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Fractional Differential Equations in Electronic Information Models

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
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Recibido: 16 Jan 2022
Aceptado: 24 Mar 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Abstract

The article first uses the fractional derivative to define a new fractional bounded variation function space. This method constructs the corresponding electronic information image model denoising mask by setting a smaller fractional integration order. The experimental results show that the image denoising algorithm based on fractional integration can not only improve the signal-to-noise ratio of the image compared with the traditional denoising method, but also can better retain the details of the edge and texture of the electronic information image.

Keywords

MSC 2010

Introduction

Synthetic Aperture Radar (SAR) is inevitably contaminated by coherent speckle noise in imaging due to its imaging mechanism. Coherent speckle noise is one of the main reasons for the degradation of SAR image quality. In SAR image denoising, traditional methods such as Lee filtering and wavelet denoising perform better in removing noise but will blur the details of the image. In recent years, methods based on partial differential equations (PDE) have become a research hotspot in SAR image denoising because they can better preserve the edges of images. Some scholars have proposed a denoising AA model for coherent speckle noise that obeys the Gamma distribution [1]. Some scholars have proposed a fast algorithm for the AA model. Although the AA model can effectively remove the coherent speckle noise and better maintain the edges, it does not maintain the image texture. It is prone to the “staircase effect” in areas where the gray level does not change much.

This paper uses fractional derivative and negative exponent Sobolev space modeling to solve the problem of texture preservation and “staircase effect” suppression in SAR image denoising. This paper proposes a new fractional multiscale SAR image denoising variational PDE model. At the same time, we use the statistical characteristics of image local variance and the relationship between wavelet coefficients after wavelet decomposition and function regularity to give an adaptive selection method of model parameters. On this basis, an adaptive denoising algorithm is proposed [2]. Theoretical analysis and experiments show that this new method can effectively remove noise and suppress the “staircase effect” in the non-textured area of the image. The algorithm can keep the edge and texture of the image better in the edge and texture area of the image.

Proposal of a fractional multiscale model

Coherent speckle noise is generally modeled as multiplicative noise. Some scholars have proposed a multiplicative noise denoising model (AA model) that takes the Gamma distribution as the noise prior. minuSΩ{F(u)=|u|BV+λΩ(logu+f/u)dxdy} \mathop {\min}\limits_{u \in S\Omega} \left\{{F\left(u \right) = {{\left| u \right|}_{BV}} + \lambda \int_\Omega {\left({\log \,u + f/\,u} \right)dxdy}} \right\}

Among them f is the observed image and u is the approximate image to be obtained. S(Ω):={uBV(Ω), u > 0)}, BV(Ω) is the bounded variation (BV) function space defined on the area Ω. The AA model maintains the edge of the image better but does not maintain the image's texture well and is prone to blocky “step effect.”

Studies have shown that the fractional derivative is good for describing the texture. Some scholars have studied the denoising method of fractional derivative based on Fourier transform based on the P-M equation for additive noise denoising. The algorithm has achieved good results in step effect suppression. However, the computational complexity of this algorithm is too large, and the model parameters are also manually selected and are not conducive to practical applications [3]. In this paper, aiming at multiplicative noise suppression, we use a new fractional derivative to model the image based on the AA model.

At present, the definition of fractional derivative is not uniform. This article adopts Grümwald-Letnikov's definition of fractional derivative. We use the α(α>0)-order derivative operator Da defined by Grümwald-Letnikov to replace the first-order derivative operator in the AA model. In this way, the following fractional variational model can be obtained: min{E(u)=Ω|Dau|dxdy+λΩ(logu+f/u)dxdy} \min \left\{{E\left(u \right) = \int_\Omega {\left| {{D^a}u} \right|dxdy + \lambda \int_\Omega {\left({\log \,u + f/\,u} \right)dxdy}}} \right\}

