Vector extension of quaternion wavelet transform and its application to colour image denoising

IET Signal ProcessingVolume 13, Issue 2 p. 133-140 Research ArticleFree Access Vector extension of quaternion wavelet transform and its application to colour image denoising Shan Gai, Corresponding Author Shan Gai gaishan@nchu.edu.cn School of Information Engineering, Nanchang Hangkong University, Nanchang, People's Republic of ChinaSearch for more papers by this authorZhongyun Bao, Zhongyun Bao School of Information Engineering, Nanchang Hangkong University, Nanchang, People's Republic of ChinaSearch for more papers by this authorKaige Zhang, Kaige Zhang Department of Computer Science, Utah State University, Logan, USASearch for more papers by this author Shan Gai, Corresponding Author Shan Gai gaishan@nchu.edu.cn School of Information Engineering, Nanchang Hangkong University, Nanchang, People's Republic of ChinaSearch for more papers by this authorZhongyun Bao, Zhongyun Bao School of Information Engineering, Nanchang Hangkong University, Nanchang, People's Republic of ChinaSearch for more papers by this authorKaige Zhang, Kaige Zhang Department of Computer Science, Utah State University, Logan, USASearch for more papers by this author First published: 01 April 2019 https://doi.org/10.1049/iet-spr.2018.5127Citations: 4AboutSectionsPDF 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 In this study, the authors study and give a new framework for colour image representation based on colour quaternion wavelet transform (CQWT). The new colour quaternion filter bank is constructed by using radon transform. Starting from link with structure tensors, the authors propose a new multi-scale tool for vector-valued signals which can provide efficient analysis of local features by using the concepts of amplitude, phase, and orientation. To demonstrate the properties of CQWT, new colour image denoising algorithm is proposed by using CQWT and bivariate shrinkage function. The performance of the proposed algorithm is experimentally verified on a variety of noise levels. Experimental results show that the proposed algorithm achieves superior performance both in visual quality and objective peak-signal-to-noise ratio, mean square error, and structure similarity values, compared with other state-of-the-art denoising algorithms. 1 Introduction With the rapid developments of mobile internet and network, the colour images and videos have been becoming the main carrier of multimedia. Colour images have become part of our everyday lives. Such images are applied in numerous fields, from the casual documentation of events to medical applications. The images are unavoidably corrupted by noise during acquisition and transmission. Such degradations can have a significant impact on the colour image quality which affects human understanding and processing by using computer. Thus, the denoising and enhancement of colour images have become the key step in many practical applications. In recent years, the denoising algorithms based on multi-scale decomposition have been widely discussed. The main process is as follows: calculate coefficients of noisy signal via some multi-resolution tool; modify the coefficients by some rules; the clean signal is obtained by inverse multi-resolution tool using modified coefficients. Sendur proposed new denoising algorithm in discrete wavelet transform domain (DWT) [1] which could capture dependencies between the coefficients and their parents in detail. Shi et al. [2] employed directional lifting transform for image denoising. Do and Vetterli [3] proposed contourlet transform which could describe the directional information of natural images (CT). In [4], a new construction was proposed based on non-subsampled pyramid structure and directional filter banks. Kingsbury proposed complex wavelet transform (CWT) [5] which was composed of two separate cascading orthogonal filter banks. Portilla et al. [6] proposed an