Markov Chain Monte Carlo-Based L 1/L 2 Regularization and Its Applications in Low-Dose CT Denoising

In this paper, a low-dose CT denoising method based on $L_1/L_2$ regularization method of Markov chain Monte Carlo is studied. Firstly, the mathematical model and regularization method of low-dose CT denoising are summarized, and then the theoretical basis of MCMC method and its application in image denoising are introduced. We evaluated the performance of various regularization strategies by comparing the denoising effects of $L_1$ , $L_2$ , and $L_1/L_2$ regularization terms in MCMC sampling at Gaussian noise levels. The experimental results show that $L_1/L_2$ regularization has the best performance in balancing noise removal and image detail retention, significantly superior to single $L_1$ and $L_2$ regularization, which proves its effectiveness for low-dose CT denoising.

• Low-Dose CT Denoising
• Regularization
• Statistical Inverse Problem
• MCMC Sampling

1. Introduction

Low-dose CT imaging is designed to reduce the radiation dose to the patient by reducing the intensity of the X-rays to obtain CT images [1] . However, this dose reduction can significantly increase the noise in the image, affecting the image quality and diagnostic value [2] . Low-dose CT denoising technology is designed to remove noise from noisy CT images to improve image quality. The inverse problem of image denoising is generally regarded as an ill-posed problem [3] , which is characterized by the non-uniqueness of the solution, instability and extreme sensitivity to the input data. For image denoising, small changes in noise can cause instability in the recovery results. In order to solve the problem of noise reduction, regularization method is introduced to impose appropriate constraints in the process of solving, so as to improve the stability and reliability of the solution. Regularization avoids over-fitting noise by incorporating prior knowledge (such as signal smoothness, sparsity, or other statistical properties) and enhances the model’s ability to recover the real image.

However, the traditional regularization method is prone to the accumulation of errors in the process of iterative solution, which leads to the problem of fuzzy boundary in the image after denoising.

Therefore, this paper proposes a low-dose CT denoising method based on $L_1/L_2$ regularization method of Markov chain Monte Carlo. By constructing a posterior probability distribution of low-dose CT image data with noise, regularization prior is incorporated to control the discomfort. Then, the posterior probability distribution is randomly sampled to ensure the image details while effectively denoising. This method effectively balances the effect of noise reduction and detail preservation in theory and shows significant advantages in practical application. Firstly, the mathematical model and regularization method of image denoising are introduced, and the implementation process of MCMC algorithm is described in detail, including initialization and sampling, and how to optimize the performance of the algorithm by adaptive step size.

Through numerical simulation experiment and real CT image experiment, we compare the denoising effect under different regularization strategies, using peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) as evaluation indexes. By comparing the experimental results of $L_1$, $L_2$ and $L_1/L_2$ regularization based on Markov chain Monte Carlo, it is observed that $L_1/L_2$ regularization has the best denoising effect on low-dose CT, especially in preserving image structure and detail.

2. Regularization Model of Low-Dose CT Image Denoising

In the process of low-dose CT denoising, the main factor to be considered is that while removing noise, the integrity of the image itself must be ensured. $u\in {\text{R}}^n$ is the original image, and the noisy image is $b\in {\text{R}}^m$. The image denoising problem can generally be expressed as the inverse problem [4] of the following problems:

$b=Iu+\eta$ (1)

where $\eta$ is noise and $I$ is the identity matrix. Due to the existence of noise, the inverse problem shows discomfort, mainly reflected in the non-uniqueness and instability of the solution. Regularization is a common method to solve ill-determined problems, which can improve the quality and stability of the solution. The main idea is to add regularization term $\text{Reg}\left(u\right)$ to the prior distribution, and after adding regularization term, the denoised CT image is obtained by solving the following minimization problem:

$u^{\text{*}}=\text{arg}\underset{u}{\text{min}}\left\{\frac{1}{2}{\Vert Au-b\Vert }_{{\ell }_2}^2+\alpha \text{Reg}\left(u\right)\right\}$ (2)

where ${\Vert Au-b\Vert }_{{\ell }_2}^2$ is the fidelity term, $\text{Reg}\left(u\right)$ is the regularization term, and $\alpha$ is the regularization coefficient, used to balance the fidelity term and the regularization term.