Where |Dau|=(Dxau)2+(Dyau)2 \left| {{D^a}u} \right| = \sqrt {{{\left({D_x^au} \right)}^2} + {{\left({D_y^au} \right)}^2}} , Dxau D_x^au and Dyau D_y^au are the α-order partial derivatives of u concerning x and y, respectively. When α=1, formula (2) is the AA model. Using the variational method, the Euler-Lagrange equation of formula (2) can be obtained as uHa(u)λfuu=0 u \cdot {H^a}\left(u \right) - \lambda \cdot {{f - u} \over u} = 0

Where Ha(u)=(1)a¯[D¯xa(Dxau/|Dau|)+D¯ya(Dxau/|Dau|)] {H^a}\left(u \right) = \overline {{{\left({- 1} \right)}^a}} \left[{\bar D_x^a\left({D_x^au/\left| {{D^a}u} \right|} \right) + \bar D_y^a\left({D_x^au/\left| {{D^a}u} \right|} \right)} \right] , D¯a {\bar D^a} is the conjugate operator of the fractional derivative operator Da. Studies have shown that the negative exponent Sobolev space is conducive to the modeling of image texture. In formula (3) (fu) / u includes the noise and texture of the image. Since the texture has multiple scales, this paper considers using different negative exponent Sobolev spaces to model the different scale parts of the texture [4]. We first use orthogonal wavelet decomposition and reconstruction to obtain sub-images [λ(fu / u]j, j = 0,1, 2,…,L of λ(fu / u at different scales. Then we use the negative exponent Sobolev space H−sj to model the sub-images of each scale. We can improve formula (3) into the following fractional multiscale variational PDE model: uHa(u)j=0L2jsj[λ(fu)u]j=0 u \cdot {H^a}\left(u \right) - \sum\limits_{j = 0}^L {{2^{- j{s_j}}}\,{{\left[{{{\lambda \left({f - u} \right)} \over u}} \right]}_j} = 0}

Where 0 ≤ sj ≤ 1, j = 0,1,2,…,L is the space parameter of the negative exponent Sobolev space H−sj corresponding to each scale. We introduce the time variable [5]. Assuming that the initial value is u(x, y; 0) = f(x,y), we use the gradient descent method to transform the solution of equation (4) into solving the following partial differential equation: ut=uHa(u)+j=0L22jsj(λfuu)j {{\partial u} \over {\partial t}} = - u \cdot {H^a}\left(u \right) + \sum\limits_{j = 0}^L {{2^{- 2j{s_j}}}} {\left({\lambda \cdot {{f - u} \over u}} \right)_j}

Parameter adaptive selection
Selection of the regularization parameter λ

The regularization parameter corresponding to each point in the image is λ. Consider using the information in the neighborhood of this point to determine and update it in iterations. Let us suppose λ(x,y)=Ωλ(x,y)ωx,y(x,y)dxdy \lambda \left({x,\,y} \right) = \int_\Omega {\lambda \left({x ,\,y} \right){\omega _{x,\,y}}\left({x ,\,y} \right)dx dy}

Where ωx,y(x,y) = ω(|xx|, |yy|) is a normalized radially symmetric smooth window function and satisfies Ωωx,y(x,y)dxdy=1 \int_\Omega {{\omega _{x,\,y}}\left({x ,\,y} \right)dx dy = 1} . This paper uses the normalized Gauss function as the window function. λ˜ \tilde \lambda is an undetermined function combining image and noise priors. We can use formula (4), formula (6), and orthogonal wavelet decomposition to get Ωj=0L22jsj[uHa(u)]j(fuu)dxdy=Ωλ(x,y)[Ωωx,y(x,y)(fuu)2dxdy]dxdy \matrix{{\int_\Omega {\sum\limits_{j = 0}^L {{2^{2j{s_j}}}} {{\left[{u \cdot {H^a}\left(u \right)} \right]}_j}\left({{{f - u} \over u}} \right)dxdy}} \hfill \cr {= \int_\Omega {\lambda \left({x,\,y} \right)\left[{\int_\Omega {{\omega _{x,\,y}}\left({x ,\,y} \right){{\left({{{f - u} \over u}} \right)}^2}dx dy}} \right]dxdy}} \hfill \cr}

