CN109447921A - A kind of image measurement matrix optimizing method based on reconstructed error - Google Patents
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Abstract
The present invention proposes a kind of Robust Method of calculation matrix optimization based on MSE.This method is on the basis of the model of tradition optimization calculation matrix, increase a regular terms, the regular terms represents the mean square error of original image and reconstructed image, it is just distributed very much by assuming that mean square deviation error obeys standard, singular value decomposition is carried out using central-limit theorem and dictionary of equal value, the Optimized model of calculation matrix is simplified well, finally use gradient descent algorithm, calculation matrix after iteratively solving out optimization, the information of the image measurement matrix optimizing model abundant application image newly proposed itself, not only reduce the cross-correlation coefficient in calculation matrix between sparse basis, also reduce the requirement to degree of rarefication, compression of images sensory perceptual system robustness is increased to a certain extent.Experiment shows that the independence between the column of the calculation matrix after optimization increases, and more favorably reconstructs the picture signal of high quality.
Description
Technical Field
The invention belongs to the field of signal processing, and particularly relates to an image measurement matrix optimization method based on reconstruction errors
Background
According to the Nyquist sampling theorem, the sampling frequency of the original signal required by the original signal is completely recovered to be more than or equal to twice the highest frequency of the signal, so that huge sampling rate pressure is brought to a hardware system, and large redundant information is acquired, thereby causing waste of a large amount of sampling resources. Compressed Sensing (CS) is a new theory proposed by Donoho, cande, etc., which breaks through the limitation of the conventional nyquist sampling theorem, realizes the simultaneous acquisition and compression of image data, avoids the acquisition of a large amount of redundant information in an image, and relieves the pressure on data storage hardware and transmission lines when the image data is stored and transmitted. The compressed sensing theory mainly comprises three aspects of sparse representation of signals, a measurement matrix and a reconstruction algorithm.
The compressed sensing is a linear measurement process, and X is set to be equal to RNFor the original signal, with length N, by correlating with a measurement matrix phi ∈ RM×NMultiplying to obtain an observed value Y with the length M,
Y＝ΦX (1)
if X is not a sparse signal, performing sparse transformation on X ═ Ψ θ, where Ψ is a sparse basis, θ is a sparse coefficient, and D ═ Φ Ψ perception matrix, an observation process of the CS image signal can be understood as a process of reducing the signal X from N dimension to M dimension, and the specific process is as shown in fig. 1.
When theta satisfies | | theta | | non-calculation0And (5) less than or equal to S, and calling X as an S sparse signal. | θ | non-woven phosphor0Indicating the number of non-zero elements. Recovering the sparse signal X may be by the following equation:
wherein,representing the noise estimate level. In order to accurately recover theta from the formula, an equivalent matrix D epsilon R is usedM×LInstead of Φ Ψ, the cross-correlation of the measurement matrix Φ and the sparse dictionary Ψ is defined as:
although μ (D) can reflect the performance of the measurement matrix to some extent, since the obtained coherence coefficient distribution has a discrete form, the average mutual coherence sparseness can be adopted, which is defined as:
the lower bound of μ (D) is:the condition that the S sparse signal X can be accurately recovered is as follows:
from the above equation, it can be seen that the smaller the value of the correlation coefficient μ (D), the stronger the incoherence, the higher the recovery rate of the original signal, and when the value of μ (D) is decreased, the expression on the right side of the inequality is increased, the upper limit of the expression on the left side is also increased, that is, the value of S is increased, which is equivalent to reducing the requirement for the sparsity of the original signal. On the other hand, if the value of μ (D) is decreased while keeping the value of S unchanged, the reconstruction rate of the original signal will increase.
