CN109447921A - A kind of image measurement matrix optimizing method based on reconstructed error - Google Patents

Abstract

The present invention proposes a robust method for MSE-based measurement matrix optimization. The method is to add a regular term based on the traditional optimized measurement matrix model, which represents the mean square error between the original image and the reconstructed image. By assuming that the mean square error obeys the standard normal distribution, the central limit theorem and the The valence dictionary performs singular value decomposition, which simplifies the optimization model of the measurement matrix. Finally, the gradient descent algorithm is used to iteratively solve the optimized measurement matrix. The newly proposed image measurement matrix optimization model fully uses the information of the image itself, not only reduces the The cross-correlation coefficient between the measurement matrix and the sparse basis is reduced, and the requirement for sparsity is also reduced, which increases the robustness of the image compressed sensing system to a certain extent. Experiments show that the increased independence between the columns of the optimized measurement matrix is more favorable for reconstructing high-quality image signals.

Description

An Optimization Method of Image Measurement Matrix Based on Reconstruction Error

technical field

The invention belongs to the field of signal processing, in particular to an image measurement matrix optimization method based on reconstruction error

Background technique

According to the Nyquist sampling theorem, the sampling frequency of the signal is required to be greater than or equal to twice the highest frequency of the signal to completely recover the original signal. Waste of sampling resources. Compressive Sensing (CS) is a new theory proposed by Donoho, Candè and others, which breaks through the limitations of the traditional Nyquist sampling theorem and realizes the acquisition and compression of image data at the same time, avoiding the need for The acquisition of a large amount of redundant information in the image relieves the pressure on the data storage hardware and transmission lines when the image data is stored and transmitted. The theory of compressed sensing mainly includes three aspects: the sparse representation of the signal, the measurement matrix and the reconstruction algorithm.

Compressed sensing is a linear measurement process. Let X∈R ^N be the original signal and the length is N. By multiplying with the measurement matrix Φ∈R ^M×N , the observation value Y of length M is obtained,

Y=ΦX (1)

If X is not a sparse signal, it can be sparsely transformed to obtain X=Ψθ, where Ψ is the sparse basis, θ is the sparse coefficient, D=ΦΨ perception matrix, the observation process of the CS image signal can be understood as the signal X is reduced from N dimension to The M-dimensional process, the specific process is shown in Figure 1.

When θ satisfies ||θ|| ₀ ≤S, X is said to be an S-sparse signal. ||θ|| ₀ represents the number of non-zero elements. The sparse signal X can be recovered by the following formula:

in, Indicates the noise estimate level. In order to accurately recover θ from the formula, the equivalent matrix D∈R ^M × ^L is used instead of ΦΨ, and the cross-correlation between the measurement matrix Φ and the sparse dictionary Ψ is defined as:

Although μ(D) can reflect the performance of the measurement matrix to a certain extent, since the obtained coherence coefficient distribution is discrete, the average mutual coherence sparse can be used, which is defined as:

The lower bound of μ(D) is: The condition that S sparse signal X can be accurately recovered is:

It can be seen from the above formula that the smaller the value of the correlation coefficient μ(D), the stronger the incoherence, and the higher the recovery rate of the original signal. When the value of μ(D) is reduced, the expression on the right side of the inequality increases, Then the upper limit of the left equation can also be increased, that is, the value of S increases, which is equivalent to reducing the requirement on the sparsity of the original signal. On the other hand, if the value of S is kept unchanged and the value of μ(D) is reduced, the reconstruction rate of the original signal will increase.

In the traditional method, when the sparse dictionary is given, the image reconstruction effect mainly depends on the performance of the measurement matrix. Therefore, it is of great significance to optimize the performance of the existing measurement matrix. Recent studies have shown that by reducing the measurement matrix and sparseness The mutual interference coefficient of the dictionary can improve the overall performance of compressed sensing. The mutual interference coefficient affects the sparsity range of signal adaptation, the number of measurements required to reconstruct the signal, and the reconstruction effect: the smaller the mutual interference coefficient, the larger the sparsity range of signal adaptation. The fewer the number of measurements required to reconstruct the signal, the better the reconstruction of the signal. This kind of method first obtains D=ΦΨ by measuring the product of the matrix Φ and the sparse dictionary Ψ, and constructs its corresponding Gram matrix. The off-diagonal elements of the matrix are the cross-correlation coefficients between the measurement matrix and the sparse dictionary, and then by studying the Gram matrix. The characteristics of the matrix improve the performance of the measurement matrix. Elad reduces the off-diagonal elements of the Gram matrix through the threshold method, thereby reducing the correlation between the measurement matrix and the sparse dictionary. The simulation shows that the quality of the restored image is improved, but the parameter selection of the algorithm Relying on experience, the number of iterations is large, and the shrinking operation may introduce new interference; on this algorithm, the gradient descent method is used to make the Gram matrix approximate to the identity matrix, and the equiangular tight frame (ETF) is gradually updated. Gram matrix, let the off-diagonal elements of the Gram matrix be equal to the maximum cross-correlation coefficient of the matrix. Although the performance of image reconstruction is improved to a certain extent, there is still a relatively large reconstruction error between the original image and the restored image. This paper While optimizing the measurement matrix, considering the reconstruction error MSE of the sparse image signal, a robust algorithm of MSE-based measurement matrix optimization is proposed.

