CN101673393B - Image de-noising method based on lattice Boltzmann model - Google Patents

Abstract

The invention discloses an image de-noising method based on a lattice Boltzmann model. The method comprises the following steps: (1) inputting an initial image I (x,0); (2) setting the initial equilibrium state function Ii<eq>(x,0) of each action direction in a two-dimensional lattice Boltzmann evolution equation; (3) determining the iteration frequency N and the iteration step-length C of the lattice Boltzmann evolution equation; (4) traversing the image to calculate the relaxation factor omega in the lattice Boltzmann evolution equation; (5) calculating the migration process of the lattice Boltzmann model; (6) calculating the action process of the lattice Boltzmann model; (7) updating the equilibrium distribution function as Ii<eq>(x,n); and (8) judging whether achieving the iteration frequency N; if achieving N times, outputting the processed image I (x,N). The method can suppress the noise of the image and effectively protect the edge of the image. The de-noising quality of the image can be enhanced, and iterative calculation of large step-length can be realized, thus enhancing the efficiency of de-noising processing effectively.

Description

Image de-noising method based on grid Boltzmann model

Technical field

The present invention relates to that a kind of (lattice boltzmann model LBM) finds the solution the method that nonlinear diffusion equations realizes image denoising, belongs to image processing field based on grid Boltzmann model.

Technical background

At present, using the nonlinear diffusion model to carry out image denoising is the important application of PDE in image processing field.It can effectively protect edge of image when suppressing picture noise.Using the nonlinear diffusion model to carry out image denoising is at first proposed in nineteen ninety by Perona and Malik; Catte etc. has made raising after 2 years on the theory of model and implementation method, and Weickert were through introducing the smooth effect that diffusion tensor has improved texture image in 1998.Yet; Because the intrinsic uncontinuity of digital picture; The resulting PDE of mathematical model have non-linear; And the factor such as huge of image data amount, the analytic solution of PDE are difficult to obtain or are non-existent at all, at this moment are necessary by means of numerical evaluation to obtain this approximate solutions of equations.In the image processing method based on theory of partial differential equations, the Numerical Implementation problem is that restriction is handled the bottleneck that image method is used based on PDE.Many nonlinear diffusion equations all are through explicit finite difference (explicit finitedifference, EFD) realization of dispersing.Though this explicit finite-difference algorithm is easy to realize that owing to receive the restriction of stability, the step-length of iteration is very little (as 1/4).In order to arrive the diffusion time of expection, just need iteration many times like this, the The whole calculations process efficiency is low, makes it be not suitable for being applied to the image real-time processing domain.

Grid Boltzmann model has been erected the bridge of microcosmic to macroscopic view.The starting point of finding the solution PDE based on the method for grid Boltzmann model is the microvisual model according to system, and design grid Boltzmann EVOLUTION EQUATION realizes the numerical solution to PDE when system is simulated.As a kind of mathematical model of physical system, has clear and clear and definite physical interpretation based on the method for grid Boltzmann model.And as a kind of numerical solution, it is with its physical thought clearly, and the advantage of simple boundary treatment and parallel computation fast is widely used in fields such as fluid mechanics, chemical reaction diffusion, seepage flow, traffic flow.Grid Boltzmann method is for realizing that handling image non-linear diffusion denoising provides the approach that realizes efficiently.

Summary of the invention

The objective of the invention is to the technical matters to the prior art existence, propose a kind of image de-noising method based on grid Boltzmann model, this method can not only improve the image denoising quality, and can improve counting yield, is particularly useful for image and handles in real time.

In order to achieve the above object, the present invention adopts following technical scheme:

Image de-noising method based on grid Boltzmann model of the present invention comprises: set up two-dimensional lattice Boltzmann model, it is made up of the computing grid of discretize, and the node of each grid is equivalent to a cellular, and its value is by the distribution function I of particle _i(i=0,1 ..., q) with the diffusion vector C _i(i=0,1 ... Q) decision.

The EVOLUTION EQUATION of finding the solution the denoising of nonlinear diffusion equations realization gray level image based on the method for grid Boltzmann model is:

$I_i(x+c_i,n+1)-I_i(x,n)=\mathrm{\&omega;}[I_i^{\mathrm{eq}}(x,n)-I_i(x,n)]$

C wherein _iBe the vector of diffusion, I _i(x n) is positioned at the x place for iterations during for n and has speed c _iThe particle density distribution function, ω is a relaxation factor in the formula, I _i ^EqBe the equilibrium state distribution function.