The sufficient conditions for the establishment of this equation are λ(x,y)=(fu)j=0L22jsj[uHa(u)]j/[uPu(x,y)] \lambda \left({x,\,y} \right) = \left({f - u} \right)\sum\limits_{j = 0}^L {{2^{2j{s_j}}}} {\left[{u \cdot {H^a}\left(u \right)} \right]_j}/\left[{u \cdot {P_u}\left({x,\,y} \right)} \right]

Where Pu(x,y)=Ωωx,y(x,y)(fu)/u]2dxdy {\left. {{P_u}\left({x,\,y} \right) = \int_\Omega {{\omega _{x,\,y}}\left({x ,\,y} \right)\left({f - u} \right)/u}} \right]^2}dx dy . Noting the nature of ωx,y(x˜,y˜) {\omega _{x,\,y}}\left({\tilde x,\,\tilde y} \right) , we can get 1|Ω|ΩPu(x,y)dxdy=1|Ω|Ω(fuu)2dxdy=kσ2 {1 \over {\left| \Omega \right|}}\int_\Omega {{P_u}\left({x,\,y} \right)dxdy = {1 \over {\left| \Omega \right|}}\int_\Omega {{{\left({{{f - u} \over u}} \right)}^2}dxdy} = \,k{\sigma ^2}}

Where σ2 is the noise variance. Because (fu)/u also contains texture information in practice, so k ≥ 1. Equation (9) shows that kσ2 is the average value of Pu (x, y). In the non-textured area Pu (x, y) ≈ σ2 < kσ2 of the image. In the texture area, Pu (x, y) >> kσ2. Pu (x, y) is affected by the texture. In the estimation of, we first obtain the denoising image uAA and the residual image υAA = (fuAA)/uAA of the AA model, and then calculate PuAA(x, y) and kAA = var(υAA)/σ2. Where var(·) is the variance. We get an estimate of λ˜ \tilde \lambda as λ¯(x,y)=(fu)j=0L22jsj[uHa(u)]ju(kAAσ2)PuAA(x,y)kAAσ2 \bar \lambda \left({x,\,y} \right) = {{\left({f - u} \right)\sum\limits_{j = 0}^L {{2^{2j{s_j}}}} {{\left[{u \cdot {H^a}\left(u \right)} \right]}_j}} \over {u \cdot \left({{k_{AA}} \cdot {\sigma ^2}} \right)}} \cdot {{{P_{uAA}}\left({x,\,y} \right)} \over {{k_{AA}} \cdot {\sigma ^2}}}

According to formula (6) ( λ˜ \tilde \lambda is is replaced by λ¯ \bar \lambda ), λ is obtained. In this way, due to PuAA < kAA · σ2 in the non-textured area, the obtained λ value is smaller, and the denoising ability is enhanced [6]. In the texture area, due to PuAA >> kAA · σ2, λ is larger, and the details are better maintained.

Selection of fractional derivative order α

When the fractional derivative order α>1, the “staircase effect” can be better suppressed. When α is too large, the texture retention is better, but the denoising effect will be reduced. In this paper, in the non-textured area of the image, α should be taken close to and slightly greater than 1 to ensure better denoising and suppress the “staircase effect.” We can take a slightly larger α to maintain the texture [7]. Since PuAA(x, y) ≥ kAA · σ2 is established in the textured area and PuAA(x, y) < kAA · σ2 is established in the non-textured area, α is taken as α={α1,PuAA(x,y)<kAAσ2α2,PuAA(x,y)kAAσ2 \alpha = \left\{{\matrix{{{\alpha _1},} \hfill & {{P_{uAA}}\left({x,\,y} \right)\, < \,{k_{AA}} \cdot {\sigma ^2}} \hfill \cr {{\alpha _2},} \hfill & {{P_{uAA}}\left({x,\,y} \right)\, \ge \,{k_{AA}} \cdot {\sigma ^2}} \hfill \cr}} \right.

Where 1 ≤ α1α2 ≤ 2. Considering denoising, suppressing “staircase effect,” and texture preservation, we recommend α1 = 1.2, α2 = 1.5.