SUMMARY OF THE INVENTION

The purpose of the present invention is to propose a new robust algorithm of measurement matrix optimization based on MSE in view of the deficiencies of the prior art. While optimizing the measurement matrix, the algorithm considers the reconstruction error MSE of the sparse image signal, which not only improves the performance of image reconstruction, reduces the reconstruction error, but also reduces the mutual coherence between the measurement matrix and the sparse basis. To a certain extent, the requirements for the compression ratio are also alleviated.

The technical solution of the present invention is a robust method of MSE-based measurement matrix optimization. This method adds a regular term based on the traditional optimization measurement matrix model, which represents the mean square error between the original image and the reconstructed image. The newly proposed image measurement matrix optimization model fully utilizes the information of the image itself, through It is assumed that the error obeys the standard normal distribution, and the singular value decomposition of the equivalent dictionary simplifies the optimization model of the measurement matrix. Finally, the gradient descent algorithm is used to iteratively solve the optimized measurement matrix. Experiments show that the optimized measurement matrix The increased independence between the matrix columns is more favorable for reconstructing high-quality image signals. The main steps are as follows:

Step 1: Set the parameters: the total number of iterations Iter, the number of iterations is t, the initial value is 1, the column correlation coefficient μ, the regular term coefficient is β, the original image signal X, the restored image signal The random variable n obeys the mean value of 0, the variance is σ ² I Gaussian distribution, the number of rows and columns of the measurement matrix is set to: M, N, and the number of rows and columns of the sparse basis is set to: N, L respectively;

Step 2: Select a reasonable identity matrix I, generate a random Gaussian measurement matrix Φ and a sparse basis Ψ, and standardize the measurement matrix Φ, where I∈R ^L×L , Φ∈R ^M×N , Ψ∈R ^N×L , where M<N<L.

Step 3: Calculate the Gram matrix G, the original image signal X and the restored image signal The mean square error of , namely G=D ^T D=Ψ ^T Φ ^T ΦΨ, where D=ΦΨ represents the perception matrix.

Step 4: Construct an image measurement matrix optimization model. On the basis of the traditional optimized image measurement matrix, a regular term is added. which is

Step 5: Model optimization. Measurement model by CS standard:

Y=ΦΨθ+n (7)

Among them, the actual observation value Y, n represents the error obeying the Gaussian distribution, which can be obtained by the OMP algorithm. Before reconstructing the image, the non-zero part of the sparse signal X should be estimated first. The specific expression is:

Among them, the reconstruction error between the original image and the restored image mainly comes from the error between the estimation of the non-zero part of the sparse signal X, so it can be used approximate replacement Then the optimization model of the image measurement matrix in (6) can be transformed into

It can be seen from the above formula that there are two computationally complex problems before optimizing the measurement matrix. ① The processing of the random variable n brings uncertainty to the calculation due to the existence of randomness. ②The matrix inversion, even if the pseudo-inverse of the matrix is solved, still has the disadvantage of high computational complexity, and cannot ensure convergence. For ①, assuming that n={n _k }, k=1, 2...P obeys mutually independent Gaussian distributions with a mean of 0 and a variance of σ ² I, let S=(D ^T D) ^-1 D ^T , S ∈R ^L×L , then

When P approaches ∞, converge on can be drawn

From equations (10) and (11), it can be known that when P approaches infinity, and proportional. (9) can be transformed into

Expand the F norm for ② and solve the trace of the matrix, as follows:

Perform singular value decomposition on D: D=UΛV ^T , U, V represent any orthogonal matrix, Λ represents a diagonal matrix, Λ=diag[λ ₁ ≥λ ₂ ≥··≥λ _i ≥··λ _n ], where λ ₁ ≥λ ₂ ≥··≥λ _i ≥··λ _n ≥0. It can be seen from (13) that the above expression can be transformed into:

(7) can be transformed into:

Since the above equation (15) is a non-convex optimization problem, according to the upper bound constraints of (14), we can get:

in represents a diagonal matrix, the first m terms of the diagonal elements are λ _max + λ _min , and the remaining terms are zero. According to D=UΛV ^T , Equation (15) can be transformed into:

Step 6: Order Solving for the f(Φ) gradient

Step 7: Iteratively calculate the measurement matrix Φ, until t>Iter, the iteration is stopped by formula (13).

