Hybrid Lattice Boltzmann Model for Nonlinear Diffusion and Image Denoising

Abstract

In the present paper, a novel approach for image denoising based on the numerical solution to the nonlinear diffusion equation is proposed. The Perona–Malik-type equation is solved by employing a hybrid lattice Boltzmann model with five discrete velocities. In this method, the regions with large values of the diffusion coefficient are modeled with the lattice Boltzmann scheme for which hyper-viscous defects are reduced, while other regions are modeled with the conventional lattice Boltzmann model. The new method allows us to solve Perona–Malik-type equations with relatively large time steps and good accuracy. In numerical experiments, the removal of salt and pepper, speckle and Gaussian noise is considered. For salt and pepper noise, the novel scheme yields better peak signal-to-noise ratios in image denoising problems compared to the standard lattice Boltzmann approach. For certain non-small values of time steps, the novel model shows better results for speckle and Gaussian noise on average.

1. Introduction

The lattice Boltzmann (LB) method has been widely used for solving interdisciplinary problems [1,2]. In particular, one of several possible applications for the LB method is in the simulation of advection–diffusion and diffusion processes with applications to image processing.

The first LB models for the convection–diffusion processes have been studied by Flekkøy [3]. Stability domains and the accuracy of several single-relaxation time LB models have been explored in a very detailed way [4,5,6]. Numerical experiments show that four and five-velocity LB models are at least as accurate as nine-velocity models for small Peclet numbers; hence, for diffusion problems, the models with small velocity sets can be preferred. In addition, for pure diffusion, LB models are stable for any time step. This property distinguishes popular LB models from the explicit difference schemes for the diffusion equation, which have an upper stability limit on time steps. Although one can take large time steps while modeling a diffusion process using conventional LB models, the limitations emerge from accuracy issues. For non-small values of the time step and diffusion coefficient, LB models suffer from hyper-viscous errors (or truncation errors) [7,8,9,10,11,12]. In order to improve the accuracy of LB schemes, several approaches can be adopted. A fourth-order accurate scheme for the diffusion equation in one dimension can be obtained by modifying the proportion of the rest particles in three velocity LB models [13]. In a multiple-relaxation-time (MRT) LB method [14,15], a combination of relaxation times can be tuned in such a way that hyper-viscosity defects are diminished and stability is improved [16,17,18,19,20]. A fourth-order accurate MRT scheme has been proposed recently [21,22], but this model operates in one spatial dimension. On the other hand, in several spatial dimensions, hyper-viscosity reduction can be achieved only for special values of diffusion coefficient, and in some cases, single-relaxation-time models show better precision than two-relaxation-time models [23].

The LB method has found several applications in image processing. The Perona–Malik diffusion [24,25], which is widely used for denoising, can be modeled using nine-velocity and five-velocity LB schemes [26,27]. Appropriate LB schemes with external source terms can be constructed for the more complicated variants of the Perona–Malik model, in which the diffusion coefficient is also governed by an additional reaction–diffusion equation [28]. The Rudin–Osher–Fatemi total variation minimization equation for image restoration [29] can be efficiently modeled using a nine-velocity LB scheme [30]. Nine-velocity MRT LB models are applied for the simulation of a nonlinear reaction–advection–diffusion equation with a constant diffusion coefficient and non-constant advection velocity, with applications in sea clutter denoising in marine radar images [31]. LB models simulating anisotropic diffusion equations can be used for the smoothing and segmentation of two-dimensional and three-dimensional medical images [32,33,34,35]. A four-velocity LB model with a non-local collision step has been applied for the development of Gaussian pyramids and feature detection [23].

