A meshfree algorithm for ROF model
 Mushtaq Ahmad Khan^{1}Email author,
 Wen Chen^{1}Email author,
 Asmat Ullah^{1} and
 Zhuojia Fu^{1}
https://doi.org/10.1186/s1363401704886
© The Author(s) 2017
Received: 8 January 2017
Accepted: 27 June 2017
Published: 18 July 2017
Abstract
The total variation (TV) denoising method is a PDEbased technique that preserves the edges well but has undesirable staircase effect in some cases, namely, the translation of smooth regions (ramps) into piecewise constant regions (stairs). This paper introduces a novel meshfree approach using TV (ROF model) regularization and radial basis function (RBF) for the numerical approximation of TVbased model to remove the additive noise from the measurements. This approach is structured on local collocation and multiquadric radial basis function. These features enable this strategy not only to eliminate noise from images and preserve the edges but also has the advantage to minimize the staircase effect substantially from real and artificial images which cause the image to look blocky. Experimental results demonstrate that the proposed meshfree approach is robust and performs well in visual improvement as well as peak signaltonoise ratio compared with the recent partial differential equation (PDE)based traditional methods.
Keywords
1 Introduction
where z is the given noisy image containing the unknown additive noise (Gaussian noise) η and u is the known actual image, all of which are defined on a domain Ω∈R ^{2}. In literature, there are many effective numerical techniques have been utilized to tackle such models connected with image denoising having additive noise, for instance, in [1, 4, 13, 37, 44, 45].
The TV filtering [38] has proved to be one of the most successful tool in image processing for the solution of variational based partial differential equation (PDE) restoration problems. In this method, it is supposed that the images are defined on a continuous domain, which results in continuous functional. This functional then leads to a EulerLagrange equation. The resulting PDEs are then discretized by existing classical numerical methods on a regular grid for smooth solutions. For more details about the TV filtering, see [22, 38, 39]. The first TVbased model for image restoration having additive noise was proposed by Rudin et al. (ROF) [38]. This model yields very satisfactory results for removing image noise while preserving edges, see [7, 37]. However, it also processes some unfavorable properties like staircase effect, loss of image contrast and in time computation due to its nonlinearity and nondifferentiability [5, 32, 33, 38]. In [38], the authors proposed an artificial time marching method to the associated EulerLagrange equation. This strategy is slow due to its strict stability constraints in the time steps. Also, the artificial time marching method computes the approximate solution, not the exact solution. Recently different procedures have been used to overcome this difficulty and hence some good results have been obtained, for instance, see [14, 18, 29–31, 43, 46]. But still, there is space for improvement. So, in this work, we adapt the meshfree BRF collocation method to reduce these issues.
During the past decade, RBFs have been observed to be active techniques for the interpolation and approximation of multivariable smooth functions on scattered data sets [3, 6, 12]. More recently, an increasing attention has been given to the development of meshfree methods using RBFs for the numerical solution of PDEs. Most PDEs results have concerned steady state problems with smooth solutions. Recently there has been a growing interest in applying RBF methods to timedependent PDE problems, again to problems with sufficiently smooth solutions. The RBF techniques have more points of interest and have exhibited superior accuracy as compared with traditional numerical strategies, for example, finite difference method (FDM) [20, 23, 47], finite element method (FEM) [23], finite volume method (FVM) [21, 25], and pseudospectral method [24]. Interested readers can refer to [2, 8, 9, 16, 17, 26–28, 40] for more details about the RBF collocation methods.
Global RBF collocation technique is also easy to implement, gives good accuracy and converges exponentially for solving the PDEs. Although, in this strategy, the interpolation matrix is fully populated and illconditioned, and thus sensitive to shape parameter. Thus, it is computationally extremely expensive to apply global collocation method to large scale problems. So in literature, there are many domain type collocation techniques, for example, Kansa technique [19, 20] etc, is to settle these issues.
The main advantages of the RBFs for interpolating multidimensional scattered data are discussed in [19, 20]. In recent decades, meshless methods have been proved to treat scientific and engineering problems efficiently. The meshfree method based on the collocation method has been dominated and very efficient. Over the last several decades RBFs have been found to be widely successful for the interpolation of scattered data. RBF methods are not tied to a grid and in turn, belong to a category of methods called meshfree methods. They apply only a cloud of points without any information about nodal connections. It is (conditionally) positive definite [3, 36, 42], rotationally and translationally invariant. The RBF approximation is an incredibly powerful tool for representing smooth functions in nontrivial geometries since the method is meshfree and can be spectrally accurate [10]. RBFs interpolations have been used to remove the Gibbs oscillations from the given arbitrary data points [41] and very useful results have been obtained.
Motivated by the applications of TVregularization in image restoration and RBF collocation methods for the solution of PDEs, we propose a new meshfree strategy, with some modifications, of the TV (ROF) filter by RBF approximation to accomplish a new algorithm to solve the associated PDE with minimization of ROF model. This strategy is entirely meshless and is not only helpful to restore the image efficiently and resolve the edges due to is discontinuous jumps but also to eliminate the staircase effect and preserve the textures during the restoration process. The numerical treatment in this approach is also easy to implement and faster because of its meshfree properties as compared to the traditional meshbased numerical methods.
The rest of the paper is organized as follows. In Section 2, some details are provided related to the applications of TVregularization and its detail use in ROF model for image restoration. This section also contains the shortcoming in ROF model. This section also includes the details of RBFs and its applications in solving PDEs and comparison with traditional methods. Two meshbased methods, i.e., implicit and Augmented Lagrangian methods utilized for the solution of ROF model are presented in Section 3. This section also contains proposed method, i.e., BRF collocation method (Kansa method) for the solution of the associated PDE with ROF model. Section 4, describes experimental results and discussion, to compare the three methods for ROF model regarding CPU times, the number of iterations, and quality (peak signal to noise ratio (PSNR)) of the restored images. This section also includes the shape parameter analysis on image restoration and comparison of the proposed method with an other recent method. Section 5, shows the tabulated discussions about the sensitivity of parameters of the proposed method. The conclusion is provided in Section 6. And finally, the details for derivatives for our proposed method are given in an Appendix.
2 Related work
2.1 Total variationbased in image denoising RudinOsherFatemi (ROF) model
Rudin et. al (ROF) proposed the first model for image restoration from given noisy image having additive noise using TV regularization in [38]. This model achieved some useful restoration results.
for the given u(x,y,0), and also \(\frac {\partial u}{\partial n}=0 \) on ∂ Ω.