image denoising algorithm based on Bayesian least squares and Gaussian scale mixture model (BLS-GSM). Dabov et al. [7] proposed a new denoising framework based on patch redundancy (BM3D). The existing denosing methods treat RGB channels as three independent grey images and process them in a monochrome way which can lose inter-relationship among the multiple channels. Fortunately, quaternion algebra provides an elegant mathematical tool to deal with vector signals. Quaternion algebra was first proposed by Hamilton [8], which is very useful for computer vision and pattern recognition. The matrix Kalman Filter was proposed based on quaternion algebra which could preserve the natural information of the matrix plant equations [9]. The quaternion image processing was introduced to design the framework of visual quality matrix which could evaluate the quality of degraded image better [10]. The concept of quaternion random variable was introduced to address the independent component analysis of quaternion random vectors [11]. The algorithm for computing the singular value decomposition of quaternion was proposed to solve the problem of diagonalisation [12]. Wang and Li [13] established new representation linear quaternion matrix. To achieve a more compact spatially spectral analysis, some researchers have investigated colour quaternion wavelet transform (CQWT) [14-16], which is shift invariant and is made of one magnitude and three phases. Two phases denote local image shifts and the third phase captures the image texture information. The quaternion wavelet matrix is widely employed in many applications such as image registration [17], edge detection [18, 19], image segmentation [20], and classification [21, 22]. Recently, quaternion-based methods have extended to process multi-channel information in parallel, such as colour image registration [23], colour image analysis [24], colour image denoising [25-27], colour image super-resolution [28], and recognition [29]. However, there are no works on designing general framework of multi-resolution analysis tool for colour images. The contributions of this paper are summarised as follows: (i) we have designed new colour quaternion filter bank in radon transform domain, and given the new framework of the colour quaternion wavelet transform; (ii) a novel sparse representation model based on colour quaternion wavelet transform for colour images is proposed in this paper; (iii) we proposed a new colour image denoising algorithm by using the new sparse representation model and bivariate shrinkage function. The rest of the paper is organised as follows. Section 2 introduces the basic concepts of quaternion wavelet transform. Section 3 designs a new quaternion filter-bank and construct the framework of colour quaternion wavelet transform. The new colour image denoising algorithm based colour quaternion wavelet is proposed in Section 4. In Section 5, we show some experimental results that demonstrate the effectiveness of this algorithm. Finally, the conclusion is given in Section 6. 2 Quaternion wavelet transform In this section, we will review the basic concepts of quaternion wavelet transform. As it was shown in [14], the quaternion wavelet transform of 2D function can be represented as (1)where the superscript q denotes quaternion signal, denotes approximation coefficient matrix, and are detailed coefficient matrices at decomposition level j for directions which are horizontal, vertical, and diagonal. Let denote wavelet functions and denote scaling functions, which are defined as follows: (2) (3)According to (2) and (3), we can obtain the each approximation function and the detail components function (4) (5)The each approximation coefficient is the inner product of f and which is and detail coefficient is the inner product of f and which is defined as . The procedure of QWT is applied scale and wavelet filters to the rows and columns of the signal by subsample factor of two. Then the real part of the approximation at decomposition level j is taken as input at the next level, and the process repeats through all the decomposition levels. 