Flow process based on the image de-noising method of grid Boltzmann model is as shown in Figure 3; The method that the present invention proposes embeds edge of image in the relaxation factor of grid Boltzmann EVOLUTION EQUATION by function; Find grid Boltzmann EVOLUTION EQUATION and macroscopical equation corresponding relation to find the solution nonlinear diffusion equations to realize image denoising, its step is following:

(1), the input initial pictures I (x, 0), the value of node is made as the gray-scale value of respective pixel;

(2), use two-dimensional lattice Boltzmann model, the initial balance state function I of each action direction in the grid Boltzmann EVOLUTION EQUATION is set _i ^Eq(x, 0);

(3), confirm the iterations N and the iteration step length C of grid Boltzmann EVOLUTION EQUATION;

(4), the relaxation factor ω in the traversal image calculation grid Boltzmann EVOLUTION EQUATION;

(5), calculate the transition process of two-dimensional lattice Boltzmann model: I _i(x+c _i, n)=I _i(x, n);

(6), calculate the mechanism of two-dimensional lattice Boltzmann model:

$I_i(x+c_i,n+1)-I_i(x,n)=\mathrm{\&omega;}[I_i^{\mathrm{eq}}(x,n)-I_i(x,n)];$

(7), to establish n be iterations, is I according to two-dimensional lattice Boltzmann model modification balanced distribution function _i ^Eq(x, n);

(8), judge whether to reach iterations N, if when reaching N time, the image I after then output is handled (x N), if when not reaching N time, then changes step (4), repeating step (4) ~ (7), the image I after after reaching N number of iteration, exporting processing (x, N).

According to diffusion vector C discrete in the two-dimensional lattice Boltzmann model _iThe difference of direction number q, can be divided into two types of D2Q5 and D2Q9 to two-dimensional lattice Boltzmann model.

Two-dimensional lattice Boltzmann's model in the above-mentioned steps (2) is the D2Q5 model, initial balance state function I _i ^Eq(x, 0) is:

$I_i^{\mathrm{eq}}{(x,0)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=I_i{(x,0)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=\frac{1}{5}I(x,0)$

Relaxation factor ω relevant in the above-mentioned steps (4) is:

${\mathrm{\&omega;}}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=\frac{2}{1+5C\&CenterDot;g}$

Wherein g is that the edge ends function, and C is a step-length,

Above-mentioned steps (7) is upgraded balanced distribution function I _i ^Eq(x n) is:

$I_i^{\mathrm{eq}}{(x,n)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=\frac{1}{5}\underset{i=0}{\overset{4}{\mathrm{\&Sigma;}}}{I}_i(x,n),n=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;N$

Two-dimensional lattice Boltzmann model is the D2Q9 model in the above-mentioned steps (2), initial balance state function I _i ^Eq(x, 0) is:

$I_i^{\mathrm{eq}}{(x,0)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=I_i{(x,0)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=\frac{1}{9}I(x,0)$

Relaxation factor ω relevant in the above-mentioned steps (4) is:

${\mathrm{\&omega;}}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=\frac{2}{1+3C\&CenterDot;g}$

Wherein g is that the edge ends function, and C is a step-length.

Above-mentioned steps (7) is upgraded balanced distribution function I _i ^Eq(x n) is:

$I_i^{\mathrm{eq}}{(x,n)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=\frac{1}{9}\underset{i=0}{\overset{8}{\mathrm{\&Sigma;}}}{I}_i(x,n),n=\mathrm{1,2}\&CenterDot;\&CenterDot;\&CenterDot;N$

The edge that when handling image, uses based on the image de-noising method of the grid Boltzmann model of D2Q5 and D2Q9 is identical by function.In the above-mentioned step (4), the mould value of employing image gradient comes the method for estimated edge position with the protection edge of image.When handling gray level image, the edge by function g is:

$g\left(\right|\&dtri;G_{\mathrm{\&sigma;}}*I\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|\&dtri;G_{\mathrm{\&sigma;}}*I|}{K}\right)}^8})$

G wherein _σBe that variance is the gaussian kernel of σ, K is level and smooth threshold value.

