A new conjugate gradient algorithm for noise reduction in signal processing and image restoration

1 Introduction

Noise reduction is an important step in signal pre-processing; it is widely applied in many fields, including underwater acoustic imaging [1, 2], pattern recognition [3], and target detection and feature extraction [4], among others [5]. In this article, a new approach based on a conjugate gradient method is derived from mathematical principles.

We consider the degradation model of signal or image such as:

$y=\mathcal{A}\omega +\epsilon ,(1)$

where $\omega \in {\mathbb{R}}^N$ is an original signal or image, $\mathcal{A}$ is the degradation operator, ɛ is the noise, $y\in {\mathbb{R}}^M$ is the observed data. The essence of noise reduction is solving Eq. 1 to obtain ω. The solving of Eq. 1 can be considered as the following problem [6]:

$\underset{\omega \in {\mathbb{R}}^N}{\min }\frac{1}{2}\Vert y-\mathcal{A}\omega {\Vert }^2\mathrm{s}.\mathrm{t}.\Vert \omega {\Vert }_1\le r,(2)$

where r > 0 and ‖ ⋅ ‖₁ is the ℓ₁ norm. Let $C=\{\omega \in {\mathbb{R}}^N:\Vert \omega {\Vert }_1\le r\}$ and Q = {y}, then Eq. 2 can be seen as a split feasibility problem (SFP) [7–10]. Thus, we translate the problem of noise reduction to SFP, which can be described as:

$\mathrm{fi}\mathrm{n}\mathrm{d}\omega \in C\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\mathcal{A}\omega \in Q,(3)$

where H₁ and H₂ are real Hilbert spaces, $\mathcal{A}$: H₁ → H₂ is a bounded linear operator, the closed and convex set C ⊂ H₁ (C ≠ ∅), and Q ⊂ H₂ (Q ≠ ∅). In order to solve the SFP, Byrne [11, 12] presented the CQ algorithm, which creates a sequence {ω_i}:

${\omega }_{i+1}=P_C\left({\omega }_i-{\tau }_i{\mathcal{A}}^{*}\left(I-P_Q\right)\mathcal{A}{\omega }_i\right),(4)$

where P_C is the projection to C, P_Q is the projection to Q, ${\tau }_i\in (0,\frac{2}{\Vert \mathcal{A}{\Vert }^2})$, and ${\mathcal{A}}^{*}$ is the adjoint operator of $\mathcal{A}$. For convex functions c and q, the definitions of C and Q are

$C=\left\{\omega \in H_1:c\left(\omega \right)\le 0\right\}\text{and}Q=\left\{u\in H_2:q\left(u\right)\le 0\right\}.$

There have been some research works devoted to solving Eq. 3. In 2004, Yang [13] presented a relaxed CQ algorithm using $P_{C_i}$ and $P_{Q_i}$ to replace P_C and P_Q. Here, we define two sets at point ω_i by

$C_i=\left\{\omega \in H_1:c\left({\omega }_i\right)\le ⟨{\zeta }_i,{\omega }_i-\omega ⟩\right\},(5)$

where ζ_i ∈ ∂c(ω_i), and

$Q_i=\left\{u\in H_2:q\left(\mathcal{A}{\omega }_i\right)\le ⟨{\vartheta }_i,\mathcal{A}{\omega }_i-u⟩\right\},(6)$

where ${\vartheta }_i\in \partial q(\mathcal{A}{\omega }_i)$. For all i > 1, clearly, C ⊆ C_i and Q ⊆ Q_i. In addition, C_i and Q_i are half-spaces. Furthermore, referring to [14], we define

$f_i\left(\omega \right)=\frac{1}{2}\Vert \left(I-P_{C_i}\right)\omega {\Vert }^2+\frac{1}{2}\Vert \left(I-P_{Q_i}\right)\mathcal{A}\omega {\Vert }^2,(7)$

where C_i and Q_i are given as in Eqs. 5, 6. In this specific case, their gradient is

