CN104242948A - Toeplitz structure measurement matrix construction method based on singular value decomposition - Google Patents

Abstract

The invention discloses a Toeplitz structure measurement matrix construction method based on singular value decomposition. The Toeplitz structure measurement matrix construction method comprises the following steps that firstly, a row vector which is composed of elements 0 and 1 and submits to random distribution is generated; secondly, a Toeplitz structure measurement matrix phi is constructed through the vector; thirdly, singular value decomposition is conducted on the Toeplitz structure measurement matrix; fourthly, singular values are optimized, in other words, the maximum value in the singular values in the Toeplitz structure measurement matrix phi is regarded as a nonzero singular value of a newly-constructed matrix; fifthly, a new measurement matrix phi' is constructed; sixthly, approximate processing is conducted on the newly-constructed measurement matrix phi', in other words, negative elements are set to be 0, non negative elements are set to be 1, and the final Toeplitz structure measurement matrix phi'' composed of the elements 0 and 1 is obtained. The Toeplitz structure measurement matrix has the advantages that the computation complexity is low, the storage space is small, the structure is simple, and hardware is easy to achieve. Furthermore, the reconstruction effect is good.

Description

A kind of Teoplitz structure measurement Matrix Construction Method based on singular value decomposition

Technical field

The present invention relates to image compression field of sensing technologies, particularly a kind of Optimal Construction method of Teoplitz structure measurement matrix in compressed sensing field.

Background technology

In traditional signal processing, in order to avoid distorted signals, sampling rate will meet nyquist sampling law.But the acquisition to the broadband signal such as image, video, carry out sampling according to nyquist sampling law and sample rate can be caused too high, hardware circuit is difficult to realize, and due to sampled data output too large, therefore substantially increase and store and the cost of transmission; Simultaneously because its data acquisition adopts the pattern of recompression of first sampling, both wasted sensing element, lost time again and memory space.

By Candes, compressive sensing theory (the Compressive Sensing that the people such as Donoho and Tao propose, CS) restriction of conventional Nyquist Sampling Theorem is breached, for IMAQ and compression provide new theories integration, compressive sensing theory is pointed out: if signal itself is sparse or can carries out rarefaction representation on certain dictionary, so just original high dimensional signal can be projected on a lower dimensional space by a kind of calculation matrix of design, then by solving a nonlinear optimal problem to projecting to a small amount of measured value that lower dimensional space obtains, just can high probability or reconstruct primary signal accurately.The advantage that compressed sensing is given prominence to the most to reduce sampled data, saves memory space, traditional data sampling and data compression are united two into one, greatly reduce acquisition time and the access space of data.

Calculation matrix is a vital link in CS theory, and it has important impact to signal sampling and restructing algorithm.In order to most of energy of signal can be kept after ensureing accidental projection, calculation matrix must meet limited equidistant character (Restricted Isometry Property, RIP).But this condition is difficult to for design calculation matrix in practice, simultaneously, RIP principle is adequate condition, instead of necessary condition, for reducing the complexity of matrix design, the correlation that Donoho proposes calculation matrix differentiates the theoretical condition of equivalence as RIP.Calculation matrix can be divided into random measurement matrix and certainty calculation matrix from the randomness of matrix element and certainty two aspect.Certainty matrix is configured to whole calculation matrix by known portions information, as Teoplitz structure matrix and circular matrix etc., this kind of calculation matrix adopts specific structuring generating mode to generate, and therefore matrix can by calculating fast, reduce complexity, and desin speed is fast.But, these calculation matrix all compare gaussian random matrix reconstruction effect on there is gap, the measurement number of requirement is more; Each element of random measurement matrix obeys independent same distribution, therefore the non-correlation between calculation matrix row can be ensured as much as possible, so just less sampled value can be used to obtain accurate reconstruction, as gaussian random matrix, sparse projection matrix, subalpine forests random matrix and Bei Nuli matrix etc.But the shortcoming that random measurement matrix has some intrinsic, as taken larger memory space, the larger amount of calculation of cost and time complexity, these are all unfavorable for hardware implementing.

