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SN Computer Science

, 1:39 | Cite as

On Multi-Quadric Based RBF–FD Method for Second-Order Diffusion Filters

  • Mahipal JettaEmail author
  • Satyanarayana Chirala
Original Research

Abstract

The usual three-node approximations for the first-order spatial derivatives in the radial basis functions-based finite difference (RBF-FD) scheme allows relatively smaller time step size than the standard scheme (forward difference in time and central difference in space) in obtaining stable results. In this paper, to better mimic the continuous model, we use nine nodal points in RBF–FD approximation of first-order spatial derivatives. We then employ these approximations to solve a two-dimensional nonlinear diffusion equation from image processing. The constructed numerical scheme is shown to be stable and allows up to obtain four times bigger time step size than the standard scheme.

Keywords

Multi-quadric RBF Stability Nonlinear diffusion filter Image processing Finite difference method 

Introduction

The method of radial basis functions (RBF) is an important tool for the interpolation of multidimensional scattered data. Franke [7] has tested the performance of various basis functions (multi-quadric, Gaussian and splines) for the interpolation of scattered data and concluded that the multi-quadric (MQ) basis produces the most accurate results when compared with other basis functions. This observation led to development of accurate numerical schemes for the partial differential equations [2, 9, 13, 14, 15, 16, 21].

In most of the existing works on RBF-FD-based numerical schemes, the spatial derivatives have been approximated with values of the underlying function at grid points on one principal direction, for example, \(u_x\) has been approximated with values of u at grid points on the X-axis. This enables the user to employ efficient algorithms for solving the resulting system of equations. However, these three node-based schemes, in general, suffer from heavy numerical dissipation which hinders their application in image processing [5, 10, 20].

To address this issue, in this paper, an RBF–FD-based scheme is employed to obtain the \(3\times {3}\) stencil approximations for the first-order spatial derivatives. These stencils have been used to approximate the spatial derivatives in the nonlinear parabolic partial differential equation, and the time derivative is discretized using the forward difference approximation.

Further, we analyze the stability of the proposed scheme and show that it is stable under some restriction on the time step size. This restriction (bound) on the time step size is a function of the grid size and shape parameter of the radial basis functions.

The rest of the paper is organized as follows. In Sect. 2, the RBF–FD scheme for solving unsteady partial differential equations is presented. In Sect. 3, the stability analysis of the proposed scheme for two-dimensional linear diffusion equation is carried out. Finally, Sect. 4 constitutes the concluding remarks.

