Mixed Noise Parameter Estimation Based on Variance Stable Transform

: The ultimate goal of image denoising from video is to improve the given image, which can reduce noise interference to ensure image quality. Through denoising technology, image quality can have effectively optimized, signal-to-noise ratio can have increased, and the original mage information can have better reflected. As an important preprocessing method, people have made extensive research on image denoising algorithm. Video denoising needs to take into account the various level of noise. Therefore, the estimation of noise parameters is particularly important. This paper presents a noise estimation method based on variance stability transformation, which estimates the parameters of variance stability transformation by minimizing the noise distribution peak, and improves the parameter accuracy of mixed peak estimation by comparing and analyzing the changes of parameters. The experimental results show that the new algorithm of noise estimation has achieved good effects, which are making the field of video denoising more extensive.

denoising algorithm and solve the problem of image quality degradation caused by noise interference. The denoising technology can effectively improve the image quality, increase the signal-to-noise ratio, and better reflect the information carried by the original image. As an important means of preprocessing, people have carried out extensive research on image denoising algorithm. Among the existing denoising algorithms, some have achieved good results in low-dimensional signal image processing, but they are not suitable for high-dimensional signal image processing. Either the denoising effect is better, but part of the image edge information is lost, or the research of detecting image edge information is devoted to preserving image details. How to find a better balance between resisting noise and retaining details has become the focus of research in recent years. The spatial noise estimation algorithm mainly relies on the weak texture region to process the noisy image, which has divided into mixed noise estimation, filter-based noise estimation and image patch-based noise estimation [Liu and Lin (2012)]. Liu et al. [Liu, Tanaka and Okutomi (2012)] obtained the weak texture region by the gradient covariance matrix, and got the level of Gaussian noise by the weak texture. Pei et al. [Pei, Tong, Wang et al. (2010)] processes the texture detail information by adaptive filter, and combines the noise image patch with its filtered image patch to estimate the noise level. Ponomarenko et al. [Ponomarenko, Lukin, Zriakhov et al. (2007)] uses DCT transform to transform the image into the frequency domain, and finally separates the noise to obtain a pure, noise-free image. The above method is easier to obtain noise estimation and obtain a noise-free image when processing some low-pixel images with less texture. Li et al. [Li, Wang, Chang et al. (2011)] proposed wavelet transform to reduce the entropy of the image, and used the relationship between the entropy detection of the noise signal and the variance to estimate the noise. Huang et al. [Huang, Dong, Xie et al. (2017)] proposed to perform an average absolute deviation estimation (MAD) to obtain the estimative standard soft threshold. In many cases, the image patch texture between the spatial domain and the transform domain is more and rough, and the video image often contains many random texture interferences. The traditional noise estimation algorithm cannot recognize the complex texture video image. The matrix domain noise estimation performs matrix decomposition on the image signal from video and the noise signal, and is suitable for removing the video image of the multi-dimensional texture [Zhu and Milanfar (2010)]. Pyatykh et al. [Pyatykh, Hesser and Zheng (2013)] uses a block-based principal component analysis (PCA) algorithm to process video images with smooth regions, and can obtain accurate noise estimation levels for video images with complex texture regions. Liu et al. [Liu and Lin (2012)] used a singular value decomposition (SVD) based noise estimation algorithm to obtain the noise level. The above noise estimation algorithm is only used to estimate Gaussian noise, and the Poisson-Gaussian noise estimation caused by hardware such as sensors is not ideal. Based on the above research, this paper proposes a method based on principal component analysis combined with variance stable transform (VST) to achieve image noise estimation. The modified algorithm can not only identify the Gaussian noise estimation caused by hardware, but also estimate the noise evaluation level caused by multi-texture interference. At the same time, this paper in Section 2 introduces Matrix domain based noise estimation, which we can know the noise distribution and mixed noise estimation parameters. Section 3 presents VST based noise estimation, which can get the Variance stable transformation and the parameters of the VST transformation. The experimental results in Section 4 show that the optimized algorithm combining the noise estimation algorithm with the conventional video denoising idea has fast response speed and wider application fields.

Matrix domain based noise estimation 2.1 PCA noise estimation
It is assumed that the noise signal is additive white Gaussian noise, and x, n, y represent the original video image, the noise signal, and the noisy video image contaminated by noise, respectively. Then, the number of image patches contained in the noisy image y is N , the calculation formula is as follows: where S1, S2 represents the number of columns and rows of the noisy image y , that is, Because the noisy image is superimposed by the original video image and the noise signal, x, n, y respectively contain corresponding image patches, and the corresponding vectors are X , N , Y , respectively. Since the noise signal is additive white Gaussian noise, that is, independent of the image signal from video, the original video image and the noise signal satisfy the following relationship: , the information of the original video image x has redundancy. The subspace has a smaller dimension − M m than the vector dimension M . Therefore, we select the required image patches according to this assumption, and select the standard formula as follows: According to the above formula, the noisy image patch i y is positively correlated with the image patch distance , so that the appropriate image patch can be selected for principal component analysis based on the noisy image patch . After obtaining the image patch model and the appropriate image patch, the noise variance is obtained by principal component analysis. Assume that the sample covariance matrix SX, SY of the vectors X , Y are respectively represented, and the eigenvalue decomposition is performed on SX, SY respectively.