In this study, a five-velocity single-nonconstant-relaxation-time LB model with non-local collision term is considered. Since the similar LB scheme with constant relaxation time shows small hyper-viscous errors while modeling the diffusion equation with non-small diffusion coefficients and time step, one expects that this model can be used for the efficient simulation of the Perona–Malik diffusion. By using the Chapman–Enskog expansion, it is shown that this model simulates the nonlinear diffusion equation, which is very similar to the Perona–Malik equation. As an application of the model, a hybrid approach is proposed: boundaries and regions with small diffusion coefficients are modeled with the conventional five-velocity LB model, while other regions are modeled with the model with a non-local collision term. Numerical experiments for the removal of salt and pepper, speckle and Gaussian noise are carried out. It is demonstrated that in several cases, the proposed hybrid method shows a better peak signal-to-noise ratio (PSNR) than the conventional five-velocity LB model applied with the same time step.

This paper is organized as follows. In Section 2, the LB model with non-local collision kernel is considered. By applying multiple-scale Chapman–Enskog expansion, it is demonstrated that the model simulates a Perona–Malik-type diffusion. In Section 3, the application of the hybrid method and boundary conditions are discussed. Numerical experiments are carried out: four test images are considered, the results of the removal of salt and pepper, speckle and Gaussian noise are presented. In the Discussion section, the main results and possible directions for the future work are outlined.

2. Lattice Boltzmann Scheme for Nonlinear Diffusion

Assume that a two-dimensional grayscale image color intensity is given by a function $I_0(x,y)$. The application of the nonlinear Perona–Malik filter is equivalent to solving the following equation [24,25]:

$$\frac{\partial I(t,x,y)}{\partial t}=div\left(D\left(\right|\nabla I(t,x,y){|}^2)\nabla I(t,x,y)\right),$$

(1)

where ${I(t,x,y)|}_{t=0}=I_0(x,y)$ and

$$D\left(s\right)=\frac{1}{1+{(s/\lambda )}^2}$$

(2)

and $\lambda$ is a positive constant (contrast parameter). It is worth mentioning that the Perona–Malik evolution is a forward diffusion in the regions of a constant or small color intensity but is backward diffusion in the regions of sharp intensity variations (larger than $\lambda$). As a result, similarly to the Gaussian filtering, the Perona–Malik filter blurs small-scale fluctuations, but in contrast, tends to preserve edges; this is obviously a very desirable property.

In order to solve Equation (1) numerically, the appropriate LB model is adopted. For modeling the advection–diffusion process in two dimensions, it is sufficient to employ LB models with four, five or nine velocities. Recent studies indicate that a five-velocity model ($D2Q5$) shows a good trade-off between accuracy, stability and computation complexity, for instance, five-velocity models with small Peclet numbers show better precision than nine-velocity LB models [6,36].

Then, in the present study, a five-velocity lattice is adopted, and the corresponding particle velocities are as follows: ${\mathit{c}}_{\mathbf{0}}=(0,0),{\mathit{c}}_{\mathbf{1}}=(0,c),{\mathit{c}}_{\mathbf{2}}=(c,0),$${\mathit{c}}_{\mathbf{3}}=(0,-c),{\mathit{c}}_{\mathbf{4}}=(-c,0)$. Each time iteration for the LB equation is solved in two steps:

$$f_i^{*}(t,\mathit{x})=f_i(t,\mathit{x})+J_i(t,\mathit{x}),i=0\dots 4,$$

(3)

$$f_i(t+\Delta t,\mathit{x}+{\mathit{c}}_{\mathit{i}}\Delta t)=f_i^{*}(t,\mathit{x}),i=0\dots 4,$$

(4)

where Equation (3) is the collision step and (4) is the streaming step; $f_i,f_i^{*}$ are the distribution functions before and after collision, respectively; and $J_i$ is the collision kernel, whose explicit form is discussed further. Moreover, $\mathit{x}=(x,y),t$ are the spatial and time variables, whereby $\Delta t$ is the time step and the lattice spacing is $\Delta x=c\Delta t$ or $c=\Delta x/\Delta t$. In this study, $\Delta x=1$.