The choice of Sobolev space index sj

The negative exponent Sobolev space exponent is a measure of the singularity of a function. The singularity of a function can also be measured with the Lipschitz exponent β. The difference between the two is not big. In the wavelet multiscale analysis theory, we often use wavelet coefficients’ amplitude to measure the function's regularity [8]. The Lipschitz exponent βj of each scale part of the function f (x, y) is estimated as follows: βj=12[12log2(max|W2j|/max|W2j1)1],j=1,2,,L {\beta _j} = {1 \over 2}\left[{{1 \over 2}{{\log}_2}\left({\max \left| {{W_{{2^j}}}\left| {/\max} \right|{W_{{2^{j - 1}}}}} \right.} \right) - 1} \right],\,j = 1,\,2,\, \ldots ,\,L

Where W2j f is the wavelet coefficient of f in the 2j scale. In this article, we first perform L + 1 orthogonal wavelet decomposition of λ(fu)/u to obtain the wavelet coefficients of each layer. Then calculate the Lipschitz index βj corresponding to each scale according to formula (12). At the same time, we use βj to estimate sj. Considering the value range (0 ≤ sj ≤ 1), so we assume sj={0,j=0or(βj)<01,j>0and(βj)>1j=0,1,2,L,Lβj,j>0and0<(βj)<1 {s_j} = \left\{{\matrix{{0,} & {j = 0\,{\rm{or}}\,\left({- {\beta _j}} \right) < 0} & {} \cr {1,} & {j > 0\,{\rm{and}}\,\left({- {\beta _j}} \right) > 1} & {j = 0,\,1,\,2,\,{\rm{L}},\,L} \cr {- {\beta _j},} & {j > 0\,{\rm{and}}\,0 < \left({- {\beta _j}} \right) < 1} & {} \cr}} \right.

Algorithm description

An image of size N × N is denoted as u=[u(i,j)]i,j=1N u = \left[{u\left({i,j} \right)} \right]_{i,\,j = 1}^N . We assume that i, j < 1 or i, j > N time ui,j = 0. Then the discretization operators Δxau \Delta _x^au and Δyau \Delta _y^au of Dxau D_x^au and yau _y^au are respectively (Δxau)i,j=k=0K1(1)k(ak)u(ik,j)(Δyau)i,j=k=0K1(1)k(ak)u(i,jk)} \left. {\matrix{{{{\left({\Delta _x^au} \right)}_{i,\,j}}} & = & {\sum\limits_{k = 0}^{K - 1} {{{\left({- 1} \right)}^k}\left({\matrix{a \cr k \cr}} \right)u\left({i - k,\,j} \right)}} \cr {{{\left({\Delta _y^au} \right)}_{i,\,j}}} & = & {\sum\limits_{k = 0}^{K - 1} {{{\left({- 1} \right)}^k}\left({\matrix{a \cr k \cr}} \right)u\left({i,\,j - k} \right)}} \cr}} \right\}

But [Ha(u)]i,j=k=0K1(1)k(ak)p1(ik,j)+k=0K1(1)k(ak)p2(i,jk) {\left[{{H^a}\left(u \right)} \right]_{i,j}} = \sum\limits_{k = 0}^{K - 1} {{{\left({- 1} \right)}^k}\left({\matrix{a \cr k \cr}} \right){p_1}\left({i - k,\,j} \right) + \sum\limits_{k = 0}^{K - 1} {{{\left({- 1} \right)}^k}\left({\matrix{a \cr k \cr}} \right){p_2}\left({i,j - k} \right)}}

