Image Denoising Using Singular Value Difference in the Wavelet Domain

Singular value (SV) difference is the difference in the singular values between a noisy image and the original image; it varies regularly with noise intensity. This paper proposes an image denoising method using the singular value difference in the wavelet domain. First, the SV differencemodel is generated for different noise variances in the three directions of the wavelet transform and the noise variance of a new image is used tomake the calculation by the diagonal part. Next, the single-level discrete 2-Dwavelet transform is used to decompose each noisy image into its low-frequency and high-frequency parts. Then, singular value decomposition (SVD) is used to obtain the SVs of the three high-frequency parts. Finally, the three denoised high-frequency parts are reconstructed by SVD from the SV difference, and the final denoised image is obtained using the inverse wavelet transform. Experiments show the effectiveness of this method compared with relevant existing methods.


Introduction
Wavelet transform [1][2][3][4][5][6] and singular value decomposition (SVD) [7] have been widely used as transform domain methods [8] in image denoising.Wavelet transform can highlight the detailed information of an image, while SVD is generally used as a kind of nonlinear filter.Recently, several image denoising methods based on SVD have been proposed.In [9], a denoising algorithm based on adaptive SVD in the wavelet domain is proposed.An adaptive representation method is proposed [10] using the K-means and singular value decomposition (K-SVD), which uses a greedy algorithm to learn an overcomplete dictionary for image representation and denoising.An image denoising method based on K-SVD is proposed [11], which finds a few atoms from the dictionary having the best linear combination to represent each subblock.In [12], a denoising algorithm using adaptive SVD (ASVD) is proposed.A patch-based weighted-SVD denoising method is proposed [13] with feature retention.In [14], an image denoising method is proposed using higher order SVD (HOSVD).A denoising algorithm using SVD and wavelet transform is proposed [15], which enhances directional features.In [16], a denoising algorithm using SVD and Ridgelet transform is proposed; Ridgelet transform is slightly inferior in homogenous regions of nontextured images.A singular value decomposition (APBSVD) [17] that preserves the edge structure and avoids blurriness is proposed.In [18], a denoising method based on spatially adaptive iterative singular value thresholding (SAIST) is proposed.A new wavelet threshold determination method considering interscale correlation in signal denoising is proposed [19].
Singular value (SV) difference, the difference in the singular values between a noisy image and the original image, varies regularly with noise variance.Therefore, we can use a standard SV difference to generate a new SV difference function using the different noise variance estimates.Then, the denoised image can be expressed as the SV of the noisy image minus the estimated SV difference.In this paper, an image denoising method is proposed using the SV difference in the wavelet domain.First, establish the SV difference model with 2 Mathematical Problems in Engineering different noise variances in the three directions after wavelet transform and estimate the noise variance of a new image.Then, use the single-level discrete 2-D wavelet transform to decompose each noisy image into a low-frequency and three high-frequency image parts, and use the SVD to obtain the SV of three high-frequency parts.Finally, reconstruct three denoised high-frequency parts by SVD with the SV difference and obtain the final denoised image by the inverse wavelet transform.
The rest of this paper is organized as follows.In Section 2, we briefly review the principles of wavelet transform and SVD denoising.In Section 3, we present the proposed denoising method in detail.In Section 4, we give some numerical experiments and performance analysis.Finally, Section 5 contains the concluding remarks.

Wavelet Transform and Variance
Estimating.Using the wavelet transform, an image can be decomposed into its low-frequency and high-frequency parts.The low-frequency part represents the approximate energy of an image, while the three high-frequency parts represent the detailed information of an image, including the horizontal, vertical, and diagonal parts.
New wavelet coefficients are estimated by thresholding the original wavelet coefficients.Common threshold functions include hard and soft threshold functions.These thresholds are determined by the varianceof the noise, which is estimated as where  ,1 is the median of the wavelet coefficients in the diagonal part.

