CN109584330A - One kind approaching L based on compressed sensing0The gradient projection image rebuilding method of norm - Google Patents

Abstract

The invention discloses one kind to approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, linear measurement observation is carried out to image first in the image reconstruction of compressed sensing, sampling had not only been completed to original signal in this way but also has completed compression, substantially reduce signal dimension, resulting measured value passes through algorithm for reconstructing again to recover original signal, therefore the superiority and inferiority of algorithm for reconstructing has been largely fixed the quality of image reconstruction quality.The present invention is in minimum L₀Norm non-convex optimization algorithm and L₂On the basis of norm convex optimized algorithm, using the feature of convex optimized algorithm and non-convex optimization algorithm respectively, propose it is a kind of by way of gradient projection from L₂Norm approaches L₀The restructing algorithm of norm.This algorithm synthesis advantage of convex optimized algorithm and non-convex optimization algorithm respectively, accelerates the speed of image reconstruction, improves the Y-PSNR (PSNR) and structural similarity (SSIM) of reconstruction image.

Description

One kind approaching L based on compressed sensing0The gradient projection image rebuilding method of norm

Technical field

The present invention relates to one kind to approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, feature are to letter Number original signal data goes out the original signal data of higher precision with lower sample rate (measured value) restoration and reconstruction, is applied to letter Number compression and recovery, image procossing and computer vision etc., belong to signal compression transmission in Signal and Information Processing with it is extensive Field is rebuild again.

Background technique

The core of compression sensing is linear measurement process, if x (n) is original signal, length N measures square by premultiplication Battle array Φ obtains y (m), and length is M (M < N).If x (n) is not sparse signal, orthogonal sparse transformation will be carried out and obtain s (k), remembered For x=Ψ s, measurement process is re-written as y=Θ s, wherein Θ=Φ Ψ (M × N), referred to as sensing matrix, process such as Fig. 2 institute Show.Compression sensing theory mainly includes three rarefaction representation of signal, the construction of calculation matrix and restructing algorithm aspects.

Image sparse expression refers to that image some biggish coefficient sets of numerical value in the coefficient on particular transform base have suffered figure The most of energy and information of picture, and other coefficients are all zero or close to zero, it means that just using a small amount of bit number It can achieve the purpose that indicate image.Natural sign in usual time domain is all non-sparse, for example, for a width natural image, Almost all of pixel value is all non-zero, but when being transformed to wavelet field, the absolute value of most of wavelet coefficients is all connect Be bordering on zero, and limited big coefficient it can be shown that original image most information.The sparsity of signal is that compression passes Feel theoretical basis and premise, this paper experiment simulation carries out rarefaction to image using wavelet transform base.

And in terms of the construction of calculation matrix, it need to meet the equidistant condition of constraint with the sensing matrix Θ that sparse basis is constituted (RIP condition, 1 formula), so that it may which original signal is recovered by the above restructing algorithm.

Wherein, δ_kMinimum value be known as RIP constant, be measure RIP property quality a standard.

RIP condition is the adequate condition that guarantees signal and can reconstruct, however to verify whether sensing matrix meets this condition It is an extremely complex problem, it is therefore desirable to have a kind of RIP condition alternative that is easy, being easily achieved.It is theoretical and real If trampling proof can guarantee that calculation matrix Φ and orthogonal basis Ψ are uncorrelated, Θ meets RIP property on very big probability.Due to Ψ be it is fixed, meet Θ=Φ Ψ and constrain equidistant condition, design calculation matrix Φ can be passed through and solved.Pass through mathematics It is theoretical and a large amount of practice have shown that, be commonly used to calculation matrix have shellfish make great efforts calculation matrix (two-value random measurement matrix), with Machine Gauss measurement matrix, Fourier random measurement matrix, Hadamard calculation matrix, these matrixes all meet RIP with high probability Condition.The calculation matrix used herein is the diagonal calculation matrix of supersparsity two-value, as shown in figure 3, and using as shown in Figure 4 Bilateral projection pattern.

Signal reconstruction algorithm, which refers to the process of, measures the sparse signal s that vector y reconstruct length is N (M < N) by M times.It is above-mentioned Unknown number number N is more than equation number M in equation group, directly can not recover s (n) from y (m), can be by solving minimum L₀ Norm non-convex optimization problem (2) is solved.