In the present study, two collision kernels are used. The first one is a standard local collision term for the $D2Q5$ model:

$$f_i^{*}(t,\mathit{x})=f_i(t,\mathit{x})+\frac{\Delta t}{\tau }(f^{eq}(t,\mathit{x}-f_i(t,\mathit{x})),i=1\dots 5,$$

(5)

where $f^{eq}=(1/5)I$. Using Chapman–Enskog expansion, it has been shown that the LB model with the collision term in the form (5) reproduces the Perona–Malik equation [27], where $D=\frac{2c^2}{5}(\tau -\Delta t/2)$.

The second considered model has a non-local collision term [23]:

$$\begin{array}{c}{f}_i^{*}(t,\mathit{x})={\left(1-\frac{\Delta t}{\tau }\right)}^2{f}_i(t,\mathit{x}-{\mathit{c}}_i\Delta t)+\\ +\frac{\Delta t}{\tau }\left(1-\frac{\Delta t}{\tau }\right)f^{eq}(t,\mathit{x}-{\mathit{c}}_i\Delta t)+\frac{\Delta t}{\tau }{f}^{eq}(t,\mathit{x}),i=1\dots 5.\end{array}$$

(6)

Note that the model (6) shows significantly better accuracy for diffusion problems than the conventional model (5) for non-small time steps $\Delta t$ due to reduced hyper-viscous errors. Compared with the conventional single-relaxation-time LB models [4,5,6], the model (6) differs only in the form of their collision step. In the streaming step, the local equilibrium (with zero advecting velocity) remains unchanged. The conventional LB models employ local collisions, i.e., the after-collision values of the distribution function depend on the data at the same spatial node. In the present approach, the collision step also depends on the values of the distribution function in the adjacent nodes. In the Chapman–Enkog expansion, this affects the proportional second-order and higher-derivative terms, this affects the second-order and higher-derivative terms and allows us to reduce hyper-viscous errors in the corresponding nonlinear diffusion equation.

One needs to evaluate the equation for I in the case of non-constant $\tau$, depending on $\nabla I(t,x,y)$. Following the Chapman–Enskog method [37], let us assume that the solution can be expanded on a small parameter $\epsilon$:

$$f_i(t,\mathit{x})=I(t_1,t_2,\dots ,{\mathit{x}}_1)/5+\epsilon f_i^{\left(1\right)}(t_1,t_2,\dots ,{\mathit{x}}_1)+\epsilon f_i^{\left(2\right)}(t_1,t_2,\dots ,{\mathit{x}}_1)+\dots ,$$

(7)

where $t_m\equiv {\epsilon }^{m}t,{\mathit{x}}_m\equiv {\epsilon }^{m}x$. The non-equilibrium components $f_i^{\left(n\right)},n>0$ of the distribution function $f_i$ satisfy the relations: (a) the functions $f_i^{\left(n\right)},n>0$ depend on time and spatial variables via I and its derivatives; and (b) ${\sum }_i{f}_i^{\left(n\right)}=0,n>0$.

By applying the Taylor expansion on $\epsilon$ up to the second-order terms, and collecting the terms standing at the same powers of $\epsilon$, one obtains a set of equations. The terms proportional ${\epsilon }^1$ yields

$$\Delta t({\partial }_{t_1}+({\mathit{c}}_{\mathit{i}},{\nabla }_{\mathbf{1}}))f^{\left(eq\right)}+f_i^{\left(1\right)}=-\Delta t\left(1-\frac{\Delta t}{\tau }\right)({\mathit{c}}_{\mathit{i}},{\nabla }_{\mathbf{1}})f^{\left(eq\right)}+{\left(1-\frac{\Delta t}{\tau }\right)}^2{f}_i^{\left(1\right)},$$

