Hybrid Noise Eliminating Algorithm for Radar Target Images Based on the Time-Frequency Domain

Abstract

The radar target imaging effect directly affects the resolution of the radar target, which affects the commander’s decision. However, the hybrid noise composed of speckle and Gaussian noise is one of the main affecting factors. The existing methods for image denoising are hard to eliminate the hybrid noise in radar images. Hence, this paper proposes a new hybrid noise elimination algorithm for the radar target image. Based on the strong correlation between wavelet coefficients, this algorithm first uses the wavelet coefficient correlation denoising algorithm (WCCDA) to filter the high-frequency information and high-frequency part of low-frequency information for different directions of the three channels of the image. Then, an improved adaptive median filtering algorithm (IAMF) is proposed to perform fine-grained filtering on each re-constructed channel. Finally, the radar target image is reconstructed. The results show that the proposed algorithm outperforms the comparison approaches in the peak signal-to-noise ratio (PSNR) and mean-square error (MSE) indexes with better denoising effects.

1 Introduction

With the rapid development of radar technology, the application fields of radar are becoming more wide, especially in the military. Hence, the requirement of the radar performance is gradually increased, and the effect of noise elimination for radar target images is one of the crucial indicators for testing the radar performance. Radar target images are mainly affected by speckle noise caused by the coherence of scattering phenomena and Gaussian noise caused by complex environments. The radar imaging effect may directly affect the resolution of radar target recognition, which will delay the time for commanders to make effective strategies, Hence, an effective method for radar target image processing is necessary.

The difficulty in image denoising is that noise, edges, and textures all belong to high-frequency components, and it is difficult to distinguish them during the denoising process. The denoised image inevitably loses some edge and detailed information. Hence, this is also a vital issue in the image processing field today. The image denoising technology is mainly divided into denoising methods based on spatial domain and denoising algorithms based on frequency domain [1].

Spatial filters primarily declare noise occupying the spectrum higher regions by low-pass filtering groups of pixels. Hence, many methods filter out noise information by processing images with high frequency in the spatial domain. However, these methods can eliminate noise to a certain extent but may cause blur, loss of edge, and detailed information about the image. The frequency domain methods convert the image into a specific transformation domain and decompose the signal. Then, the suitable filter is designed for noise elimination based on the different statistical characteristics of the natural features of the image [2]. The frequency domain methods mainly include the method based on Fast Fourier Transformation (FFT) [3], Discrete Cosine Transform (DCT) [4], and Wavelet Transform (WT) [5]. Since WT can perform time-frequency analysis simultaneously and quickly discover the features of data mutation points, the algorithms based on WT are also one of the vital research contents in the image and signal processing field.

However, the WT only has information in the horizontal, vertical, and diagonal directions because it lacks directionality, so it cannot optimally represent two-dimensional images containing singular lines or surfaces. As a result, WT produces distortion and cannot reconstruct the original image well. In order to overcome these drawbacks, this paper proposes a hybrid noise elimination algorithm based on improved adaptive median filtering and wavelet coefficient correlation algorithm.

The proposed algorithm first uses the improved Wavelet Coefficient Correlation Denoising Algorithm (WCCDA) to perform wavelet transformation on the three channels R, G, and B of the image. Then, the high-frequency information and the high-frequency part of the low-frequency information are extracted in the horizontal, vertical and diagonal directions for each channel. After that, coarse-grained filtering on all high-frequency information is performed. To solve the color and line/surface distortion problems caused by image reconstruction, an improved adaptive median filtering algorithm (IAMF) is proposed to perform fine-grained filtering on each reconstructed channel and calibrate the reconstructed image. Finally, the original radar target image is reconstructed.

2 Wavelet Coefficient Correlation Denoising Algorithm

2.1 Wavelet Decomposition and Reconstruction of Radar Target Images

The two-dimensional discrete wavelet decomposition of digital images $f(x,y)$ can be expressed as the following formulas (1)~(5) [6]:

$$ cA_{0} = f(m,n) $$

(1)

$$ cA_{j + 1} (m,n) = \sum\limits_{k} {\sum\limits_{l} {h_{k - 2m} \times h_{l - 2n} \times cA_{j} (k,l)} } $$

(2)

$$ cH_{j + 1} (m,n) = \sum\limits_{k} {\sum\limits_{l} {h_{k - 2m} \times g_{l - 2n} \times cA_{j} (k,l)} } $$

(3)

$$ cV_{j + 1} (m,n) = \sum\limits_{k} {\sum\limits_{l} {g_{k - 2m} \times h_{l - 2n} \times cA_{j} (k,l)} } $$