$L_1$ regularization [5] and $L_2$ regularization on a given gradient:

${\Vert \nabla u\Vert }_1={\Vert {\nabla }_{x}u\Vert }_1+{\Vert {\nabla }_{y}u\Vert }_1$

${\Vert \nabla u\Vert }_2=\sqrt{{\Vert {\nabla }_{x}u\Vert }_2^2+{\Vert {\nabla }_{y}u\Vert }_2^2}$

$L_1/L_2$ regularization term:

$\frac{{\Vert \nabla u\Vert }_1}{{\Vert \nabla u\Vert }_2}$

where $\nabla$ is the gradient operator, let $D_{x}u,D_{y}u\in R^n$ be the first-order forward difference operators of the horizontal and vertical directions of the image respectively, then $D=\left[D_{x}u,D_{y}u\right]=\nabla$.

3. Solution of Low-Dose CT Denoising Problem and Sampling Method

3.1. Estimation of the Posterior Density Function

The prior density function corresponding to the regularization term is

${\pi }_{\text{pr}}\left(u\right)\propto {\text{e}}^{-\alpha \text{Reg}\left(u\right)},$

suppose that the $\eta$ is a Gaussian random vector with a mean of 0, and the covariance matrix Γ which is positive definite, i.e.

$\eta ~\text{N}\left(0,\Gamma \right)$.

The likelihood density function is

$\pi \left(b|u\right)\propto \text{exp}\left\{-\frac{1}{2}{\left(Au-b\right)}^{\text{T}}{\Gamma }^{-1}\left(Au-b\right)\right\}.$ (3)

The posterior probability density of $u$ obtained by Bayes’ formula is

$\pi \left(u|b\right)\propto \text{exp}\left\{-\frac{1}{2}{\left(Au-b\right)}^{\text{T}}{\Gamma }^{-1}\left(Au-b\right)-\alpha \text{Reg}\left(u\right)\right\}.$ (4)

Assuming that the noise covariance matrix is $\Gamma ={\beta }^{2}I$, the standard form of the posterior probability density function of $u$ is

$\pi \left(u|b\right)\propto \text{exp}\left\{-\frac{1}{2{\beta }^2}{\Vert Au-b\Vert }_{l_2}^2-\alpha \text{Reg}\left(u\right)\right\},$ (5)

CM method is used to estimate the posterior density function, and the calculation formula of CM method is

$u_{\text{CM}}={\displaystyle {\int }_{{ℝ}^n}u\pi }\left(u|b\right)\text{d}u.$ (6)

3.2. MCMC Sampling Method

The MCMC algorithm is used to sample the posterior density function [6] , and sample sequence $u^{\left(1\right)},u^{\left(2\right)},\cdots ,u^{\left(M\right)}$ is obtained. Suppose that the number of samples in the pre-burning period of the probe posterior probability density function is $m_0$, when the number of samples sampled is large enough, the remaining number after removing the samples in the pre-burning period is $M-m_0$, and the above integral can be approximated as the average of $M-m_0$ samples, that is

$u_{\text{CM}}={\displaystyle {\int }_{{ℝ}^n}u\pi \left(u|b\right)\text{d}u}\approx \frac{1}{M-m_0}{\displaystyle {\sum }_{m=m_0+1}^M{u}^{\left(m\right)}}.$ (7)

The MCMC sampling algorithm is used to sample the posterior density function, and a proposed distribution $Q\left(u^{\prime }|u\right)=\text{N}\left(u,{\sigma }^{2}I\right)$ is obtained, where u is the current state, $u^{\prime }$ is the proposed new state, ${\sigma }^2$ is the variance, $I$ is the identity matrix, new state $u^{\prime }=u+\delta \times \epsilon$, $\epsilon ~\text{N}\left(0,I\right)$, the proposed distribution is symmetric, i.e. $Q\left(u^{\prime }|u\right)=Q\left(u|u^{\prime }\right)$, since the jump probabilities from $u$ to $u^{\prime }$ and from $u^{\prime }$ to $u$ are the same. The adaptive step size adjustment is used to make the acceptance rate close to 0.234 [7] . Every 10,000 sampling times, the current acceptance rate is checked and the step size is adjusted accordingly. The rule of step size adjustment can be expressed as