But minimum L₀Norm problem is a NP-hard problem, needs all of nonzero value in exhaustive xKind arrangement can Can, thus be difficult to solve, although the solution found be it is most sparse, the sparse signal of reconstruct generally can not be obtained closest to original The global optimum of sparse signal.Thus it commonly uses near-optimal solution algorithm to be solved, mainly includes convex optimized algorithm (minimum L₁Model Number method and minimum L₂Norm method), match tracing serial algorithm, iteration method and special disposal two dimensional image problem minimum Full calculus of variations etc..New Algorithm proposed in this paper is then by L₂Norm is to L₀A kind of algorithm that norm is approached, and thrown by gradient The mode of shadow realized, be one from convex optimization constantly to the iterative solution process of non-convex optimization, can preferably take into account this two The advantage of kind algorithm.

Summary of the invention

The invention solves technical problems are as follows: is difficult to combine for original algorithm in compression sensed image signal reconstruction The problem of globally optimal solution and sufficiently low degree of rarefication, devises a kind of novel by L₂Norm is to L₀The algorithm that norm is approached, this Kind New Algorithm has preferably taken into account global optimum and sufficiently low degree of rarefication.In reconstruction condition of the same sample rate as Under, this method improves the precision and quality of reconstruction signal well.

The present invention solves the technical solution that above-mentioned technical problem uses are as follows: one kind approaches L based on compressed sensing₀The ladder of norm Backprojection image reconstruction method is spent, this method is based on minimum L₂The convex optimized algorithm and minimum L of norm₀The non-convex optimization algorithm of norm, It is proposed it is a kind of from L₂Norm approaches L₀The New Algorithm of norm makes full use of the respective advantage of the two algorithm, and it is multiple to reduce algorithm The miscellaneous precision that image reconstruction is improved while spend.This method comprises the following steps:

Step 1, first define an approximating function, by the parameter value in adjustment function make the functional value of approximating function from Approximate L₂Norm is to L₀Norm is approached, this approximating function model is used for the sparse signal most rarefaction representation of approximate substitution reconstruct Solution；

Step 2, then using the sparse representation model of this approximating function be used as regular terms, pass through gradient projection solution side It is minimum that formula makes the error of reconstructed image Cephalometry and actual measured value approach item, and then reconstructs sparse signal；

Step 3, during constantly iterative solution reconstruction signal, parameter value in approximating function according to reconstructed error into The adaptive adjustment of row, to realize sparse table aspect from L₂To L₀Constantly approaching for norm, finally reaches more accurate reconstruction signal Purpose.

Wherein, due to L₂The objective function of norm convex optimized algorithm be it is convex, be easily found globally optimal solution, algorithm is complicated Spend it is relatively low, but reconstruct sparse signal degree of rarefication it is not high enough, cause the reconstruction accuracy of sparse signal inadequate, to make image weight The quality built is relatively low.

Wherein, about L₀The non-convex optimization algorithm objective function of norm be it is non-convex, be easily trapped into locally optimal solution, so It is not easily found globally optimal solution, algorithm complexity is higher, but required pendulous frequency is less, and reconstruction signal has most sparse Solution, therefore possess the reconstruction accuracy of better sparse signal, to improve image reconstruction quality.

Wherein, L₂Norm convex optimized algorithm is more suitable for rebuilding the structure and profile information of the low frequency of image, and most Small L₀The non-convex optimization algorithm of norm is more conducive to rebuild the details and texture image of image high frequency, and by L₂Norm to L₀The approximate procedure of norm is then that a continuous iterative approach is optimal and the process of reconstruction is constantly refined to image.

Wherein, in L₂Norm approaches L₀During norm, falls into the instable risk of local optimum reconciliation and dropped A possibility that low, and solution falls into the globally optimal solution environs of our needs, is greatly increasing, therefore in this way Approach minimum L₀The obtained solution of the algorithm of norm can preferably approach global optimum, so that the precision of reconstruction signal be made to obtain To improve.

Since invention increases the sparsities and accuracy for rebuilding sparse signal, this is increased by times of signal sampling compression Rate reduces transmission and the storage cost of signal, will bring very big application value, thus compressed sensing be widely used in Signal transmits and handles the fields such as relevant field, such as communication, detection, medical treatment, military affairs, aerospace.