SVD Denoising.
Every two-dimensional imageof size  × ( ≥ ) can be decomposed into three matrices by SVD: where  and  are the left and right singular matrices of , with column vectors  ⇀   and  ⇀ V  , respectively.The rank of  is ( ≤ ), and when the diagonal SV matrix  = diag( 1 ,  2 , . . .,   ) is a nonnegative matrix, the nonzero singular values can be arranged as These SVs reflect the energy distribution of the image, and   can be considered the representation coefficient.Therefore, image  can be expressed by ignoring SVs having value zero as follows: Wang et al. proposed an improved image denoising method based on wavelet and SVD transforms using the directional features [15]: use the SVD to filter the noise of the high-frequency parts with image rotations and the enhancement of the directional features; then rotate it back after filtering.

The Features of SV Difference.
For a two-dimensional image , a noisy image  can be expressed as where  represents the random noise.SV difference represents the difference in SVs between the noisy and original images, defined as where   and   are the SVs of noisy image  and denoised image , respectively.When noise is added to a clear image, it can be regarded as disturbance.This disturbance is bounded and is only related to the features of the noise, independent of the original image.Therefore, SV difference varies regularly with the noise variance.That is, if we know a certain SV difference with one variance and this SV difference has a regular relationship with variance, we can establish the SV difference model for an arbitrary variance.Moreover, for denoising, we can use the SV of the noisy image minus the SV difference to represent the SV of the denoised image.However, this denoising method based on SV difference applied to the entire image is suitable for flat images but very insensitive to detailed features.Therefore, we can consider applying SV difference on the detail-retained images after wavelet transform to remove the noise.
To the original images, shown in Figures 1(a) and 1(e), white Gaussian noise of various intensities are added, generating a series of SV differences of the three high-frequency parts.Figure 1 shows that the shape of these SV difference curves varies regularly with the intensity of the noise.
With the upper left corner of Figure 1(e) as a reference, we extract the first image (of size 200×200) and then increase the length and width by 50 pixels, thus obtaining seven images to be considered as original images; we then add white Gaussian noise with variance of 0.02 to these original images.Figure 2 shows that the shape of the SV difference curves changes only with the size of the image, and it also varies regularly.
Figure 3 contains seven SV difference curves, where the noisy images (with variance of 0.04) originate from Figure 1(e), extracting seven images (the size of 200 × 200) from upper left to lower right corner of each shift 50 pixels.Figure 3 shows that the SV difference does not change with the position of the image.

Proposed Image Denoising Procedure
The steps of the proposed image denoising procedure using SV difference in the wavelet domain are given below (Figure 4): (1) Generate the SV difference model with different noise variance in the three high-frequency parts of the wavelet transform (Section 3.1).(2) Use the single-level discrete 2-D wavelet transform to decompose the new noisy image into its lowfrequency part and three high-frequency parts, and then calculate the size and estimate the variance of the noise to get the new SV differences for this noisy image.
(3) Use SVD to get the noisy SV of the high-frequency parts, and subtract the above SV difference to obtain the denoised SV (Section 3.2).
(4) Reconstruct the denoised high-frequency parts from the estimated denoised SVs and the corresponding eigenvectors by SVD.
(5) Finally, reconstruct the final denoised image by the inverse wavelet transform.
3.1.Establish the SV Difference Model.Figures 1 and 2 show that the SV difference curve is sensitive to the size of the image and the added noise.We introduce two parameters related to the size of the image and the intensity of the added noise to adjust the SV difference function: where The longitudinal shrinkage coefficients () associated with the size are fitted by the maximum values in Figure 2, as shown in Figure 7.According to the shape of SV difference curves vary regularly with the size of the image of Figures 2(a These coefficients of functions ( 6)-( 9) are calculated from Figure 1(e) as the test image; because the noise is random, these coefficients will change slightly.Therefore, we can choose a random clear figure as the new testing standard figure to determine the coefficients of fitting function according to the above steps, or we can take the mean of the coefficients from multiple images as the final factor.

Estimate the Denoised SV.
We get the approximate SV sequence of the original image through subtraction of the SV difference from the SVs of the noisy image: where    and    represent the SVs of the denoised and noisy images, respectively;   represents the SV difference;  ∈ {H, V, D} represents the horizontal, vertical, or diagonal direction.We then reconstruct the denoised image by SVD.

