1. Introduction
In many applications of image processing, the solution of the large-scale linear discrete ill-posed equation or the least squares solution are required. Denoising and deblurring are some of the most important and challenging tasks in image restoration. For recent works on image denoising and deblurring for large scale linear discrete ill-condition equations and least squares, see [
1,
2,
3].
Consider the restoration of two-dimensional gray-scale images [
4],
where
is a blurry and noisy image,
is the point spread function (PSF),
represents additive noise in the available data
, and
is the clean image.
The discretized form of Equation (
1) can be expressed as
where the noise is obtained by adding random a normally distributed vector
e with mean 0 and standard deviation 1, and
.
H represents the fuzzy operator, which is usually a real symmetric block Teoplitiz matrix,
is the vector generated by the original image, and
f represents the blur- and noise-contaminated image vector.
We would like to determine an approximate value of
when the blurry and noisy image
and the fuzzy operator
H are known but the noise vector
e is unknown. Ignoring the noise vector
e, Equation (
2) can be expressed as,
where
u is the solution we want to find.
Since the matrix
H is a large-scale singular matrix, the linear discretization Equation (
3) is usually an ill-posed problem. Although we can solve Equation (
3) directly by some iterative methods, due to the loss of high-frequency information, these iterative methods cannot accurately recover edges, which seriously affects the quality of image restoration.
In order to determine meaningful values of
, one of the classical methods of replacing Equation (
2) is the Tikhonov regularization model,
where
is a regularization parameter and
is a regularization operator. The choice of different regularization operators creates different image restoration models.
For the nonlinear Equation (
4), refs. [
5,
6] propose the explicit and semi-implicit discretization schemes, which have a good preserving effect on the edges of images and can provide high quality restoration of images; however, the amount of computation is extremely high. In addition, the selection of regularization parameters and time intervals is not easy.
In [
4,
7], the restoration Equation (
3) is discussed in detail by using the cascadic multigrid method, which is effectively applied to the restoration of small- and medium-sized images. For large-scale images, linear and nonlinear often require a lot of storage space and computation time, which make the recovery process very difficult. In recent years, in order to solve these problems, many scholars have extended domain decomposition methods [
8] to image restoration, achieving remarkable results for many kinds of restoration models.
Ref. [
9] proposes a simple domain decomposition algorithm based on the waveform relaxation of nonlinear diffusion equations for denoising images. It shows the feasibility of the domain decomposition algorithm in image restoration. Ref. [
10] proposes parallel overlapping domain decomposition methods and shows the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, but the CPU time of this method is very long. Refs. [
11,
12] combine the domain decomposition method with the graph cuts algorithm for total variation minimization. This method improves the computation efficiency and greatly reduces the memory cost and enables us to solve very large-scale image problems effectively. Ref. [
13] combines the domain decomposition method and the Bregman algorithm for image inpainting and image deblurring. It accelerates the computation of subproblems by the nested Bregman iteration. It turned out that the proposed new solution is up to three times faster than the iterative algorithm currently used in domain decomposition methods. In [
14], the meshfree finite point method with parallel domain decomposition is investigated for image denoising. The domain decomposition technique, which can be implemented in parallel, decreases the size of the system of equations and significantly reduces the computational cost. In [
15,
16], a fast algorithm based on overlapping domain decomposition technology is proposed to solve larger image problems and perform more efficiently than traditional method in terms of the restored image quality and CPU time. In [
17], the image is denoised by the overlapping domain decomposition method based on the successive subspace correction method and parallel subspace correction method. This method shows the ability of processing large-scale images. Ref. [
18] presents a fast domain decomposition method for global image smoothing. This method converges quickly after a few iterations and the runtime is much shorter than conventional methods. Ref. [
19] presents a parallel domain decomposition algorithm for large scale image denoising, which, compared with [
10], greatly reduces the calculation time.
The above methods have successfully solved the problem of large-scale image processing, but most of them only denoise or deblur the input image. For large-scale images that contain both noise and blur, we research a kind of quick and effective method.