An RBF–FD Scheme for Unsteady Problems

Consider a well-posed linear unsteady partial differential equation:
$$\begin{aligned} \dfrac{\partial {u}}{\partial {t}}(\mathbf{x },t)& = \mathscr {L}u (\mathbf x ,t)~~ \text{ in }~~ \varOmega , \end{aligned}$$
(1)
$$\begin{aligned} u_{0}(\mathbf x )& = u(\mathbf x ,t=0),~ \end{aligned}$$
(2)
$$\begin{aligned} \dfrac{\partial {u}}{\partial {n}}& = 0~~ \text{ on }~ \partial \varOmega , \end{aligned}$$
(3)
where u is an unknown function, \(\mathbf x \in \mathbb {R}^{d}\), \(\mathscr {L}\) is a linear partial differential operator, and \(\dfrac{\partial {u}}{\partial {n}}\) is a directional derivative of u in the outward normal direction n. To solve this equation, we employ RBF–FD scheme [4, 21], which is briefly described as follows.
Let N be the total number of centers in the computational domain, and let
$$\begin{aligned} {\mathbf {C}}_{i} = \{\mathbf{x }_{1}, \mathbf{x }_{2}, \ldots , \mathbf{x }_{n_{i}}\} \end{aligned}$$
be the local support of the center \(\mathbf x _{i}\) having \(n_i\) neighboring centers. Then \(\mathscr {L}u(\mathbf{x _{i}})\) is expressed approximately as a linear combination of u values at \(n_{i}(<< N)\) centers of the local support \(\mathbf {C}_{i}\). Symbolically, this is given by
$$\begin{aligned} \mathscr {L}u(\mathbf x _{i})\approx \displaystyle \sum _{j=1}^{n_{i}}c_{j}u(\mathbf x _{j}) ,~~ \text{ for } \text{ each }~~ \mathbf x _{i} \in \varOmega . \end{aligned}$$
(4)
The weights, \(\mathbf c = \{c_j \}^{n_i}_{j = 1}\) in (4), are obtained by solving the linear system,
$$\begin{aligned} A \begin{bmatrix} \mathbf c \\ \gamma \end{bmatrix} = (\mathscr {L}B(\mathbf x _{i}))^{T},\;\; \end{aligned}$$
(5)
$$\begin{aligned} \text{ where }\;\;A=\left[ \begin{array}{rcl}\varXi &{}\mathbf 1 \\ \mathbf 1 ^{T}&{} {0}\\ \end{array} \right] \text {with}~~~ \mathbf 1 ^{T}= [1,1, \ldots ~1]_{1\times {n_i}}, {\gamma } \text { is a dummy variable} ,\\ \text {and} ~~ B(\mathbf x ) = [\phi (\Vert \mathbf x -\mathbf x _{1}\Vert ,\varepsilon ),\ldots ,\phi (\Vert \mathbf x -\mathbf x _{n_{i}}\Vert ,\varepsilon ),~ 0 ]. \end{aligned}$$
The elements of the sub-matrix \(\varXi\) are given by
$$\begin{aligned} \varXi _{i,j}:=\phi (\Vert \mathbf x _{i}-\mathbf x _{j}\Vert ,\varepsilon ),~~ i,j=1,2,\ldots ,n_{i}. \end{aligned}$$
(6)
Michelli [11] showed that the choice of multi-quadric basis produces a non-singular interpolation matrix (A) in Eq. (5). The last row in the matrix A is used to have the zero sum of the coefficients.
Now, the discretized form of Eq. (1), with the obtained weights \(c_j\), reads as
$$\begin{aligned} \dfrac{u (\mathbf x _{i},t+\varDelta {t}) - u (\mathbf x _{i},t)}{\varDelta {t}}= \sum _{j = 1}^{n_i} c_{j} u(\mathbf x _{j},t). \end{aligned}$$
(7)
The accuracy of this approximation depends on the choice of the underlying basis functions.

In this work, the computations are carried out with multi-quadric and Gaussian basis functions, which are defined as \(\phi (r) = \sqrt{1+\epsilon ^2 r^2}\) and \(\phi (r) = e^{-\epsilon ^2 r^2}\), respectively. Here, r is the Euclidean distance between the center \(\mathbf x\) and \(\mathbf x _{i}\), and \(\epsilon\) is the (scaling) shape parameter of the basis function. It has been realized that this parameter plays an important role in the accuracy of the numerical solution [1, 2].

Note that throughout this article, we take uniform distribution of the nodes to compare our scheme with the finite difference method, which can be easily applied as the structure of the underlying nodal points (pixels) is uniform.

Diffusion Filters in Image Processing

Linear Diffusion Filter

In this section, we demonstrate the performance of finite difference and RBF–FD schemes for a two-dimensional heat equation with an application in image processing.

Consider a partial differential equation of the form
$$\begin{aligned} \dfrac{\partial {u}}{\partial {t}}(\mathbf x ,t)= div(D(|\nabla {u}|)\nabla {u}(\mathbf x ,t)), ~ \mathbf x \in {\varOmega } \subset {\mathbb {R}^2} \end{aligned}$$
(8)
with initial condition \(u_{0}(\mathbf x )=u(\mathbf x ,t=0)\) and homogeneous Neumann boundary condition: \(\dfrac{\partial {u}}{\partial {n}}=0\) on \(\partial {\varOmega }\). Here, we evolve the noisy image \(u(\mathbf x ,0)\) according to this PDE to remove noise from it. The diffusion coefficient D is a function of the edge detector \(|\nabla {u}|\) which aims to accelerate the diffusion process in the homogeneous regions and slows down the diffusion in the non-uniform regions [12]. A typical D which possesses the required characteristics is
$$\begin{aligned} D(|\nabla {u}|)=\dfrac{1}{1+\dfrac{|\nabla {u}|^2}{\lambda ^2}}, \end{aligned}$$
(9)
where \(\lambda\) is a contrast parameter used to distinguish the important features and the noise. We analyze the stability of multi-quadric-based RBF-FD, hereinafter abbreviated as MQ, and Gaussian-based RBF–FD [2, 6], hereinafter abbreviated as Gaussian, schemes for the linear filter, obtained with \(D=1\) in (8). For more denoising techniques, one may refer to [17, 18, 19, 22].

