Generally, the practical images always hold the noise that does not only undermine
the display but also affect the subsequent treatment results of the higher-level
image. It is a big challenges to remove the noise of images with the maintainance
of geometric characters during the scientific research and engineering practical
activities. Therefore, denoising of image denoising is one of the important
issues in the study of image processing and computer vision. Image denoising
based on nonlinear diffusion equation is an effective method, about which many
research achievements have been obtained and applied in many fields (Chan
and Shen, 2005; Lysaker and Tai, 2006; Perona
and Malik, 1990). The basic idea is to use different smooth policies at
the target edge, namely at the edge area, the smooth process will be controlled
but accelerated in the other regions.
Based on the nonlinear diffusion equation, the complex filtering process can
be divided into two simple ones: one along the image gradient direction and
the other perpendicular to the image gradient direction. The equations with
better denoising results should have various diffusion rates in both directions,
namely, diffusion process is anisotropics. This method can also retain the image
geometry while removing the noise. There are some classic and anisotropics diffusion
models. There are some classic and anisotropics diffusion models such as Perona-Malik
model, mean curvature motion model, total variation model, among which total
variation model (Rudin et al., 1992) give the
following energy functional equation:
In the model, BV energy terms of the image Function model (based on the image gradient pattern energy term with L1 norm determined) determine the corresponding evolution equation that has a good non linear (has good non-linear) diffusion properties. In fact, the diffusion is unidirectional with non-zero diffusion velocity only in the tangent direction of horizontal lines of images which determines no demolishment of the important features of image structure but a certain effect of denoising during the evolution of the equation. However, in the local area of unimportant characteristics, the unidirectional diffusion speed becomes too slow and too single so as to affect the denoising effect and efficiency. If the energy term of I the equation s replaced by the L2 norm of the gradient mod, a thermal diffusion equation will be obtained which will cause excessive smoothing and loss of image features.
p-Laplace equation (Dibendetto, 1993), with a profound
physical meaning, could be related to lots of natural phenomena. Until now,
there are lots of theoretical achievements about p-Laplace equation. In this
study, based on the total variation model, we firstly discuss the diffusion
behaviors driven by p-Laplace which is applied to image denoising. A well-pose
proof of the model is implemented with a discussion of numerical solutions.
Finally, some related experimental results and conclusions are provided.
DIFFUSION BEHAVIORS DRIVEN BY p-LAPLACE
For the gray image function u(x): u(x): Ω⊂R2→R, analyzing
the local anisotropic diffusion, as |∇u(x)|≠0, at the x point, we determined
these two local directional vectors. In the gradient direction:
and the tangent direction:
subsequently computed and the following equation is obtained:
Consider the following energy term:
This is general p-Dirichlet calculus, the variation about υεC0∞(Ω):
From δEp = 0 and υεC0∞(Ω), p-Laplace equation is available:
It can be remarked as -Δpu = 0. In virtue of the direction vector N and T, p-Dirichlet calculus is rewritten as:
Theorem 1: Euler-Lagerange of Functional 6 is:
Proof: Whose proof is ignored.
An evolutive PDE is as follows in virtue of the gradient decline method:
When p = 2, this is equal to the heat equation. The diffusion term corresponds
to the minimum energy term of the total variation model, namely when p→∞,
corresponds to the infinite Laplace Δ∞u which can be defined
as uNN (Aronsson et al., 2004) or
u2NuNN (Oberman, 2005).
Those were applied to the image fixing (Caselles et al.,
1996) and the transforming problems (Cong et al.,
MODEL OF IMAGE DENOISING BASED ON THE p-LAPLACE EQUATION
Chambolle and Lions use the heat diffusion term to accelerate the total variation
model partially (Chambolle, 1995). Chen
et al. (2006) studied the diffusion behaviours of variational exponentiate.
With the inspiration of these studies, this paper proposes the following function based on the p-Dirichlet Calculus and total variation of energy function:
where, u0 refers to the images that is needed to be denoised and the nonnegative function F(s) is defined as the following:
Here, when β>0, p>1, F (s) is constant.
