This paper focuses on the problem of multiplicative noise removal. Using a gray level indicator, we derive a new functional which consists of the adaptive total variation term and the global convex fidelity term. We prove the existence, uniqueness, and comparison principle of the minimizer for the variational problem. The existence, uniqueness, and long-time behavior of the associated evolution equation are established. Finally, experimental results illustrate the effectiveness of the model in multiplicative noise reduction. Different from the other methods, the parameters in the proposed algorithms are found dynamically.
Multiplicative noise occurs while one deals with active imaging system, such as laser images, microscope images, and SAR images. Given a noisy image
The goal of this paper is to propose a globally strictly convex functional well-adapted to removing multiplicative noise, which is as follows:
Various adaptive filters for multiplicative noise removal have been proposed. In the beginning, variational methods for multiplicative noise reduction deal with the Gaussian multiplicative noise [
In the additive noise case, the most classical assumption is that the noise is a white Gaussian noise. So one case in which dealing with multiplicative noise is the white Gaussian noise. Using the framework in [
In order to make sure the two constraints (
If by the gradient projection method the values of
Generally, the speckle noise is treated as Gamma noise with mean equal to one. The probability distribution function
Gamma noise is more complex than Gaussian noise [
The simplest idea is to take the log of both sides of (
Recently, Huang et al. [
Notice that the fidelity term
The paper is organized as follows. In Section
The goal of this section is to propose a new variational model for multiplicative noise removal. First, we proposed a new fidelity with global convexity, which can always satisfy the constraint (
Based on the idea in [
Note that
(1) It is easy to check that the function
(2) Let us denote by
We have
Assume
The relation between the influence of the noise and the gray levels. (a) 1D signal
Noise-free signal
Speckled noise
Noise signal
Original/noise signal
In [
The previous analysis leads us to propose a convex adaptive total variation model for multiplicative noise removal,
The evolution of the Euler-Lagrange equation for (
In this section, we study the existence and uniqueness of the solution to the minimization problem (
If
We always assume that
A function
(1) If
(2) From (
Now, we directly show some lemmas on
Assume that
Assume that
By a minor modification of the proof of Lemma 1 in Section 4.3 of [
Let
In this subsection, we show that problem (
Let
Let us rewrite
In fact, we remark that
Finally, from Remark
In this subsection, we state a comparison principle for problem (
Let
Let us denote
From Theorem
In this section, by an approach from the theory used in both [
Denote
On the other hand, let
A function
In this subsection, we consider the approximating problem
Let us denote
Let
Based on this fact, we have the following existence and uniqueness result for the problems (
Let
Let us fix
Next, let us verify that the truncated function
Moreover, multiplying (
Finally, we will verify that the above weak solution will be the pseudosolution for the problems (
In this subsection, we will prove the main theorem for the existence and uniqueness for the solution to the problems (
Suppose
Let
In fact, from (
Note that, for any
Since
To see (
From (
Next we show that, for all
Replacing
Then, by a processing similar to the one for getting (
Replacing
Moreover, replacing
Finally, the uniqueness of pseudosolutions to the problems (
At last, we will show the asymptotic limit of the solution
As
Take a function
By dividing
We present in this section some numerical examples illustrating the capability of our model. We also compare it with the known model (AA). In the next two subsections, two numerical discrete schemes, the
Numerically we get a solution to the problem (
The numerical algorithms for the problems (
Here the MATLAB function “conv2" is used to represent the two-dimensional discrete convolution transform of the matrix
From the proof of Theorem
In this section, we used the similar way for numerical experiments as [
The denoising algorithms were tested on three images: a synthetic image (
For a noise-free image
For fair comparison, the parameters of SO and AA were tweaked manually to reach their best performance level. Their values are summarized in Table
Parameters used in the comparison study.
Algorithm | Parameters | ||
---|---|---|---|
|
|
|
|
The synthetic image ( |
|||
| |||
|
|
|
|
|
|
|
|
SO |
|
|
|
AA |
|
|
|
| |||
The aerial image ( |
|||
| |||
|
|
|
|
|
|
|
|
SO |
|
|
|
AA |
|
|
|
| |||
The cameraman image ( |
|||
| |||
|
|
|
|
|
|
|
|
SO |
|
|
|
AA |
|
|
|
The results are depicted in Figures
Synthetic image (
Noisy:
Original
SO: PSNR = 4.14, MAE = 23.94
AA: PSNR = 16.33, MAE = 4.50
Synthetic image (
Noisy:
Original
SO: PSNR = 15.18, MAE = 6.15
AA: PSNR = 20.14, MAE = 2.94
Synthetic image (
Noisy:
Original
SO: PSNR = 21.00, MAE = 2.84
AA: PSNR = 23.94, MAE = 1.49
Aerial image (
Noisy:
Original
SO: PSNR = 17.82, MAE = 25.44
AA: PSNR = 22.34, MAE = 13.35
Aerial image (
Noisy:
Original
SO: PSNR = 24.00, MAE = 11.05
AA: PSNR = 24.20, MAE = 10.30
Aerial image (
Noisy:
Original
SO: PSNR = 27.27, MAE = 7.42
AA: PSNR = 26.04, MAE = 8.13
Cameraman image (
Noisy:
Original
SO: PSNR = 14.15, MAE = 38.92
AA: PSNR = 18.81, MAE = 22.14
Cameraman image (
Noisy:
Original
SO: PSNR = 21.61, MAE = 14.64
AA: PSNR = 21.91, MAE = 14.25
Cameraman image (
Noisy:
Original
SO: PSNR = 24.96, MAE = 8.36
AA: PSNR = 24.25, MAE = 9.97
In the numerical experiments, we can see that for the nontexture image, our methods and AA method work well (see Figure
PSNR and MAE.
PSNR | MAE | ||||||
---|---|---|---|---|---|---|---|
|
1 | 4 | 10 |
|
1 | 4 | 10 |
The synthetic image (300 × 300) | |||||||
| |||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SO | 4.14 | 15.81 | 21.00 | SO | 23.94 | 6.15 | 2.84 |
AA | 16.33 | 20.14 | 23.94 | AA | 4.50 | 2.94 | 1.49 |
| |||||||
The aerial image ( |
|||||||
| |||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SO | 17.82 | 24.00 | 27.27 | SO | 25.44 | 11.05 | 7.42 |
AA | 22.34 | 24.20 | 26.04 | AA | 13.35 | 10.30 | 8.13 |
| |||||||
The cameraman image ( |
|||||||
| |||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
SO | 14.15 | 21.61 | 24.96 | SO | 38.92 | 14.64 | 8.36 |
AA | 18.81 | 21.91 | 24.25 | AA | 22.14 | 14.25 | 9.97 |
The authors would like to express their sincere thanks to the referees for their valuable suggestions for the revision of the paper which contributed greatly to this work. The authors would also like to thank Jalal Fadili for providing them the MATLAB code of his algorithm. They would like to express their deep thanks to the referees for their suggestions while revising the paper, yet again. This work was partially supported by the Fundamental Research Funds for the Central Universities (Grant nos. HIT.NSRIF.2011003 and HIT.NSRIF.2012065), the National Science Foundation of China (Grant no. 11271100), the Aerospace Supported Fund, China, under Contract no. 2011-HT-HGD-06, China Postdoctoral Science Foundation funded project, Grant no. 2012M510933, and also the 985 project of Harbin Institute of Technology.