(8)

where ${\nabla }_{\mathbf{1}}\equiv {\partial }_{{\mathit{x}}_{\mathbf{1}}}$. Taking the sum over the index i and remembering that ${\sum }_i{f}_i^{\left(1\right)}=0$, one concludes that ${\partial }_{t_1}{f}^{eq}=0$, and hence, ${\partial }_{t_1}{f}_i^{\left(n\right)}=0,n>0$; and then one obtains from (8):

$$f_i^{\left(1\right)}=-\tau ({\mathit{c}}_{\mathit{i}},{\nabla }_{\mathbf{1}})f^{eq}.$$

(9)

Collecting the ${\epsilon }^2$ terms, one obtains

$$\begin{array}{c}{f}_i^{\left(2\right)}+\Delta t({\mathit{c}}_{\mathit{i}},{\nabla }_{\mathbf{1}})f_i^{\left(1\right)}+\left(\Delta t{\partial }_{t_2}+\frac{1}{2}\Delta t^2{({\mathit{c}}_{\mathit{i}},\nabla )}^2\right)f^{\left(eq\right)}={\left(1-\frac{\Delta t}{\tau }\right)}^2\times \\ \times \left(f_i^{\left(2\right)}-\Delta t({\mathit{c}}_{\mathit{i}},\nabla )f_i^{\left(1\right)}+\frac{1}{2}\Delta t^2{({\mathit{c}}_{\mathit{i}},\nabla )}^2{f}^{\left(eq\right)}\right)+\frac{\Delta t^2}{2}\left(\frac{\Delta t}{\tau }\right)\left(1-\frac{\Delta t}{\tau }\right){({\mathit{c}}_{\mathit{i}},\nabla )}^2{f}^{\left(eq\right)}.\end{array}$$

(10)

Using (9) and taking the sum over i, the following equation is derived from (10):

$${\partial }_{t}I=-\frac{c^2\Delta t^2}{5\tau }div\left(\nabla I\right)+\frac{2c^2}{5}\left(2-\frac{2\Delta t}{\tau }+{\left(\frac{\Delta t}{\tau }\right)}^2\right)div\left(\tau \nabla I\right),$$

(11)

where ${\partial }_t\equiv \epsilon {\partial }_{t_1}+{\epsilon }^2{\partial }_{t_2}$ and $\nabla \equiv \epsilon {\nabla }_{\mathbf{1}}$.

In the case of constant $\tau$, one obtains the diffusion equation

$${\partial }_{t}I=\frac{4c^2\tau }{5}{\left(1-\frac{\Delta t}{2\tau }\right)}^{2}div\left(\nabla I\right),$$

(12)

where the diffusion coefficient D in (12) equals

$$D=\frac{4c^2\tau }{5}{\left(1-\frac{\Delta t}{2\tau }\right)}^2.$$

(13)

Using (13), one can express $\tau$ in terms of D by solving a quadratic equation (only the largest root $\tau$ is used since the other one leads to $\Delta t/\tau >1$, which can cause instability):

$$\tau =\frac{\Delta t}{2}\left(1+\frac{5D}{4c^2\Delta t}+\sqrt{{(\frac{5D}{4c^2\Delta t}+1)}^2-1}\right).$$

(14)

Formula (14) will be also used in the case of D depending on $\nabla I$.

Note that (11) is not exactly the Perona–Malik Equation (1) but one can demonstrate that its properties are very similar to (1). First, one needs to introduce an orthonormal frame consisting of two vectors: one vector defined by $\mathit{T}$ is parallel to $\nabla I$ and another one $\mathit{N}$ is perpendicular to $\nabla I$. These vectors have the components $\mathit{T}=\frac{1}{\left|\nabla I\right|}(I_x,I_y)$ and $\mathit{N}=\frac{1}{\left|\nabla I\right|}(-I_y,I_x)$. By straightforward computations, one can convince that the second-order derivatives of I along $\mathit{T}$ and $\mathit{N}$ are as follows:

$$I_{TT}=\frac{1}{{\left|\nabla I\right|}^2}(I_x^2{I}_{xx}+2I_x{I}_y{I}_{xy}+I_y^2{I}_{yy}),I_{NN}=\frac{1}{{\left|\nabla I\right|}^2}(I_y^2{I}_{xx}-2I_x{I}_y{I}_{xy}+I_x^2{I}_{yy}).$$

