A sparse total variation image reconstruction method based on primal-dual algorithm

Abstract

The present invention discloses a sparse total variation image reconstruction method based on the primal-dual algorithm. First, a mathematical optimization problem is established by combining image prior information; then, the dual norm is introduced to transform the composite regularized optimization problem into a saddle point problem; then, the primal-dual problem framework is constructed, and the solution of the primal problem provides an estimate of the reconstructed image; the proximal mapping operator is used to obtain an iterative scheme of the primal-dual variables; finally, the primal variables and the dual variables are updated alternately until the error between the primal variables and the true value tends to converge, and the final reconstructed image is obtained. Based on the image reconstruction background, the present invention uses the primal-dual scheme to process the composite regularized optimization problem, expresses the optimization problem as a saddle point problem, and finally solves it step by step by decomposing it into the primal problem and the dual problem. It can not only handle the non-differentiability of the non-smooth regularization function, but also improve the computational efficiency by decomposing it into sub-problems, and ensure the stability and accuracy of the solution.

Description

Technical Field

[0001] The present invention relates to the technical field of image signal processing, and in particular to a sparse total variation image reconstruction method based on a primal-dual algorithm. Background Art

[0002] Image reconstruction technology plays a crucial role in image processing. Its primary task is to recover missing image information by processing and analyzing partial known information. This technology has broad applications in medical imaging, radar imaging, astronomy, and geophysics. Image reconstruction is often formulated as a linear inverse problem: reconstructing a complete image from partial observational data.

[0003] Traditional image reconstruction methods include linear least squares, back-projection, and regularization-based optimization methods. These methods can solve the image reconstruction problem to a certain extent, but when faced with complex practical application scenarios, it is often difficult to strike a balance between computational efficiency and reconstruction effect. Due to the non-differentiability of the sparsity and total variation (TV) regularization terms, the optimization process becomes more complicated. In addition, parameter selection is also an important challenge. How to reasonably select the regularization parameters to balance sparsity and smoothness is crucial to the reconstruction effect, but this process usually requires a lot of experiments and debugging. Especially when the observation data is noisy or the sampling rate is low, the quality of the reconstructed image may be significantly reduced. Summary of the Invention

[0004] To achieve high-quality image reconstruction under noisy and incomplete data conditions, this paper proposes a sparse total variation (CTPD) compressed sensing total variation primal dual (CTV) image reconstruction method based on the primal-dual algorithm. Due to the excellent performance of sparse representation and total variation regularization methods in denoising and preserving image details, this paper introduces the l1 norm and total variation (TV) norm to construct a composite regularization model. To facilitate the solution, the composite regularization optimization problem is formulated as a saddle point optimization problem. By introducing the dual norm, the primal problem is decomposed into several interrelated subproblems. Each subproblem is solved independently to reduce computational complexity. In each iterative step, the primal and dual variables are updated using the proximal gradient method. Simulation results demonstrate the effectiveness of the proposed algorithm, which is verified by simulation experiments on a publicly available image dataset. The proposed CTV algorithm outperforms existing methods in terms of reconstructed image accuracy, convergence, and image entropy.

[0005] A sparse total variation image reconstruction method based on a primal-dual algorithm comprises the following steps:

[0006] Step 1: Combine image prior information to establish a mathematical optimization problem;

[0007] Step 2: Introduce the dual norm to transform the composite regularized optimization problem into a saddle point problem;

[0008] Step 3: Construct the original dual problem framework, where the solution to the original problem provides the reconstructed image x * estimates;

[0009] Step 4: Use the proximal mapping operator to obtain the iterative scheme of the primal-dual variables;

[0010] Step 5: Alternately update the original variables and the dual variables until the error between the original variables and the true value converges and the final reconstructed image x is obtained. * .