${\text{stepsize}}_{\text{new}}=\{\begin{array}{ll}\text{stepsize}\times 1.1,\hfill & \text{if acceptance rate}>0.234\hfill \\ \text{stepsize}÷1.1,\hfill & \text{otherwise}\hfill \end{array}$ (8)

in practical applications, 1.1 is a common adjustment factor that provides a reasonable gradual adjustment amplitude (about 10% step change), without causing too drastic a change in sampling step size, and can quickly adapt to the characteristics of the state space, in the experiment, the adjustment factor 1.1 keeps the sampling acceptance rate in the range of 20% - 30%, which is close to the theoretical optimal value of 23.4%.

The acceptance rate is calculated as

$\alpha =\min \left(1,\frac{Q\left(u^{\prime }|b\right)Q\left(u|u^{\prime }\right)}{Q\left(u|b\right)Q\left(u^{\prime }|u\right)}\right)=\min \left(1,\frac{Q\left(u^{\prime }|b\right)}{Q\left(u|b\right)}\right).$ (9)

Since the sampling method usually deals with the probability density in logarithmic form, in order to simplify the calculation, the posterior probability is logarithmic and set as

$Q\left(u|b\right)=\log \pi \left(b|u\right)+\log \pi \left(u\right)+C,$ (10)

where $C$ is a normalized constant and is independent of $u$, so it can be ignored.

The adaptive step size can be dynamically adjusted according to the acceptance rate in the sampling process. When the acceptance rate is higher than the target value, increasing the step size can accelerate the speed of exploring the state space, and thus reach the equilibrium state faster. On the contrary, if the acceptance rate is too low, reducing the step size can improve the acceptance rate, ensure that the algorithm effectively explores the state space, and enable the algorithm to adjust to the scale suitable for the current target distribution more quickly, so as to accelerate the smooth distribution.

We summarize the MCMC:

Step 1: Set the initial value $u^{\left(1\right)}=b=\left[b_1,b_2,\cdots ,b_n\right]$, $m=1$, the sample of the combustion period is $m_0$, the total number of samples is M, and the initial step size is $\delta$.

Step 2: Update candidate samples $u=u^{\left(m\right)}+\delta \cdot \epsilon$, $\epsilon ~\text{N}\left(0,I\right)$.

Step 3: The logarithmic posterior probability distribution after the regularization term is

$Q\left(u|b\right)=-\frac{1}{2{\beta }^2}{\Vert Au-b\Vert }^2-\text{Reg}\left(u\right)$

Calculate the receiving probability $\alpha$ as

$\alpha =\min \left(1,\frac{Q\left(u|b\right)}{Q\left(u^{\left(m\right)}|b\right)}\right)=\min \left(1,\exp \left(Q\left(u|b\right)-Q\left(u^{\left(m\right)}|b\right)\right)\right)$.

Step 4: If $\alpha \ge t$, $t~U\left(0,1\right)$, then $u^{\left(m+1\right)}=u$, otherwise reject candidate sample order $u^{\left(m+1\right)}=u^{\left(m\right)}$.

Step 5: When m = M, sampling stops. Otherwise, continue sampling and make m = m + 1. Repeat the second and third steps.

Step 6: After every 10,000 samples, adjust Step 1 according to the current acceptance rate $\delta$

$\delta =\{\begin{array}{ll}\delta \times 1.1,\hfill & \alpha >0.234\hfill \\ \delta ÷1.1,\hfill & \text{otherwise}\hfill \end{array}$

go to Step 2 update $\delta$.