$\nabla f_i\left(\omega \right)=\left(I-P_{C_i}\right)\omega +{\mathcal{A}}^{*}\left(I-P_{Q_i}\right)\mathcal{A}\omega .(8)$

Yang [13] presented a relaxed CQ algorithm in a finite-dimensional Hilbert space:

${\omega }_{i+1}=P_C\left({\omega }_i-{\tau }_i\nabla f_i\left({\omega }_i\right)\right),(9)$

where ${\tau }_i\in (0,\frac{2}{\Vert \mathcal{A}{\Vert }^2})$. Notice that calculating $\Vert \mathcal{A}\Vert$ is complex and costly when $\mathcal{A}$ is a high-dimensional dense matrix. In 2005, Yang [15] presented a new adaptive step size τ_i, which is defined as:

${\tau }_i=\frac{{\rho }_i}{\Vert \nabla f_i\left(x\right)\Vert },(10)$

where

$\sum _{i=1}^{\infty }{\rho }_i=\infty ,\sum _{i=1}^{\infty }{\rho }_i^2<+\infty .$

However, Yang’s step size (Eq. 10) requires that Q_i is bounded and the matrix $\mathcal{A}$ is full rank. Recently, Wang [16] absolutely eliminated these problems. Considering the CQ algorithm, López [17] introduced a novel step size to overcome these problems; this is defined as:

${\tau }_i=\frac{{\rho }_i{f}_i\left(x_i\right)}{\Vert \nabla f_i\left(x_i\right){\Vert }^2},(11)$

where ρ_i ∈ (0, 4). With Lopez’s step size (Eq. 11), it was proved that {ω_i} in Eq. 9 weakly converges to the solution of the SFP.

In 2005, Qu and Xiu [18] introduced a relaxed CQ algorithm that is improved by using an Armijo line search in Euclidian space. In 2017, on the basis of the above application, Gibali [19] extended this to Hilbert spaces, which proved that {ω_i} weakly converges to a solution of the SFP as follows:

$\begin{array}{c}\hfill y_i=P_{C_i}\left({\omega }_i-{\tau }_i\nabla f_i\left({\omega }_i\right)\right),\\ \hfill {\omega }_{i+1}=P_{C_i}\left({\omega }_i-{\tau }_i\nabla f_i\left(y_i\right)\right),\end{array}(12)$

where ${\tau }_i=\gamma {\ell }^{l_i}$, γ > 0, ℓ ∈ (0, 1), l_i is the smallest nonnegative integer, and ν ∈ (0, 1) satisfies:

${\tau }_i\Vert \nabla f_i\left({\omega }_i\right)-\nabla f_i\left(y_i\right)\Vert \le \nu \Vert {\omega }_i-y_i\Vert .$

In 2020, Kesornprom et al. [20] introduced a gradient-CQ algorithm that derived a weak-convergence theorem for solving the SFP in the framework of Hilbert spaces. This is described as:

$\begin{array}{ll}\hfill y_i& ={\omega }_i-{\tau }_i\nabla f_i\left({\omega }_i\right),\hfill \\ \hfill {\omega }_{i+1}& =P_{C_i}\left(y_i-{\phi }_i\nabla f_i\left(y_i\right)\right),\hfill \end{array}$

where C_i, f_i, and ∇f_i are given in Eqs 5, 7, 8, respectively, and

${\tau }_i=\frac{{\rho }_i{f}_i\left({\omega }_i\right)}{\Vert \nabla f_i\left({\omega }_i\right){\Vert }^2+{\theta }_i},\text{and}{\phi }_i=\frac{{\rho }_i{f}_i\left(y_i\right)}{\Vert \nabla f_i\left(y_i\right){\Vert }^2+{\theta }_i},0<{\rho }_i<4,0<{\theta }_i<1.$

The conjugate gradient method [21] is a commonly used acceleration scheme in the steepest descent method. The conjugate gradient direction of f at ω_i is

$d_{i+1}=-\nabla f_i\left({\omega }_i\right)+{\beta }_i{d}_i,$

where d₀ = −∇f(ω₀) and β_i ∈ (0, ∞). In this article, motivated by previous works [22–24], a new viscosity approximation method based on the conjugate gradient method is introduced. Many other iterative methods of solving the SFP have been proposed [25–29].