Stability of the RBF–FD Scheme

Five-Node Approximation

Approximating the time derivative with forward difference, and spatial derivatives with central differences using three nodal points in X and Y directions, we obtain the following scheme for (8)
$$\begin{aligned} \dfrac{U_{i,j}^{n+1}-U_{i,j}^{n}}{ \varDelta {t}}=d_1 U_{i,j}^{n}+d_2 U_{i-1,j}^{n}+d_3 U_{i+1,j}^{n}+d_4 U_{i,j+1}^{n}+d_5 U_{i,j-1}^{n}, \end{aligned}$$
(10)
where \(\varDelta {t}\) is the time step size and \(U_{i,j}^{n}\) denotes the approximation of u(xyt) at the pixel (ij) at time \(n\varDelta {t}\). Here, the coefficients in Eq. (10) can be obtained by solving the local system (5) with \(n_i=5\). It can be observed that, in case of MQ, \(d_2=d_3=d_4=d_5=\dfrac{1}{h^2}+\dfrac{5}{6}\epsilon ^2\) and \(d_1=-\dfrac{4}{h^2}-\dfrac{10}{3}\epsilon ^2\), where h is the grid size in both X and Y directions [1]. Equation (10) can be rewritten as
$$\begin{aligned} U_{i,j}^{n+1}=d_2 \varDelta {t}\left( U_{i-1,j}^n+U_{i+1,j}^n+U_{i,j-1}^n+U_{i,j+1}^n \right) + (1+d_1 \varDelta {t}) U_{i,j}^n. \end{aligned}$$
(11)
The maximum–minimum principle will hold for (11) if each coefficient of U is non-negative and the sum of all the coefficients of U on the right hand side of (11) is equal to 1. These conditions enforce the restriction on time step size given by
$$\begin{aligned} \varDelta {t}\le \min \left\{ \dfrac{-1}{d_1}, \dfrac{-1}{d_1+ d_2} \right\} . \end{aligned}$$
In similar lines, one can prove that the stability condition, in case of Gaussian, is
$$\begin{aligned} \varDelta {t}\le \dfrac{1}{4h \epsilon ^2\left( cosech (h^2\epsilon ^2)+sech(h^2 \epsilon ^2) \right) }. \end{aligned}$$
Fig. 1

Stability bound for FD and RBF schemes for two-dimensional heat equation for \(0<\epsilon \le {1}\). a Five-node approximation, b nine-node approximation

In Fig. (1), the time step size bound for (11) is plotted against the shape parameter \(\epsilon\) with \(h=\frac{1}{2}\). One can observe from this figure that the time step size bound for finite difference scheme dominates both the MQ and Gaussian scheme bounds for all the values of the shape parameter. Therefore, the RBF–FD schemes with five-node approximation are relatively less efficient than the standard finite difference scheme.