At present, although random measurement matrix is higher than certainty calculation matrix reconstruction precision on the whole, itself intrinsic uncertainty determines the limitation in its hardware implementing.So certainty calculation matrix is the main direction of studying of following calculation matrix, but also there is the not high shortcoming of reconstruction precision in certainty calculation matrix.Because Teoplitz structure measurement matrix utilizes row vector to generate whole matrix by cyclic shift, in actual applications, this cyclic shift is easy to hardware implementing, and Teoplitz structure measurement matrix is a kind of matrix of highly structural, the structuring of matrix makes it possible to by quick calculating, reduces computation complexity.Therefore, the present invention studies around toeplitz matrix, by carrying out singular value decomposition to toeplitz matrix, and is optimized process to the singular value of this matrix, and then improves row non-correlation or the row independence of calculation matrix, thus improves its reconstruction property.At present, due to the random matrixes such as gaussian random matrix, Bernoulli Jacob's random matrix and most orthogonal basis or orthogonal dictionary uncorrelated, and measurement number needed for Accurate Reconstruction is fewer, therefore, this type of random matrix is suitable as the calculation matrix in compressive sensing theory very much.But there is following bottleneck in this kind of random matrix: the generation of (1) random number is very high to hardware requirement; (2) compression projection and signal reconstruction process need carry out storage and transmission measurement matrix, and this is very high to the requirement of system; (3) random matrix only meets RIP and weak coherence with very high probability under statistical significance, namely can not ensure that the matrix of each random generation meets specific RIP or coherence's condition, thus can not ensure to recover primary signal all accurately at every turn.Because above-mentioned random measurement matrix exists above shortcoming in actual applications, therefore the research of certainty calculation matrix is even more important for the propagation and employment of compressive sensing theory.But certainty calculation matrix compares the random measurement matrixes such as Gaussian matrix, reconstruction effect exists gap, and the measurement number of requirement is more, also has higher requirements to the degree of rarefication of signal simultaneously, calculation matrix in practical application requires that simple structure is effective, compression and rebuild efficiency all higher, therefore to be easy to hard-wired calculation matrix very meaningful in design, the construction process of Teoplitz calculation matrix goes to generate whole matrix with vector, the process that this vector generates whole matrix is realized by cyclic shift, this cyclic shift is easy to hardware implementing, this is also the main cause that toeplitz matrix is widely studied, the same with other most of certainty calculation matrix, Teoplitz calculation matrix also also exists when ensureing high reconstruction precision, need the shortcoming that the number of measured value is more, and the acquisition of measured value is very expensive in a lot of practical application.Therefore, for random measurement matrix not easily hardware implementing and be easy to the undesirable problem of hard-wired toeplitz matrix quality reconstruction, the present invention proposes a kind of Teoplitz structure measurement matrix design method based on singular value decomposition.

Summary of the invention

For above deficiency of the prior art, one is the object of the present invention is to provide to improve perception efficiency better and reduce computation complexity, reduce memory space, and matrix element is made up of at random 0 and 1, structure is simply sparse, be easy to hardware implementing, the Teoplitz structure measurement Matrix Construction Method that reconstruction property is good, technical scheme of the present invention is as follows:

Based on a Teoplitz structure measurement Matrix Construction Method for singular value decomposition, it comprises the following steps:

101, measure the original image signal A that obtains, generates an element and obey random distribution and by 0, the 1 row vector u=(u formed ₁, u ₂..., u _n, u _n+1..., u _n+M-1), N represents the columns of wanted structural matrix, the line number of M representing matrix;

102, then Teoplitz structure measurement matrix Φ is constructed by the row vector in step 101;

103, next singular value decomposition is carried out to this Teoplitz structure measurement matrix Φ, and process is optimized to singular value, obtain new calculation matrix Φ ',

104 ,/the calculation matrix Φ ' of neotectonics is set to 0 as negative element, non-negative element puts the process of 1, obtain final by 0, the structure of the Teoplitz structure measurement matrix Φ "; complete Teoplitz structure measurement matrix Φ " of 1 element composition, measures step 101 the original image signal A obtained and utilizes Teoplitz structure measurement matrix Φ and " projecting to lower dimensional space, then passing through projecting to the measured value that lower dimensional space obtains; solve a nonlinear optimization equation, then just reconstruct primary signal.