(15)

Employing (15) and remembering that $\tau \equiv \tau \left(s\right)\equiv \tau \left(\right|\nabla I\left|\right)$, one eventually obtains

$$\begin{array}{c}{\partial }_{t}I=\frac{4c^2\tau }{{5\left|\nabla I\right|}^2}{\left(1-\frac{\Delta t}{2\tau }\right)}^2{I}_{NN}+\\ +\frac{1}{{\left|\nabla I\right|}^2}\left(\frac{4c^2\tau }{5}{\left(1-\frac{\Delta t}{2\tau }\right)}^2+\frac{2c^2}{5}\left|\nabla I\right|\frac{d\tau }{d\left|\nabla I\right|}\left(2-\frac{2\Delta t}{\tau }+\frac{\Delta t^2}{{\tau }^2}\right)\right)I_{TT},\end{array}$$

(16)

or

$${\partial }_{t}I=D_{NN}{I}_{NN}+D_{TT}{I}_{TT},$$

(17)

where the diffusion coefficients $D_{NN},D_{TT}$ account for the diffusion processes in the directions perpendicular and parallel to $\nabla I$, respectively. The relaxation time $\tau$ is expressed via $D\left(\right|\nabla I\left|\right)$ using (14). The exact forms of $D_{NN},D_{TT}$ are as follows:

$$D_{NN}=\frac{4c^2\tau }{{5\left|\nabla I\right|}^2}{\left(1-\frac{\Delta t}{2\tau }\right)}^2,D_{TT}=\frac{4c^2\tau }{{5\left|\nabla I\right|}^2}{\left(1-\frac{\Delta t}{2\tau }\right)}^2+\frac{2c^2}{5\left|\nabla I\right|}\frac{d\tau }{d\left|\nabla I\right|}\left(2-\frac{2\Delta t}{\tau }+\frac{\Delta t^2}{{\tau }^2}\right).$$

(18)

If $\lambda$ in (2) is smaller than 1, then the coefficients $D_{NN}=D=1/(1+{(\nabla I/\lambda )}^2)$, and $D_{TT}$ behaves similarly to the ones for the Perona–Malik equation. For the Perona–Malik equation, the corresponding coefficients are [25]:

$$D_{NN}=D=\frac{1}{1+{(\nabla I/\lambda )}^2},D_{TT}=2\left|\nabla I\right|\frac{dD}{d\left(\right|\nabla I\left|\right)}+D.$$

(19)

In Figure 1, the coefficients $D_{NN},D_{TT}$ computed from (18) and (19) are presented against the values of $s\equiv \nabla I$, $\lambda =0.2$. One can see that the diffusion in the direction perpendicular to $\nabla I$ is the same for both models. In the direction tangential to $\nabla I$, the diffusion is qualitatively similar: the regions with relatively small $\nabla I$ (“ramps”) are smoothed ($D_{TT}>0$), while the regions with non-small $\nabla I$ (edges) are sharpened since $D_{TT}<0$. Thus, one can conclude that the model (6) leads to Perona–Malik-type diffusion.

3. Applications of Image Denoising

3.1. The Hybrid Method

The previous study [23] shows that the LB model with non-local collision term (6) has significantly smaller hyper-viscous errors than the scheme with the local collision step (5). This means that the model (6) has better accuracy than (5) in the case of non-small time step and diffusion coefficient. Then, one expects that the denoising algorithm based on (6) can be applied for a larger time step, and hence, a smaller number of time iterations are required to model the diffusion process efficiently.