Nine-Node Approximation

To improve the efficiency of the RBF–FD scheme, we consider nine nodes in the approximation of first-order spatial derivatives. Considering \(n_{i}=9\) in (4) and solving the resulting system gives the following approximations, in stencil form, for the first-order spatial derivatives
$$\begin{aligned} u_{x}& = \left( \begin{matrix} \beta &{} ~~0~~ &{} -\beta \\ -\alpha &{} ~~0~~ &{} \alpha ~ \\ \beta &{} ~~0~~ &{} -\beta ~ \end{matrix} \right) , \, \text {and} \nonumber \\ u_{y}& = \left( \begin{matrix} -\beta &{} \alpha &{} -\beta \\ 0 &{} 0 &{} 0\\ \beta &{} -\alpha &{} \beta ~ \end{matrix} \right) , \end{aligned}$$
(12)
where \(\alpha\) and \(\beta\) are functions of the shape parameter \(\epsilon\).
Note that the explicit form of the approximation of \(u_x\) at a mesh point \((x_i,y_j)\), with the given stencil, is given by
$$\frac{{\partial u}}{{\partial x}}{\mid }_{{i,j}} \approx \beta U_{{i - 1,j + 1}} - \beta U_{{i + 1,j + 1}} - \alpha U_{{i - 1,j}} + \alpha U_{{i + 1,j}} + \beta U_{{i - 1,j - 1}} - \beta U_{{i + 1,j - 1}} .$$
Similarly, the approximation for \(u_y\) at \((x_i,y_j)\) is
$$\frac{{\partial u}}{{\partial y}}{\mid }_{{i,j}} \approx - \beta U_{{i - 1,j + 1}} + \alpha U_{{i,j + 1}} - \beta U_{{i + 1,j + 1}} + \beta U_{{i - 1,j - 1}} - \alpha U_{{i,j - 1}} + \beta U_{{i + 1,j - 1}} .$$
The approximation of second-order spatial derivatives can be obtained by consecutive application of first-order stencils. For example, the approximating stencil for \(u_{xx}\) is
$$\begin{aligned} L=\begin{aligned} \left( \begin{matrix} \beta ^2 &{} 0 &{} -2 \beta ^2 &{} 0 &{} \beta ^2 \\ -2\alpha \beta &{} 0 &{} 4\alpha \beta &{} 0 &{} -2 \alpha \beta ~ \\ 2\beta ^2+\alpha ^2 &{} 0 &{} -4 \beta ^2-2\alpha ^2 &{} 0 &{} 2\beta ^2+\alpha ^2 \\ -2\alpha \beta &{} 0 &{} 4\alpha \beta &{} 0 &{} -2 \alpha \beta ~ \\ \beta ^2 &{} 0 &{} -2 \beta ^2 &{} 0 &{} \beta ^2 \end{matrix} \right) . \end{aligned} \end{aligned}$$
(13)
For \(u_{t}=u_{xx}\) in the two-dimensional domain, the resulting explicit scheme will take the following form:
$$\begin{aligned} \frac{U_{i,j}^{n+1}-U_{i,j}^{n}}{\varDelta {t}}= L_{i,j}* U_{i,j}^{n}. \end{aligned}$$
(14)
Here, the convolution \(L_{i,j}* U_{i,j}^{n}\) is a discretization of \(u_{xx}\) at the n-th time level.
The column-wise arrangement of U, which can be obtained using single index \(l={(i-1)N+j}\) with N being the number of grid points in a principal axis direction, enables us to rewrite (14) as
$$\begin{aligned} U_l^{n+1}=(I+S)U_l^{n}, \end{aligned}$$
(15)
where I is the identity matrix and S is the sparse matrix representing the convolution of U with small convolution kernel of \(\varDelta {t}\times L\). The stability of (15) is guaranteed if spectral radius of the iteration matrix \((I+S)\) is less than 1 [3, 8]. One may note that this condition also ensures the stability of
$$\begin{aligned} U_l^{n+1}=-(I+S)U_l^{n}. \end{aligned}$$
(16)
We rewrite (16) as
$$\begin{aligned} U_l^{n+1}+((2I+S)-I)U_l^{n}=0. \end{aligned}$$
(17)
The stability of (17) is guaranteed if \((2I+S)\) is diagonally dominant [8]. Considering the approximations in the inner part of the image domain (obtained by removing two boundary rows and two boundary columns), the non-zero matrix elements of S at each row are given in the following order:
$$\begin{gathered} {\mathbf{\Delta }}t \times [\beta ^{2} ~~ - 2\beta ^{2} ~~\beta ^{2} ~~ - 2\alpha \beta ~~4\alpha \beta ~~ - 2\alpha \beta ~~2\beta ^{2} + \alpha ^{2} ~~ - 4\beta ^{2} \hfill \\ \qquad - 2\alpha ^{2} 2\beta ^{2} + \alpha ^{2} ~~ - 2\alpha \beta ~~4\alpha \beta ~~ - 2\alpha \beta ~~\beta ^{2} ~~ - 2\beta ^{2} ~~\beta ^{2} ]. \hfill \\ \end{gathered}$$
Here, the diagonal entry of S is \(\varDelta {t}(-4\alpha ^2-2\beta ^2)\). Therefore, the diagonal dominant condition will be satisfied if
$$\begin{aligned} |2+\varDelta {t}(-4\beta ^2-2\alpha ^2)|> \varDelta {t}(12\beta ^2+2\alpha ^2+16\alpha \beta ). \end{aligned}$$
(18)
By simplifying this inequality, one can see that the non-negative bound on \(\varDelta {t}\) as
$$\begin{aligned} \varDelta {t}<\frac{1}{8\beta ^2+2\alpha ^2+8\alpha \beta }. \end{aligned}$$
(19)
Note that, in general, the bound on \(\varDelta {t}\) is bigger for the rows corresponding to boundary nodal points than the internal nodal points [8].
It is easy to extend these ideas to two-dimensional linear diffusion equation, and one can obtain the time step size bound on \(\varDelta {t}\) as
$$\begin{aligned} \varDelta {t}<\frac{1}{16\beta ^2+4\alpha ^2+16\alpha \beta }. \end{aligned}$$
(20)
The plot of the time step size bound against the shape parameter for multi-quadric and Gaussian RBFs has been provided in Fig. 1.b. In case of multi-quadric, for \(\epsilon \ge 1\), the \(\varDelta {t}\) bound increases up to 3; see Fig. 2. In case of Gaussian RBF, it has been observed that the stencil weights \(\alpha\) and \(\beta\) tend to zero as the shape parameter \(\epsilon\) increases, which makes the time step size bound increase rapidly.
Fig. 2