(4)

$$ cD_{j + 1} (m,n) = \sum\limits_{k} {\sum\limits_{l} {g_{k - 2m} \times g_{l - 2n} \times cA_{j} (k,l)} } $$

(5)

where $j$ is the wavelet decomposition scale, $\left\{ {h_{k} } \right\}$ and $\left\{ {g_{k} } \right\}$ represent low-pass and high-pass filters, the $j$-scale layer image $cA_{j}$ is decomposed by one layer of wavelets to: low frequency coefficient $cA_{j + 1}$, horizontal high frequency coefficient $cH_{j + 1}$, vertical high frequency coefficients $cV_{j + 1}$ and diagonal high frequency coefficients $cD_{j + 1}$.

After that, the system can process different information according to actual requirements. Finally, the image is reconstructed based on the processed high-frequency information in each direction and low-frequency information to reconstruct the original image. The reconstruction process is as follows:

$$ cA_{j} (m,n){ = }A_{j + 1} + H_{j + 1} + V_{j + 1} + D_{j + 1} $$

(6)

where $A_{j + 1}$, $H_{j + 1}$, $V_{j + 1}$ and $D_{j + 1}$ are sub-images of j scale images reconstructed by the low frequency coefficients and high frequency coefficients in three directions.

2.2 Radar Target Image Denoising Based on Wavelet Coefficient Correlation

There is a strong correlation between the upper and lower layers of the wavelet coefficients but the noise has no such correlation. The WCCDA is to compare the normalized correlation coefficients at each location for each layer and determine whether each data is a pixel or a noise point [7, 8]. The specific process can be expressed through Eqs. (7) and (8):

$$ CW_{j,k} = W_{j,k} W_{j + 1,k} $$

(7)

$$ \widetilde{{W_{j,k} }} = CW_{j,k} \sqrt {\frac{{PW_{j} }}{{PCW_{j} }}} $$

(8)

where $W_{j,k}$ is the wavelet coefficient of the high-frequency, $CW_{j,k}$ is the correlation coefficient at point $k$ of scale$j$, $\widetilde{{W_{j,k} }}$ is the normalized wavelet correlation coefficient, $PW_{j} = \sum\limits_{k} {W_{j,k}^{2} }$ represents the energy of wavelet coefficients of scale$j$, $PCW_{j} = \sum\limits_{k} {CW_{j,k}^{2} }$ represents the correlation coefficients energy of scale$j$.

Then, calculate the wavelet coefficient of the high-frequency $W_{j,k}$ for each layer first and compare $W_{j,k}$ and $\widetilde{{W_{j,k} }}$. If $\widetilde{{W_{j,k} }} \ge W_{j,k}$, the system considers that the point is the real pixel, takes $\widetilde{{W_{j,k} }} = W_{j,k}$, and sets $W_{j,k} = 0$. If $\widetilde{{W_{j,k} }} < W_{j,k}$, the system thinks that the pixel is controlled by noise, then leaving $W_{j,k}$, set $\widetilde{{W_{j,k} }} = 0$. Then, the system repeats the above process and recalculates $\widetilde{{W_{j,k} }}$ on each scale. Finally, the real pixel points are kept in $\widetilde{{W_{j,k} }}$, and the noise points are kept in $W_{j,k}$.

Normally, for single noise, most of the noise information is concentrated in the high-frequency part of the signal. However, when there has hybrid noises in the image, the low-frequency information is likely to contain residual large-amplitude noise. On the other hand, low-frequency and high-frequency information have different features. If they are processed together, their respective features are destroyed, thereby reducing the image denoising efficiency, increasing errors. Hence, the denoising process for low-frequency information is necessary. The proposed algorithm improves the traditional WCCDA, not only denoising the high-frequency information but also denoising the high-frequency part of the low-frequency information to achieve a more comprehensive noise elimination.

3 Improved Adaptive Median Filtering Algorithm

The median filter algorithm is widely used in the image processing fields. It sorts the pixels in the neighborhood by grayscale and selects the middle value of the group as the output pixel value to remove noise. The two-dimensional median filtering algorithm [9] is mainly used, as shown in Eq. (9):

$$ P(i,j) = {\text{median(}}s(m{))} $$

(9)

where $m$ is the number of pixels, $s$ is a gray sequence which is already sorted, $i$ and $j$ are the horizontal and vertical coordinates of the pixels respectively. The output pixels are determined by the image median, which makes the median filter less sensitive to the limit pixel value than the mean. Hence, it not only eliminates the isolated noise points but also gets better image clarity.