SVD estimation
The noisy estimation of singular value decomposition is estimated by the singular level, and is mainly divided into three parts. Assume that the noisy video image is y , the original video image is x , and the noise signal is n , which satisfies the following formula: (1) Singular value decomposition (SVD): Based on linear algebra, r is a matrix A of rank, it can be decomposed into three different matrices as follows: Where U, S, V denote an orthogonal matrix, a diagonal matrix and another orthogonal matrix, respectively, m, n, I are representing the size of the matrix and the unit square matrix A , respectively. ( )( 0,1,..., ) = s i i r is a singular value, and it has arranged in descending order as follows: (2) Additive Gaussian white noise N analysis: Additive white Gaussian noise has a mean of 0 and a standard deviation of σ , the standard deviation and singular value decomposition (SVD) are expressed as follows: In the above formula, the standard deviation between the scale factor and the noise signal is respectively indicated by 1 , k σ . Only when the noise signal 1 N has the same distribution for N , that is, the additive white Gaussian noise can obtain a linear relationship. Therefore, it can be obtained that when the process of additive white Gaussian noise is the same, a linear relationship is satisfied with M P and σ .
When the action of the noise signal is different, the linear relationship between M P and σ does not apply to all singular values, but a large number of experiments prove that, if M is large enough, it is still linear, M P and σ the relationship is as follows: where α is the slope of the linear function, the degree of linear correlation, related to the choice of M . In practical applications, the relationship between the last mean value M P and the standard deviation σ of the noise signal is as follows: (3) Estimation of noise variance: The noise signal of video image is Gaussian noise N with mean value and standard deviation σ , and noise 1 N has added to the noisy image, and the standard deviation is 1 σ , so a new noisy video image is obtained. The standard deviation is 2 2 1 σ σ + such that the following two formulas are true.
The final noise variance, the expression is as follows: The noise estimation in the matrix domain has good noise estimation performance. The noise estimation algorithm of PCA can only deal with Gaussian noise. In addition, It is used in the noise estimation of images, so this paper improves on the basis of this algorithm, it not only can accurately estimate the Gaussian noise level, but also it can be used for the estimation of sensor noise, Gauss-Poisson mixed noise.

VST based noise estimation 3.1 Variance stable transformation
Defining the expectation of the random variable, the variance and the standard deviation represents the row j and column k element of matrix B . and T v represents the transpose of vector v . For the original image x , the pixel value ( ) x q at the position p and the pixel value ( ) y q of the noisy image y at the point q , there is the following model expression: representing the electrothermal noise signal independent of the signal, introduced by the hardware of the sensor. The variance of the noisy image model of Eq. (16) to get the following formula: It can be seen that the noise variance is linear with the pixel value of the original image.
According to the characteristics of the Poisson distribution, when ( ) q λ is large enough, ( ) q ω approximates the mean ( ) q λ , distribution with a variance is ( ) q λ . Therefore, it can be known from Eq. (16) that the pixel value ( ) y q of the noisy image approximates the mean distribution is ( ) x q . The pixel value ( ) y q of the noisy image and the original image pixel value ( ) x q satisfy the following relationship.
In order to find the noise level of the noisy image, the noise parameter a, b has first obtained. Therefore, the noise level problem is transformed into the parameter problem of the noise model.
The expression looks is as follows.
According to the expansion formula, the approximate expression of the formula (19) can be obtained as follows: The two sides of the formula (22) are integrated to obtain the following expression: The random variable is represented by τ , which is a variance stability transformation of a random variable τ .

PCA-based image patch transformation
It can be seen from the above-described variance stability conversion characteristic that 3) The normalized feature vectors 1 ,..., K a a of the sample covariance S are obtained.
These feature vectors obey the following relationship: Therefore, the distribution of weight ω K is the same as the noise distribution [Liu, Tanaka and Okutomi (2012)], so in practice, the distribution of noise signals can be replaced by the distribution characteristicω K of the weights. From Eq. (29), in the case , the noise variance can be approximated as a weight variance, the expression is as follows:

VST correction
The noise level of the distribution characteristics with the noise signal must be considered.
Since the true value of the noise parameter is unknown, the parameters ' ' a b 、 obtained by the VST deviates from the real parameter, we use excessive peaks for detection, and the excess peak calculation for a random variable is as follows: The noise variance of the pixel value 1 < ⋅⋅⋅ < M x x of the image x can be obtained.
Among them, the parameter σ is a non-negative number, and the parameter φ has a value range of [0,2] π . The following formula is as follows: We can get the calculation as follows.
According to the above formula, the noisy variance of final average has obtained as the noise variance value. The expression is as follows: Therefore, the optimized method proposed in this paper can accurately estimate Gaussian noise and mixed noise, and it has not affected by camera hardware and video image texture. The test results are more stable than the PCA estimates.