The cascadic multigrid method is a unidirectional computation method from the coarse grid level to the fine grid level. The solution to the coarse grid provides a better initial value for the fine grid by interpolation methods, during which coarse grid correction is not required. The solution of the fine grid is obtained only after interpolation and iteration operations are used. The overlapping domain decomposition method has a strong performance in dealing with large-scale problems. Ref. [
20] introduces a new parallel cascadic multigrid method. Ref. [
21] introduces the multigrid domain decomposition method. Inspired by their methods, combining the overlapping domain decomposition method with the cascadic multigrid method, we propose a new algorithm for image restoration. Our algorithm has the following advantages:
(1) The algorithm can not only improve the computational efficiency and achieve good denoising and deblurring effects but also protect the image edge.
(2) The algorithm has a significantly faster convergence compared to when solving the problem directly at the finest level. The image is subdivided into multiple overlapping subimages, and each of the subimages is solved by the cascadic multigrid method or the new extrapolation cascadic multigrid method.
(3) The algorithm is parallel, which greatly saves storage cost and improves the computing efficiency.
The structure of this paper is as follows. In
Section 2, we introduce the domain decomposition method with cascadic multigrids and the domain decomposition method with new extrapolated cascadic multigrids.
Section 3 introduces the form and application of the edge-preserving denoising operator.
Section 4 introduces how to locally weight the overlapped part of the subdomain. In
Section 5, some numerical results are given. In
Section 6, a summary is presented.
2. Overlapping Domain Decomposition Method with Cascadic Multigrid
In this section, we propose an overlapping domain decomposition method with the cascadic multigrid algorithm to solve the problem of image deblurring and denoising. The advantages of this method are that it has a lower cost and can be extended on parallel computers.
First, we divide the image domain into
N non-overlapping subdomains denoted as
with
and
. Then, we extend each subdomain into a larger subdomain
, including
columns or
rows pixels from its neighboring subdomains. We can denote the overlapping size
as the distance between the boundaries of
and
. For any
, we define the set of neighboring subdomains as
. For any
,
denotes the overlapping domain.
denotes the non-overlapping domain. See
Figure 1 for an example.
Then, we apply the domain decomposition method to Equation (
3). We can parallelly solve the following subproblems,
Let us denote the solution of Equation (
5) as
. After solving the problem on each subdomain, we will carry out the weighted processing (Equation (
6)) for overlapped parts. More details will be given in
Section 4.
Next, we obtain the iteration solution
in the following way,
where
is a restriction operator from
to
and
is an extension operator from
to
. It can extend the subdomain to the entire domain. Either
or
is a matrix with only elements 0 and 1.
Then, the above process is repeated until the following stopping rules were satisfied,
where
and is a given constant.
The framework of the domain decomposition method is given in Algorithm 1.
Algorithm 1: Overlapping domain decompsition method (DDM) for image restoration |
|
In Step (1), we can also solve the subproblem by the cascadic multigrid method.
For the sake of discussion, we let every be an column vector, let m be an integer, and set . Let be a nested subspace. The subscript l indicates the number of the grid level, denotes the coarsest grid level, and indicates the finest grid level. Let be the representation of in . Accordingly, we have . When M is odd, ; when M is even, .
In the cascadic multigrid algorithm, the following linear system needs to be solved iteratively at each
,
where
is the representation of the fuzzy operator
in
.
Set
as the kth element of
, and we let
We define the restriction operator
which satisfies
when
,
.
is generated in the following way,
when
,
.
The prolongation operator is from level to level for each l. Generally, linear interpolation or quadratic interpolation is used as the prolongation operator.
We use to represent the smoother MR on level l in the subdomain. The smoothing is terminated when the following stopping rule is satisfied.
Stopping rule: For a given
,
is a constant and independent of
. When
the iteration is stopped. The number of iterations is denoted as
.