Set β = 2, p = 2.5 and Fig. 1 shows a comparison of
the function F(s) and the function
(heat diffusion model), the function f(s) = s (total variation model) when s
is in smaller and bigger conditions.
As is shown in the figures, with a smaller s, the value of function F(s) in this model decreases more quickly than that in the model of diffusion during the process of s→0; with a bigger value of s, it increases more slowly in the process of s→+∞. What is more, the increase in the value of the function F(s) in the model approaches that in total variation model which endows the evolutionary Eq. 8 minimized from the energy function with the expected diffusion attribute. When |Δu| is smaller, the diffusion will be accelerated but when |Δu| is larger, diffusion behaviors approach and then will be equal to that of the total variation model.
|| Comparison F(s) of TV model and with that of heat diffusion
model; (a) 0≤s≤β and (b) s>β
The variation of E(u) is computed when υεC0∞(Ω):
When, δEp = 0, the evolution equation is obtained:
From the above equation, as |∇u|≤β, the diffusion behaviors of
Eq. 11 is driven by the term ∇. (|∇u|p-2Δu)
of p-Laplace. If only |∇u|>1 and p>2 or |∇u|<1 and p<2,
then the diffusion results show the exceeding effect of heat diffusion equation.
When |∇u|>β behaviors of Eq. 11 is driven by
the curvature term .
Consequently, another form of evolutionary Eq. 11 is thus
Theorem 2: The Eq. 11 is equal to the following form:
Proof: The proof is ignored (Use the conclusion of Theorem 1).
Proof is ignored (Use the conclusion of theorem 1).
Utilizing the above mentioned model, we propose the preliminary boundary problem:
where, the definition of:
is similar to that in Eq. 11. Through the adaptation of
the method of Rudin et al. (1992), both sides
of the first formula are multiplied by u-u0 and the integrations
are also performed in Ω. Since t→∞, ,
We can get the following after using the Green function:
Then it yields:
Here, the denominator dx
is usually twice over noise variance which is usually predictable. After the
numerical solution of the problem (14), the equation proposed in this study
could be used in the field of denoising image.
|| Results of transaction of phoenix tree leaves with noise;
(a) Original image, (b) Noise image, (c) p = 1.0, (d) p = 1.6, (e) p = 2.0
and (f) p = 2.2
|| Results of rice-grains with noise; (a) Original image, (b)
noise image, (c) p = 1.0, (d) p = 1.6, (e) p = 2.0 and (f) p = 2.2
As shown in Fig. 2a and b, given original
images of phoenix tree leaves and denoised images, different p values are chosen
to perform numerical solution which produces corresponding results.
Test 2.2: Figure 3a and b dividedly
the original rice-grains images and denoising images. We utilize the proposed
model to implement the denoising procedures with various p values.
Figure 2c-f and Fig. 3c-2f
show the different results in the two experiments with various p-values. p-value,
iterative numbers n of model solution, the solved results of Peak-signal-to-noise
ratio (PSNR) are indicated in Table 1 and 2.
|| P and n and PSNR data
Given the constant p-value, with the iterative evolution, PSNR is gradually
increased from values of 18.9763 and 18.7481 to the final results. The process
|| Specific data of p, n and PSNR
When, n is dramatically decreased by p-value, the PSNR value of final results
is changed with a tiny visual effection. When, p = 1 and p = 2.0, the results
are total variation model and Chambolle-lions model.
In this study, we utilize the p-Dirichlet calculus and TV energy to build p-Laplace equation model applied in the denoising process of images. The test results show that, according to the reasonably adjusting parameter p values, the presented model can dramatically decrease the iterative numbers with better denoising effects. It is very meaningful of vital significance for the issues of denoising in the complicated environment.
We would like to thank the anonymous reviewers for their valuable comments. This study is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No.11JK0504).