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 field of signal processing, specially a kind of image measurement matrix optimizing method based on reconstructed error
Background technique
The sample frequency of signal is required to be more than or equal to letter it is found that recovering original signal completely by Nyquist sampling thheorem
Twice of number highest frequency, this not only brings huge sampling rate pressure to hardware system, but also collects the superfluous of Dalian
Remaining information causes the waste for largely sampling resource.Compressed sensing (Compressive Sensing, CS) is by Donoho, Cand
E et al. advances a new theory, and the theoretical breakthrough limitation of traditional nyquist sampling theorem realizes image data
Acquisition and compression at the same carry out, avoid and the bulk redundancy information in image acquired, alleviate image data and storing
With transmission when to data storage hardware and transmission line bring pressure.Compressive sensing theory mainly includes the sparse table of signal
Show, three aspects of calculation matrix and restructing algorithm.
Compressed sensing is a linear measurement process, if X ∈ RNFor original signal, length N, by with calculation matrix Φ
∈RM×NIt is multiplied, obtains the observation Y that length is M,
Y=Φ X (1)
If X is not sparse signal, the available X=Ψ θ of sparse transformation is carried out, wherein Ψ is sparse basis, and θ is dilute
Sparse coefficient, D=Φ Ψ perceive matrix, and the observation process of CS picture signal can be understood as signal X and be reduced to the mistake that M is tieed up from N-dimensional
Journey, detailed process such as Fig. 1.
When θ meets | | θ | |0≤ S, claiming X is S sparse signal.||θ||0Indicate the number of nonzero element.Restore sparse signal X
Following formula can be passed through:
Wherein,Indicate that noise estimation is horizontal.In order to accurately recover θ from formula, with equivalent matrice D ∈ RM
×L, instead of Φ Ψ, the cross correlation of calculation matrix Φ and sparse dictionary Ψ is defined as:
Although μ (D) is able to reflect the performance of calculation matrix to a certain extent, have due to obtaining coherence factor distribution
Discrete type, can be sparse using average mutual coherence, is defined as:
The lower bound of μ (D) are as follows:Wherein S sparse signal X can
Accurately it is resumed condition are as follows:
The value of related coefficient μ (D) is smaller as can be seen from the above equation, and incoherence is stronger, then the recovery rate of original signal is got over
Height, the formula on the right of the value for reducing μ (D), inequality increase, then left side formula can also increase the upper limit, i.e. the value of S increases, phase
When in reducing the requirement to the degree of rarefication of original signal.On the other hand, if keeping the value of S constant, reduce the value of μ (D), then
The reconstruct rate of original signal will increase.
In conventional methods where, in the case where sparse dictionary is given, image reconstruction effect depends on calculation matrix
Performance, therefore, the performance for optimizing existing calculation matrix are of great significance, recently research have indicated that, by reduce calculation matrix with
The overall performance of compressed sensing can be improved in the mutual coherence factor of sparse dictionary, and mutual coherence factor affects the sparse of signal adaptation
Spend measurement number needed for range and reconstruction signal and rebuild effect: mutual coherence factor is smaller, the degree of rarefication range that signal adapts to
Bigger, the number for the measured value that reconstruction signal needs is fewer, and the reconstruction effect of signal is also better.Such method passes through measurement first
The product of matrix Φ and sparse dictionary Ψ obtain D=Φ Ψ, construct its corresponding Gram matrix, the matrix off diagonal element
It is the cross-correlation coefficient of calculation matrix and sparse dictionary, calculation matrix is then improved by the characteristic of research Gram matrix
Property, Elad reduces Gram matrix off diagonal element by threshold method, and then it is related to sparse dictionary to reduce calculation matrix
Property, emulation shows to improve the quality for restoring image, but algorithm parameter selection relies on experience, and the number of iterations is more, Er Qieshou
Contracting operation may introduce new interference；Derive on this algorithm makes Gram matrix approximation in unit square using gradient descent method
Battle array, and by isogonism tight frame (Equiangular Tight Frame, ETF) progressive updating Gram matrix, allows Gram matrix
The off diagonal element of matrix is equal to the maximum cross-correlation number of matrix, although improving the property of image reconstruction to a certain extent
Can, but original image and recovery image still remain bigger reconstructed error, and it, will be dilute herein while optimizing calculation matrix
The reconstructed error MSE for dredging picture signal takes into account, and proposes a kind of Robust Algorithms of calculation matrix optimization based on MSE.