Step 7: Based on the sampling results, the denoised image estimated by conditional mean (CM) is calculated

$u_{\text{CM}}\approx \frac{1}{M-m_0}\underset{m=m_0+1}{\overset{M}{{\displaystyle \sum }}}{u}^{\left(m\right)}.$

In order to compare the denoising effects of $L_1$, $L_2$ regularization terms and $L_1/L_2$ regularization terms, and compare the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) [8] of the denoised images, the higher the PSNR value is, the smaller the error between the denoised image and the original image, and the better the denoised effect is. The closer the SSIM value is to 1, the higher the similarity between the denoised image and the original image. The calculation of PSNR and SSIM is given below:

$\text{PSNR}=10\cdot {\text{log}}_{10}\left(\frac{{\text{MAX}}_{\text{x}}^2}{\text{MSE}}\right)$ (11)

where $\text{MSE}=\frac{1}{mn}{\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=1}^n{\left(x\left(i,j\right)-y\left(i,j\right)\right)}^2}}$, where $x$ is the original image, $y$ is the denoised image, m and n are the size of the image, and ${\text{MAX}}_{\text{I}}$ is the maximum possible pixel value of the image.

$\text{SSIM}=\frac{\left(2{\mu }_x{\mu }_y+c_1\right)\left(2{\sigma }_{xy}+c_2\right)}{\left({\mu }_x^2+{\mu }_y^2+c_1\right)\left({\sigma }_x^2+{\sigma }_y^2+c_2\right)}$ (12)

where x represents the original image, y represents the denoised image, ${\mu }_x$ and ${\mu }_y$ are the mean values of the images x and y, ${\sigma }_x^2$ and ${\sigma }_y^2$ are the variances of the images x and y, ${\sigma }_{xy}$ is the covariance of the images x and y, $c_1={\left(k_{1}L\right)}^2$, $c_2={\left(k_{2}L\right)}^2$ is a small constant to avoid denominators of zero, where $L$ is the pixel value of the dynamic range, and $k_1=0.01$ and $k_2=0.03$ are the default values.

4. Experiment Results

4.1. Low-Dose Noise Image

In the study of low-dose CT image denoising algorithm, in order to objectively evaluate the effectiveness of the denoising algorithm, people often choose the recognized Sheep-Logan model as the research object.

Noise was added to the projection data of Sheep-Logan head model to mimic low-dose CT images and the noise approximately follows the non-stationary Gaussian distribution with the mean of 0. The noise variance formula [9] is as follows:

${\sigma }_i^2=f_i\times \text{exp}\left(\frac{{\mu }_i}{\phi }\right)$ (13)

where ${\mu }_i$ and ${\sigma }_i^2$ are the projected data mean and noise variance obtained on the i-th detector, $f_i$ and $\phi$ are two configuration parameters, set here $f_i=0.3$, $\phi =30$.

Inverse Radon transform is used to invert the projected data after noise, and the noise image is obtained. Figure 1 shows the comparison between the original image and the noise image.

Figure 1

Figure 1. The left image is the original image without noise, and the right image is the noisy image.

4.2. Comparison of the Denoising Effect of L 1 , L 2 and L 1 / L 2 Regularization Terms

Next, the regularization terms $\text{Reg}\left(u\right)={\Vert \nabla u\Vert }_1$, $\text{Reg}\left(u\right)={\Vert \nabla u\Vert }_2$ and $\text{Reg}\left(u\right)=\frac{{\Vert \nabla u\Vert }_1}{{\Vert \nabla u\Vert }_2}$ are respectively used to carry out MCMC sampling on the projected data containing Gaussian noise, set the sampling to 400,000 times, $\beta =0.05$, $\alpha ={10}^5$ and obtain three denoised images, as shown in Figure 2 .

Figure 2

Figure 2. The first one is the denoised image using L 1 regularization, the second one is the denoised image using L 2 regularization, and the third one is the denoised image using L 1 / L 2 regularization.

Table 1

Table 1. Comparison of PSNR and SSIM for noisy images and denoised images using L 1 , L 2 , and L 1 / L 2 regularization.

	PSNR	SSIM
Noisy	23.98	0.7124
$L_1$ regularization	24.02	0.8886
$L_2$ regularization	22.54	0.8059
$L_1/L_2$ regularization	25.39	0.9130

Figure 3

Figure 3. Local comparison results, select the local area in the red box in Figure 2 (50 - 79 columns, 80 - 109 rows, a total of 30 × 30 pixels) for analysis.

