Image denoising based on wavelet thresholding and Wiener filtering in the wavelet domain

The Journal of EngineeringVolume 2019, Issue 19 p. 6012-6015 IET International Radar Conference (IRC 2018)Open Access Image denoising based on wavelet thresholding and Wiener filtering in the wavelet domain Wen-quan Fan, Wen-quan Fan The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this authorWen-shu Xiao, Wen-shu Xiao The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this authorWen-shu Xiao, Corresponding Author Wen-shu Xiao 704498191@qq.com The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this author Wen-quan Fan, Wen-quan Fan The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this authorWen-shu Xiao, Wen-shu Xiao The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this authorWen-shu Xiao, Corresponding Author Wen-shu Xiao 704498191@qq.com The Overall Research and Development Department, First Sector, The Fourteenth Institute CETC (China Electronics Technology Group Corporation), No. 8 Guo Rui Road, Yuhuatai District, Jiangsu Province, Nanjing, People's Republic of ChinaSearch for more papers by this author First published: 05 July 2019 https://doi.org/10.1049/joe.2019.0194Citations: 5AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Wavelet transform has become a very important tool in the field of image denoising. A wavelet transform is a localised analysis of time (space) frequency. It uses a telescopic translation operation to gradually multi-scale refine the signal function and finally adapt to time frequency. One popular approach involves thresholding the wavelet coefficients by using the soft or hard threshold. Another method of image denoising is the Wiener filtering in the wavelet domain. In this study, Gaussian white noise has been added to two grey scale images and the two different denoising methods have been used. By comparing the performance of the two methods, it can be found that the Wiener filtering in the wavelet domain is more prowerful. 1 Introduction There will be various noises in the process of image acquisition and transmission. To keep the image at a high-quality level, we must make sure that the denoised image keeps most of important signal characteristics. Image denoising techniques based on wavelet theory has drawn more attention to scholars in recent years. This is because of several advantages of a wavelet transform: (i) great local time-frequency, (ii) great multi-scalability, and (iii) high multi-resolution. In the wavelet telescopic translation function space, wavelet image denoising is set to find the best approximation of the original image. So the wavelet transform is also called the ‘mathematical microscope’. Working in a wavelet domain is strongly preferred, because the discrete wavelet transform (DWT) can convert the signal energy into a smaller number of coefficients, which thus can have a higher signal-to-noise ratio (SNR). After noisy coefficients (the coefficients with low SNR) are removed, the image will be reconstructed by inverse DWT. The newly constructed image will have better quality because most of noise is removed or filtered. The wavelet method is more efficient to characterise signals compared with either the original domain or transforms with global basis elements (e.g. Fourier transform). Wavelet thresholding, as a signal-estimation technique, is an effective way to remove noise by killing the coefficients which are irrelevant relative to the threshold. It is usually applied for signal denoising in a wavelet transform. The efficiency of the method largely depends on the selection of the threshold parameter, and the choice of the threshold parameter largely determines the efficiency of the denoising. Different from a way to remove noise by killing wavelet coefficients, Wiener filtering is another image-denoising method that assumes that wavelet coefficients are conditionally independent of Gaussian random variables. So the Wiener filter works best for image filtering with white noise. In this report, we first analyse the definition of the two methods and propose an improved soft threshold in Section 2. According to theory, we used the MATLAB to simulate the process of image denoising. In Section 3, two grey scale images were analysed by comparing the wavelet thresholding methods to the Wiener filtering in the wavelet domain. In Section 4, we draw a conclusion that the Wiener filtering in the wavelet domain is more powerful [1]. 2 Principle of image denoising 2.1 Wavelet thresholding methods 2.1.1 Soft thresholding The denoising method of such an additive white Gaussian noise-affected image includes thresholding the wavelet coefficients. Soft threshold processing wavelet coefficients are almost minimax because the percentage of the worst-case risk to minimax risk is very small. When we use the Haar wavelet transform, the measures applying soft threshold techniques to achieve denoising are as follows: (i) Adopt the wavelet transform to the raw image (ii) Adopt soft thresholding techniques (Fig. 1 a) to the wavelet coefficients y (i, j); the soft thresholding formula is defined as (1) where are the wavelet coefficients which are deniosed and T is the general threshold which is defined as . where n is the count of the wavelet coefficients and σ is the standard deviation of the noise. (iii) Adopt inverse wavelet transform and obtain the results we expect [2]. Fig. 1Open in figure viewerPowerPoint Wavelet thresholding function (a) Soft thresholding, (b) Hard thresholding 2.1.2 Hard thresholding The denoising method mentioned in Section 2.1.1 (soft thresholding) could be performed to use a hard threshold to take the place of a soft threshold on the wavelet coefficients in step (ii). The hard threshold formula is as follows: (2) where T represents the threshold described in Section 2.1.1 (see Fig. 1 b). 