The advantages of the present invention over the prior art are that:

(1) present invention is to combine original L₂Norm algorithm and minimum L₀Norm algorithm, and asking in the way of gradient projection Solution advantage, the one kind designed approach smooth L₀The New Algorithm of norm.Neutralized the advantage of two kinds of norms respectively, make solve by Step is optimized from convex to non-convex optimization transition, is dexterously avoided and is solved minimum L₀The np hard problem of norm, thus with higher effect Rate and bigger probabilistic approximation globally optimal solution, and guarantee to solve enough degree of rarefications.

(2) of the invention from approximate L₂Norm approaches L₀Norm, without considering L₁The problem of norm does not parse at zero point, And approximating function model f_σ(s) structure type is simple, it is easy to accomplish.With the existing frequently-used greedy algorithm based on match tracing and L₁Norm is that the minimum norm class algorithm of representative is compared, and under identical sample rate, reduces algorithm complexity and sampling number, Improve the reconstruction accuracy of picture signal.

(3) in general compressed sensing image reconstruction is all poor for the image reconstruction effect of high grain details class, compares In the more image of low-frequency component, weight of the New Algorithm used in the present invention to the image containing high frequency texture and abundant details It builds precision improvement effect to become apparent, making the details of image rebuild ability has certain enhancing, therefore to a certain extent more Problem not high to high frequency texture class image reconstruction accuracy in existing compression sensing technology is mended.

Detailed description of the invention

Fig. 1 is the implementation flow chart that the method for the present invention is used for that compressed sensing data-signal to be rebuild；

Fig. 2 is the basic principle block diagram of compressed sensing linear measurement process in the present invention；

Fig. 3 is the diagonal calculation matrix of supersparsity two-value used in the present invention；

Fig. 4 is bilateral projection pattern structural block diagram used in invention；

Fig. 5 is approximating function f defined in the present invention_σ(s_i) and s_i, σ two and three dimensions relational graph, wherein Fig. 5 (a) For f_σ(s_i) and s_i, σ two-dimentional relation figure, Fig. 5 (b) be f_σ(s_i) and s_i, σ three-dimensional relationship figure；

Difference algorithm for reconstructing rebuilds Mandrill image effect comparison diagram when Fig. 6 is sample rate 1/9, wherein Fig. 6 (a) is SP algorithm reconstruction image, Fig. 6 (b) are IRLS algorithm reconstruction image, and Fig. 6 (c) is OMP algorithm reconstruction image, and Fig. 6 (d) is ROMP algorithm reconstruction image, Fig. 6 (e) are GOMP algorithm reconstruction image, and Fig. 6 (f) is L1_BP algorithm reconstruction image, and Fig. 6 (g) is L1_LS algorithm reconstruction image, Fig. 6 (h) are L0GP algorithm reconstruction image, and Fig. 6 (i) is Mandrill original image；

Difference algorithm for reconstructing rebuilds Fingerprint image effect comparison diagram when Fig. 7 is sample rate 1/4, wherein Fig. 7 (a) For SP algorithm reconstruction image, Fig. 7 (b) is IRLS algorithm reconstruction image, and Fig. 7 (c) is OMP algorithm reconstruction image, and Fig. 7 (d) is ROMP algorithm reconstruction image, Fig. 7 (e) are GOMP algorithm reconstruction image, and Fig. 7 (f) is L1_BP algorithm reconstruction image, and Fig. 7 (g) is L1_LS algorithm reconstruction image, Fig. 7 (h) are L0GP algorithm reconstruction image, and Fig. 7 (i) is Fingerprint original image.

Specific embodiment

Opinion specific embodiment further illustrates the present invention with reference to the accompanying drawing.