[0011] Preferably, in step 1, the purpose of image reconstruction is to recover the signal vector from the missing data, and the observed signal is expressed in vector form as:

[0012] y=Hx * +v

[0013] Where y is the observation vector of dimension M×1, x * Represents the N×1 dimensional unknown image vector to be reconstructed, H∈{0,1} is the M×N dimensional sampling matrix (M≤N). And M=N×sampling rate, v is the M×1 dimensional noise vector. The reconstructed image x is recovered from the observation value y * , the formula is as follows:

[0014]

[0015] In practical scenarios, most images are sparse and smooth. These prior characteristics can be used to enhance the performance of image reconstruction. By introducing regularization constraints, the image reconstruction problem is reformulated as a composite regularized optimization problem involving compressed sensing and total variation:

[0016]

[0017] where λ1 and λ2 are regularization parameters, ||x||1 is the l1 norm as a sparsity regularization term, and ||x|| TV Represents the total variation of the image, and the calculation formula is as follows:

[0018]

[0019] In the formula is the gradient operator, definition W ij Represents pixel x i and xj (i,j∈[1,2,…,N]), in the image, if the pixel x j Located at pixel x i The lower or right side of the adjacent, then define W ij =1; otherwise define W ij = 0. This constitutes the weight matrix W.

[0020] Preferably, in step 2, a dual norm is introduced to transform the composite regularization optimization problem into a saddle point optimization problem;

[0021] Let ||·|| represent an arbitrary norm, and its dual norm is defined as ||a|| * =max ||b||≤1 <a,b> According to the theorem, we can get ||x|| 1* =||x|| ∞ ,||x|| ∞* =||x||1,||x|| 2* =||x||2, we can get the following expression:

[0022]

[0023]

[0024] where z and h are dual variables. TV Expressed as the sum of the dual norms of N l2 norms:

[0025]

[0026] where Z is a composite vector matrix: z i express The dual variable of the dual norm of , and Represents a constrained set, Each column vector z i All satisfy their l2 norm is less than or equal to 1. This constraint set It ensures that the l2 norm of each column vector of Z is within a given range, thereby limiting the solution space of the problem. According to the dual norm of the l1 norm and the TV norm, the composite regularized optimization problem is transformed into a saddle point optimization problem:

[0027]

[0028] in, x * ,h * ,Z * Indicates the optimal solution.

[0029] As a preferred method, in step 3, since it is difficult to directly solve the saddle point solution, a primal-dual problem framework is proposed to decompose the saddle point optimization problem into interrelated subproblems (primal problem and dual problem); the constructed primal-dual problem framework is as follows:

[0030]

[0031]

[0032]

[0033] in Indicator function The definition is as follows:

[0034]

[0035]

[0036] As a preference, in step 4, the objective functions of the above sub-problems can be regarded as the sum of two convex functions. In order to solve the problem of minimizing the sum of two convex functions, the problem form is as follows: min x We use the proximal gradient method to solve q(x)+g(x). This method is implemented through the proximal operator iteration formula:

[0037]

[0038] where x (k+1) Indicates the result of the k+1th iteration, prox μg (·) is a proximal operator defined for a given convex function g:

[0039]

[0040] It can be concluded that the original dual problem framework can be solved step by step by the proximal gradient method, and the problem solving framework is:

[0041]

[0042] Among them, υ>0, μ>0 are both step sizes, which are used to update the original variables and the dual variables respectively. represents the intermediate variable. Note that The negative adjoint operator of produces the divergence operator because Then there is Map the matrix Z into a vector divZ whose i-th element is equal to:

[0043]

[0044] As a preference, in step 5, the initial value x of the given variable (0) , h (0) ,Z (0) , alternately calculating and updating the original variable x (k) and the dual variable h (k) ,Z (k) .