In the traditional method, under the condition that a sparse dictionary is given, the image reconstruction effect mainly depends on the performance of a measurement matrix, so that the optimization of the performance of the existing measurement matrix has important significance, recent researches show that the overall performance of compressed sensing can be improved by reducing the mutual interference coefficients of the measurement matrix and the sparse dictionary, and the mutual interference coefficients influence the sparsity range of signal adaptation, the measurement number required by signal reconstruction and the reconstruction effect: the smaller the mutual interference coefficient is, the larger the sparsity range of signal adaptation is, the fewer the number of measurement values required for signal reconstruction is, and the better the signal reconstruction effect is. According to the method, firstly, the product of a measurement matrix phi and a sparse dictionary psi is obtained, D & ltphi & gt psi & lt, a corresponding Gram matrix is constructed, off-diagonal elements of the matrix are cross-correlation coefficients of the measurement matrix and the sparse dictionary, then the property of the measurement matrix is improved by researching the characteristics of the Gram matrix, Elad reduces off-diagonal elements of the Gram matrix through a threshold method, and further reduces the correlation between the measurement matrix and the sparse dictionary, and simulation shows that the quality of a recovered image is improved, but the algorithm parameter selection depends on experience, the iteration times are multiple, and new interference may be introduced by contraction operation; a gradient descent method is derived on the algorithm to enable the Gram matrix to approach an identity matrix, the Gram matrix is gradually updated through an Equiangular Tight Frame (ETF), non-diagonal elements of the Gram matrix are equal to the maximum cross-correlation number of the matrix, although the performance of image reconstruction is improved to a certain extent, a larger reconstruction error still exists between an original image and a recovered image, the reconstruction error MSE of a sparse image signal is considered while the measurement matrix is optimized, and a robustness algorithm based on MSE measurement matrix optimization is provided.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a novel MSE-based measurement matrix optimization robustness algorithm. According to the algorithm, the reconstruction error MSE of the sparse image signal is taken into consideration while the measurement matrix is optimized, so that the image reconstruction performance is improved, the reconstruction error is reduced, the coherence of the measurement matrix and the sparse basis is reduced, and the requirement on the compression ratio is also reduced to a certain extent.
The technical scheme of the invention is as follows: a robust method for MSE-based measurement matrix optimization. The method is characterized in that a regular term is added on the basis of a traditional model for optimizing a measurement matrix, the regular term represents the mean square error of an original image and a reconstructed image, the newly proposed optimization model for the image measurement matrix fully applies the information of the image, the error obeys standard positive-false distribution, singular value decomposition is carried out on an equivalent dictionary, the optimization model for the measurement matrix is well simplified, finally, a gradient descent algorithm is used, the optimized measurement matrix is solved in an iterative mode, and experiments show that the independence among optimized measurement matrix columns is increased, and high-quality image signals are better reconstructed. The method mainly comprises the following steps:
setting parameters including total iteration number Iter, iteration number t, initial value 1, column correlation coefficient mu, regular term coefficient β, original image signal X, restored image signalThe random variable n obeys a mean value of 0 and the variance is σ2I Gaussian distribution, and the number of rows and columns of the measurement matrix is respectively set as: m, N, the number of rows and columns of the sparse base is respectively set as: n, L, respectively;
step two: selecting a reasonable identity matrix I, generating a random Gaussian measurement matrix phi and a sparse basis psi, and standardizing the measurement matrix phi, wherein I belongs to RL×L，Φ∈RM×N，Ψ∈RN×LWherein M is less than N and less than L.
Step three: calculating a Gram matrix G, an original image signal X and a restored image signalMean square error of (i.e. G ═ D)TD＝ΨTΦTΦΨ，Where D ═ Φ Ψ denotes the perceptual matrix.