However, the window type and size directly affect the algorithm’s performance. In theory, the larger the window size, the image processing effect is better, but if the window size is unrestricted to increase, it may damage the details of the image. Hence, one of the main problems is the selection of the window. In addition, the hybrid noise is with a large amplitude, and the image edge burr and distortion may occur. It is because the median point may take the noise point as the real pixel when the noise amplitude is large, which leads to the algorithm being invalid.

Many researchers have proposed the improved schemes. For instance, González et al. [10] propose a dynamic weighted adaptive median filtering algorithm for impulse noise, which can adaptively adjust the window size according to the noise amplitude. However these methods cannot restore the color and edge information well. Hence, this paper proposes an improved adaptive median filtering algorithm (IAMF). By selecting an appropriate filter window for each pixel point, the system can accurately determine whether the point is a noise point. The specific steps are as follows.

3.1 Expand the Pixel Matrix of the Original Image Pixel Matrix

Fig. 1.

Pixel matrix expansion process of the image

To ensure the data in the filter window is not empty and valid, the original image pixel matrix needed to be expanded. The system sets a matrix expansion radius $N$ for the algorithm. The overall performance of the system can be optimal when the matrix expansion radius $N$ does not exceed 10 after experimental verification. The system assumes the original image matrix is $X = m \times n$. Then, the original image pixel matrix is expanded from the upper, right, lower, and left directions with the expansion amplitudes of $N$, $N + 1$, $N + 1$ and $N$. The expanded image matrix dimension reaches $X_{1} = \left( {m + 2*N + 1,\,\,n + 2{*}N + 1} \right)$, where $m$ and $n$ represent the number of rows and columns of the original image pixel matrix.

The specific process is shown as follows: Assumes $m = n = 5$, $N = {2}$, the expanded pixel matrix is $X_{{1}} = {10} \times 10$ as shown in Fig. 1. The data enclosed by dashed circles represents the expanded part. The expanded pixel data are represented by different markers. After 4 expansions, a new image pixel data matrix is obtained.

3.2 Adaptive Filter Window Size Computation

After the pixel matrix is expanded, the system selects an appropriate filter window size for each point in the original pixel matrix. For instance, the process of the filter window size for point (7, 7) is shown in Fig. 2.

Fig. 2.

Size calculation of filtering window

The system first sets filter radius $r = 1$ and calculates the neighborhood $I$ of each pixel in the original image along the row direction of the matrix, as shown in Eq. (10).

$$ I{ = }X_{1} (i - r:i + r,j - r:j + r) $$

(10)

where $i$ and $j$ represent the coordinates of the pixel point. Then, the pixels in the neighborhood $I$ are re-sorted to abtain $I{\prime}$, makes the pixels in each column are arranged in ascending order. Finally, the maximum $I_{\max }{\prime}$ and minimum $I_{\min }{\prime}$ are found, and the central pixel is used as the median point $I_{{{\text{med}}}}{\prime}$. If the center pixel value is 0, set $I_{{{\text{med}}}}{\prime} { = 1}$, and then determine whether $I_{{{\text{med}}}}{\prime}$ belongs to the range of $(I_{\min }{\prime} ,I_{\max }{\prime} )$. If the above conditions are satisfied, the neighborhood size is considered to meet the actual filtering requirements. Otherwise, the system continues to expand the filter radius, let $r = r + 1$ ($r \le N + 1$), and repeats the above calculation process until a suitable filter window size is found. In addition, the system can adjust the size of the matrix expansion radius $N$ according to the features of the noise.

3.3 Noise Point Judgement

First, determine whether the pixel in the original image is between the maximum and minimum value points within the filter window range. If this condition is met, the point is considered to be a real pixel and is retained. On the contrary, it means that the pixel is controlled by noise and needs to be replaced by the median point $I_{{{\text{med}}}}{\prime}$ within the filter window. Finally, the system repeats the above process to complete the image filtering.

4 Radar Target Image Denoising Algorithm Based on WCCDA and IAMF Algorithm

For the hybrid noise, the denoising methods for the single noise is not ideal and may easily cause image edges missing and color distortion. Hence, this paper combines the WCCDA and the IAMF algorithm to denoise the image. This algorithm can eliminate the noise while retaining the edges, details, and color information and can effectively prevent image distortion. The specific process is shown as Fig. 3.

Fig. 3.

Flow of the proposed method

The proposed algorithm is mainly divided into the following 5 steps:

• 1) Decompose the radar target image into R, G, and B channels, then extract the high-frequency information of each channel. In this step, the system extracts the high-frequency information in the horizontal, vertical, and diagonal directions and the high-frequency part of the low-frequency information.