When handling coloured image, fully utilize three kinds of colors (R, G; B) information of component image; In above-mentioned steps (4), ask for through with the gaussian kernel convolution after the difference of eigenwert of gradient factor matrix of image as the estimated value at edge, the gradient of vector image square be:

${\left|\mathrm{dI}\right|}^2=(\mathrm{dI},\mathrm{dI})=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;x}\mathrm{dx}+\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;y}\mathrm{dy})$

That is:

${\left|\mathrm{dI}\right|}^2={\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]}^T\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]={\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]}^{T}A\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]$

Wherein: $\left\{\begin{array}{c}E=(I_x,I_x)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\left(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;x}\right)}^2\\ F=(I_x,I_y)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}\left(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;x}\right)\left(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;y}\right)\\ G=(I_y,I_y)=\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{\left(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;y}\right)}^2\end{array}\right.$

Ask two eigenvalue of coefficient matrices A ₁And λ ₂, then the edge of vector image can be taken as by function:

$g\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2|}{K}\right)}^8})$

K is level and smooth threshold value.

The present invention compared with prior art; Have following conspicuous outstanding substantive distinguishing features and remarkable advantage: above-mentioned image de-noising method based on grid Boltzmann model can not only be realized image non-linear diffusion denoising; Obtain high-quality image denoising effect; And guaranteeing to carry out the computing of big step-length under the stable situation of algorithm, thereby improve the efficient of calculating effectively.Being particularly useful for image handles in real time.

Description of drawings

Fig. 1 is the two-dimensional lattice grid Boltzmann model structure synoptic diagram that the computing grid by discretize constitutes;

Fig. 2-the 1st, among Fig. 1 based on the grid Boltzmann model structure synoptic diagram of D2Q5;

Fig. 2-the 2nd, among Fig. 1 based on the grid Boltzmann model structure synoptic diagram of D2Q9;

Fig. 3 is the process flow diagram of the image de-noising method based on grid Boltzmann model of the present invention;

Fig. 4-the 1st, the former figure of gray level image;

Fig. 4-2 gray level image adds the design sketch after making an uproar;

Fig. 4-3 is based on the grid Boltzmann method of D2Q5 model and handles gray level image denoising effect figure;

Fig. 4-4 is based on the grid Boltzmann method of D2Q9 model and handles gray image denoising effect figure;

Fig. 5-the 1st, the former figure of coloured image;

Fig. 5-the 2nd, coloured image add the design sketch after making an uproar;

Fig. 5-3 is based on the grid Boltzmann method of D2Q5 model and handles coloured image denoising effect figure;

Fig. 5-4 is based on the grid Boltzmann method of D2Q9 model and handles coloured image denoising effect figure.

Embodiment

Embodiment in the face of the image de-noising method based on grid Boltzmann model of the present invention elaborates down: present embodiment is to implement under the prerequisite with technical scheme of the present invention; Provided detailed embodiment, but protection scope of the present invention is not limited to following embodiment.

Embodiments of the invention are described with reference to the accompanying drawings as follows:

Like Fig. 1, shown in 2, set up two-dimensional lattice Boltzmann model, it is made up of the computing grid of discretize, and the node of each grid is equivalent to a cellular, and its value is by the distribution function I of particle _i(i=0,1 ..., q) with the diffusion vector C _i(i=0,1 ... Q) decision.Each renewal (iteration) of nodal value can be divided into two stages: migration phase and effect stage.Migration phase is transmitted particle by the neighborhood node to Centroid, and the effect stage then determines the quantity transmitted.During the digital picture of utilization grid Boltzmann models treated M * N, each pixel and computing node can natural being mapped, and the gray-scale value of pixel is the population on the corresponding node then.Each pixel of node among Fig. 1 (round dot) representative image, arrow has shown the migration and the action direction of model.