Step four: an image measurement matrix optimization model is constructed, a regular term is added on the basis of the traditional optimized image measurement matrix,namely, it is
Step five: and (6) optimizing the model. Measurement model by CS standard:
Y＝ΦΨθ+n (7)
the actual observation value Y, n represents an error obeying Gaussian distribution, and can be obtained by an OMP algorithm, before an image is reconstructed, firstly, a non-zero part of a sparse signal X is estimated, and a specific expression is as follows:
wherein the reconstruction error between the original image and the recovered image is mainly from the error between the non-zero estimation of the sparse signal X, so that the method can be usedSubstitution of approximationsThe optimization model for the image measurement matrix in equation (6) can be converted into
It can be seen from the above equation that there are two problems of complicated calculation before optimizing the measurement matrix, the processing of ① random variable n brings uncertainty to the calculation due to the randomness ② matrix inversion, even if the pseudo-inverse of the matrix is solved, there is still a disadvantage of high calculation complexity and convergence cannot be ensured, and for ①, assume n ═ { n ═ nkP obeys a gaussian distribution mean of 0 and a variance of σ, independent of each other2I, let S ═ DTD)-1DT，S∈RL×LThen, then
When P approaches to ∞ as well,converge onCan be derived from
As can be seen from the formulas (10) and (11), when P approaches infinity,andis in direct proportion. (9) Can be converted into
Expanding the F norm for ②, and solving the trace of the matrix, specifically as follows:
d is subjected to singular value decomposition: d ═ U Λ VTU, V denotes an arbitrary orthogonal matrix, Λ denotes a diagonal matrix, and Λ ═ diag [ λ [ lambda ] ]1≥λ2≥··≥λi≥··λn]Wherein λ is1≥λ2≥··≥λi≥··λnIs more than or equal to 0. As shown in equation (13), the above expression can be converted into:
(7) the formula can be converted into:
since equation (15) above is a non-convex optimization problem, the upper bound constraint according to (14) can be derived:
whereinRepresenting a diagonal matrix with the first m terms of the diagonal elements being λmax+λminAnd the remaining terms are zero. According to D ═ U Λ VT，Equation (15) can be converted to:
step six: order toSolving for f (phi) gradient
Step seven: and (4) iteratively calculating the measurement matrix phi until t is greater than Iter, and stopping iteration by the formula (13).
Φk+1＝Φk-γ▽f(Φ) (19)
The invention has the advantages that: compared with the prior optimized measurement matrix, (1) under the condition of adding image reconstruction errors, the information of the image is fully considered, and the PSNR of the reconstructed image is well improved. (2) The method greatly reduces the mutual coherence of the measurement matrix and the sparse matrix, reduces the relative error between the original image and the reconstructed image, and increases the independence between the measurement matrix columns.
Drawings
FIG. 1 is a signal observation process for compressed sensing
FIG. 2 is a histogram of cross-correlation coefficient distribution before and after optimization of a measurement matrix
FIG. 3 is a graph of relative error between original and recovered signals before and after optimization of a measurement matrix
FIG. 4 is a graph of PSNR of restored images before and after measurement matrix optimization as a function of sparsity
FIG. 5 is a drawing of the abstract in the embodiment of the present invention
Detailed Description
The invention provides an image measurement matrix optimization method based on mean square error, the experiment of the invention is realized on an MATLAB platform, and the specific operation comprises the following steps:
setting parameters, wherein the total iteration number Iter is 100, the iteration number is t, the initial value is 1, the regular term coefficient is α is 1.1, m is 10, the original image signal X is lena256, the random variable n obeys the mean value 0, and the variance is sigma2I Gaussian distribution, and the number of rows and columns of the measurement matrix is respectively set as: m is 20, N is 64, and the number of the rows and the columns of the sparse base are respectively set as: n-64, L-100
Step two: selecting a 100X 100 identity matrix I, generating a 20X 64 random Gaussian measurement matrix phi, standardizing the measurement matrix phi, and training by KSVD to obtain psi of sparse basis 64X 100.
Step three: and (3) iteratively calculating the measurement matrix phi until t is greater than Iter, and stopping iteration by the formula (19). Multiplying phi after iteration by a sparse basis psi, respectively calculating an overall correlation coefficient mu (D) and an average cross correlation coefficient by the following formulaThe cross-correlation coefficient histogram, as shown in Table 1, is calculated as shown in FIG. 2
TABLE 1
Step four: constructing a one-dimensional 100 multiplied by 1 sparse signal sequence with sparsity of 4 and standard normal distribution of each column, and recording the sparse signal sequence as { sk1000, by z, 1,2k＝Φyk＝ΦΨskObtain a test signal sequence, denoted as { ykGet the observation value sequence { z }kAnd (4) reconstructing a sparse sequence through an OMP algorithmRecovery of signal sequencesNsThe performance of the optimized measurement matrix can be checked using the relative error formula as 1000:
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.
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