• 2) Perform WCCDA on the high-frequency part of the low-frequency information of each channel, as well as the high-frequency information of each channel in three directions. This way can better retain the data features of different parts [11].

• 3) Use the new low-frequency and high-frequency information to reconstruct the three channels of the image to complete the first coarse-grained filtering.

• 4) The IAMF algorithm is used to perform secondary fine-grained filtering on the three channels of the reconstructed image.

• 5) Integrate the denoised three channels to reconstruct the radar target image.

5 Experiments and Evaluation

5.1 Experimental Parameter Settings

To verify the effectiveness of the proposed algorithm, this paper adds the hybrid noise composed of Gaussian white noise with variances $\sigma_{1} = 0.01$, $\sigma_{2} = 0.02$, and $\sigma_{3} = 0.03$ and multiplicative speckle noise with noise densities $d_{1} = 10\%$, $d_{2} = 20\%$, and $d_{3} = 30\%$ to the radar target image, and conducts multiple experiments.

5.2 Wavelet Function Selection

There is another parameter that directly affects the system performance, which is the choice of wavelet function. Since the WCCDA algorithm mainly uses the correlation between wavelet coefficients of image data to eliminate noise, a suitable wavelet function can optimize the algorithm’s performance. This experiment uses different wavelet functions in the WCCDA algorithm to process images, then calculates the correlation coefficients between the high-frequency wavelet coefficients before and after denoising. The larger the correlation coefficient value, the denoising effect is better [12]. The calculation process is shown in Eq. (11):

$$ CW_{{{\text{j}},k}}{\prime} = W_{j,k} \cdot W_{j,k}{\prime} $$

(11)

where $CW_{{{\text{j}},k}}{\prime}$ is the correlation coefficient at point $k$ of scale $j$, $W_{j,k}$ and $W_{j,k}{\prime}$ are the high-frequency wavelet coefficient before and after denoising, respectively.

Specifically, a hybrid noise composed by Gaussian white noise with variance $\sigma_{2} = 0.02$ and multiplicative speckle noise with density $d_{2} = 20\%$ is added to the image. Then, the three wavelet functions ‘haar’, ‘symlet’, and ‘dbn’ are selected for testing. After experiments verification, it is found that the three wavelet functions can obtain the maximum correlation coefficient when the wavelet scale is 8. Hence, the scale of the wavelet function in this paper is set to 8. The experiment results are shown in Table 1.

Table 1. Comparison of correlation coefficients under different wavelet functions

As Table 1 shown, the correlation coefficient value reaches the highest when using the ‘haar’ wavelet. It also proves the algorithm gets the best denoising effect when using the ‘haar’ wavelet function. Hence, the system uses the ‘haar’ wavelet function in the following experiments.

5.3 Hybrid Noise Eliminating

To verify the denoising effectiveness of the proposed algorithm (M4), the denoising effect for hybrid noise should be evaluated. This experiment adds Gaussian white noise and multiplicative speckle noise of different intensities to the radar target image and compares the denoising performance with different methods such as M1 [5], M2 [13], and M3 [14] in related fields.

Performance of the Algorithm When the Intensity of Gaussian Noise and Speckle Noise are $\sigma_{1} = 0.01$ and $d_{1} = 10\%$. After using different algorithms to denoise radar target images, the details, edges, and color recovery and the overall reconstruction effect are demonstrated, as shown in Fig. 4.

Fig. 4.

Results of each method when the hybrid noise intensity is $\sigma_{1} = 0.01$ and $d_{1} = 10\%$

It can be seen that under the low-intensity condition of hybrid noise, all methods can effectively eliminate noise. M1 and M2 can eliminate noise and better retain the edge information. However, the image color has a distortion problem, and detailed information is seriously missing. In addition, the target orientation information and part of the image marks have been blurred. Though M3 is better than M1 and M2 in retaining image edges and detail information, the image color is completely distorted after denoising. The proposed method can better retain the image edges, details, and color information while eliminating noise.

Performance of the Algorithm When the Intensity of Gaussian Noise and Speckle Noise are $\sigma_{2} = 0.02$ and $d_{2} = 20\%$. To better verify the above analysis and the proposed algorithm, this experiment increased the noise intensity to $\sigma_{2} = 0.02$ and $d_{2} = 20\%$, and the results are shown in Fig. 5.

It can be seen that as the noise increases, M1 and M2 algorithms can still better eliminate noise and retain the edge information. However, the image blur is intensified, and the detailed information is seriously lost. Especially for M1, the resolution are seriously reduced, and the target with weak signal strength already cannot be identified from the image. Though the M3 algorithm retains the details and edge information well, it cannot eliminate noise absolutely, and the color distortion is serious. The proposed algorithm can retain the image edges, details, and color information better, which restores all the features of the original image with high clarity. In summary, compared with the other methods, the performance of the proposed method is better than other methods.