The principle of the present invention and innovation the improvement is that: one kind approaching L in the sparse reconstruction of compressed sensing₀The ladder of norm Spend the design constructing method of projection algorithm.Usually, non-convex optimization algorithm especially minimum L₀Norm method has most sparse Solution, required pendulous frequency is minimum, but algorithm complexity highest；Based on minimum L₁The convex optimized algorithm of norm has very strong heavy Guarantee is built, required pendulous frequency is only slightly more than non-convex optimization algorithm, but is nonanalytic (can not derivation) at zero point, and algorithm is multiple Miscellaneous degree is also usually higher, is difficult to be suitable for large scale problem；Greedy algorithm based on match tracing has rebuilds speed well, But the theoretical guarantee of Exact Reconstruction is weaker, reconstruction precision is lower.And L₂It is also a kind of convex optimized algorithm that norm, which seeks pseudoinverse technique, Global optimum can be preferably found, is also parsing at zero point, but required pendulous frequency is more, reconstructs the sparse of sparse signal Property generally can not be met the requirements.In conjunction with L₂Norm and L₀The respective advantage and disadvantage of norm algorithm, since (2) formula is that an iteration is asked most The process of excellent solution considers to be used in an iterative process from L₂Norm approaches L₀The mode of norm, so that the two norm be made full use of to ask Solve the respective advantage of corresponding least square solution, realize one from the convex solution optimized to non-convex optimization gradually switch transition, thus Realize more accurate ground reconstruction signal.

One kind approaching L in the sparse reconstruction of compressed sensing₀The design constructing method of the gradient project algorithms of norm, this method Based on L₂The convex optimized algorithm and minimum L of norm₀The non-convex optimization algorithm of norm, proposition it is a kind of from L₂Norm approaches L₀Norm New Algorithm, and realized by way of gradient projection from L₂Norm approaches L₀The iterative solution of norm, takes full advantage of two The respective advantage of person's algorithm improves the precision of image reconstruction while reducing algorithm complexity.This method includes such as Lower step:

Step 1 defines an approximating function model first, and the function of approximating function is made by the parameter value in adjustment function It is worth from L₂Norm is to smooth L₀Norm is approached, this approximating function model is used for the sparse signal rarefaction representation of approximate substitution reconstruct Solution；

Step 2, then using the sparse representation model of this approximating function as regular terms, make reconstructed image Cephalometry Approach that item is minimum with the error of actual measured value, i.e. solution error approaches the least square solution of item, and then reconstructs sparse signal；

Step 3, constantly iterative solution reconstruction signal during, realize solution with the mode of gradient projection, approach letter Parameter value in number is adaptively adjusted according to reconstructed error, to realize sparse table aspect from L₂Norm is to L₀Norm is not It is disconnected to approach, finally reach the purpose of more accurate reconstruction signal.

Wherein, due to seeking minimum L₂The convex optimized algorithm objective function of norm be it is convex, be easily found globally optimal solution, calculate Method complexity is relatively low, but the degree of rarefication for reconstructing sparse signal is not high enough, causes the reconstruction accuracy of sparse signal inadequate, to make The quality of image reconstruction is relatively low.Due to seeking minimum L₀Norm non-convex optimization algorithm objective function be it is non-convex, be easily trapped into part Optimal solution, so being not easily found globally optimal solution, algorithm complexity is higher, but required pendulous frequency is less, reconstruction signal tool There is most sparse solution, therefore possess the reconstruction accuracy of better sparse signal, to improve image reconstruction quality.L₂Norm is convex excellent Change algorithm to be more suitable for rebuilding the structure and profile information of the low frequency of image, and minimum L₀The non-convex optimization algorithm of norm More conducively the details and texture image of image high frequency are rebuild, and by L₂Norm is to L₀The approximate procedure of norm is then one The process of continuous iterative approach optimal solution.In L₂Norm approaches L₀During the continuous iterative approach optimal solution of norm, part is fallen into A possibility that optimal instable risk of reconciliation has been lowered, and solution falls into the globally optimal solution environs of our needs It is greatly increasing, therefore is approaching minimum L in this way₀The obtained solution of the algorithm of norm can preferably approach global optimum Value, so that the precision of reconstruction signal be made to be improved.