According to discrete diffusion vector C _iThe difference of direction number q, can be divided into two types to two-dimensional lattice Boltzmann model DnQq (n dimension q direction):

The D2Q5 model, shown in Fig. 2-1, it has 5 discrete speed

$c_i=\left\{\begin{array}{c}\left(\mathrm{0,0}\right),i=0\\ (\cos (i-1)\mathrm{\&pi;}/2,\sin (i-1)\mathrm{\&pi;}/2)c,i=\mathrm{1,2,3,4}\end{array}\right.$

The D2Q9 model, shown in Fig. 2-2, it has 9 discrete speed

$c_i=\left\{\begin{array}{ccc}{c}_0=0& & i=0\\ c_i={\mathrm{\&lambda;}}_1(\cos \frac{\mathrm{\&pi;}(i-1)}{2},\sin \frac{\mathrm{\&pi;}(i-1)}{2})& {\mathrm{\&lambda;}}_1=1& i=\mathrm{1,2,3,4}\\ c_i={\mathrm{\&lambda;}}_2(\cos (\frac{\mathrm{\&pi;}(i-5)}{2}+\frac{\mathrm{\&pi;}}{4}),\sin (\frac{\mathrm{\&pi;}(i-5)}{2}+\frac{\mathrm{\&pi;}}{4}))& {\mathrm{\&lambda;}}_2=\sqrt{2}& i=\mathrm{5,6,7,8}\end{array}\right.$

Flow process based on the image de-noising method of grid Boltzmann model is as shown in Figure 3; The method that the present invention proposes embeds edge of image in the relaxation factor of grid Boltzmann EVOLUTION EQUATION by function, finds grid Boltzmann EVOLUTION EQUATION and macroscopical equation corresponding relation to find the solution nonlinear diffusion equations to realize image denoising.

Embodiment 1: the gray level image denoising

The effect of handling gray level image is as shown in Figure 4, and 4-1,4-2,4-3,4-4 are followed successively by original Lena image; Adding average is 0, the white noise of variance 0.01 add the Lena image of making an uproar; Based on the image after denoising method (parameter is: threshold value 25, step-length 5, the iterations 10) processing of D2Q5 grid Boltzmann model; Based on the image after denoising method (parameter is: threshold value 25, step-length 5, the iterations 8) processing of D2Q9 grid Boltzmann model.Denoising method with based on D2Q9 grid Boltzmann model is an example, and its step is following:

(1), the input initial pictures I (x, 0), its gray-scale value is made as population on the node,

(2), according to the neighborhood direction number of model, all directions initial balance state function I is set _i ^Eq(x, 0),

$I_i^{\mathrm{eq}}{(x,0)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=I_i{(x,0)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=\frac{1}{9}I(x,0)$

(3), confirm iterations be 8 and iteration step length be 5,

(4), the relaxation factor in the calculating D2Q9 model evolution equation:

$\mathrm{\&omega;}{\left(\right|\&dtri;G_{\mathrm{\&sigma;}}*I\left|\right)}_{D2\mathrm{<figure-callout\; id="9"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}9}=\frac{2}{1+3\&times;5\&CenterDot;g\left(\right|\&dtri;G_{\mathrm{\&sigma;}}*I\left|\right)}$

Wherein the edge by function is:

$g\left(\right|\&dtri;G_{\mathrm{\&sigma;}}*I\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|\&dtri;G_{\mathrm{\&sigma;}}*I|}{25}\right)}^8})$

(5), calculate the transition process of grid Boltzmann model: I _i(x+c _i, n)=I _i(x, n),

(6), calculate the mechanism of grid Boltzmann model: $I_i(x+c_i,n+1)-I_i(x,n)=\mathrm{\&omega;}[I_i^{\mathrm{Eq}}(x,n)-I_i(x,n)],$

(7), to establish n be iterations, upgrading the balanced distribution function is I _i ^Eq(x, n),

$I_i^{\mathrm{eq}}(x,n)=\frac{1}{9}\underset{i=0}{\overset{8}{\mathrm{\&Sigma;}}}{I}_i(x,n)$

(8), judge whether to reach iterations 8, if when reaching 8 times, if the image I (x, 8) after then output is handled when not reaching 8 times, is then changeed step (4), repeating step (4) ~ (7) reach at 8 o'clock up to iterations, the image after output is handled: I (x, 8).