This model yields staircase effect, in restoring the smooth images in applications where edges are not the main features.

This model also generates to the loss of image contrast during the restoration process.

This model also contains the difficulty with the nondifferentiability term in the total variation norm.
2.2 Radial basis functions approximation
Name of RBF  Definition  CPD order (m) 

Multiquadric (MQ)  ϕ(r,c)=(r ^{2}+c ^{2})^{ k } i f k>0, k∉N  [k]+1 
Inverse multiquadric (IMQ)  ϕ(r,c)=(r ^{2}+c ^{2})^{−k } i f k>0, k∉N  0 
Gaussian (GA)  \(\phi (r,c)=e^{\frac {r^{2}}{c^{2}}}\)  0 
Polyharmonic spline  \(\phi (r) = \left \{ \begin {array}{lr} r^{2k1} & if \quad k\in N\\ r^{2k1}log(r) & if \quad k\in N.\\ \end {array} \right.\)  [k/2]+1 
Thin plate splines (TPS)  ϕ(r)=r ^{2} l n(r)  0 
Moreover, details of positive definite (PD) and conditionality positive definite (CPD) RBFs are discussed in [3, 36] and listed in Table 1. For RBFs containing the shape parameter c, such as as in Table 1, small shape parameters produces more accurate results, but also associated with poorly conditioned interpolation matrix [3, 36].
3 Numerical methods for solution of ROF model
3.1 Implicit method (M1)
Rudin et al. (ROF) proposed the implicit scheme to solve the Eq. (4) in [38].
Where \(m[a,b]=\left (\frac {sign(a)+sign(b)}{2}\right).min\left (a,b\right).\) For further details, see [38].
3.2 Augmented Lagrangian method (M2)
The augmented Lagrangian process is used to solve (9), which is stated in the given Algorithm 1.
To solve the minimization problem (??), we split (??) into two subproblems.
where w=r∇u−χ ^{ k }. The given Algorithm 2 shows the solution of subproblem (??).
For further details, we refer the readers [43].
3.3 Proposed method (M3)
which gives approximate solution at any point in Ω. Where u is N×1 matrix.
where \(L(u)=\sqrt {u^{2}_{x}+u^{2}_{y}}, u_{x}=H_{x}z, u_{xx}=H_{xx}z, u_{y}=H_{y}z, u_{yy}=H_{yy}z, \frac {\partial u}{\partial n}=u_{n}=H_{n}z,\) and z ^{(0)}=z.
Since the RBF in the Kansa scheme does not necessarily satisfy the governing Eq. (24), so we have more freedom to choose a RBF. The most popular RBF in the Kansa method is the multiquadric (MQ) [20, 34], which usually shows spectral accuracy if an appropriate shape parameter c is chosen. Here, the shape parameter c used in RBF is also one of the most important parameters for the smoothness in our method M3. For the optimal value of c, our proposed methodology gives more accurate and smooth results in image denoising having additive noise. In this technique the shape parameter c and fitting parameter λ depend on the size of the image and the noise level in the image.
4 Results and discussion
where ε indicates the maximum permissible error. Here, it is set to be 10^{−5}.
Comparison of methods M1, M2, and proposed method M3 in terms of PSNR
Image  Size  Method M1  Method M2  Method M3 

PSNR  PSNR  PSNR  
Lena  400^{2}  24.96  25.47  26.01 
Parrot  400^{2}  26.89  27.55  28.04 
SynImag1  400^{2}  28.31  29.11  29.87 
SynImag2  400^{2}  26.15  27.33  27.95 
Comparison of methods M1, M2, and proposed method M3 in terms of number of iterations (Iters) and CPUtime (Time) in seconds
Image  Size  Method M1  Method M2  Method M3  