[0045] ①Update the original variable x (k) :

[0046]

[0047] Taking the derivative of the cost function and setting it equal to 0 gives:

[0048]

[0049] Get Update x (k) The expression:

[0050]

[0051] ②Update the dual variable h (k) :

[0052]

[0053]

[0054] Update h (k) The elements of are expressed as follows:

[0055]

[0056] ③Update the dual variable Z (k) :

[0057]

[0058]

[0059] Get Update Z (k) The expression of the i-th row vector is:

[0060]

[0061] Repeat the above iterative process ①②③ until the original variable x (k+1) The error between the true value and the image tends to converge steadily, and the final reconstructed image x is obtained. * .

[0062] The beneficial effects of the present invention are as follows:

[0063] Based on the context of image reconstruction, this paper uses a primal-dual scheme to address composite regularized optimization problems, formulating the optimization problem as a saddle point problem and then gradually solving it by decomposing it into a primal problem and a dual problem. This method not only handles the non-differentiability of nonsmooth regularized functions, but also improves computational efficiency by decomposing it into subproblems, ensuring the stability and accuracy of the solution. Numerical experimental results show that compared with existing methods, the proposed algorithm performs better in image reconstruction, and the entropy of the reconstructed image is lower, stabilizing between 0.6 and 0.9, verifying the effectiveness of the algorithm in preserving image information and details. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 : A specific implementation flow chart of the embodiment of this method;

[0065] Figure 2 :(a) is the original image, (b) is the noisy image, and the reconstructed image is obtained after (c) CTPD, (d) LASSO, (e) TV, and (f) LS. The signal-to-noise ratio is 15dB and the sampling rate is 100%;

[0066] Figure 3 : A graph showing the relationship between relative error and the number of iterations and signal-to-noise ratio;

[0067] Figure 4 : Reconstruction results of CTPD, FISTA, BP and ADMM algorithms under different target models, where SNR = 18dB and sampling rate is 80%;

[0068] Figure 5 : A graph showing the relationship between relative error and sampling rate under different target models;

[0069] Figure 6 : For different regularization parameters, the relative error of CTPD and FISTA algorithms changes with the number of iterations. DETAILED DESCRIPTION

[0070] The technical solution of the present invention will be further described in detail below through specific embodiments in conjunction with the accompanying drawings.

[0071] like Figure 1 As shown, a sparse total variation image reconstruction method based on the primal-dual algorithm includes the following steps:

[0072] Step 1: Combine image prior information and use l1 and TV norm to construct a composite regularized optimization problem;

[0073]

[0074] The regularization parameters are set to: λ1 = 0.2, λ2 = 0.05.

[0075] Step 2: Introduce ||x||1 and ||x|| TV The dual norm of , transforms the composite regularization optimization problem into a saddle point optimization problem:

[0076]

[0077] Step 3: Since it is difficult to directly solve the saddle point solution, a primal-dual problem framework is proposed to decompose the saddle point optimization problem into interrelated sub-problems (primal problem and dual problem):

[0078]

[0079]

[0080]

[0081] Step 4: Use the proximal mapping operator to obtain the primal-dual iterative scheme:

[0082]

[0083] The iteration step size is set to: μ = 0.1, υ = 0.003.

[0084] Step 5: Alternately calculate and update the original variable x (k) and the dual variable h (k) ,Z (k) :

[0085] ① Update x (k) : in Taking the derivative of the cost function and setting it equal to 0 gives:

[0086]

[0087] Get Update x (k) The expression:

[0088]

[0089] ② Update h (k) : in Get h (k+1) The elements of are expressed as follows:

[0090]

[0091] ③ Update Z (k) : in Z (k+1) The i-th row vector expression of is:

[0092]

[0093] Repeat the above iterative process ①②③ until the iterative error converges and the image to be reconstructed x is obtained. * .

[0094] This example uses numerical results to evaluate the performance of the proposed sparse total variation image reconstruction method based on the primal-dual algorithm. Simulations are performed using a publicly available image dataset. All simulations are performed in a Gaussian white noise environment, and all simulation results are averaged over 100 independent Monte Carlo experiments. The relative error (RE) is used to evaluate the accuracy of the reconstructed image: where x and represent the real image and the estimated image respectively.