Where p1(i,j)=(Δxau)i,j(Δxau)i,j2+(Δyau)i,j2+ε2 {p_1}\left({i,\,j} \right) = {{{{\left({\Delta _x^au} \right)}_{i,\,j}}} \over {\sqrt {\left({\Delta _x^au} \right)_{i,\,j}^2 + \left({\Delta _y^au} \right)_{i,\,j}^2 + {\varepsilon ^2}}}} , p2(i,j)=(Δyau)i,j(Δxau)i,j2+(Δyau)i,j2+ε2 {p_2}\left({i,\,j} \right) = {{{{\left({\Delta _y^au} \right)}_{i,\,j}}} \over {\sqrt {\left({\Delta _x^au} \right)_{i,\,j}^2 + \left({\Delta _y^au} \right)_{i,\,j}^2 + {\varepsilon ^2}}}} . ε2 It is a small constant, and the main guarantee is that the denominator is not zero. Because the generalized binomial coefficients in the 1 ≤ α ≤ 2 interval decay very quickly and tend to zero as k increases, k does not need to be too large in actual calculations. This article takes K=20. Thus the discrete iteration format of equation (5) is u(n+1)=u(n)+Δt[u(n)Ha(u(n))+j=0L22jsj(λfu(n)n(n)+ε2)j] {u^{\left({n + 1} \right)}} = {u^{\left(n \right)}} + \Delta t \cdot \left[{- {u^{\left(n \right)}} \cdot {H^a}\left({{u^{\left(n \right)}}} \right) + \sum\limits_{j = 0}^L {{2^{- 2j{s_j}}}{{\left({\lambda \cdot {{f - {u^{\left(n \right)}}} \over {{n^{\left(n \right)}} + {\varepsilon ^2}}}} \right)}_j}}} \right]

In the iterative process of equation (16), we update the parameters a, λ and sj = 0, j = 0,1,…,L adaptively in the iteration, and the adaptive algorithm steps for solving the fractional multiscale variational PDE model can be obtained as follows: Step 1: We take the initial value u(0) = f, the parameter a =1, sj = 0 (j ≥ 0), and the fixed parameter λ. We perform iterative calculations according to equation (16). After M iterations, the AA model denoising and the residual image uAA = u(M), υAA = (fuAA) / uAA is obtained. In the article, υAA is zero-averaged and then convolved with the normalized Gauss template to obtain PuAA. At the same time, we calculate α according to equation (11).

Step 2: Re-set the initial value u(0) = f, λ(0) = 0, sj(0)=0 s_j^{\left(0 \right)} = 0 , j = 0,1,2,…,L, to set parameters Δt and ε. Assume that u(n), λ(n), sj(n)=0 s_j^{\left(n \right)} = 0 , j = 0,1,2,…,L, has been obtained.

Step 3: We use an orthogonal wavelet to decompose u(n) · Ha(u(n)) into L times wavelet. Use each scale factor to perform single-branch reconstruction to get [Ha(u(n))]j, j = 0,1,2,…,L. Calculate λ¯(n+1) {\bar \lambda ^{\left({n + 1} \right)}} , and λ(n+1) according to formula (6) and formula (10).

Step 4: We use orthogonal wavelets to perform wavelet decomposition on λ(n+1)(fu(n) / (u(n) + ε2), according to formula (12) and formula (13) update to obtain sj(n+1) s_j^{\left({n + 1} \right)} . Use wavelet single-branch reconstruction to get [λ(n+1)(fu(n) / (u(n) + ε2)]j, j = 0,1,2,…,L.

Step 5: Calculate u(n+1)=u(n)+Δt[u(n)Ha(u(n))+j=0L22jsj(n+1)(λ(n+1)fu(n)u(n)+ε2)j] {u^{\left({n + 1} \right)}} = {u^{\left(n \right)}} + \Delta t \cdot \left[{- {u^{\left(n \right)}} \cdot {H^a}\left({{u^{\left(n \right)}}} \right) + \sum\limits_{j = 0}^L {{2^{- 2js_j^{\left({n + 1} \right)}}}{{\left({{\lambda ^{\left({n + 1} \right)}}{{f - {u^{\left(n \right)}}} \over {{u^{\left(n \right)}} + {\varepsilon ^2}}}} \right)}_j}}} \right] . The iteration ends when u(n + 1) meets the given iteration termination condition. Otherwise, let n: n + 1 go to step 3.