Herein, combining the relaxed CQ algorithm with a new step size and the conjugate gradient method, we find the solution of noise reduction problem in Eq. 1 by solving the SFP in Hilbert spaces with a novel approach. Section 2 gives some basic definitions and lemmas. In Section 3, the theorem for proving the weak convergence of our method is presented. In Section 4, we present experimental results and compare them with the relaxed CQ algorithms of López [17], Yang [15], and Sakurai and Iiduka [20]. Finally, conclusions are given in Section 5.

2 Preliminaries

Throughout this article, to obtain our results, some technical lemmas are used.

3 Algorithm and convergence

A novel gradient-CQ algorithm is established in this section. Furthermore, we prove that the sequence created by our approach is convergent.

4 Experimental results

In this section, we describe numerical simulations to demonstrate the applications of Yang’s algorithm [15], López’s algorithm [17], Sakurai and Iiduka’s algorithm [20], and the proposed algorithm (Algorithm 1) in signal processing and image recovery. The results of our simulations show that the proposed method has higher efficiency than the well-known methods in the literature. The experiments were carried out in the environment of Matlab2016 and the CPU is Intel(R) Core(TM) i5-8265U with @1.60GHz 1.80 GHz.

4.1 Signal processing

In the test, let original signal has m nonzero components, we choose N = 4,096, M = 2048, and m = 128 according to Eq. 1 . The mean value and variance of Gaussian noise are 0 and 10^–4, respectively. The initial point ω₁ = (1,1,…,1)^T, ω₀ = (0,0,…,0)^T, α₁ = 0.8, α₂ = 0.9, ρ_i = 1.1, ${\theta }_i=\frac{1}{i^3}$ and r = m. The mean squared error (MSE) can be chosen as the evaluation criterion, which is defined as:

$\text{MSE}=\frac{1}{N}\Vert {\omega }^{*}-\omega {\Vert }^2,$

where ω is the original signal, ω* is the recovered signal. We set the stopping criteria MSE ≤10^–5. Figure 1 shows the results of this experiment. These indicate that the number of iterations and CPU time required by our approach are the best of the four methods.

FIGURE 1

FIGURE 1. From top to bottom: original signal, observed signal, and signals recovered by López’s algorithm, Yang’s algorithm, Sakurai and Iiduka’s algorithm, and Algorithm 1.

4.2 Image recovery

The value of each pixel in a grayscale image is in the range [0.255]. The image restoration can be described as the minimizer:

$\underset{\bar{s}\in C}{\min }\Vert \mathcal{A}\bar{s}-y{\Vert }_2,$

where ‖⋅‖₂ is the standard Euclidean norm, y is the observed image, $\bar{s}$ is the approximation of the original image, and $\mathcal{A}$ is a blurring operator. When a color image is processed, we divide it into three channels: red, green, and blue. Supposing the size of the image in each channel is M × N, we have the formula for the MSE:

$\text{MSE}=\frac{1}{MN}\sum _{i=0}^{M-1}\sum _{j=0}^{N-1}\Vert \bar{s}\left(i,j\right)-s\left(i,j\right){\Vert }^2,$

where $\bar{s}$ and s are the restored and original images, respectively.