The PSNR and SSIM of the three images are shown in Table 1 , it can be seen that $L_1/L_2$ regularization has the highest PSNR and SSIM values, while $L_2$ regularization has the lowest PSNR and SSIM values. It can be seen that $L_1/L_2$ regularization has the best effect on image denoising and detail retention. In comparison, $L_1$ regularization can also effectively improve image quality. However, it is inferior to $L_1/L_2$ regularization in terms of denoising and detail retention, while $L_2$ regularization performs the worst in terms of denoising and detail retention, and from a local zooming in image, as shown in Figure 3 , $L_1/L_2$ is also superior to $L_1$ and $L_2$ regularization in terms of detail retention and denoising.

4.3. Explore the Denoising Effect of Real Low-Dose CT Image

In order to further prove the effectiveness of this algorithm in practical application, this experiment selected full dose CT and quarter dose CT images from data set files on the Kaggle platform. Kaggle is a globally renowned data science and machine learning competition platform that provides rich datasets for users to use. In order to study the effectiveness of the algorithm, this paper selected full dose CT and quarter dose chest CT images. The section thickness was 1mm, and the quarter dose CT was low-dose CT. As shown in Figure 4 , it can be seen that there was obvious noise in the quarter dose CT images.

Figure 4

Figure 4. The left image is a full dose CT image, and the right image is a quarter dose image.

The algorithm in this paper was used to de-noise low-dose CT, the sampling times were still set to 400,000 times, and the parameters were set to $\alpha =0.02$ and $\beta ={10}^3$ to obtain three different kinds of CT images after regularization, as shown in Figure 5 .

Figure 5

Figure 5. The first one is the denoised image using L 1 regularization, the second one is the denoised image using L 2 regularization, and the third one is the denoised image using L 1 / L 2 regularization.

According to the experimental results, this is shown in Figure 5 and Table 2 , the PSNR of CT images after $L_1$, $L_2$ and $L_1/L_2$ regularization has been improved, among which $L_1/L_2$ regularization has been improved the most, and SSIM has also been improved in addition to $L_2$ regularization decrease, which is also the most obvious improvement in $L_1/L_2$ regularization. In addition, by observing the local amplification area, as shown in Figure 6 , $L_1/L_2$ regularized and denoised CT images also have the best performance in noise removal and detail retention.

Table 2

Table 2. Comparison of PSNR and SSIM for noisy images and denoised images using L 1 , L 2 , and L 1 / L 2 regularization.

	PSNR	SSIM
Quarter dose	28.21	0.9088
$L_1$ regularization	29.75	0.9152
$L_2$ regularization	28.33	0.8907
$L_1/L_2$ regularization	30.30	0.9238

Figure 6

Figure 6. Local comparison results, select the local area in the red box in Figure 5 for analysis.

Through the real clinical low-dose CT image denoising analysis, the effectiveness and advantages of the $L_1/L_2$ regularization denoising effect based on MCMC sampling algorithm are further proved.

5. Conclusions

In this paper, the low-dose CT denoising algorithm is analyzed and verified, and the regularization model based on MCMC sampling is introduced in this field. By employing MCMC sampling to process noisy image data, the denoising effect of three regularization strategies— $L_1$ regularization, $L_2$ regularization, and $L_1/L_2$ regularization—was evaluated.

The results demonstrated that $L_1/L_2$ regularization outperformed both $L_1$ and $L_2$ regularizations in multiple evaluation metrics. Particularly, $L_1/L_2$ regularization excelled in balancing noise removal with the preservation of image details, offering superior denoising results. This highlights the significance of using regularization in denoising models to effectively balance image fidelity and noise suppression. In addition, the adaptive step size in MCMC sampling further improves the stability and convergence speed of the algorithm, making the denoising process more efficient and practical. The experimental results show that the proposed method is not only feasible in theory, but also can achieve the expected denoising performance in practical application.

In summary, this study demonstrated the effectiveness and superiority of combining regularization models with MCMC sampling algorithms for image denoising. Future research could focus on optimizing computational efficiency and exploring its applications in broader image-processing tasks.

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