Embodiment 2: the coloured image denoising

The effect of handling coloured image is as shown in Figure 5, and 5-1,5-2,5-3,5-4 are followed successively by the pepper original image; Adding average is 0, the white noise of variance 0.01 add the pepper image of making an uproar; Based on the image after denoising method (parameter is: threshold value 100, step-length 5, the iterations 5) processing of D2Q5 grid Boltzmann model; Based on the image after denoising method (parameter is: threshold value 100, step-length 5, the iterations 4) processing of D2Q9 grid Boltzmann model.Denoising method with based on D2Q5 grid Boltzmann model is an example, and its step is following:

(1), each component image of establishing coloured image is I _j(x, 0) (j=1,2,3)

(2), the gray scale of the initial all directions of component image is I _Ji(x, 0) is provided with the equilibrium state function of all directions to component image:

$I_{\mathrm{ji}}^{\mathrm{eq}}{(x,0)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=I_{\mathrm{ji}}{(x,0)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=\frac{1}{5}{I}_j(x,0)$

(3), confirm iterations 5 and iteration step length 5

(4), traversal image calculation relaxation factor:

$\mathrm{\&omega;}{\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)}_{D2\mathrm{<figure-callout\; id="5"\; label="Q"\; filenames="H2009101965129E00013.png"\; state="\{\{state\}\}">Q</figure-callout>}5}=\frac{2}{1+5\&times;5\&CenterDot;g\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)}$

Wherein the edge by function is:

$g\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2|}{100}\right)}^8})$

(5), calculate the transition process of grid Boltzmann model: I _Ji(x+c _i, n)=I _Ji(x, n)

(6), calculate the mechanism of grid Boltzmann model:

$I_{\mathrm{ji}}(x+c_i\mathrm{\&Delta;t},t+\mathrm{\&Delta;t})-I_{\mathrm{ji}}(x,t)=\mathrm{\&omega;}\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)[I_{\mathrm{ji}}^{\mathrm{eq}}(x,t)-I_{\mathrm{ji}}(x,t)]$

(7), to establish n be iterations, upgrade each component image balanced distribution function respectively to be:

$I_{\mathrm{ji}}^{\mathrm{eq}}(x,n)=\frac{1}{5}\underset{5=0}{\overset{4}{\mathrm{\&Sigma;}}}{I}_{\mathrm{ji}}(x,n)$

(8), judge whether to reach iterations 5, if when reaching 5 times, the image I after then output is handled _jStep (4) if when not reaching 5 times, is then changeed in (x, 5), and repeating step (4) ~ (7) reach at 5 o'clock up to iterations, the image after output is handled: I _j(x, 5)

Experiment shows; Relatively with additive operator splitting-up method (additive operator splitting; AOS) be the method for finite difference of representative, can obtain better denoising quality based on the image de-noising method of grid Boltzmann model, its counting yield also is superior to the AOS algorithm.Adopt the objective measurement denoising of Y-PSNR (PSNR) quality.The PSNR that handles the back image is big more, and its more approaching desirable denoising result is described, corresponding denoising method is effective more.Use association's rising sun 410A notebook (CPU T2080, Memory 1G) during simulation run, running environment is MATLAB 2008b.Table 1 is 0 for use method and the AOS method based on D2Q5, D2Q9 grid Boltzmann model of the present invention to handle having added average, and variance is respectively the result of the Lena image of 0.01 and 0.1 white Gaussian noise.The variances sigma unification of Gaussian convolution nuclear is taken as 1 in function on the edge of, and threshold value is 25.Choose the highest iteration result that time of PSNR, can find out that the method based on grid Boltzmann model can more effectively suppress noise than AOS.Here be the serial computing time computing time of grid Boltzmann method, because the effect and the transition process of grid Boltzmann model all directions are independently, can further use parallel algorithm to reduce computing time.