Iters  Time (s)  Iters  Time (s)  Iters  Time (s)  
Lena  400^{2}  32  27.11  21  16.30  15  11.82 
Parrot  400^{2}  29  23.71  18  13.79  12  9.36 
SynImag1  400^{2}  35  31.50  24  20.57  19  14.47 
SynImag2  400^{2}  31  28.35  22  18.52  16  12.91 
Comparison of PSNR value of the restored image “Cameraman” for different additive noise values for three algorithms M1, M2, and M3
Image  σ=24  σ=22  σ=20  

M1  M2  M3  M1  M2  M3  M1  M2  M3  
Cameraman (PSNR)  19.83  20.90  21.29  20.57  21.42  21.94  21.76  22.47  23.01 
4.1 Shape parameter analysis
In this subsection, we compare the image restoration (PSNR) by our proposed method M3 for the different values of shape parameter c for real and artificial images.
Comparison of the image quantity (PSNR values) for different values (increase and decrease) in shape parameter c with the optimal value of shape parameter c of the proposed method M3 for real and artificial images
Image  Size  Optimal value c  PSNR  Increase c  PSNR  Decrease c  PSNR 

Lena  400^{2}  1.69  26.01  1.78  25.57  1.61  25.29 
SynImage1  400^{2}  1.76  29.87  1.84  29.63  1.68  29.37 
4.2 Comparison with an other recent model
5 Sensitivity analysis of parameters
To comment briefly on the choice of the shape parameter (c) and fitting parameter (λ) used in the algorithm M3 described above, it is recommended from our experience that all the two parameters c and λ are more complicated to choose. However, its optimal values are adjusted and tuned according to the noise variance, image size, etc. It has been observed that the range of values allowed are c∈ [1.63,1.78] and λ∈ [13.2,16.3], for natural and artificial images. It indicates that all the parameters c and λ are more important for improving denoising performance. Similarly, the number of iterations required for convergence are taken to be in the range [7,27] for results with improved PSNR. Thus, the availability of information about the uncertainty of the denoising result on the userchosen parameters (by Trial and Error Method) is helpful to avoid incorrect decisions.
 1.(·)% i n c r e a s e−↑, and (·)% d e c r e a s e−↓Table 7
PSNR value of the restored image “Lena” with optimal values of c and λ is 26.01. Parameter sensitivity analysis for our proposed method M3 by percentage increased in values of the parameters c and λ, with the resultant percentage increase or decrease in PSNR of the denoised image of size (400^{2})
Image
30% (↑)
60% (↑)
c
λ
PSNR
c
λ
PSNR
Lena
2.20
20.15
2.26 (↓)
2.71
24.80
3.93 (↓)
Table 8PSNR value of the restored image “Lena” with optimal values of c and λ is 26.01. Parameter sensitivity analysis for our proposed method M3 by percentage decreased in values of the parameters c and λ, with the resultant percentage increase or decrease in PSNR of the denoised image of size (400^{2})
Image
30% (↓)
60% (↓)
c
λ
PSNR
c
λ
PSNR
Lena
1.18
10.85
2.57 (↓)
0.68
6.20
4.81 (↓)
 2.
For example (0.17)↓ stands for 0.17% decrease in PSNR
 3.
(0.21)↑ stands for 0.21% increase in PSNR
6 Conclusions
In this paper, a new TV based meshfree algorithm for additive noise removal is presented in which TV regularization (ROF model) is employed in conjunction with MQRBF approximation. This algorithm is exploited for the solution of nonlinear PDE arisen from the minimization of the associated TV functional of ROF model. The proposed algorithm (Kansa method) is mathematically simple and robust compared with the classical meshbased methods and hence provide more optimal results because of meshfree applications of MQRBF associated with his algorithm.
This approach is tested on different artificial and real images for additive noise, and the results are compared with the existing methods. Our experimental results have shown that the quality of the restoration of images, the number of iterations, and the CPU times with the use of the proposed method are quite good, and the proposed algorithm is quite efficient. We have also noticed that the performance of our proposed method is far better than that of the existing methods regarding restoration quality (PSNR), the number of iterations, and CPU times because of the meshfree properties of RBF used in our algorithm. The choice of shape parameter c also plays a significant role in this algorithm, which affects the image restoration. The shape parameter analysis has also been discussed here. A comparison with another method in this field is provided as well.
However, this method produces an unsymmetrical interpolation matrix. Additionally, sometimes, this approach suffers relatively lower accuracy in boundaryadjacent regions. These problems are under intense study and results will be reported in the subsequent paper.
7 Appendix
The differentiation matrix is welldefined since it is known that the system matrix A is invertible.
Declarations
Acknowledgements
The work described in this paper is supported by the China Scholarship Council (CSC), the National Science Funds of China (Grant Nos. 11572111, 11372097) and the 111 Project (Grant No. B12032). This paper does not necessarily reflect the views of the funding agencies.
Authors’ contributions
All the three authors have contributed equally to the text, while MAK has implemented the algorithms and performed most of the tests. All the three authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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