Seeking to illustrate the effects of image recovery, we use the signal-to-noise ratio (SNR) and peak SNR (PSNR), which are defined:

$\text{SNR}≔20{\log }_{10}{\frac{\Vert \bar{s}{\Vert }_2}{\Vert s-\bar{s}\Vert }}_2,\text{PSNR}≔20{\log }_{10}\frac{255}{\sqrt{\text{MSE}}}.$

In short, larger SNR and PSNR values indicate better restoration of the image. Figure 2 show the results of different color images recovery. Figure 3 shows a comparison of the SNR and PSNR values for images recovered using the four algorithms. The experimental results show that the proposed algorithm always has the largest SNR and PSNR values for different images, which clearly indicates that the proposed algorithm is more effective in recovery than other algorithms.We next applied our method to medical images. Figures 4, 5 show computed tomography (CT) images of a knee joint and a head, and Figure 6 shows a comparison of the SNR values resulting from recovery using each algorithm for these images. From Figure 6, it can be seen clearly that the SNR of our method(the red line) is significantly higher than other methods.

FIGURE 2

FIGURE 2. Comparison of recovered color images of Lena, peppers, house, panda using different algorithms with 1,000 iterations. From left to right: original image, noised image, López’s algorithm, Yang’s algorithm, Sakurai and Iiduka’s algorithm, and Algorithm 1.

FIGURE 3

FIGURE 3. Comparison of SNR (left) and PSNR (right) values resulting from image recovery using the four algorithms with 1,000 iterations.

FIGURE 4

FIGURE 4. Comparison of CT images of a knee joint recovered using different algorithms with 500 iterations. From left to right: original image, noised image, López’s algorithm, Yang’s algorithm, Sakurai and Iiduka’s algorithm, and Algorithm 1.

FIGURE 5

FIGURE 5. Comparison of CT images of a head recovered using different algorithms with 500 iterations. From left to right: original image, noised image, López’s algorithm, Yang’s algorithm, Sakurai and Iiduka’s algorithm, and Algorithm 1.

FIGURE 6

FIGURE 6. Comparison of SNR values of the knee joint (left) and head (right) images resulting from image recovery using the four algorithms with 500 iterations.

Figure 7 shows an original grayscale image. In Figure 8, we investigate the use of our method on this image with different sampling rates. In Figure 9, we show the image recovered by the four algorithms with different sampling rates. Figure 10 shows a comparison of the SNR values of these images. It can clearly be seen that the performance of our method is the best.Finally, it can clearly be seen that our method provides higher SNR and PSNR values than López’s algorithm, Yang’s algorithm, or Sakurai and Iiduka’s algorithm.

FIGURE 7

FIGURE 7. Original image.

FIGURE 8

FIGURE 8. Versions of the image in Figure 7 with sampling rates of 30%, 40%, 50%, and 60% from left to right.

FIGURE 9

FIGURE 9. Comparison of images recovered using the four algorithms. From top to bottom: López’s algorithm, Yang’s algorithm, Sakurai and Iiduka’s algorithm, and Algorithm 1; from left to right: sampling rates of 30%, 40%, 50%, and 60%.

FIGURE 10

FIGURE 10. Comparison of the SNR values of the images in Figure 9.

5 Conclusion

In this article, we propose a new conjugate gradient method for signal recovery. The superiority of our method lies in its employment of the ideas of accelerated conjugate gradient methods with a new adaptive way of choosing the step size. Under some assumptions, the weak convergence of the designed method was established. As application demonstrations, we implemented our method to solve signal-processing and image-restoration problems. The results of our numerical simulations verify the effectiveness and superiority of the new approach. However, in the numerical experiments in this paper, we always assume that the noise is known. In the future work, we will devote to signal and image recovery research without prior knowledge of noise by optimization method.

Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author contributions

PH and KL designed the work, wrote the manuscript, and managed communication among all the authors. PH and KL analyzed the data. All authors contributed to the article and approved the submitted version.

Funding

This project is supported by the Shandong Provincial Natural Science Foundation (Grant No. ZR2019MA022).

Acknowledgments

The authors would like to thank the reviewers and editors for their useful comments, which have helped to improve this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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