Stability bound for FD and RBF–FD (MQ) schemes for two-dimensional heat equation with nine-node approximation for \(\epsilon >1\)

Validation

In Fig. 3, we considered a cameraman image and corrupted it with Gaussian noise of mean zero and variance 0.05. This noisy image is evolved under (8), with various values for D, to get a noise-free image. The parameters used in this process are \(\varDelta {t}=0.2\), \(\varDelta {x}=\varDelta {y}=1\), and the number of iterations is 2, where the high-quality filtered image is obtained. One can see from this figure that MQ is more stable, for larger D, than the finite difference and Gaussian schemes.

The quantitative performance of this linear diffusion filter with finite difference and RBF (MQ, Gaussian) schemes is illustrated in Fig. 4. It is clear from this figure that the RBF (MQ) scheme produces relatively high-quality filtered image, when \(\epsilon =0.4\), than the finite difference and RBF (Gaussian) schemes. One can also observe that the relative error, defined in this case as \(\dfrac{||{u-u_0}||_{2}}{||{u_0}||_{2}}\) with u being the filtered image and \(u_0\) being the reference image, is minimum when \(\epsilon =0.4\). These simulations show that RBF-based approximations may produce relatively high accurate solutions than with the standard finite difference scheme. This result encourages us to employ RBF-based schemes for the approximation of nonlinear diffusion filters.
Fig. 3

Numerical realization of (8) through FD and RBF–FD schemes. Row 1 a Original Cameraman image. b Image corrupted with Gaussian noise of mean zero and variance 0.05. Row 2 (\(D=1\)) c Finite difference scheme. d RBF–FD (Gaussian) scheme. e RBF–FD (MQ) scheme. Row 3 (\(D=2\)). f Finite difference scheme. g RBF–FD (Gaussian) scheme. h RBF–FD (MQ) scheme. Row 4 (\(D=3\)) i Finite difference scheme. j RBF–FD (Gaussian) scheme. k RBF–FD (MQ) scheme. Parameters \(\epsilon =0.8\) for Gaussian, and \(\epsilon =2\) for MQ, no of iterations \(=2\) with \(\varDelta {t}=0.2\)

Fig. 4

Performance of linear filter using FD and RBF (MQ, Gaussian) schemes, in terms of a PSNR, b relative error

Fig. 5

Numerical realization of nonlinear filter through finite difference and RBF schemes: a original cameraman image, b image corrupted with Gaussian noise of mean zero and variance 0.05. c Finite difference scheme with \(\varDelta {t}=0.2\) and 20 iterations, d RBF (Gaussian) scheme with \(\epsilon =0.8\), \(\varDelta {t}=0.3\) and 10 iterations, e RBF (MQ) scheme with \(\epsilon =2\), \(\varDelta {t}=0.7\) and 7 iterations

Fig. 6

Performance of nonlinear filter using FD and RBF (MQ, Gaussian) schemes, in terms of a PSNR, b relative error

Table 1

Comparison of numerical schemes

Image

Noise level

Scheme

\(\varDelta {t}\)

\(\epsilon\)

PSNR

\(\varDelta {t}_{max}\)

PSNR index

\(T_{\varDelta {t}}\)