The simulations are performed with the following hybrid LB approach. For the regions with a small diffusion coefficient, they are defined by the condition $\Delta t/\tau >1$, where $\tau$ is given by (14), and for the boundary nodes, the collision step is local (5). Internal regions with a non-small diffusion coefficient ($\Delta t/\tau \le 1$), which are prone to hyper-viscous errors are modeled using the non-local collision step (6). Note that the condition $\Delta t/\tau \le 1$ guarantees the stability of the LB scheme with the non-local collision step (6). For instance, for $\Delta t=1,\Delta x=1$, the condition $\Delta t/\tau <1$ is satisfied when $D>0.2$. In practice, approximately 80–95% nodes are covered with the non-local collision term if $\Delta t=1$, and 70–90% if $\Delta t=0.5$.

At the moment $t=0$, the values of the distribution functions $f_i$ are taken as $I_0(x,y)/5$. Now, consider the formulation of the boundary conditions. Assume that one needs to find $f_i(t+1,{\mathit{x}}_0)$, where index i corresponds to the lattice velocity, such that the node ${\mathit{x}}_0-{\mathit{c}}_{\mathit{i}}\Delta t$ lies beyond the considered spatial domain (ghost node); therefore, the value of $f_i^{*}({\mathit{x}}_0-{\mathit{c}}_{\mathit{i}}\Delta t)$ is undefined. In this case, the extrapolation scheme is used

$$f_i^{*}(t,{\mathit{x}}_{-1})=2f_i^{*}(t,{\mathit{x}}_0)-f_i^{*}(t,{\mathit{x}}_1),$$

where $-1,0,1$ define the ghost node (${\mathit{x}}_0-{\mathit{c}}_{\mathit{i}}\Delta t$), boundary node $\mathit{x}$, and the first node in the interior of the spatial domain (${\mathit{x}}_0+{\mathit{c}}_{\mathit{i}}\Delta t$), respectively.

In addition, the gradient $\nabla I$ is approximated using central differences. It is worth mentioning that it is also possible to approximate first spatial derivatives using the non-equilibrium part of the distribution function $f_i^{neq}=f_i-f_i^{eq}$ [28]. The contrast parameter $\lambda$ is taken to equal $0.1max\left(\right|\nabla \mathit{I}\left|\right)$.

3.2. Numerical Experiments

In the simulations, peak signal-to-noise ratio (PSNR) is measured over time variable t, and PSNR is defined as

$$PSNR\left(t\right)=10{log}_{10}\left(\frac{max{\left(I\right)}^2}{\frac{1}{NxNy}{\sum }_{xy}{(I(t,x,y)-\overline{I}(x,y))}^2}\right),$$

where $Nx,Ny$ is image resolution, $\overline{I}$ is the image gray color intensity without noise, and ${I|}_{t=0}\equiv I_0$ is the considered image with noise. Salt and pepper, speckle and Gaussian noise are considered. The salt and pepper noise distorts images in such way that some pixels take maximal or minimal color intensity values (randomly); in this study, the proportion of such pixels is set to $5\%$. The speckle noise modifies the image by adding the term $nI$, where n takes values from the uniform distribution, with zero mean and variance equal to $0.05$ in this study. The applied Gaussian noise has zero mean and the variance is taken to equal $0.05$. The simulations are performed in Matlab (version 2022b), where four Matlab built-in gray-colored demo images are considered, and noise is added using the imnoise function.

The visual difference between the results of the hybrid method based on the collision terms (5), (6), and the standard model (5) is mostly pronounced for the synthetic images and salt and pepper noise. In Figure 2 and Figure 3, the applications of the hybrid method and the model $D2Q5$ for the denoising of the synthetic images “circlesBrightDark” ($512\times 512$) and ”threads” ($500\times 500$) are presented. Clearly, the hybrid approach provides significantly better results for non-small time steps. For the other images—“Flamingos” ($1296\times 972$), “Parkavenue” ($2048\times 1536$), and speckle and Gaussian noise (Figure 4 and Figure 5)—the results of both methods are very similar. Hence, in order to assess noise reduction quantitatively, PSNR values are recorded for the different time steps $\Delta t$. It is important to mention that the reduction in $\Delta t$ leads to larger values of $\Delta t/\tau$ and therefore to a smaller proportion of nodes covered by the non-local collision term (6). In addition, for small $\Delta t$, hyper-viscous errors are negligible. As a result, for $\Delta t<0.5$, both LB methods present similar results. Then, the simulations are performed for $\Delta t=0.5,1$. Also, PSNR data are provided for the $D2Q5$ model, where $\Delta t=0.1$; the smaller values of $\Delta t$ present a small improvement in PSNR values.