Thus, the cascadic multigrid method (CMG) in this paper first determines an approximate solution of in using MR. Then, this iterative value is mapped from to by the prolongation operators . A correction of this mapping in is iterated by and terminated with the stopping rule. The approximate solution in is mapped into by the prolongation operators . The computations are continued in this manner until an approximate solution has been determined in the finest grid level .
In our algorithm, quadratic interpolation is used as the prolongation operator
. The edge-preserving denoising operator is denoted as
. The framework of the domain decomposition method with cascadic multigrids is given in Algorithm 2.
Algorithm 2: Overlapping domain decomposition method with cascadic multigrid (DDM-CMG) for image restoration |
|
In [
22], the new extrapolation cascadic multigrid method (NECMG) is proposed, which can provide better initial values for the next level and has better convergence by using new extrapolation formulas and quadratic interpolation.
Let us take the one-dimensional triplet grid
as an example to understand the new extrapolation formulas and quadratic interpolation. The pixel at the
node on
is denoted as
, and the corresponding pixel value is denoted by
. Let
be the node cell on the grid level
and they are represented as follows,
The value at the corresponding node can be expressed as,
Our aim is to try to use the values at five nodes on and to provide a good initial value on the finest grid .
(1) New extrapolation,
.
(2) Quadratic interpolation,
.
Based on Algorithm 2 with new extrapolation formulas and quadratic interpolation, we present Algorithm 3.
In Algorithms 2 and 3, the smoothing is terminated also according to a stopping rule, see [
7,
23] for more information.
Algorithm 3: Overlapping domain decomposition method with new extrapolation cascadic multigrid (DDM-NECMG) for image restoration |
|
3. Edge-Preserving Denoising Operator
In this section, we introduce the edge-preserving denoising operator used in Algorithms 2 and 3.
In nonlinear model
4, the discrete regularization term
is denoted by
, where the appropriate selection of the regularization operator
is crucial for the quality of image restoration. The most classical model is the following nonlinear anisotropic diffusion Equation (
16),
where
is the image at moment
t,
is the diffusion coefficient, and the diffusion coefficients are chosen as follows,
Set
, then the finite difference discrete scheme of Equation (
16) can be expressed as:
where
represents the pixel value at
in the finite difference scheme.
Equation (
18) can be expressed in matrix form as,
where
and
respectively denote the
m and
time moment images.
Since the nonlinear Equation (
19) has a good preserving effect on the edge of the image, we apply it to the restoration of linear Equation (
3) as an edge-preserving denoising operator
. Specifically, let us take
and implement Equation (
19) for ten iterations.
5. Numerical Examples
In this section, we give several numerical examples to show the effectiveness of our algorithms.
The elements in the fuzzy operator
are defined by the following function,
where ⊗ denotes the Kronecker product,
M represents the dimensions of the subimage matrix,
represents the variance of the Gaussian point diffusion function, and the value of band represents the half-bandwidth of the Toplitz matrix
. The larger the values of
and band, the fuzzier the image generated.
The image restoration effect is measured by the following peak signal ratio,
The three original images are shown in
Figure 3. Let
and band = 11; then, we obtain noise and blur images as shown in
Figure 4. In our experiments, let
. We check the effectiveness of Algorithms 1–3.
This shows that Algorithm 3 not only can achieve a good recovery effect in a very short time, but also solves large resolution images effectively.
Next, we discuss the influence of the overlap size on image restoration. Taking Pirate 1083 × 1083 as an example, we introduce six different overlapping sizes and analyze them by PSNR and CPU time. The numerical results are shown in
Table 4.
From
Table 4 and
Figure 9, the overlapping domain decomposition method is superior to the non-overlapping domain decomposition method on PSNR, see
Figure 9a. However, the CPU time will increase with the increase in overlap size, see
Figure 9b. We use the efficiency value (PSNR/CPU time) to represent the performance of the algorithms. By
Figure 9c, when
, restoration efficiency is suggested.