No of iterations

Lena

0.01

FD

0.25

29.43

0.25

13

3.25

13

Gaussian

0.25

0.8

28.14

0.46

14

3.5

8

MQ

0.25

0.4

28.60

0.37

22

5.5

15

0.04

FD

0.25

25.77

0.25

12

3

12

Gaussian

0.25

0.8

25.11

0.46

13

3.25

7

MQ

0.25

0.4

25.30

0.37

19

4.75

13

0.08

FD

0.25

21.18

0.25

10

2.5

10

Gaussian

0.25

0.9

21.24

0.44

14

3.5

8

MQ

0.25

0.4

21.55

0.37

21

5.25

14

Peppers

0.01

FD

0.25

27.93

0.25

12

3

12

Gaussian

0.25

0.9

26.80

0.44

12

3

7

MQ

0.25

0.4

27.05

0.37

20

5

14

0.05

FD

0.25

23.75

0.25

10

2.5

10

Gaussian

0.25

0.8

23.22

0.46

11

2.75

6

MQ

0.25

0.4

23.45

0.37

17

4.25

12

0.1

FD

0.25

19.24

0.25

8

2

8

Gaussian

0.25

0.9

19.05

0.44

9

2.25

6

MQ

0.25

0.4

19.18

0.37

15

3.75

11

House

0.01

FD

0.25

29.39

0.25

15

3.75

15

Gaussian

0.25

0.8

28.36

0.46

16

4

9

MQ

0.25

0.4

28.81

0.37

25

6.25

17

0.05

FD

0.25

24.27

0.25

13

3.25

13

Gaussian

0.25

1

23.97

0.44

13

3.25

8

MQ

0.25

0.4

24.14

0.37

24

6

17

0.1

FD

0.25

19.49

0.25

12

3

12

Gaussian

0.25

1

19.73

0.44

24

6

14

MQ

0.25

0.4

20.18

0.37

45

11.25

31

Nonlinear Diffusion Filter

Algorithm

Here, we provide the algorithm to find the numerical solution, U, of (8). We can rewrite (8) as
$$\begin{aligned} u_{t}=\partial _{x}(D {u_x})+\partial _{y}(D {u_y}). \end{aligned}$$
(21)
Now, we employ forward difference for time derivative, and \(3\times {3}\) stencils (13) for first-order spatial derivatives. The rest of the steps of the algorithm are as follows.
  1. (i)

    Calculate \(f^{n}_1:=(D u_{x})^{n}\), \(f_2:=(D u_{y})^{n}\) by using the first-order derivate approximating stencils, for a given \(\epsilon\). Here, \((D u_{x})^{n}\) denotes the approximation of \((D u_{x})\) at the n-th time level.

     
  2. (ii)

    Calculate \(g^{n}:=\partial _{x}f_1^{n}+\partial _{y}f_2^{n}\) by applying the same stencils for \(f_1^{n}\) and \(f_2^{n}\).

     
  3. (iii)

    Update U: \(U^{n+1}=U^{n}+\varDelta {t} g^{n}\), where \(U^n\) denotes the numerical solution at the n-th time level.

     
Since we are applying \(3\times {3}\) stencils consecutively in the approximation, the final scheme involves \(5\times {5}\) stencil.

It can be easily seen, by assuming D is non-negative and bounded above by 1, that the stability of this new scheme is guaranteed if (20) is satisfied.

The performance of the nonlinear diffusion filter with standard finite difference scheme, Gaussian and MQ schemes is shown in Fig. 5.

It is worth noting here that the new 25 node scheme, obtained using (13) which is generated through (12), gives a modest quality filtered image in much shorter number of iterations with a larger time step size than the finite difference scheme. The quantitative results regarding the filtered image quality in terms of PSNR and relative error have been presented in Fig. 6. This figure justifies the use of RBF (MQ) scheme in obtaining reasonable quality processed image. To further assess the performance of the RBF scheme, with both MQ and Gaussian basis functions, we conduct several numerical experiments on various images with different noise levels; see Table 1. In this table, the shape parameter (\(\epsilon\)) is chosen so that the the obtained filtered image, using (8) with (9), has maximum PSNR. \(\varDelta {t}_{\max }\) denotes the maximum possible step size for the given \(\epsilon\). PSNR index represents the iteration number at which maximum PSNR is obtained for the chosen \(\epsilon\) with step size \(\varDelta {t}=0.25\). \(T_{\varDelta {t}}\) denotes the time required to obtain the filtered image with maximum PSNR. This is obtained by multiplying PSNR index with \(\varDelta {t}\). The last column of the table denotes the required number of iterations with the improved time step size \(\varDelta {t}_{max}\): \(\displaystyle \textit{Number of Iterations}=\lceil {\frac{ T_{\varDelta {t}}}{\varDelta {t}_{max}}}\rceil\), where \(\lceil \rceil\) denotes the ceiling function. One can infer from this table that the RBF (Gaussian) with \(\epsilon \in {(0.8,1)}\) is up to two times more efficient than the finite difference and RBF (MQ) schemes, since RBF (Gaussian) requires up to half the number of iterations as that of FD scheme. However, this scheme is inaccurate for larger values of \(\epsilon\) as the stencil weights tend to zero as \(\epsilon\) increases. To check the efficiency of RBF (MQ), for \(\epsilon >1\), we performed simulations on Lena image for various noise levels; see Table 2. One can infer from this table that the RBF (MQ) is more stable as it allows up to 12 times bigger time step size than FD. Moreover, from the last column one can see that the RBF (MQ) is up to four times more efficient than FD, with a reasonable quality processed image. It is important to note here that the higher \(\epsilon\) improves efficiency while compromising the image quality; see Fig. 7.
Fig. 7