For an image of size N × N, when the model parameters are fixed, the calculation amount of each iteration of the gradient descent method for calculating the AA mode is O(N2). The method in this paper considers the parameter adaptation, so the calculation amount is larger than the gradient descent method with fixed parameters. The increased amount of calculation is mainly reflected in the following aspects:

(1) The amount of calculation caused by the adaptation of the parameter λ increases. This part is mainly produced by step 1 of the algorithm and the convolution operation with the normalized Gauss template involved in the algorithm. Since in step 1, only a few finite iterations need to be performed using the gradient descent method with fixed parameters, the increase in calculation order is still O(N2). When the normalized Gauss template window is relatively small, the amount of calculation added to the convolution operation is O(N2). The overall order of these two parts is still O(N2).

(2) The amount of calculation caused by the adaptation of parameter sj increases. This part of the calculation is mainly produced by the algorithm's two wavelet decomposition and reconstruction [9]. When sj = 0, j = 0,1,2,…L does not need to perform wavelet decomposition and reconstruction calculations. Since the calculation amount of each two-dimensional wavelet transform is O(N2), the total increase in this part of calculation is still O(N2).

(3) The amount of calculation caused by the calculation of the fractional derivative increases. In the formula (15) for calculating the fractional difference, when α=1.0, K≡2, and the calculation amount is O(N2) at this time. When α is a non-integer, the calculation amount for calculating the fractional difference is K · O(N2), but the calculation amount is still O(N2). Although the calculation amount of the algorithm in this paper is more O(N2) than that of the gradient descent algorithm with fixed parameters in each iteration, the total amount of calculation and the calculation amount of the gradient descent method with fixed parameters are still of the same order.

Numerical experiment and analysis

We use the adaptive Lee filter, AA model, and method in this paper to process and compare an artificially noised image and a real SAR image. The experimental image to be processed is shown in Figure 1. The parameter Δt = 0.1, L = 3, ε = 10−10 in the algorithm of this paper [10]. Orthogonal wavelets are Db4 orthogonal wavelets, and the normalized Gauss template window size is 9×9. In the adaptive Lee filter, we take the window size as 5×5.

Figure 1

Experimental image to be processed

The clean image and noise variance of the image are known. This paper uses the peak signal-to-noise ratio (PSNR) as a quantitative indicator to measure the denoising effect. In the three methods PSNR(u(n+1) < PSNR(u) is the termination condition of the iteration. Denoising and as shown in Figure 2. Table 1 lists the comparison of the peak signal-to-noise ratio of the denoising image corresponding to Figure 2.

Figure 2

Experimental image 1 Comparison of denoising image effects

PSNR comparison of experimental image 1 denoising image

Original image Adaptive Lee filter AA Method of this article
PSNR 19.0959 26.7535 27.7593 27.8881

From the comparison of denoising and residual images, it can be seen that the Lee filter has the best denoising effect in non-textured areas, but the edges and textures are blurred. The AA model can maintain the edges well, but the texture information is not well maintained. The method in this paper is better in terms of edge and texture preservation. From the comparison of the peak signal-to-noise ratio, it can be seen that the peak signal-to-noise ratio obtained by the method in this paper is the largest. From the comparison of the partially enlarged images, it can be seen that there is no “staircase effect” in Lee filtering [11]. The AA model has a more obvious “step effect.” The method in this paper can keep the details better while effectively suppressing the “staircase effect.”

The real SAR image noise of experimental image 2 is unknown. This paper uses Donoho's noise estimation method to estimate the standard noise deviation. The equation is as follows: σ˜=MidHH/0.6745 \tilde \sigma = Mi{d_{HH}}/0.6745

Where MidHH is the median amplitude value of the wavelet coefficients of the highest frequency HH subband of the image wavelet decomposition. In this paper MidHH is the median amplitude of the highest frequency HH subband wavelet coefficients of the wavelet decomposition of the AA model denoising residual image. Here, the equivalent visual number is used to compare the denoising effect. The equivalent visual number is defined as ENL = E(Is)2/var(Is)