Table 1

Claims (1)

1. image de-noising method based on grid Boltzmann model; It is characterized in that; Through the mode in the relaxation factor that embeds edge of image by function grid Boltzmann microcosmic EVOLUTION EQUATION; In two-dimensional lattice Boltzmann model, find microcosmic EVOLUTION EQUATION and macro non-linear diffusion equation corresponding relation to find the solution nonlinear diffusion equations to realize image denoising, its step is following:

(1), the input initial pictures I (x, 0), the value of node is made as the gray-scale value of respective pixel;

(2), use two-dimensional lattice Boltzmann model, the initial balance state function

of each action direction in the grid Boltzmann microcosmic EVOLUTION EQUATION is set

(3), confirm the iterations N and the iteration step length C of grid Boltzmann microcosmic EVOLUTION EQUATION;

(4), the relaxation factor ω in the traversal image calculation grid Boltzmann EVOLUTION EQUATION;

(5), calculate the transition process of two-dimensional lattice Boltzmann model: I _i(x+c _i, n)=I _i(x, n);

(6), calculate the mechanism of two-dimensional lattice Boltzmann model:

$I_i(x+c_i,n+1)-I_i(x,n)=\mathrm{\&omega;}\left[(I_i^{\mathrm{eq}}(x,n)-I_i(x,n))\right];$

(7), to establish n be iterations, is

according to two-dimensional lattice Boltzmann model modification balanced distribution function

(8), judge whether to reach iterations N, if when reaching N time,, the image I after then output is handled (x N), if when not reaching N time, then changes step (4), repeating step (4)～(7), the image I after after reaching N number of iteration, exporting processing (x, N);

When two-dimensional lattice Boltzmann model was the D2Q5 model in the above-mentioned steps (2), initial balance state function

was:

$I_i^{\mathrm{eq}}{(x,0)}_{D2Q5}=I_i{(x,0)}_{D2Q5}=\frac{1}{5}I(x,0)$

Relaxation factor ω relevant in the above-mentioned steps (4) is:

${\mathrm{\&omega;}}_{D2Q5}=\frac{2}{1+5C\&CenterDot;g}$

Wherein g is that the edge ends function, and C is a step-length,

Above-mentioned steps (7) is upgraded balanced distribution function

:

$I_i^{\mathrm{eq}}{(x,n)}_{D2Q5}=\frac{1}{5}\underset{i=0}{\overset{4}{\mathrm{\&Sigma;}}}{I}_i(x,n),n=\mathrm{1,2}...N;$

When above-mentioned steps (2) two-dimensional lattice Boltzmann's model was the D2Q9 model, initial balance state function

was:

$I_i^{\mathrm{eq}}{(x,0)}_{D2Q9}=I_i{(x,0)}_{D2Q9}=\frac{1}{9}I(x,0)$

Relaxation factor ω relevant in the above-mentioned steps (4) is:

${\mathrm{\&omega;}}_{\mathrm{DaQ}9}=\frac{2}{1+3C\&CenterDot;g}$

Wherein g is that the edge ends function, and C is a step-length,

Above-mentioned steps (7) is upgraded balanced distribution function

:

$I_i^{\mathrm{eq}}{(x,n)}_{D2Q9}=\frac{1}{9}\underset{i=0}{\overset{8}{\mathrm{\&Sigma;}}}{I}_i(x,n),n=\mathrm{1,2}...N;$

When handling gray level image, the edge by function g is among the relaxation factor ω of said step (4):

$g\left(\right|\&dtri;G_{\mathrm{\&sigma;}}*I\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|\&dtri;G_{\mathrm{\&sigma;}}*I|}{K}\right)}^8})$

G wherein _σBe that variance is the gaussian kernel of σ, K is level and smooth threshold value;

When handling coloured image, the edge by function g is among the relaxation factor ω of said step (4):

$g\left(\right|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2\left|\right)=1-\exp (-\frac{3.315}{{\left(\frac{|{\mathrm{\&lambda;}}_1-{\mathrm{\&lambda;}}_2|}{K}\right)}^8})$

Wherein K is level and smooth threshold value, λ ₁And λ ₂Be gaussian kernel G through variances sigma _σWith the gradient factor matrix after the coloured image convolution be:

${\left|\mathrm{dI}\right|}^2=(\mathrm{dI},\mathrm{dI})=\underset{i=1}{\overset{8}{\mathrm{\&Sigma;}}}(\frac{{\&PartialD;I}^{\left(i\right)}}{\&PartialD;x}\mathrm{dx}+\frac{{\mathrm{dI}}^{\left(i\right)}}{\&PartialD;y}\mathrm{dy})={\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]}^T\left[\begin{array}{cc}E& F\\ F& G\end{array}\right]\left[\begin{array}{c}\mathrm{dx}\\ \mathrm{dy}\end{array}\right]$

Eigenwert.

Images