Φ _k+1 = Φ _k -γ▽f(Φ) (19)

The advantages of the present invention: compared with the previous optimized measurement matrix, (1) in the case of adding the image reconstruction error, the information of the image itself is fully considered, and the PSNR of the reconstructed image is well improved. (2) The mutual coherence between the measurement matrix and the sparse basis is greatly reduced, the relative error between the original image and the reconstructed image is reduced, and the independence between the columns of the measurement matrix is increased.

Description of drawings

Figure 1 shows the signal observation process of compressed sensing

Figure 2 is the histogram of the cross-correlation coefficient distribution before and after the optimization of the measurement matrix

Figure 3 is the relative error diagram of the original signal and the recovered signal before and after the optimization of the measurement matrix

Figure 4 is a graph showing the variation of PSNR with sparsity of the restored image before and after optimization of the measurement matrix

FIG. 5 is a drawing of the abstract in the embodiment of the present invention

Detailed ways

An image measurement matrix optimization method based on mean square error proposed by the present invention, the experiment of the present invention is realized on the MATLAB platform, and the specific operation includes the following steps:

Step 1: Set parameters, the total number of iterations Iter=100, the number of iterations is t, the initial value is 1, the regular term coefficient is α=1.1, m=10, the original image signal X is lena256*256, the random variable n obeys the mean value of 0 , the variance is σ ² I Gaussian distribution, the number of rows and columns of the measurement matrix is set as: M=20, N=64, and the number of rows and columns of the sparse basis is set as: N=64, L=100

Step 2: Select a 100×100 identity matrix I, generate a 20×64 random Gaussian measurement matrix Φ, and standardize the measurement matrix Φ, and obtain a sparse base 64×100 Ψ by KSVD training.

Step 3: Calculate the measurement matrix Φ by iteration until t>Iter, and the iteration is stopped by the formula (19). Multiply the Φ after iteration by the sparse basis Ψ, calculate the overall correlation coefficient μ(D) by the following formulas, and the average cross-correlation coefficient As shown in Table 1, the cross-correlation coefficient histogram, the calculation results are shown in Figure 2

Table 1

Step 4: Construct a one-dimensional 100×1 sparse signal sequence with a sparsity of 4 for each column and a standard normal distribution, denoted as {s _k }(k=1,2,...1000), through z _k =Φy _k =ΦΨs _k to obtain the test signal sequence, denoted as {y _k }, and similarly, the observation value sequence {z _k } can be obtained. Through the OMP algorithm, the reconstructed sparse sequence recovery signal sequence N _s = 1000 can check the performance of the optimized measurement matrix using the relative error formula:

Claims (4)

1. An image measurement matrix optimization method based on reconstruction errors is characterized by comprising the following steps: firstly, constructing a random measurement matrix which follows Gaussian distribution, standardizing the random measurement matrix, then obtaining sparse basis from an image training set by using KSVD, constructing a Gram matrix through the product of the measurement matrix and the sparse basis, solving the F norm of the Gram matrix and an identity matrix to obtain an original measurement matrix optimization model, adding a regular term on the basis of the model, wherein the regular term is formed by the mean square error of an original image signal and a restored image signal, and finally selecting a proper regular term coefficient to obtain the image measurement matrix model based on the reconstruction error.

2. The reconstruction error based image measurement matrix model of claim 1, wherein: the model is composed of two parts, and the specific expression is as follows:

the first part mainly aims at reducing the cross correlation coefficient between the measurement matrix and the sparse basis, the second part adds the mean square error of the image signal into the model, and the error is used as available image information, so that the performance of the measurement matrix can be better improved.

3. The regularization term according to claim 1 comprised of the mean square error of the original image signal and the recovered image signal, wherein: the mean square error is mainly from the non-zero part estimation of the sparse signal X and can be obtained by an OMP algorithmSubstitution of approximationsSimplified optimization model

4. The method as claimed in claims 1 and 2, wherein the closer the Gram matrix is to the unit matrix, the better the non-correlation between the measurement matrix and the sparse basis, and the lower the requirement for sparsity, and the mean square error between images is added to further optimize the performance of the measurement matrix, not only improving the PSNR of the restored image, but also increasing the robustness of the image compression sensing system to a certain extent.