3 Colour quaternion wavelet transform 3.1 Quaternion filter design Let denote 2D function, and its radon transform at direction is defined as follows: (6)According to the Fourier slice theorem [30], the 1D Fourier in radon domain is equivalent to 2D Fourier transform in the space domain. Then it has (7)where denote the Fourier transform of f. This property is applied in image reconstruction from projection methods. The isotropic 2D filtering is performed as (8)where denotes 1D symmetric filter, f denotes 2D signal, and denotes 1D convolution. Then, the new framework of quaternion filter is proposed in this paper. It is well known that Fourier sample will be extracted from the best possible slice . Hence, we would downsample the radon projections at each scale. The perfect reconstruction of 2D signal is defined as (9)where and are quaternion high-pass filter and quaternion low-pass filter, respectively, which are symmetric in the space domain. is the reconstruction signal of . Then, they are defined in Fourier domain as (10) (11)Inspired by Soulard and Carre [31], we applied Fourier implementation for high-order filters design. The Hilbert transform is approximated by and dyadic dilation rule . At each scale, the sub-bands are normalised by the energy of equivalent filters. Then the final framework of the 2D quaternion filter is illustrated in Fig. 1. Fig. 1Open in figure viewerPowerPoint Framework of 2D quaternion filter for one-level decomposition The quaternion wavelet can describe the 2D analytic signal, which is the extension of wavelet transform and CWT. The analytic signal at different scale is obtained by embedding Hilbert transform directly into the quaternion wavelet transform framework. Hence, quaternion wavelet is able to make multi-resolution analysis for vector-valued signal such as colour image signal. Compared with complex wavelet, quaternion wavelet filter has no direct current response and avoid the forward bias effectively. The complex wavelet has translation invariant. Hence, the quaternion wavelet is translation invariant due to the fact that it can be viewed as tensor product of complex wavelet. 3.2 Colour quaternion representation Suppose is multidimensional signal by employing the quaternion algebra. The colour signal is expressed as sum of sine waves by using Fourier atoms [31], which applied Fourier analysis on each channel, respectively. Then, we can obtain (12)where the phase , and the amplitude matrix is defined as (13)In [32], it can be seen that always draws an ellipse in the colour space. The maximum and minimum radius of the apogees are defined as (14) (15)Next, we define the apogee by identifying the ellipse's position (16)Then the perigee is obtained by calculating the appropriate arc-tangent operations (17)where and are two rotation matrices, they are given by (18)In order to make analysis of the colour image signal, we combined the quaternion formalism with phase shifted version of each colour channel in 2D case. Hence, the can be rewritten as (19)Thanks to the strong theoretical link between the Riesz transform and gradient operator [33], the Riesz-based colour structure tensor is defined as (20)Then the proposed quaternion representation of colour image is given by (21)The decomposition framework of colour image at scale two is illustrated in Fig. 2. Fig. 2Open in figure viewerPowerPoint Illustration of colour quaternion wavelet transform at level two. The amplitude and three phases of colour image Airplane are listed from left to right (row) and top to down (column) 4 Colour image denoising algorithm based colour quaternion wavelet There are strong dependencies between a decomposition coefficient and its parent decomposition coefficient in the CQWT domain. Inspired by Sendur and Selesnick [1], the bivariate shrinkage