Simulations
To validate the proposed method, we conducted several experiments on two test images with added different white Gaussian noise; the results are presented in this section.The noise mean is 0, and the normalized variance is from 0.001 to 0.1.The size of both test images is 512×512.(i) the denoising method proposed in this paper, (ii) the conventional wavelet hard threshold method, (iii) the conventional wavelet soft threshold method, (iv) the conventional SVD filtering method, (v) the denoising method proposed in [15], (vi) the denoising method proposed in [19] for hard and soft thresholding.
We used peak signal to noise ratio (PSNR) and structural similarity index (SSIM) to evaluate the denoising effectiveness.
Figures 8-14 show that the denoising method proposed in this paper significantly outperforms PSNR and SSIM and provides a better denoising effect with greater detail preservation compared to those methods.(w) the denoised image using the method in [19] for hard thresholding; (x) the denoised image using the method in [19] for soft thresholding.(v) the denoised image using the method in [15]; (w) the denoised image using the method in [19] for hard thresholding; (x) the denoised image using the method in [19] for soft thresholding.

Conclusions
This paper provides an image denoising method using the singular value difference in the wavelet domain.Experimental results show that this proposed method is effective in denoising and has a better denoising performance compared with relevant existing methods.This method can be used to process images from remote sensing, from the medical domain, and other common color-image denoising problems.Because the longitudinal shrinkage coefficients (), (V), and () are calculated by the polynomial fitting, the degree of polynomial selected and the testing value of variance and size all affect the coefficients of polynomial fitting function, which affects the final accuracy of image denoising.Therefore, further research still needs to be done on how to optimize the polynomial and choose the best standard image.Denoising in [15] Hard thresholding in [19] Conventional SVD Soft thresholding in [19] (l) SSIM 0.9

Soft thresholding
Denoising in [15] Hard thresholding in [19] Conventional SVD Soft thresholding in [19] (m) (i) the denoised image using the method in [15]; (j) the denoised image using the method in [19] for hard thresholding; (k) the denoised image using the method in [19] for soft thresholding; (l) PSNR results for different methods; (m) SSIM results for different methods.(f) the denoised image using the method in [15]; (g) the denoised image using the method in [19] for hard thresholding; (h) the denoised image using the method in [19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.(f) the denoised image using the method in [15]; (g) the denoised image using the method in [19] for hard thresholding; (h) the denoised image using the method in [19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.
Mathematical Problems in Engineering (f) the denoised image using the method in [15]; (g) the denoised image using the method in [19] for hard thresholding; (h) the denoised image using the method in [19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.
like to thank American Journal Experts [https://www.aje.cn] for English language editing.

Figure 1 :
Figure 1: SV difference curves of different images with different added noise: (a) first original image; (b) SV difference curves of (a) of the horizontal part; (c) SV difference curves of (a) of the vertical part; (d) SV difference curves of (a) of the diagonal part; (e) second original image; (f) SV difference curves of (e) of the horizontal part; (g) SV difference curves of (e) of the vertical part; (h) SV difference curves of (e) of the diagonal part.

( 7 )Figure 2 :
Figure 2: Effects of different image sizes on SV difference: (a) SV difference curves of the horizontal part; (b) SV difference curves of the vertical part; (c) SV difference curves of the diagonal part.

Figure 3 :Figure 4 :
Figure 3: SV difference curves at different positions in the image: (a) SV difference curves of the horizontal part; (b) SV difference curves of the vertical part; (c) SV difference curves of the diagonal part.

Figure 5 :Figure 6 :Figure 7 :
Figure 5: The standard SV difference functions in three directions: (a) SV difference curves of the horizontal part; (b) vertical SV difference curves of the vertical part; (c) diagonal SV difference curves.