[0095] Attachment Figure 2 Figure 2 shows the reconstruction results of the CTPD algorithm, LASSO algorithm (when λ2=0), TV algorithm (when λ1=0), and LS under the conditions of a signal-to-noise ratio of 15dB and a sampling rate of 100%. The results show that the CTPD algorithm has the best reconstructed image performance, while the LS algorithm has the worst reconstruction effect. Compared with the CTPD algorithm, the performance of the LASSO and TV algorithms is also relatively poor, mainly because they ignore the sparsity or piecewise smoothness in the image x. In addition, the curve of RE changing with the number of iterations and signal-to-noise ratio is shown in the attached figure. Figure 3 As shown in the figure, the CTPD algorithm shows lower relative error than other algorithms at different signal-to-noise ratio levels.

[0096] Attachment Figure 4 The reconstruction results obtained by CTPD, FISTA, BP and ADMM algorithms for different reconstruction targets are shown under the conditions of SNR of 18dB and sampling rate of 80%. Figure 5 The trend graph of the relative error of the reconstruction results changing with the sampling rate under different reconstruction targets is presented. It is observed that the proposed CTPD algorithm performs better in reconstruction performance, FISTA ignores some important details, such as the edge information of the image, and the performance of BP and ADMM is relatively poor. As the sampling rate increases, the relative error improvement of the CTPD algorithm is better than that of the FISTA and BP algorithms. This shows that the CTPD algorithm has stronger adaptability in maintaining the quality of image reconstruction, especially at higher sampling rates, the performance improvement of the CTPD algorithm is 2 to 5dB higher than that of the FISTA and BP algorithms. Figure 6The relationship between the relative error of the CTPD and FISTA algorithms and the number of iterations is shown for different penalty parameter settings. Clearly, the convergence rate of CTPD remains nearly constant for different penalty parameters, demonstrating good convergence stability for different reconstruction objectives. In contrast, FISTA is sensitive to parameter changes, with its convergence significantly affected by these changes.

[0097] For the three reconstruction targets mentioned above, the average entropy value (IE, image entropy) of image reconstruction at different sampling rates is calculated. The entropy of image X is defined as: Among them, bin i Represents the discrete interval of the i-th pixel value, p i represents the probability distribution of the i-th pixel value interval, and C is the total number of discrete intervals. The entropy results are presented in Table 1. We can observe that the CTPD algorithm consistently maintains a low entropy value, indicating a higher concentration of image pixels and less randomness. In comparison, the FISTA and BP algorithms are slightly inferior, while the ADMM algorithm reconstructs the most complex image pixel value distribution.

[0098] Table 1 Entropy values ​​of reconstructed images at different sampling rates

[0099]

[0100] It should be understood that the above description of the preferred embodiment is relatively detailed and cannot be regarded as limiting the scope of protection of the patent of the present invention. Under the guidance of the present invention, ordinary technicians in this field can make substitutions or modifications without departing from the scope of protection of the claims of the present invention, which fall within the scope of protection of the present invention. The scope of protection requested by the present invention shall be based on the attached claims.

