Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 1542509, 19 pages

https://doi.org/10.1155/2018/1542509

## Image Denoising Using Singular Value Difference in the Wavelet Domain

^{1}College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101, China^{2}Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China

Correspondence should be addressed to Min Wang; moc.361@1080uy

Received 15 July 2017; Revised 27 October 2017; Accepted 14 November 2017; Published 17 January 2018

Academic Editor: Raffaele Solimene

Copyright © 2018 Min Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 difference model is generated for different noise variances in the three directions of the wavelet transform and the noise variance of a new image is used to make the calculation by the diagonal part. Next, the single-level discrete 2-D wavelet 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.

#### 1. Introduction

Wavelet transform [1–6] 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 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.

#### 2. Mathematical Preliminaries

##### 2.1. 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 aswhere is the median of the wavelet coefficients in the diagonal part.

##### 2.2. 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 , respectively. The rank of is , and when the diagonal SV matrix 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.

##### 2.3. The Features of SV Difference

For a two-dimensional image , a noisy image can be expressed aswhere represents the random noise.

SV difference represents the difference in SVs between the noisy and original images, defined aswhere 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.