Further, Teoplitz structure measurement matrix Φ ∈ R in step 102 ^{m × N}(M<N), namely

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}{u}_N& u_{N-1}& ...& u_2& u_1\\ u_{N+1}& u_N& ...& u_3& u_2\\ .& .& & .& .\\ .& .& ...& .& .\\ .& .& & .& .\\ u_{N+M-1}& u_{N+M-2}& ...& u_{M+1}& u_M\end{array}\right].$

Further, M>=CK ³/ ln (NK), C>0 is constant, and K is degree of rarefication

Suppose that length is N=256, degree of rarefication is the signal of K, gets different measurement number M, wherein 0<M<256.Adopt OMP algorithm for reconstructing and Teoplitz calculation matrix to carry out signal reconstruction, experimental result shows: when degree of rarefication K fixes, then the measured value M required for Exact Reconstruction is larger, as K=4, the value of M close to 50 time, signal energy Exact Reconstruction.

Further, in step 103 to matrix Φ ∈ R ^{m × N}(M<N) carry out singular value decomposition, obtain

$\mathrm{\&Phi;}=U\left[\begin{array}{cc}\mathrm{\&Sigma;}& 0\\ 0& 0\end{array}\right]V^H={\mathrm{USV}}^H$

Wherein, matrix U, V are respectively M × M, and the unitary matrice of N × N, meets UU ^h=E, VV ^h=E, Σ=diag (σ ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m, σ ₁, σ ₂..., σ _mfor the singular value of matrix Φ; If Φ ^hthe characteristic value of Φ is λ ₁>=λ ₂>=... λ _r> λ _r+1=... λ _n=0, then ${\mathrm{\&sigma;}}_i=\sqrt{{\mathrm{\&lambda;}}_i}(i=\mathrm{1,2},...,M).$

Further, in step 103 to the step that singular value is decomposed be: Σ=diag (σ ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m>0, therefore obtain σ _max=max (σ ₁, σ ₂..., σ _m)=σ ₁, make S '=diag (σ ' ₁, σ ' ₂..., σ ' _m), σ ' ₁=σ ' ₂=σ ' _m=σ _max.Does diag represent the diagonal matrix be made up of the non-zero singular value of matrix Φ? after S ' expression is optimized process to the non-zero singular value of matrix Φ, the diagonal matrix be made up of new non-zero singular value obtained? and the calculation matrix Φ ' new according to formula construction, and Φ '=US ' V ^h.

Advantage of the present invention and beneficial effect as follows:

The present invention is directed to gaussian random matrix randomness strong, take memory space large, computation complexity is high, is difficult to the shortcoming that realizes and the deficiency of certainty matrix in reconstruction property such as toeplitz matrix instantly in hardware circuit; The present invention is optimized process by the method increasing singular value to toeplitz matrix, and then improves the linear independent between rectangular array vector; The Teoplitz structure measurement matrix of neotectonics can improve perception efficiency better and reduce computation complexity, and reduce memory space, and matrix element is made up of at random 0 and 1, structure is simply sparse, is easy to hardware implementing, and reconstruction property is good.

Accompanying drawing explanation

Fig. 1 is the overall flow figure of the Teoplitz structure measurement matrix design based on singular value decomposition that the present invention proposes;

Fig. 2 is the PSNR curve chart of Lena image reconstructed image under different sample rate;

Fig. 3 be Lena image when sample rate is 0.5, the Teoplitz calculation matrix, gaussian random matrix and the toeplitz matrix do not optimized that adopt this patent to propose carry out the quality reconstruction figure after compressing observation;

Fig. 4 be Lena image when sample rate is 0.6, the Teoplitz calculation matrix, gaussian random matrix and the toeplitz matrix do not optimized that adopt this patent to propose carry out the quality reconstruction figure after compressing observation.