2.1.3 Improved soft thresholding The soft threshold method not only makes the denoised signal smoother, but it also throws away some features of the image, whereas the hard threshold preserves the features of the image, but the image is not smooth enough. We use a shape coefficient to control the degree of attenuation, thereby making the soft threshold function in the image continuous and deviation better than the traditional soft threshold method [3]. 2.2 Wiener filtering in the wavelet domain For this algorithm, we make an assumption that wavelet coefficients are independent of Gaussian random variables. Simultaneously, noise is also set as a fixed independent zero-mean Gaussian variable. Wiener filtering is a commonly used algorithm for recovering degraded images. It is also the earliest and most well-known linear image restoration method. The design idea of this method is to multiply the input signal by the output after the response, and the mean squared error of the desired output is minimum. The specific denoising step includes the wavelet transform decomposition of the image, mean filtering of value of each high-frequency component in the wavelet decomposition according to the Wiener filtering technique, and reconstructing images. The coefficients of the image disturbed by a fixed independent zero-mean Gaussian noise are defined as (3) where indicate the coefficients of the undegraded image in the wavelet domain, indicate the coefficients of the disturbed image in the wavelet domain, and indicate the coefficients of the noise. We assume that the noise sequence are normal, independent random variables with the zero mean and variance σ 2. Owing to the decorrelation properties of the orthogonal wavelet transform, we make an assumption that the signal components are not correlated. If the noise and signal are independent from each other, we can obtain (4) (5) Without loss of generality, we make an assumption that the can be determined in a window centred at . This assumption can be defined as (6) (7) At last, the coefficients of the Wiener filtering can be defined as (8) Limiting the values of to only positive values, we perform this Wiener filtering process according to two assumptions: one is the noise variance can be known, and the other is the noise variance is estimated by applying the mean absolute deviation algorithm. The estimated noise formula of the mean absolute deviation algorithm is defined as (9) where (10) and wi indicates the wavelet coefficients [4]. 3 Results of image denoising In this section, we state a lot about simulation achievements worked on the hat and the loco images. We use the methods such as hard thresholding, soft thresholding and Wiener filtering in the wavelet domain to deal with these images added to white Gaussian noise. We choose two initial images, that is, the original hat image and the loco image. Fig. 2 shows the simulation results of the different denoising techniques stated in this article. Fig. 2 shows the raw images and noised images. Fig. 2Open in figure viewerPowerPoint Image with white Gaussian noise (a) Raw hat image, (b) Hat image with white Gaussian noise, (c) Raw loco image, (d) Loco image with white Gaussian noise 3.1 Results of wavelet thresholding We use the wavelet thresholding method to estimate the performance of denoising. This method includes hard thresholding and soft thresholding. Another improved soft thresholding method is proposed to enhance the denoising effect. Figs. 3 and 4 give the results of two denoised images. Fig. 3Open in figure viewerPowerPoint Denoised hat image using wavelet thresholding method (a) Hat image with white Gaussian noise, (b) Denoised hat image using the wavelet thresholding hard thresholding method, (c) Denoised hat image using wavelet thresholding soft thresholding method, (d) Denoised hat image using the improved wavelet thresholding soft thresholding method Fig. 4Open in figure viewerPowerPoint Denoised loco image using wavelet thresholding method (a) Loco image with white Gaussian noise, (b) Denoised loco image using the wavelet thresholding hard thresholding method, (c) Denoised loco image using the wavelet thresholding soft thresholding method, (d) Denoised loco image using the improved wavelet thresholding soft thresholding method From Fig. 3, we can see that after wavelet thresholding, no matter which thresholding (soft or hard), the performance of the noised image is better. Some of the white noise is removed. However, different thresholdings have different effects (Figs. 3b–d). We can find that the effect of hard thresholding is better than that of soft thresholding. So, for this situation, we use an improved soft thresholding to improve the performance, and the effect is obvious. For Fig. 4, we use the same method for another loco image. Just like the first image, the effect and regularity are similar. 