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

- fractional differential equation
- Electronic information model
- Image denoising

#### MSC 2010

- 34A08

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.

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.

Among them

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 ^{a}

Where

Where
^{a}_{j}, ^{−sj}

Where 0 ≤ _{j}^{−sj}

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

Where _{x,y}(

The sufficient conditions for the establishment of this equation are

Where

Where σ^{2} is the noise variance. Because (^{2} is the average value of _{u}_{u}^{2} < ^{2} of the image. In the texture area, _{u}^{2}. _{u}_{AA}_{AA}_{AA}_{AA}_{uAA}_{AA}_{AA}^{2}. Where var(·) is the variance. We get an estimate of

According to formula (6) (
_{uAA}_{AA}^{2} in the non-textured area, the obtained λ value is smaller, and the denoising ability is enhanced [6]. In the texture area, due to _{uAA}_{AA}^{2}, λ is larger, and the details are better maintained.

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 _{uAA}_{AA}^{2} is established in the textured area and _{uAA}_{AA}^{2} is established in the non-textured area, α is taken as

Where 1 ≤ _{1} ≤ _{2} ≤ 2. Considering denoising, suppressing “staircase effect,” and texture preservation, we recommend _{1} = 1.2, _{2} = 1.5.

_{j}

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 _{j}

Where _{2j} ^{j} scale. In this article, we first perform _{j}_{j}_{j}_{j}

An image of size _{i,j} = 0. Then the discretization operators

But

Where
^{2} It is a small constant, and the main guarantee is that the denominator is not zero. Because the generalized binomial coefficients in the 1 ≤

In the iterative process of equation (16), we update the parameters _{j}^{(0)} = _{j}_{AA} = ^{(M)}, _{AA} = (_{AA}_{AA} is obtained. In the article, _{AA} is zero-averaged and then convolved with the normalized Gauss template to obtain _{uAA}

Step 2: Re-set the initial value ^{(0)} = ^{(0)} = 0,
^{(n)}, ^{(n)},

Step 3: We use an orthogonal wavelet to decompose ^{(n)} · ^{a}^{(n)}) into L times wavelet. Use each scale factor to perform single-branch reconstruction to get [^{a}^{(n)})]_{j}, ^{(n+1)} according to formula (6) and formula (10).

Step 4: We use orthogonal wavelets to perform wavelet decomposition on ^{(n+1)}(^{(n)} / (^{(n)} + ^{2}), according to formula (12) and formula (13) update to obtain
^{(n+1)}(^{(n)} / (^{(n)} + ^{2})]_{j},

Step 5: Calculate
^{(n + 1)} meets the given iteration termination condition. Otherwise, let

For an image of size ^{2}). 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 ^{2}). When the normalized Gauss template window is relatively small, the amount of calculation added to the convolution operation is ^{2}). The overall order of these two parts is still ^{2}).

(2) The amount of calculation caused by the adaptation of parameter _{j}_{j}^{2}), the total increase in this part of calculation is still ^{2}).

(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 ^{2}) at this time. When α is a non-integer, the calculation amount for calculating the fractional difference is ^{2}), but the calculation amount is still ^{2}). Although the calculation amount of the algorithm in this paper is more ^{2}) 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.

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 Δ^{−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.

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 ^{(n+1)} < ^{(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.

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:

Where _{HH} is the median amplitude value of the wavelet coefficients of the highest frequency HH subband of the image wavelet decomposition. In this paper _{HH} 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 _{s}^{2}/var(_{s}

Where _{s}_{s}_{s}

Among them _{s}_{0} is the pixel value of the original image. In the definition of _{s}

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

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.

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.

#### 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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