Embodiment

The invention will be further elaborated to provide an infinite embodiment below in conjunction with accompanying drawing.But should be appreciated that, these describe just example, and do not really want to limit the scope of the invention.In addition, in the following description, the description to known features and technology is eliminated, to avoid unnecessarily obscuring concept of the present invention.

In order to design calculation matrix, Donoho provides three features that compressed sensing calculation matrix will meet: the minimum singular value of the submatrix that (1) is made up of the column vector of calculation matrix must be greater than certain constant, and also namely the column vector of calculation matrix meets certain linear independent; (2) column vector of calculation matrix embodies the independent random of certain similar noise; (3) solution meeting degree of rarefication is the vector meeting 1-Norm minimum.Above-mentioned first feature is pointed out, the column vector of calculation matrix will meet certain linear independent, and the minimum singular value of the linear independent of matrix and matrix is closely bound up, and namely minimum singular value is less, then the linear dependence of matrix is larger, and independence is more weak.When minimum singular value close to 0 time, the linear dependence of matrix is tending towards maximum, and independence disappears thereupon.Therefore, the present invention proposes a kind of Teoplitz structure measurement matrix design method based on singular value decomposition, object is the linear independent strengthening matrix by increasing singular values of a matrix;

As shown in Figure 1, Fig. 1 is the overall flow figure of a kind of Teoplitz structure measurement matrix design method based on singular value decomposition that the present invention proposes, and first the method generates element and obey random distribution and by 0, the row vector u=(u of 1 composition ₁, u ₂..., u _n, u _n+1..., u _n+M-1), then vector constructs Teoplitz structure measurement matrix Φ thus, next singular value decomposition is carried out to this calculation matrix, and process is optimized to singular value, obtain new calculation matrix Φ ', according to the stability for disturbance of singular value, the calculation matrix Φ ' of neotectonics is set to 0 as negative element, non-negative element puts the process of 1, obtains final by 0, the Teoplitz structure measurement matrix Φ of 1 element composition ".The method concrete steps are as follows:

Step one: generate an element by 0, the row vector u=(u of 1 composition ₁, u ₂..., u _n, u _n+1..., u _n+M-1);

Step 2: construct element by 0, the Teoplitz structure measurement matrix of 1 composition;

Be easy in hardware circuit to make the calculation matrix of structure realize, structure is simple, and so, entry of a matrix element by 0,1 composition meeting random distribution, therefore can generate Teoplitz structure measurement matrix Φ ∈ R by vectorial u ^{m × N}(M<N),

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}{u}_N& u_{N-1}& ...& u_2& u_1\\ u_{N+1}& u_N& ...& u_3& u_2\\ .& .& & .& .\\ .& .& ...& .& .\\ .& .& & .& .\\ u_{N+M-1}& u_{N+M-2}& ...& u_{M+1}& u_M\end{array}\right]---\left(1\right)$

Step 3: to matrix Φ ∈ R ^{m × N}(M<N) carry out singular value decomposition, can obtain

$\mathrm{\&Phi;}=U\left[\begin{array}{cc}\mathrm{\&Sigma;}& 0\\ 0& 0\end{array}\right]V^H={\mathrm{USV}}^H$

(2)

Wherein, U and V is respectively M × M, the unitary matrice of N × N,

Σ=diag (σ ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m, σ ₁, σ ₂..., σ _mfor the singular value of matrix Φ;

Step 4: process is optimized to the singular value of calculation matrix Φ;

Due to Σ=diag (σ ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m>0, therefore can obtain σ _max=max (σ ₁, σ ₂..., σ _m)=σ ₁, make S '=diag (σ ' ₁, σ ' ₂..., σ ' _m), σ ' ₁=σ ' ₂=σ ' _m=σ _max;

Step 5: construct new calculation matrix Φ ', namely

Φ′＝US′V ^H

(3)

Step 6: carry out approximate processing to the element of the matrix Φ ' of neotectonics, namely non-negative element puts 1, and negative element sets to 0, obtains final calculation matrix Φ ".