Summary of the invention
The purpose of the present invention is excellent in the calculation matrix based on MSE new with one kind in view of the deficiencies of the prior art, is proposed
The Robust Algorithms of change.The algorithm takes into account the reconstructed error MSE of sparse picture signal while optimizing calculation matrix
It goes, not only increases the performance of image reconstruction, reduce reconstructed error, and reduce the mutual relevant of calculation matrix and sparse basis
Property, also mitigate the requirement to compression ratio to a certain degree.
Technical solution of the present invention: the Robust Method of the calculation matrix optimization based on MSE.This method is optimized in tradition
On the basis of the model of calculation matrix, increase a regular terms, which represents the mean square error of original image and reconstructed image,
The information of the image measurement matrix optimizing model abundant application image newly proposed itself is just divided very much by assuming that error obeys standard
Cloth and parity price dictionary carry out singular value decomposition, simplify the Optimized model of calculation matrix well, finally using under gradient
Algorithm drops, and the calculation matrix after iteratively solving out optimization, experiment shows that the independence between the column of the calculation matrix after optimization increases
Add, more favorably reconstructs the picture signal of high quality.Key step is as follows:
Step 1: setting parameter: iteration total degree Iter, the number of iterations t, initial value 1, column relative coefficient μ, just
Then term coefficient is β, original image signal X, restores picture signalStochastic variable n obeys mean value 0, variance σ2I Gauss point
The ranks number of cloth, calculation matrix is respectively set are as follows: the ranks number of M, N, sparse basis are respectively set are as follows: N, L；
Step 2: choosing reasonable unit matrix I, generates random Gaussian calculation matrix Φ and sparse basis Ψ, and standardize
Calculation matrix Φ, wherein I ∈ RL×L, Φ ∈ RM×N, Ψ ∈ RN×L, wherein M < N < L.
Step 3: Gram matrix G, original image signal X and recovery picture signal are calculatedMean square error, i.e. G=DTD
=ΨTΦTΦ Ψ,Wherein D=Φ Ψ indicates perception matrix.
Step 4: construction image calculation matrix Optimized model increases on the basis of tradition optimizes image measurement matrix
One regular terms,I.e.
Step 5: model optimization.By the measurement model of CS standard:
Y=Φ Ψ θ+n (7)
Wherein actual observed value Y, n indicate Gaussian distributed error, can be obtained by OMP algorithm, reconstruct image it
Before, it first has to estimate sparse signal X non-zero, expression are as follows:
Wherein, original image and the reconstructed error restored between image estimate it mainly from sparse signal X non-zero
Between error, so can useIt is approximate to replaceThe then optimization in (6) formula about image measurement matrix
Model can be converted into
As can be seen from the above formula that there is a problem of that two calculate complexity, 1. stochastic variable before optimizing calculation matrix
The processing of n brings uncertainty to calculating since there are randomnesss.2. matrix inversion, even if the pseudoinverse of solution matrix, still
Have the shortcomings that computation complexity is high, and does not ensure that convergence.For 1., it is assumed that n={ nk, k=1,2...P is obeyed
Mutually independent Gaussian Profile mean value is 0, variance σ2I enables S=(DTD)-1DT, S ∈ RL×L, then
When P levels off to ∞,It converges onIt can be concluded that
By (10) (11) formula it is found that when P levels off to infinity,WithIt is directly proportional.(9) formula
It can be converted into
For 2. F norm is unfolded, the mark of solution matrix is specific as follows:
Singular value decomposition: D=U Λ V is carried out to DT, any orthogonal matrix of U, V expression, Λ expression diagonal matrix, Λ=
diag[λ1≥λ2≥··≥λi≥··λn], wherein λ1≥λ2≥··≥λi≥··λn≥0.By (13) formula it is found that above formula
Expression formula can be with a conversion are as follows:
(7) formula can convert are as follows:
Since above formula (15) is a non-convex optimization problem, can be obtained according to the upper boundary constraint condition of (14):
WhereinIndicate diagonal matrix, preceding m of diagonal entry are λmax+λmin, remainder zero.According to D=U Λ VT,(15) formula can be converted are as follows:
Step 6: it enablesSolve f (Φ) gradient
Step 7: iterative calculation calculation matrix Φ, until t > Iter, (13) formula stops iteration.
Φk+1=Φk-γ▽f(Φ) (19)
Advantages of the present invention: compared with optimizing calculation matrix before, (1) is filled in the case where image reconstruction error is added
Divide the information for considering image itself, improves the PSNR of reconstructed image very well.(2) the mutual of calculation matrix and sparse basis is greatly reduced
Coherence reduces the relative error between original image and reconstructed image, increases independence between calculation matrix column.