Fig. 5.

Results of each method when the hybrid noise intensity is $\sigma_{{2}} { = }0.0{2}$ and $d_{{2}} { = }20\%$

Performance of the Algorithm When the Intensity of Gaussian Noise and Speckle Noise are $\sigma_{3} = 0.03$ and $d_{3} = 30\%$. In order to test the maximum denoising performance of the proposed algorithm, this experiment continues to increase the intensity of the hybrid noise to $\sigma_{3} = 0.03$ and $d_{3} = 30\%$. The experiment results are shown in Fig. 6.

Fig. 6.

Results of each method when the hybrid noise intensity is $\sigma_{3} = 0.03$ and $d_{3} = 30\%$

As the noise further increases, the performance of each algorithm declines. The image processed by the M1 and M2 algorithms has a high degree of blur, while the detailed and edge information in the image cannot be distinguished. In addition to the above issues, the image processed by M3 has more serious color distortion problems, and the algorithm has failed. Though the denoising performance of the proposed method begins to decline, compared with the other three algorithms, the denoising effect is the best, in which the edge and detailed information of the image are well preserved and the clarity is high.

Evaluation of Denoising Performance for Each Algorithm.

The above experiments show the results and analysis of each algorithm’s processing of hybrid noise images with different strengths from an intuitive visual perspective. To more accurately display the denoising effect of each algorithm and verify the analysis results of the above experiments. This paper uses peak signal-to-noise ratio (PSNR) and mean square error (MSE) [15] as the measurement criteria for objective evaluation. The equations for MSE and PSNR are as follows:

$$ MSE = \frac{1}{M \times N}\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{N} {[f(i,j) - g(i,j)]^{2} } } $$

(12)

$$ PSNR = 10\log_{10} (\frac{{255^{2} }}{MSE}) $$

(13)

where, $f(i,j)$ is the reconstructed image after denoising, $g(i,j)$ is the original image without noise, $M$ and $N$ are the number of rows and columns of the image matrix. The larger the PSNR, the smaller the MSE, indicating that the image after denoising is closer to the original image.

Based on the above three experiments, the intensity of Gaussian noise is varied between $\sigma = 0.01\sim 0.03$, and the speckle noise is kept at $d_{{2}} = 20\%$ in this experiment. Then, the noise image is processed and reconstructed by each algorithm and compared with the original image to obtain the relevant PSNR and MSE values. The specific experimental results are shown in Table 2.

Table 2. The PSNR and MSE values after image processing under different noise

It can be seen from Table 2 that the proposed method in this paper gets the largest PSNR and the smallest MSE value compared to the other three algorithms, which also proves the consistency with the results and analysis of the above experiments. To more intuitively analyze the varies of PSNR and MSE with the intensity of hybrid noise, this paper gives the curves of PSNR and MSE values changing with noise intensity. As the intensity of Gaussian noise changes, while keeping the intensity of speckle noise stayed, the changes in PSNR and MSE can be more clearly observed. The results are shown in Figs. 7a and 7b.

Figure 7 can more intuitively show the varying of PSNR and MSE values for the four methods after processing noisy images. Even when the noise intensity reaches a, the method proposed in this article still maintains the maximum PSNR and minimum MSE value. It proves that the noise image processed by the proposed algorithm is closest to the original image and has high reliability. In addition, the performance of each algorithm decreases as the noise intensity increases, which is consistent with the above experimental results and analysis.

Fig. 7.

Curves of PSNR and MSE values with the various noise intensity

6 Conclusion

This paper proposes a new denoising algorithm for the radar target image based on wavelet transform to solve the problems of image blur, loss of edge and detail information, and image color distortion in the hybrid noise processing of radar target images. The proposed algorithm first uses the WCCDA algorithm to perform coarse filtering on the noisy image in different directions of R, G, and B channels to eliminate most of the noise in the image. Then, the IAMF algorithm is proposed to perform secondary fine-grained filtering on each channel after denoising. The IAMF algorithm expands the original target image matrix and adaptively selects an appropriate filter window size for each pixel in the image. It can quickly and accurately identify whether the pixel is a real pixel point and eliminate the remaining noise in the image to the maximum extent, which can better retain the image edges, details, and color information. Many simulation experiments prove that the proposed method in this paper has better denoising effect, higher image information recovery, and stronger stability than other related algorithms, which is suitable for radar target images denoising.