Wherein, described in step 4, singular value decomposition is carried out to matrix Φ, and process is optimized to singular value; The singular value of matrix Φ is after optimization process, and the number not change of matrix non-zero singular value, namely this optimizing process does not change rank of matrix.

New calculation matrix Φ ' is constructed according to step 5, because singular value has stability for disturbance and stability, therefore in order to make constructed calculation matrix be easy to hardware implementing, structure is simple, in step 6, the element in calculation matrix Φ ' can be carried out approximate processing, obtain final required calculation matrix Φ ".

In order to verify that this patent designs by 0, the validity of Teoplitz structure measurement matrix in image compression perception processing procedure of 1 element composition, present embodiment adopts three width international standard gray scale test patterns to carry out emulation experiment, the resolution of image is 256 × 256, first to Lena, Boat, this three width image of Peppers adopts wavelet transformation to carry out rarefaction representation, and then utilize respectively by 0, the Teoplitz structure measurement matrix of 1 element composition, gaussian random matrix, this patent propose optimization after by 0, the Teoplitz structure measurement matrix of 1 element composition carries out compression observation to image, finally utilize reconstructed velocity faster OMP algorithm measured value is reconstructed.Table 1 is, after adopting different calculation matrix to carry out compression measurement, reconstruct the comparative result of the Y-PSNR (PSNR) of image.

Table 1 simulation result compares (dB)

As can be seen from Table I, under identical sample rate, the toeplitz matrix that its PSNR value reconstructing image of Teoplitz structure matrix after optimization designed by this patent is higher than gaussian random matrix and does not optimize under identical sample rate, for comprising the Lena image enriching detailed information and textural characteristics, the matrix of this patent design has more advantage on reconstructed image quality.

With reference to the accompanying drawings 2, the performance of reconstructed image when the Teoplitz structure measurement matrix compared further after optimization Teoplitz calculation matrix, gaussian random calculation matrix and this patent proposed is used for the compressed sensing of image; Wherein, Fig. 2 is under different sample rates, and the PSNR value of reconstructed image is with the situation of change of sample rate.Can more significantly find out from Fig. 2, along with the increase of sample rate, the PSNR value reconstructing image also increases thereupon, and namely the quality of reconstructed image also increases thereupon.And under identical sample rate, the Teoplitz structure measurement matrix based on singular value decomposition of this patent structure, the PSNR value of its reconstructed image is all higher than the toeplitz matrix do not optimized and gaussian random matrix, and especially when sample rate is greater than 0.5, its reconstruct advantage is more obvious.

With reference to the accompanying drawings 3, accompanying drawing 4 compare intuitively toeplitz matrix, gaussian random matrix, this patent the quality reconstruction of calculation matrix is proposed, Fig. 3 and Fig. 4 is sample rate respectively adopts the reconstruct visual effect figure of different measuring matrix when being 0.5,0.6.From Fig. 3 and Fig. 4 can find out the quality reconstruction of gaussian random matrix than the toeplitz matrix do not optimized good, and the reconstruct visual effect of the calculation matrix utilizing this patent to propose is better than gaussian random matrix, especially the observing matrix adopting this patent to propose carries out compression observation to image, the edge and the profile that reconstruct image are more clear, and blocking effect is not obvious, this is also consistent with objective evaluation index value.

Comprehensive Experiment effect can draw: no matter be visual effect or objective evaluation index value, the present invention has superiority.This patent, by being optimized process to toeplitz matrix, had both remained that the computation complexity that certainty matrix has is low, memory space is few, and structure is simple, and is easy to hard-wired characteristic, overcame again its deficiency in reconstruction property.