Figure 8 :
Figure 8: (a) The noisy image ( 2 = 0.001); (b) the low-frequency part of (a); (c) the horizontal part of (a); (d) the vertical part of (a); (e) the diagonal part of (a); (f) the SV of noisy horizontal part; (g) the SV of noisy vertical part; (h) the SV of noisy diagonal part; (i) the SV difference of the horizontal part; (j) the SV difference of the vertical part; (k) the SV difference of the diagonal part; (l) the denoised SV and original SV of the horizontal part; (m) the denoised SV and original SV of the vertical part; (n) the denoised SV and original SV of the diagonal part; (o) the reconstructed horizontal part; (p) the reconstructed vertical part; (q) the reconstructed diagonal part; (r) the denoised image using the method in this paper; (s) the denoised image based on the conventional wavelet hard threshold; (t) the denoised image based on the conventional wavelet soft threshold; (u) the denoised image based on the conventional SVD; (v) the denoised image using the method in [15];(w) the denoised image using the method in[19] for hard thresholding; (x) the denoised image using the method in[19] for soft thresholding.

Figure 9 :
Figure 9: (a) The noisy image ( 2 = 0.01); (b) the low-frequency part of (a); (c) the horizontal part of (a); (d) the vertical part of (a); (e) the diagonal part of (a); (f) the SV of noisy horizontal part; (g) the SV of noisy vertical part; (h) the SV of noisy diagonal part; (i) the SV difference of the horizontal part; (j) the SV difference of the vertical part; (k) the diagonal SV difference; (l) the denoised SV and original SV of the horizontal part; (m) denoised SV and original SV of the vertical part; (n) the diagonal denoised SV and original SV; (o) the reconstructed horizontal part; (p) the reconstructed vertical part; (q) the reconstructed diagonal part; (r) the denoised image using the method in this paper; (s) the denoised image based on the conventional wavelet hard threshold; (t) the denoised image based on the conventional wavelet soft threshold; (u) the denoised image based on the conventional SVD;(v) the denoised image using the method in[15]; (w) the denoised image using the method in[19] for hard thresholding; (x) the denoised image using the method in[19] for soft thresholding.

Figure 10 :
Figure 10: (a) PSNR results with four methods of Figures 8 and 9; (b) SSIM results with four methods of Figures 8 and 9; (c) PSNR results with three methods of Figures 8-9; (d) SSIM results with three methods of Figures 8 and 9.

Figure 11 :
Figure 11: (a) The noisy image ( 2 = 0.01); (b) the horizontal denoised SV and original SV; (c) denoised SV and original SV of the vertical part; (d) the diagonal denoised SV and original SV; (e) the denoised image using the method in this paper; (f) the denoised image based on the conventional wavelet hard threshold; (g) the denoised image based on the conventional wavelet soft threshold; (h) the denoised image based on the conventional SVD;(i) the denoised image using the method in[15]; (j) the denoised image using the method in[19] for hard thresholding; (k) the denoised image using the method in[19] for soft thresholding; (l) PSNR results for different methods; (m) SSIM results for different methods.

Figure 12 :
Figure 12: (a) The noisy image ( 2 = 0.1); (b) the denoised image using the method in this paper; (c) the denoised image based on the conventional wavelet hard threshold; (d) the denoised image based on the conventional wavelet soft threshold; (e) the denoised image based on the conventional SVD;(f) the denoised image using the method in[15]; (g) the denoised image using the method in[19] for hard thresholding; (h) the denoised image using the method in[19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.

Figure 13 :
Figure 13: (a) The noisy image ( 2 = 0.1); (b) the denoised image using the method in this paper; (c) the denoised image based on the conventional wavelet hard threshold; (d) the denoised image based on the conventional wavelet soft threshold; (e) the denoised image based on the conventional SVD;(f) the denoised image using the method in[15]; (g) the denoised image using the method in[19] for hard thresholding; (h) the denoised image using the method in[19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.

Figure 14 :
Figure 14: (a) The noisy image ( 2 = 0.01); (b) the denoised image using the method in this paper; (c) the denoised image based on the conventional wavelet hard threshold; (d) the denoised image based on the conventional wavelet soft threshold; (e) the denoised image based on the conventional SVD;(f) the denoised image using the method in[15]; (g) the denoised image using the method in[19] for hard thresholding; (h) the denoised image using the method in[19] for soft thresholding; (i) PSNR results for different methods; (j) SSIM results for different methods.