We propose a hybrid total-variation-type model for the image restoration problem based on combining advantages of the ROF model with the LLT model. Since two
Image restoration is of momentous significance in coherent imaging systems and various image processing applications. The goal is to recover the real image from the deteriorated image, for example, image denoising, image deblurring, image inpainting, and so forth; see [
For the additive noisy image, many denoising models have been proposed based on PDEs or variational methods over the last decades [
However, as we all know the
Another topic for image restoration is to find some efficient methods to solve the above proposed models. In fact, there are many different methods based on PDE or convex optimization to solve the minimization problem (
The paper is organized as follows. In Section
Let us describe some notations and definitions used in this paper. For simplifying, we use
The operator
Let
Assume that
Let
Let
The projection operator
The shrinkage operator
It is obvious that the function
Variable splitting methods such as the ADMM [
We now consider the following constrained problem:
(1) Choose the original values:
(2) Compute
(3) If the stop criterion is not satisfied, set
Since
Assume that
Notice that Algorithm
Though the ADMM in Algorithm
(1) Choose the original values:
(2) Compute
(3) If the stop criterion is not satisfied, set
Set
Assume that
In Section
The related examples in the following subsections are performed using Windows 7 and Matlab 2009(a) on a desktop with Intel Core i5 processor at 2.4 GHz and 4 GB memory. All of the parameters for related models are chosen by trial and empirically which can yield better restored images. On the other hand, we should notice that it is not very expensive when we use the ADMM and the PPM to get
In this subsection, we consider to use the ADMM and the PPM to solve (
(1) Choose the original
(2) Compute
(3) If the stop criterion is not satisfied, set
For the first subproblem (
(1) Choose the original
(2) Compute
(3) If the stop criterion is not satisfied, set
For Algorithm
If
For the above two algorithms, we can also set
In this example, we compare the ADMM with the PPM for solving the ROF model (
The related results in Example
|
The ROF model | ||||||
---|---|---|---|---|---|---|---|
Stopping | ADMM | PPM | |||||
Size | Conditions |
Time (s) | Ite. | SNR | Time (s) | Ite. | SNR |
|
|
0.2184 | 86 | 24.7949 | 0.1248 | 83 | 24.7043 |
|
|
0.9204 | 77 | 16.2934 | 0.2652 | 38 | 16.3540 |
|
|
4.3368 | 64 | 15.7146 | 1.9656 | 53 | 15.9353 |
| |||||||
|
The LLT model | ||||||
Stopping | ADMM | PPM | |||||
Size | Conditions |
Time (s) | Ite. | SNR | Time (s) | Ite. | SNR |
| |||||||
|
|
0.4368 | 60 | 24.5688 | 0.3432 | 80 | 25.0287 |
|
|
1.7004 | 55 | 16.9065 | 0.9360 | 53 | 16.9083 |
|
|
15.4441 | 55 | 15.9432 | 7.2384 | 50 | 15.9512 |
| |||||||
|
The hybrid model | ||||||
Stopping | ADMM | PPM | |||||
Size | Conditions |
Time (s) | Ite. | SNR | Time (s) | Ite. | SNR |
| |||||||
|
|
0.7176 | 69 | 25.7224 | 0.5304 | 70 | 25.4982 |
|
|
2.5584 | 55 | 16.9699 | 1.4664 | 52 | 16.9701 |
|
|
19.6561 | 56 | 15.9503 | 9.3445 | 50 | 15.9512 |
The original images and the noisy images with three different sizes in Example
In this example, the noisy image is added to the Gaussian white noisy with the standard deviation
The related data in Example
Model | R.P. | Time (s) | SNR | MSE |
---|---|---|---|---|
ROF | 4.5 | 0.8268 | 17.7677 | 44.0400 |
LLT | 3.0 | 2.0904 | 17.9998 | 42.7826 |
Convex | 3.5 | 2.6832 | 18.2399 | 39.7756 |
Hybrid | 4.5 | 2.5272 | 18.3694 | 39.1447 |
The original and the noisy image in Example
The related restored images in Example
In this subsection, we extend the hybrid model to other classes of restoration problems. As we can see in Section
Assume that the functional
It should be noticed that the following three models satisfy the conditions of Theorem
Now we apply the hybrid model to the image deblurring problems with the following formula:
In this experiment, we use the image Lena, which is blurred with a Gaussian kernel of “hsize = 3” and in which is added the Gaussian white noise with the standard deviation
The related SNRs in Example
Original image
ROF model
LLT model
Hybrid model
Here we consider to use the hybrid model for the image inpainting problem with the following form:
In this example, we show the results of real image inpainting in Figures
The related results in Example
Model | Image (c) in Figure |
Image (d) in Figure |
||||
---|---|---|---|---|---|---|
|
|
SNR |
|
|
SNR | |
ROF | 220 | 0.005 | 18.5533 | 32 | 0.008 | 15.4055 |
LLT | 150 | 0.005 | 16.2934 | 60 | 0.005 | 16.4336 |
Hybrid | 220 | 0.005 | 20.2713 | 70 | 0.0025 | 16.6034 |
The related images in Example
Original image
Mask image
Painting image
Noisy image
The related restored images corresponding to the painting image in Example
The related restored images corresponding to the noisy image in Example
Based on the hybrid model (
In this example, we restore the multiplicative noisy image. The noisy Lena image shown in Figure
The related images and the local zooming images in Example
In this paper, based on the edge detector function, we proposed a hybrid model to overcome some drawbacks of the ROF model and the LLT model. Following the augmented Lagrangian method, we can employ the ADMM of multipliers to solve this hybrid model. In this paper, we mainly proposed the PPM to solve this model due to the fact that the PPM unnecessarily solves a PDE compared with the ADMM so that it is more effective than the ADMM. The convergence of the proposed method was also given. However, the convergence rate of the proposed method is only
Following the assumption, we can find that the saddle point
Set
Let
Setting
On the other hand, if
From the assumption, the functional
Now we show that the sequence
The second author would like to thank the Mathematical Imaging and Vision Group in Division of Mathematical Sciences at Nanyang Technological University in Singapore for the hospitality during the visit. This research is supported by Singapore MOE Grant T207B2202, Singapore NRF2007IDM-IDM002-010, the NNSF of China (no. 60835004, 60872129), and the University Research Fund of Henan University (no. 2011YBZR003).