These embodiments are interpreted as only being not used in for illustration of the present invention limiting the scope of the invention above.After the content of reading record of the present invention, technical staff can make various changes or modifications the present invention, and these equivalence changes and modification fall into the inventive method claim limited range equally.

Claims (5)

1., based on a Teoplitz structure measurement Matrix Construction Method for singular value decomposition, it is characterized in that, comprise the following steps:

101, measure the original image signal A that obtains, generates an element and obey random distribution and by 0, the 1 row vector u=(u formed ₁, u ₂..., u _n, u _n+1..., u _n+M-1), N represents the columns that will construct calculation matrix, the line number of M representing matrix;

102, then Teoplitz structure measurement matrix Φ is constructed by the row vector in step 101;

103, next singular value decomposition is carried out to this Teoplitz structure measurement matrix Φ, and process is optimized to singular value, obtain new calculation matrix Φ ';

104, the calculation matrix Φ ' of neotectonics is set to 0 as negative element, non-negative element puts the process of 1, obtain final by 0, the structure of the Teoplitz structure measurement matrix Φ "; complete Teoplitz structure measurement matrix Φ " of 1 element composition, measures step 101 the original image signal A obtained and utilizes Teoplitz structure measurement matrix Φ and " projecting to lower dimensional space, then passing through projecting to the measured value that lower dimensional space obtains; solve a nonlinear optimization equation, then just reconstruct primary signal.

2. the Teoplitz structure measurement Matrix Construction Method based on singular value decomposition according to claim 1, is characterized in that, Teoplitz structure measurement matrix Φ ∈ R in step 102 ^{m × N}(M<N), namely

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}{u}_N& u_{N-1}& ...& u_2& u_1\\ u_{N+1}& u_N& ...& u_3& u_2\\ .& .& & .& .\\ .& .& ...& .& .\\ .& .& & .& .\\ u_{N+M-1}& u_{N+M-2}& ...& u_{M+1}& u_M\end{array}\right].$

3. a kind of Teoplitz structure measurement Matrix Construction Method based on singular value decomposition according to claim 2, is characterized in that, M>=CK ³/ ln (NK), C>0 is constant, and K is degree of rarefication.

4. a kind of Teoplitz structure measurement Matrix Construction Method based on singular value decomposition according to claim 1, is characterized in that, to matrix Φ ∈ R in step 103 ^{m × N}(M<N) carry out singular value decomposition, obtain

$\mathrm{\&Phi;}=U\left[\begin{array}{cc}\mathrm{\&Sigma;}& 0\\ 0& 0\end{array}\right]V^H={\mathrm{USV}}^H$

Wherein, matrix U, V are respectively M × M, and the unitary matrice of N × N, meets UU ^h=E, VV ^h=E, Σ=diag (σ ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m, σ ₁, σ ₂..., σ _mfor the singular value of matrix Φ; If Φ ^hthe characteristic value of Φ is λ ₁>=λ ₂>=... λ _r> λ _r+1=... λ _n=0, then ${\mathrm{\&sigma;}}_i=\sqrt{{\mathrm{\&lambda;}}_i}(i=\mathrm{1,2},...,M).$

5. a kind of Teoplitz structure measurement Matrix Construction Method based on singular value decomposition according to claim 4, is characterized in that, be: Σ=diag (σ in step 103 to the step that singular value is decomposed ₁, σ ₂..., σ _m), σ ₁>=σ ₂>=σ _m>0, therefore obtain σ _max=max (σ ₁, σ ₂..., σ _m)=σ ₁, make S '=diag (σ ' ₁, σ ' ₂..., σ ' _m), σ ' ₁=σ ' ₂=σ ' _m=σ _maxdiag represents the diagonal matrix be made up of the non-zero singular value of matrix Φ, after S ' expression is optimized process to the non-zero singular value of matrix Φ, the diagonal matrix be made up of new non-zero singular value obtained, and the calculation matrix Φ '=US ' V new according to formula construction ^h.