Where Is is the selected flat area in the image. E(Is), var(Is) is the mean and variance of the flat area, respectively. The edge retention index is used here for comparison [12]. The edge retention index EPI is defined as EPI=(us(i,j)us(i+1,j))2+(us(i,j)us(i,j+1))2(u0(i,j)u0(i+1,j))2+(u0(i,j)u0(i,j+1))2 EPI = {{\sum {\sqrt {{{\left({{u_s}\left({i,\,j} \right) - {u_s}\left({i + 1,\,j} \right)} \right)}^2} + {{\left({{u_s}\left({i,\,j} \right) - {u_s}\left({i,\,j + 1} \right)} \right)}^2}}}} \over {\sqrt {{{\left({{u_0}\left({i,\,j} \right) - {u_0}\left({i + 1,\,j} \right)} \right)}^2} + {{\left({{u_0}\left({i,\,j} \right) - {u_0}\left({i,\,j + 1} \right)} \right)}^2}}}}

Among them us is the image after denoising and u0 is the pixel value of the original image. In the definition of EPI, us(i, j) are the sample points selected in the pixel in the region where the gradient of the image changes rapidly. The maximum value of EPI is 1, and the minimum value is 0. The larger the value of EPI, the stronger the edge retention capability.

Figure 3 lists the denoising of experimental image 2 and the comparison of residual images. Table 2 lists the equivalent visual numbers corresponding to the A, B, and C3 regions in experimental image 2. The margin of the three methods keeps the comparison of the exponent EPI and the number of iterations.

Figure 3

Experimental image 2 denoising image comparison

Comparison of ENL and EPI of experimental image 2 denoising effect

Noise pollution image Adaptive Lee filter AA Method of this article
Regional AENL 10.0479 167.0048 199.0837 204.641
Regional BENL 4.1646 10.1255 9.8474 9.2243
Regional CENL 4.1583 13.2139 14.4077 11.7949
EPI 1 0.5033 0.5989 0.7229
Number of iterations 7 500 500

From the comparison of the results in Table 2 and Figure 3, it can be seen that the Lee filter has a better denoising effect in the flat area after multiple iterations. Still, the blurring of the edges is serious [13]. The AA model can achieve a better denoising effect while maintaining the edge better when iterating 500 times. The method in this paper has the highest equivalent apparent in area A when iterated 500 times and slightly lower in areas B and C. But this method has the best effect on the edges and details of the image. From the quantitative comparison of the edge retention index EPI, we can see that the edge retention effect of the method in this paper is relatively good.

Conclusion

This paper proposes a fractional multiscale denoising model for SAR image denoising. This method studies the adaptive selection of model parameters and proposes an adaptive algorithm. The article compares with the classic adaptive Lee filter and AA model. Compared with the fixed-parameter gradient descent algorithm for solving the AA model, the calculation amount of the method in this paper has increased. However, the calculation amount is still of the same order. The simulation experiment shows that the fractional multiscale adaptive method proposed in this paper can better distinguish the image's texture area and non-textured area in the denoising process. In the non-textured area of the image, a better denoising effect can be achieved, and the “staircase effect” can be effectively suppressed. In the texture area of the image, this method can better maintain the texture information of the image. Therefore, the denoising method proposed in this paper effectively supports coherent speckle suppression in SAR images.

Figure 1

Experimental image to be processed
Experimental image to be processed

Figure 2

Experimental image 1 Comparison of denoising image effects
Experimental image 1 Comparison of denoising image effects

Figure 3

Experimental image 2 denoising image comparison
Experimental image 2 denoising image comparison

Comparison of ENL and EPI of experimental image 2 denoising effect

Noise pollution image Adaptive Lee filter AA Method of this article
Regional AENL 10.0479 167.0048 199.0837 204.641
Regional BENL 4.1646 10.1255 9.8474 9.2243
Regional CENL 4.1583 13.2139 14.4077 11.7949
EPI 1 0.5033 0.5989 0.7229
Number of iterations 7 500 500

PSNR comparison of experimental image 1 denoising image

Original image Adaptive Lee filter AA Method of this article
PSNR 19.0959 26.7535 27.7593 27.8881

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