function which is jointly non-Gaussian model can capture the decomposition coefficients in the CQWT domain more efficiently. Then, we describe the proposed colour image denoising algorithm in detail. The colour image in the CQWT domain can be formulated as , where is clean data, xk denotes CQWT decomposition coefficient at scale k, and xk+1 denotes the parent CQWT coefficient at scale k + 1. Similarly, are the noisy observations of xk and xk+1, respectively, and is the additive Gaussian noise with zero mean and variance σ. The aim is to estimate x from the noisy observation y. Then, the MAP estimator is defined by (22)where denotes the conditional pdf of x, and the probability density function is defined as follows: (23)According to [1], xk and xk+1 are independent Laplacian distribution and the joint probability density function can be given as follows: (24)Then, we can rewrite the estimator by using (22) as follows: (25)where , let is differentiable and convex, the solution of (25) is given as follows: (26) (27)In order to estimate the noise variance from the noisy CQWT coefficients, the robust median estimator [34] used the CQWT coefficients at the finest scale (28)where denotes the finest scale of CQWT. and denote the derivative of corresponding to xk and xk+1, respectively. Then, can be rewritten as (29)By using (29), and are defined as (30)Combined with (26), (27), and (30), the clean data is obtained by (31)where is defined as follows: (32)Then, the detail steps of the proposed algorithm based on CQWT can be summarised as below. Algorithm.CQWT denoising Input: Noisy image: x Output: De-noised image: y Step 1: set initial decomposition scale values to three; Setp 2: compute colour image representation by CQWT using (21); Step 3: estimate noise variance of colour image using (28); Step 4: calculate joint probability density function using (24); Step 5: compute MAP estimator using (25); Step 6: calculate using (29) and (30); Step 7: estimate true CQWT coefficient using (31) Step 8: denoised colour image is obtained by applying inverse of CQWT to . 5 Experimental results and discussions In this section, denoising experiments are conducted on eight popularly utilised test images with size of pixels as shown in Fig. 3. The noise is assumed to be zero-mean white Gaussian with . Although there are many objective indicators; in this paper, peak-signal-to-noise ratio (PSNR), mean square error (MSE), and structure similarity (SSIM) are selected as the objective indicators to evaluate the denoising performance, which are defined as follows: (33)where , and are the mean and variance of the original image and denoised image, respectively, σfg is the covariance of images f and g, and are two variables to stabilise the division with a weak denominator (34) (35)where M and N denote image size, and are original image and denoised image, respectively. Fig. 3Open in figure viewerPowerPoint Test colour images used in experiments, referred to as ‘Baboon’, ‘Bird’, ‘Sailboat’, ‘Airplane’, ‘Peppers’, ‘Baby’, ‘Parrot’, ‘Barbara’ from left to right and top to bottom First, we can evaluate the visual quality of the test images by using objective indicator MSE, which can describe the average error difference between the original image and denoised image by employing different algorithms. A comparison between DWT [1], CWT [5], CT [3], BLS-GSM [6], BM3D [7], and CQWT is illustrated in Fig. 4. The experimental results show that the MSE value of the proposed method CQWT is smallest, and the error difference is also the smallest among the other denoising methods. Fig. 4Open in figure viewerPowerPoint MSE values for different test images by using various denoising algorithms Figs. 5-8 show the denoised results on images ‘Bird’, ‘Sailboat’, ‘Airplane’, ‘Peppers’, and ‘Parrot’ by using different denoising algorithms which can illustrate the visual quality and effectiveness of the proposed algorithm. From Figs. 5-8, we can see that the proposed