Performance of nonlinear filter using FD and MQ schemes for \(\epsilon >1\), in terms of PSNR

Table 2

Comparison of FD, MQ and Gaussian schemes

Image

Noise level

Scheme

\(\varDelta {t}\)

\(\epsilon\)

PSNR

\(\varDelta {t}_{max}\)

PSNR index

\(T_{\varDelta {t}}\)

No of Iterations

Lena

0.01

FD

0.25

-

29.44

0.25

13

3.25

13

MQ

0.25

1

28.34

0.41

19

4.75

12

MQ

0.25

21

27.26

2.58

31

7.75

3

MQ

0.25

41

27.19

2.87

33

8.25

3

Gaussian

0.25

0.6

28.49

0.57

16

4

8

Gaussian

0.25

0.8

28.45

0.46

13

3.25

8

Gaussian

0.25

1

28.39

0.44

12

3

7

0.04

FD

0.25

25.83

0.25

13

3.25

13

MQ

0.25

1

25.28

0.41

20

5

13

MQ

0.25

21

24.73

2.58

32

8

4

MQ

0.25

41

24.70

2.87

33

8.25

3

Gaussian

0.25

0.6

25.13

0.57

16

4

8

Gaussian

0.25

0.8

25.14

0.46

13

3.25

8

Gaussian

0.25

1

25.14

0.44

12

3

7

0.08

FD

0.25

21.28

0.25

11

2.75

11

MQ

0.25

1

21.27

0.41

17

4.25

11

MQ

0.25

21

20.94

2.58

23

5.75

3

MQ

0.25

41

20.91

2.87

23

5.75

2

Gaussian

0.25

0.6

21.04

0.57

22

5.5

10

Gaussian

0.25

0.8

21.04

0.46

18

4.5

10

Gaussian

0.25

1

21.04

0.44

17

4.25

10

Conclusions

In this paper, we have analyzed multi-quadric and Gaussian-based RBF–FD schemes for two-dimensional linear and nonlinear diffusion equations. It has been shown that the stability of RBF–FD scheme, derived using three-node approximation for first-order spatial derivatives, is guaranteed for a relatively smaller time step size than that of the standard finite difference scheme.

We have improved the time step size bound for RBF–FD scheme by considering nine nodes in the approximation of first-order spatial derivatives. We then analyzed the stability of the resulting \(5\times {5}\) stencil scheme for two-dimensional parabolic equations, and shown that this scheme allows up to four times larger time step size than the standard finite difference scheme.

To analyze the practical applicability of the proposed RBF–FD scheme, we have considered a second-order nonlinear partial differential equation from image processing. We showed that the Gaussian-based RBF–FD scheme for \(\epsilon \in {(0.8,1)}\) is twice efficient than multi-quadric and finite difference schemes. Further, for larger values of the shape parameter \(\epsilon\), we demonstrated that the MQ scheme is up to four times more efficient than the standard finite difference scheme. We emphasize here that this gain in efficiency with MQ scheme comes at the expense of filtered image quality. Therefore one should be cautious about trade-off between efficiency and the processed image quality while employing the MQ-based RBF–FD scheme for a nonlinear diffusion filter.

In future, we wish to work on the selection of optimal shape parameter for multi-quadric and Gaussian-based RBF–FD schemes to solve the time-dependent higher-order nonlinear partial differential equations.

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions and comments for improving the quality of the paper.

Conflict of interest

Conflicts of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Springer Nature Singapore Pte Ltd 2019

Authors and Affiliations

  1. 1.Mahindra École CentraleHyderabadIndia

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