Test results
This paper verifies the effectiveness of the noise estimation method in the optimized algorithm, and selects four groups of video such as akiyo, supervisor, salesperson, and football [Xiao, Li, Jiang et al. (2015)]. The estimation effects of Gaussian noise or Poisson-Gaussian noise are compared. The error of the noise estimate is defined as ( ) σ σ σ ∆ = − . This article measures the accuracy of noise estimation by estimating the error of each algorithm. The algorithm-running environment is Windows 7, CPU-Intel Core I 7-2300 K, clocked at 4.70 GHz, and memory is 4 GB, 64 bit. The 5th frame of the test video is compared. The third frame of the four sets of noise-free video sequences is as follows:  It can be seen from Tab. 1, the case of simply adding Gaussian noise, the difference between the noise estimation and the real noise variance estimated is small, and in most cases estimated the noise level by comparing the two methods.
(2) The results of error comparison when the parameters are added Poisson-Gaussian noise have shown in Tab. 2. From the comparison of Tab. 2, by introducing Gauss-Poisson mixed noise, the average variance obtained by the noise estimation algorithm is very close to the average variance of the actual noise, and the accuracy of the comparison algorithm is better. Especially for the PCA noise estimation algorithm has significant accuracy. According to the above comparison results, the noise estimation algorithm can obtain relatively accurate noise level, which can not only accurately estimate simple Gaussian noise, but also obtain more accurate results of Gaussian-Poisson mixed noise. Therefore, the noise estimation algorithm in this paper has good applicability. In order to test and verify the effect of the Text Algorithm, a space-time joint filtering algorithm based on three-dimensional block matching (VBM3D) [Dabov, Foi and Egiazarian (2007)] is combined with the noise estimation (VBM3D+est). Compared with the accelerated near-end gradient method (APG) [Ji, Liu, Shen et al. (2010)], VBM3D algorithm, and dual-domain filtering algorithm (DDID) [Knaus and Zwicker (2013)], Gaussian noise and Gauss-Poisson noise have added to the original video for comparison. The comparison effect is as follows.
(1) Gaussian noise is added, the average value is zero, and the standard deviation is 20. The noise reduction effect of each algorithm is shown in Figs. 2-5. Adding a Gaussian white noise with an average value has a standard deviation of 20, from the comparison results of the noise reduction in the above figure, it can be seen that the image brightness after noise reduction by the APG algorithm is generally reduced, and the picture is blurred. The DDID algorithm has a high brightness, and the details are well preserved, but the picture has some oil painting and can only have used for image denoising, which has certain limitations. The VBM3D algorithm also has better brightness, but ringing and blockiness distortion occur. Finally, the processing result of the algorithm is not only better in brightness, but also better in detail. Compared with the VBM3D algorithm, the ringing and blockiness distortions are optimized, and noise estimation has added, and the application range is wider.
(2) Add Gauss-Poisson mixed noise, and compare the noise reduction algorithms as follows: In Figs. 6-7, when the Poisson-Gaussian noise mixed noise is added, the comparison algorithms APG, DDID and VBM3D noise reduction algorithms do not know the noise level. Therefore, the artificial setting makes the image after noise reduction too smooth, as shown in Fig. 6, or the noise reduction is not complete, and a large amount of noise is reserved. However, the algorithm of this paper obtains the noise level according to the noise estimation and then performs noise reduction, and the obtained noise image has better effect. This paper adopts PSNR and SSIM [Zhang, Yu, Ding et al. (2013)] as objective indicators of noise reduction. The comparison results are as follows: From the objective indicators in Tab. 3, by adding pure Gaussian noise, the difference between the proposed algorithm and the comparison algorithm DDID and VBM3D is small, which is obviously better than the APG noise reduction algorithm. When adding Poisson-Gaussian mixed noise, the optimized method is obviously better than the contrast method, and the noise reduction effect is excellent.

Conclusion
With the development of science and technology and the need of work and life, the application of digital image filtering will be more and more extensive, and the requirements will be higher and higher. So far, there are still many new ideas and methods in denoising, and constantly enrich image denoising methods. Moreover, the research scope of noise is also expanding, from Gaussian noise to non-Gaussian noise. Denoising technology has a wide range of applications and research prospects, and the research field is constantly expanding. Noise interference has puzzled by video image acquisition, which affects the accuracy of recognition. This paper is a brief introduction to image denoising technology. In this paper, image-denoising technology is summarized, including the concept of noise and denoising principle, and some basic image denoising methods are introduced. The optimized noise estimation method is proposed for the above phenomena and actual needs by PCA and variance stable transform. In addition, the innovative concept of introducing excessive noise peaks in this paper greatly improves the precision of noise estimation by judging the proportion of noise distribution. This paper estimates the VST transform parameters by the excessive peak minimization, the optimized noise estimation can better obtain the denoising effect and suppress the interference of hardware, which can be widely used in actual production in the future.