algorithms CQWT and BM3D [7] have the outstanding performances that are superior to the other denoising algorithms. DWT [1] and CWT [5] cannot remove the noise effectively, and CT [3] and BLS-GSM [6] have better performances than DWT [1] and CWT [5], but still have some artifacts. Although BM3D [7] can obtain quite good overall effects, it can lose some detailed information of the image due to the excessive smoothing. In conclusion, our proposed algorithm cannot produce artifacts, and can better reserve the detailed information of the test image. Fig. 5Open in figure viewerPowerPoint Visual quality comparison of denoising results on colour image Bird. Original image; noisy image ; DWT [1] (PSNR = 22.67 dB); CWT [5](PSNR = 26.35 dB); CT [3](PSNR = 26.73 dB); BLS-GSM [6] (PSNR = 27.12 dB); BM3D [7] (PSNR = 27.65 dB); CQWT (PSNR = 27.91 dB) Fig. 6Open in figure viewerPowerPoint Visual quality comparison of denoising results on colour image Sailboat. Original image; noisy image ; DWT [1] (PSNR = 26.65 dB); CWT [5] (PSNR = 26.87 dB); CT [3] (PSNR = 27.09 dB); BLS-GSM [6] (PSNR = 27.57 dB); BM3D [7] (PSNR = 27.98 dB); CQWT (PSNR = 28.05 dB) Fig. 7Open in figure viewerPowerPoint Visual quality comparison of denoising results on colour image Airplane. Original image; noisy image ; DWT [1] (PSNR = 21.94 dB); CWT [5] (PSNR = 23.82 dB); CT [3] (PSNR = 23.91 dB); BLS-GSM [6] (PSNR = 24.16 dB); BM3D [7] (PSNR = 24.59 dB); CQWT (PSNR = 24.97 dB) Fig. 8Open in figure viewerPowerPoint Visual quality comparison of denoising results on colour image Peppers. Original image; noisy image ; DWT [1] (PSNR = 24.81 dB);CWT [5] (PSNR = 24.94 dB); CT [3] (PSNR = 25.32 dB); BLS-GSM [6] (PSNR = 25.73 dB); BM3D [7] (PSNR = 25.94 dB); CQWT (PSNR = 26.21 dB) Fig. 9 shows the PSNR values of the evaluated algorithms operating on the corrupted partial parrot. The information of partial parrot structure can preserve better with higher value of PSNR for image de-noising problem. At noise standard deviation , the PSNR values are 24.58, 25.92, 26.34, 27.74, and 27.86 for different denoising algorithms, respectively. The results show that the performance in terms of PSNR with BM3D [7] and CQWT are very close, which outperform than the other methods such as DWT [1], CWT [5], CT [3], and BLS-GSM [6]. Fig. 9Open in figure viewerPowerPoint Visual quality comparison of denoising results on part of colour image Parrot. Original image; noisy image ; DWT [1] (PSNR = 21.97 dB);CWT [5] (PSNR = 24.58 dB); CT [3] (PSNR = 25.92 dB); BLS-GSM [6] (PSNR = 26.34 dB); BM3D [7] (PSNR = 27.74 dB);CQWT (PSNR = 27.86) The PSNR and SSIM values for test colour images with different noise levels are listed in Tables 1 and 2. The experimental results show that the PSNR and SSIM values of the proposed method have the highest values, which demonstrate the effectiveness of the proposed colour quaternion wavelet-based image denoising algorithm. Table 1. Denoising results (PSNR) by different methods Test image Noisy image DWT [1] CWT [5] CT [3] BLS-GSM [6] BM3D [7] CQWT Baboon 10 28.53 30.78 31.75 31.95 32.05 32.38 32.52 20 22.51 26.94 27.55 27.85 28.91 29.37 29.45 30 19.06 25.87 26.24 26.78 27.46 27.85 27.98 40 16.59 24.92 25.26 25.87 25.91 26.35 26.57 Bird 10 28.51 28.69 32.75 32.91 33.10 33.58 33.85 20 22.62 27.68 28.79 28.95 29.23 29.37 29.68 30 18.87 22.67 26.35 26.73 27.12 27.65 27.91 40 16.48 20.76 24.34 24.89 25.69 25.85 25.98 Sailboat 10 28.51 29.68 29.85 30.15 30.49 30.71 31.03 20 22.51 27.64 27.83 28.27 28.67 29.18 29.49 30 18.87 26.65 26.87 27.09 27.57 27.98 28.05 40 16.47 25.53 25.81 25.96 26.19 26.46 26.92 Airplane 10 23.53 25.82 26.53 27.28 27.46 27.85 27.92 20 22.51 23.61 24.78 24.96 25.68 25.93 26.18 30 18.87 21.94 23.82 23.91 24.16 24.59 24.97 40 16.49 19.95 22.45 22.76 22.89 23.27 23.49 Baby 10 28.63 30.02 35.63 35.83 35.89 36.25 36.58 20 22.41 29.34 29.57 31.68 31.82 32.36 32.57 30 18.87 26.89 27.36 28.29 28.45 29.29 29.35 40 16.37 22.42 24.63 26.37 27.58 