Claims

1. A sparse total variation image reconstruction method based on the primal-dual algorithm, characterized in that: The following steps are involved: Step 1: Combine image prior information to establish a mathematical optimization problem; Step 2: Introduce the dual norm to transform the composite regularized optimization problem into a saddle point problem; Let ||·|| represent an arbitrary norm, and its dual norm is defined as ||a|| * max ||b||≤1 <a,b> ; According to the theorem, we can get ||x|| 1* =||x|| ∞ ,||x|| ∞* =||x||1,||x|| 2* =||x||2, we can get the following expression: where z and h are dual variables; re-||x|| TV Expressed as the sum of the dual norms of N l2 norms: where Z is a composite vector matrix: z i express The dual variable of the dual norm of , and Represents a constrained set, Each column vector z i All satisfy their l2 norm is less than or equal to 1; this constraint set It ensures that the l2 norm of each column vector of Z is within a given range, thereby limiting the solution space of the problem; according to the dual norm of the l1 norm and the TV norm, the composite regularized optimization problem is transformed into a saddle point optimization problem: in, x * ,h * ,Z * represents the optimal solution; Step 3: Construct the original dual problem framework, where the solution of the original problem provides an estimate of the reconstructed image; Step 4: Use the proximal mapping operator to obtain the iterative scheme of the primal-dual variables; Step 5: Alternately update the original variables and the dual variables until the error between the original variables and the true values ​​converges and the final reconstructed image is obtained.

2. The sparse total variation image reconstruction method based on the primal-dual algorithm according to claim 1, characterized in that: In step 1, the purpose of image reconstruction is to recover the signal vector from the missing data. The observed signal is expressed in vector form as: y=Hx * +v Where y is the observation vector of dimension M×1, x * Represents the N×1 dimensional unknown image vector to be reconstructed, H∈{0,1} is the M×N dimensional sampling matrix, M≤N; and M=N×sampling rate, v is the M×1 dimensional noise vector; by recovering the reconstructed image x from the observation value y * , the formula is as follows: Regularization constraints are introduced to reformulate the image reconstruction problem as a composite regularized optimization problem involving compressed sensing and total variation: where λ1 and λ2 are regularization parameters, ||x||1 is the l1 norm as a sparsity regularization term, and ||x|| TV Represents the total variation of the image, and the calculation formula is as follows: In the formula is the gradient operator, definition W ij Represents pixel x i and x j The association strength, i,j∈[1,2,…,N], in the image, if the pixel x j Located at pixel x i The lower or right side of the adjacent, then define W ij =1; otherwise define W ij =0.

3. The sparse total variation image reconstruction method based on the primal-dual algorithm according to claim 2, characterized in that: In step 3, since it is difficult to directly solve the saddle point solution, a primal-dual problem framework is proposed to decompose the saddle point optimization problem into interrelated subproblems; the constructed primal-dual problem framework is as follows: in Indicator function δ S (Z),δ Q (h) is defined as follows:

4. The sparse total variation image reconstruction method based on the primal-dual algorithm according to claim 3, characterized in that: In step 4, the objective functions of the above sub-problems can be regarded as the sum of two convex functions. In order to solve the problem of minimizing the sum of two convex functions, the problem form is as follows: min x q(x)+g(x) is solved using the proximal gradient method; this method is implemented through the proximal operator iteration formula: where x (k+1) Indicates the result of the k+1th iteration, prox μg (·) is a proximal operator defined for a given convex function g: It can be concluded that the original dual problem framework can be solved step by step by the proximal gradient method, and the problem solving framework is: Among them, υ>0, μ>0 are both step sizes, which are used to update the original variables and the dual variables respectively. represents the intermediate variable; note that The negative adjoint operator of produces the divergence operator because Then there is Map the matrix Z into a vector divZ whose i-th element is equal to:

5. The sparse total variation image reconstruction method based on the primal-dual algorithm according to claim 4, characterized in that: In step 5, the initial value x of the given variable (0) , h (0) ,Z (0) , alternately calculating and updating the original variable x (k) and the dual variable h (k) ,Z (k) ; ①Update the original variable x (k) : Taking the derivative of the cost function and setting it equal to 0 gives: Get Update x (k) The expression: ②Update the dual variable h (k) : Update h (k) The elements of are expressed as follows: ③Update the dual variable Z (k) : Get Update Z (k) The expression of the i-th row vector is: Repeat the above iterative process ①②③ until the original variable x (k+1) The error between the image and the true value tends to converge stably, and the final reconstructed image is obtained.

Images