Detailed description of the invention
Fig. 1 is the signal observation process of compressed sensing
Fig. 2 is calculation matrix optimization front and back cross-correlation coefficient distribution histogram
Fig. 3 is calculation matrix optimization front and back original signal and recovery signal relative error figure
Fig. 4 is that the PSNR of image is restored with degree of rarefication variation diagram in calculation matrix optimization front and back
Fig. 5 is Figure of abstract in the embodiment of the present invention
Specific embodiment
A kind of image measurement matrix optimizing method based on mean square error proposed by the present invention, the present invention experiment be
Realized on MATLAB platform, concrete operations the following steps are included:
Step 1: setting parameter, iteration total degree Iter=100, the number of iterations t, initial value 1, regularization coefficient
It is lena256*256 for α=1.1, m=10, original image signal X, stochastic variable n obeys mean value 0, variance σ2I Gauss point
The ranks number of cloth, calculation matrix is respectively set are as follows: the ranks number of M=20, N=64, sparse basis are respectively set are as follows: N=64, L=
100
Step 2: choosing 100 × 100 unit matrix I, generates 20 × 64 random Gaussian calculation matrix Φ, and standardize
Calculation matrix Φ obtains the Ψ of sparse basis 64 × 100 by KSVD training.
Step 3: by iterating to calculate calculation matrix Φ, until t > Iter, (19) formula stops iteration.After iteration
Φ is multiplied with sparse basis Ψ, is calculate by the following formula whole related coefficient μ (D), average cross correlation coefficient respectivelySuch as table 1, mutually
Related coefficient histogram, calculated result such as Fig. 2
Table 1
Step 4: the degree of rarefication for constructing each column is 4 and obeys one-dimensional 100 × 1 sparse signal sequence of standardized normal distribution
Column, are denoted as { sk(k=1,2 ... 1000), pass through zk=Φ yk=Φ Ψ skTest signal sequence is obtained, { y is denoted ask, similarly
The sequence of observations { z can be obtainedk, by OMP algorithm, reconstruct the sparse sequence comeRestore signal sequenceNs=1000
The performance of calculation matrix after relative error formula inspection optimization can be used:
Claims (4)
1. a kind of image measurement matrix optimizing method based on reconstructed error, it is characterised in that: construction Gaussian distributed first
Random measurement matrix, and it is standardized, then obtains sparse basis from training set of images using KSVD, pass through calculation matrix
With the product of sparse basis, Gram matrix is constructed, Gram matrix and unit matrix are solved into F norm, obtain original measurement square
Battle array Optimized model increases regular terms on the basis of the model, which is believed by original image signal and the image recovered
Number mean square error constitute, finally select suitable regularization coefficient, obtain the image proposed by the present invention based on reconstructed error
Calculation matrix model.
2. the image measurement matrix model according to claim 1 based on reconstructed error, it is characterised in that: the model has two
Part is constituted, and expression is as follows:
First part's main purpose reduces the cross-correlation coefficient between calculation matrix and sparse basis, second part by picture signal from
The mean square error of body is added in model, using error as a kind of available image information, can more preferably promote calculation matrix
Performance.
3. regular terms according to claim 1 is by the mean square error structure of original image signal and the picture signal recovered
At, it is characterised in that: it can be obtained by OMP algorithm, mean square error mostlys come from the estimation of sparse signal X non-zero, Ke YiyongIt is approximate to replaceSimplify Optimized model
4. the optimization method for requiring a kind of image measurement matrix based on reconstructed error with 2 according to claim 1, special
Sign is that non-correlation of the Gram matrix in unit matrix, calculation matrix between sparse basis is better, is wanted to degree of rarefication
It asks lower, while the mean square error between image is added, advanced optimize the performance of calculation matrix, not only improve and restore image
PSNR, and since adding somewhat to compression of images sensory perceptual system robustness.
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| CN110163821A (en) * | 2019-05-09 | 2019-08-23 | 安徽大学 | A kind of terrible imaging method and system of singular value decomposition compression based on nonnegativity restrictions |
| CN110266318A (en) * | 2019-06-24 | 2019-09-20 | 中国科学院光电技术研究所 | A kind of calculation matrix optimization method in compressed sensing signal reconstruction based on gradient project algorithms |
| CN110400258A (en) * | 2019-07-27 | 2019-11-01 | 南阳理工学院 | A kind of HEVC rate control algorithm of view-based access control model perception |
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| CN112767509A (en) * | 2021-01-18 | 2021-05-07 | 中南大学 | Image reconstruction method for correlated imaging system |
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| CN112836708A (en) * | 2021-01-25 | 2021-05-25 | 绍兴图信物联科技有限公司 | Image feature detection method based on Gram matrix and F norm |
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