3.2 Results of Wiener filtering in the wavelet domain Another method called Wiener filtering in the wavelet domain is used in this subsection. By changing the size of the Wavelet filter, we can obtain that different values have different results. It should be obviously seen that the background of the processed hat image (Fig. 5 b) seems smoother than the hat image with white Gaussian noise (Fig. 5 a). However, the Wiener filter in the wavelet domain has an advantage in removing the noise in the smooth area, but it performs poorly in certain aspects such as the edges. In Figs. 5 and 6, it can be observed that after Wiener filtering in the wavelet domain, images universally have black edges. It's noticeable that the case images are denoised by wavelet thresholding is similar with the above methods. Fig. 5Open in figure viewerPowerPoint Denoised hat image using Wiener filtering in the wavelet domain method (a) Hat image with white Gaussian noise, (b) Denoised hat image using the Wiener filtering in the wavelet domain, the size of which is 2 × 2, (c) Denoised hat image using Wiener filtering in the wavelet domain, the size of which is 3 × 3, (d) Denoised hat image using Wiener filtering in the wavelet domain, the size of which is 4 × 4 Fig. 6Open in figure viewerPowerPoint Denoised loco image using Wiener filtering in the wavelet domain method (a) Loco image with white Gaussian noise, (b) Denoised loco image using Wiener filtering in the wavelet domain, the size of which is 2 × 2, (c) Denoised loco image using Wiener filtering in the wavelet domain, the size of which is 3 × 3, (d) Denoised loco image using Wiener filtering in the wavelet domain, the size of which is 4 × 4 From the comparison of the above eight images, we can also conclude that with the different size of the wiener filter, different results appear. The larger the size is, the better the image is. 3.3 Comparison of the two denoising methods In this section, we compare Wiener filtering with the wavelet thresholding method, and obtain a conclusion that the Wiener filtering method in the wavelet domain (see Fig. 5 for the hat image and see Fig. 6 for the loco image) performs better than all wavelet thresholding methods (see Fig. 3 for the hat image and Fig. 4 for the loco image). More details are lost when using the thresholding methods, particularly for soft thresholds where the background of the image is not well denoised. When Wiener filtering is used as a threshold function, we find that it behaves better. This is because its shape function is smoother than the all other wavelet thresholding methods. The denoising advantage of wavelet thresholding is that high-order statistic is insensitive to Gaussian noise, which can eliminate the influence of covariance matrix Gaussian white noise and colored noise, so it can more accurately reflect the details of the original image while smoothing the noise. The disadvantage is that the denoising effect of the Wiener filter is good, but the edge of the image is blurred owing to the shrinking of the wavelet coefficient with a larger amplitude. Therefore, combined with the characteristics of the wavelet transform and high-order statistic, the triple-correlation coefficient of the wavelet function and signal correlation function is used to replace the wavelet coefficient to calculate the threshold, and then the image denoising effect appears better. 4 Conclusion In this report, we analysed the principle of two different denoising methods. Then we use MATLAB to simulate the two methods. We draw a conclusion from the comparison among results that the Wiener filtering method is better than the other thresholding methods that were mentioned in the previous article ‘Wavelet Soft Thresholding and Wavelet Hard Thresholding’. We hope to improve on the following aspects. One aspect is to improve performance along the edges because the methods we used did not perform well along the edges. Another aspect is to develop better optimality standards. In the field of signal analysis, a wavelet transform can be used not only for noise removal, but also for boundary processing and filtering, signal-to-noise separation, fractal index and multi-scale edge detection. It can also be extended to computer vision, computer graphics, curve design, the remote universe and biomedical research fields. To use the wavelet transform and combine other algorithms for the detection of the edges of image noise separation according to the real situation establish new target and direction for research. From the perspective of image processing, the wavelet transform has the following advantages: Wavelet decomposition can cover the entire frequency domain (providing a mathematically complete description) Wavelet transform can greatly reduce or remove the correlation between the extracted different features by selecting appropriate filters. The wavelet transform has a ‘zoom’ feature, a high-frequency resolution, and low time resolution (wide analysis window), and it is available in a low-frequency band. There is a fast algorithm on the implementation of the wavelet transform (Mallat wavelet decomposition algorithm) [5]. 5 References 1Kazubek M.: ‘Wavelet domain image denoising by thresholding and Wiener filtering’, IEEE Signal Proc. Lett., 2013, 10, (11), pp. 324 – 326 2Xiangpei Q.: ‘ MATLAB image processing and interface programming’ ( Publishing House of Electronics Industry, Beijing, China, 2009) 3Zhong S., Cherkassky V: ‘Image denoising using wavelet thresholding and model selection zhong’. Proc. 2000 Int. Conf. On Image Processing 2000, Vancouver, Canada, 10–13 September 2000, vol. 3, pp. 262 – 265 4Donoho D.L: ‘De-noising by soft-thresholding’, IEEE Trans. Inf. Theory, 1995, 41, (3), pp. 613 – 627 5Zhang H., Nosratinia A.: ‘Image denoising via wavelet-domain spatially adaptive FIR Wiener filtering’. Proc. 2000 IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, 2000. ICASSP ‘00, Istanbul, Turkey, 5–9 June 2000, vol. 6, pp. 2179 – 2182 Citing Literature Volume2019, Issue19October 2019Pages 6012-6015 FiguresReferencesRelatedInformation