From Figure 6, Figure 7, Figure 8 and Figure 9, one can observe that the hybrid model shows significantly better results than the $D2Q5$ model when salt and pepper noise is considered. For speckle noise and Gaussian noise, the hybrid model yields noticeably greater PSNR values if $\Delta t=1$, except for the “Parkavenue” image and speckle noise (Figure 9b). In the case of $\Delta t=0.5$ and the speckle and Gaussian noise, the hybrid LB method is only slightly better than the $D2Q5$ model. Moreover, one should mention that for salt and pepper, speckle noise PSNR values for the hybrid method reaches the maximum value faster.

4. Discussion

In the present paper, the hybrid LB method for solving Perona–Malik-type diffusion with applications to image noise removal is proposed. Obviously, it is desirable to solve diffusion equations with as small a number of time steps as possible. Although there are no stability restrictions on the upper bound of the time step, the LB scheme’s accuracy still suffers from hyper-viscous errors when the time step and diffusion coefficients are not small. Then, boundary nodes and regions with small diffusion coefficients are modeled with the conventional five-velocity LB model (5), while other regions are modeled with that which contains a non-local collision term (6), for which hyper-viscous errors are reduced. By employing the multiple-scale Chapman–Enskog expansion, it is demonstrated that the model (6) simulates the nonlinear diffusion Equation (11), which can also be rewritten in the form (17), (18). Similarly to the Perona–Malik equation for Equations (17) and (18), the regions with relatively small intensity gradients are smoothed, while the regions with non-small intensity gradients are sharpened. In the numerical experiments, the removal of salt and pepper, speckle, and Gaussian noise from four test images is considered. On average, the hybrid approach yields better PSNR values for non-small time steps than the standard $D2Q5$ model for the Perona–Malik equation—the improvement is very pronounced for salt and pepper noise, and $\Delta t=1$. In addition, PSNR profiles for the hybrid method tends to achieve maximum values faster. The time is kept constant during any particular simulation, although it is possible to apply a variable time step; for example, at initial moments of time, the diffusion process can be modeled with small time steps, while in later moments, $\Delta t$ can be increased since the image is smoothed further. Compared to deep learning algorithms [38], the LB method (and other methods based on numerical solution to the diffusion equations) does not require large training samples of real noisy images and is able to filter either multiplicative or additive noise, while several types of neural networks are aimed to remove only a special type of noise like Gaussian with known a priori variance. In addition, the training process of neural networks requires significant memory resources.

As a final remark, one should mention that the Perona–Malik diffusion has some drawbacks. Its solutions are susceptible to “staircase” effects, i.e., the solutions form piecewise constant profiles. To alleviate this effect, several approaches have been proposed. For instance, in a ramp-preserving Perona–Malik model [39], the contrast parameter $\lambda$ in (2) depends on the edge indicator function. In ramp regions, $\lambda$ is increased, which results in a smoothing effect, while for the edge regions, $\lambda$ takes smaller values and this yields a sharpening effect. Fourth-order diffusion equations for noise removal, on the other hand, are free from staircase effects [40,41]. In the directional Laplacian Perona–Malik model [42,43], smoothing is performed along the edge direction. Due to the flexibility of the LB method for anisotropic diffusion–advection [44,45,46,47], applications of LB schemes for the modifications of the Perona–Malik equation (especially for anisotropic extensions like the directional Laplacian method [42,43]) are of interest for future study.