27.78 27.93 Peppers 10 28.52 29.76 29.85 30.15 30.42 30.74 30.92 20 22.40 25.74 25.93 26.24 26.39 26.68 27.06 30 18.98 24.81 24.94 25.32 25.73 25.94 26.21 40 16.59 22.76 22.87 23.28 23.75 24.37 24.76 Barbara 10 28.83 28.91 33.05 33.21 33.42 33.93 34.05 20 22.94 27.98 28.59 29.15 29.53 29.87 29.95 30 19.27 22.87 27.25 27.68 27.92 28.35 28.71 40 17.26 20.92 23.25 24.49 24.89 25.36 25.65 Parrot 10 27.39 29.26 32.83 33.27 33.36 33.68 33.89 20 23.72 27.42 28.39 28.63 29.05 29.52 29.79 30 19.05 21.97 24.58 25.92 26.34 27.74 27.86 40 15.46 20.35 23.27 24.76 25.48 25.82 25.93 Table 2. Denoising results (SSIM) by different methods Test image Noisy Image DWT [1] CWT [5] CT [3] BLS-GSM [6] BM3D [7] CQWT Baboon 10 0.716 0.923 0.945 0.952 0.961 0.958 0.963 20 0.621 0.904 0.915 0.917 0.942 0.916 0.920 30 0.572 0.861 0.873 0.876 0.881 0.884 0.886 40 0.431 0.831 0.852 0.863 0.871 0.869 0.874 Bird 10 0.805 0.907 0.912 0.936 0.951 0.945 0.954 20 0.780 0.892 0.896 0.901 0.902 0.901 0.905 30 0.692 0.860 0.874 0.881 0.890 0.885 0.891 40 0.615 0.841 0.853 0.851 0.854 0.852 0.857 Sailboat 10 0.829 0.930 0.952 0.952 0.957 0.954 0.963 20 0.735 0.905 0.917 0.919 0.923 0.924 0.926 30 0.684 0.881 0.890 0.893 0.902 0.904 0.905 40 0.603 0.840 0.855 0.861 0.865 0.869 0.873 Airplane 10 0.815 0.923 0.937 0.943 0.953 0.956 0.958 20 0.730 0.906 0.913 0.915 0.916 0.918 0.921 30 0.685 0.876 0.885 0.887 0.891 0.893 0.896 40 0.604 0.852 0.855 0.858 0.861 0.864 0.865 Baby 10 0.742 0.934 0.947 0.952 0.963 0.968 0.972 20 0.675 0.896 0.913 0.917 0.9223 0.925 0.927 30 0.593 0.876 0.885 0.887 0.891 0.893 0.895 40 0.508 0.828 0.842 0.843 0.852 0.856 0.858 Peppers 10 0.813 0.945 0.952 0.954 0.961 0.964 0.969 20 0.785 0.912 0.923 0.926 0.930 0.933 0.936 30 0.708 0.891 0.905 0.908 0.915 0.917 0.919 40 0.659 0.864 0.871 0.882 0.895 0.897 0.899 Barbara 10 0.837 0.926 0.934 0.956 0.975 0.978 0.979 20 0.805 0.903 0.915 0.917 0.924 0.928 0.930 30 0.769 0.863 0.874 0.878 0.883 0.886 0.889 40 0.652 0.842 0.851 0.856 0.859 0.862 0.865 Parrot 10 0.745 0.941 0.965 0.967 0.975 0.978 0.982 20 0.706 0.923 0.929 0.935 0.938 0.941 0.945 30 0.653 0.895 0.903 0.906 0.917 0.919 0.921 40 0.592 0.863 0.876 0.879 0.883 0.886 0.889 It is important that the various denoising algorithms are also evaluated from a practical point of view, such as the running time. The approximate running time is calculated using Matlab on personal computer with an Intel Core I7 2.4-GHz CPU and 8 GB running windows 7 (Table 3). Table 3. Running time (second) of different denoising algorithms with Algorithms DWT [1] CWT [5] CT [3] BLS-GSM [6] BM3D [7] CQWT Baboon 10.45 14.58 19.21 21.35 23.68 15.32 Bird 9.26 11.08 17.36 19.25 21.72 12.49 Sailboat 7.18 10.36 19.42 22.68 26.37 14.63 Airplane 13.18 15.97 21.35 23.67 24.85 17.75 Peppers 8.93 9.84 13.97 15.20 19.63 10.94 Barbara 8.61 10.94 18.75 21.38 23.58 16.47 Baby 15.25 17.09 20.25 24.18 26.34 18.31 Parrot 9.58 14.26 18.73 20.07 23.19 15.39 6 Conclusion and future work In this paper, we have designed new quaternion filter bank in the radon domain for colour images. Based on the strong theoretical link between the quaternion signal and the structure tensor, we have constructed new framework of quaternion wavelet transform for colour image representation. We have developed an efficient colour image denoising algorithm for Gaussian white noise reduction by combining the proposed quaternion wavelet framework and bivariate shrinkage function. Experiments have been conducted using different test colour images which were corrupted by Gaussian white noise. The experimental results demonstrate that the proposed algorithm can remove noise effectively while keeping sharp edges and regular textures. It can achieve the best performance compared with other state-of-the-art denoising algorithms. In addition, the computational cost of the proposed algorithm is modest comparing with the algorithms studied in this paper. 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