image, f is the recovery and f is its mean value.

(2) Signal to noise ratio (PSNR) is defined by:

where

where f denotes the denoised image and u denotes the original image.

Image | The PM model | The LLT model | My model |
---|---|---|---|

SNR (dB) | 21.3700 | 22.6804 | 22.7970 |

PSNR (dB) | 27.9767 | 29.2871 | 29.4037 |

In

This paper proposes a convex combination model by the P-M model and the LLT model, and α = 0.625 is found to balance these two models. Numerical results represent the competitive performance of the new model, for noise removal while maintaining the jump discontinues better, and suppressing the speckles.

QianYang, (2015) Image Denoising Combining the P-M Model and the LLT Model. Journal of Computer and Communications,03,22-30. doi: 10.4236/jcc.2015.310003