One kind of the invention approaches L in the sparse reconstruction of compressed sensing₀The design construction side of the gradient project algorithms of norm Method.Usually, non-convex optimization algorithm especially minimum L₀Norm method has most sparse solution, and required pendulous frequency is minimum, but It is algorithm complexity highest；Based on minimum L₁The convex optimized algorithm of norm guarantees that required pendulous frequency is only with very strong reconstruction Compare L₀The non-convex optimization algorithm of norm is slightly more, but is nonanalytic (can not derivation) at zero point, algorithm complexity also usually compared with Height is difficult to be suitable for large scale problem；Greedy algorithm based on match tracing has rebuilds speed well, but Exact Reconstruction It is theoretical guarantee weaker, reconstruction precision is lower.And minimum L₂Norm method is also a kind of convex optimized algorithm, can preferably find the overall situation It is optimal, it is also parsing at zero point, but required pendulous frequency is more, the sparsity for reconstructing sparse signal, which generally can not meet, to be wanted It asks.In conjunction with minimum L₂Norm and minimum L₀The respective advantage and disadvantage of norm algorithm, since (2) formula is the mistake that an iteration seeks optimal solution Journey considers to be used in an iterative process from L₂Norm approaches L₀The mode of norm, to make full use of the two norm respective excellent Point, realize one from the convex solution optimized to non-convex optimization gradually switch transition.

By (2) formula it is found that Θ is sensing matrix, | | s | |₀It is L₀Norm is estimated, and indicates the number of vector nonzero term.Due to L₀Norm is without polynomial NP difficult problem, and direct solution is time-consuming combinatorial problem.

We define approximating function first:

To haveWherein, n is the length of sparse signal vector s, s_iFor sparse signal vector s Corresponding i-th of element, σ are to approach L₀The modulation parameter of norm, F_σ(s) number of the approximate representation s non-zero compared with sport.Approach letter Number f_σ(s_i) and s_i, σ relational graph as shown in figure 5, when σ is larger, F_σIt (s) can be L with approximate representation₂Norm, in Fig. 5 (a) Solid black lines shown in.According to progressive thought, when σ value is gradually reduced, F_σ(s) L is gradually approached₀Norm, the L of vector s₀Model Number criterion optimization problem can approximate representation be | | s | |₀=F_σ(s).As shown in Figure 5, the L due to approaching₀The function curve of norm It is smooth guidable, therefore also referred to as approaches smooth L₀Norm algorithm.The Discontinuous Function of rarefaction representation problem as a result, | | s | |₀It is minimum, be converted into continuous function F_σ(s) minimum.To which sparse representation model (2) formula is converted into

Model (4) is typically suitable for common rarefaction representation, for the specific inverse problem of compressed sensing image reconstruction, sparse table Show priori as regular terms, so the optimal solution of compressed sensing signal reconstruction problem system is solved, it is also contemplated that approaching the pact of item Shu Youhua.For this purpose, adding the item that approaches of reconstruct on the basis of model (4), and then forms and approach smooth L₀Norm sparse table The compressed sensing image reconstruction model shown:

Formula (5) is objective function required for us, and the final purpose of this paper algorithm is exactly to ask to make the minimization of object function Sparse vector s estimated value, wherein λ be balance of weights parameter, to adjust sparse table aspect weight be distributed, reconstruction signal Error term adds oneConstant is the calculating after derivation for convenience.Rarefaction representation priori item in above-mentioned model is with sparse letter Number s is process content, and error approach item be reconstructed image Cephalometry and actual measured value residual error it is minimum, can be considered as Global optimization to entire image.In order to ask the minimum value of J (s) objective function, F_σ(s) the σ parameter in is gradually reduced to make F_σ (s) L is approached₀Norm needs constantly to reduce the value of σ parameter in an iterative process.The iterative solution of gradient projection is used herein Optimization algorithm solves, and carries out derivation to s in (10) formula, obtains the gradient delta J (s) of objective function:

Minimum L is approached based on gradient projection₀The rudimentary algorithm frame of norm algorithm is as follows:

1) physical meaning of variable name: sensing matrix Θ, measured value y, the rarefaction representation s of original signal；

2) it initializes:Θ^⊥For the pseudo inverse matrix of Θ, Θ^⊥=(Θ^TΘ)^-1Θ^T, the decaying sequence of parameter σ σ=[σ₀, α σ₀, α²σ₀..., α^kσ₀], weight parameter λ and gradient decline step-length γ；

3) gradient projection the number of iterations k, that is, circulation σ sequence k:

(b) gradient descent direction:

(c) gradient direction updates:

(d) rectangular projection is constrained:

(e)

4) meet and stop iterated conditional, end loop: outputFor the sparse signal s finally acquired.

It is worth noting that, in algorithm solution procedure, with iterations going on, errorIt is smaller and smaller, σ's Reduction makes rarefaction representation regular termsIncreasingly approach L₀Norm.Original is reconstructed in order to be more accurate Beginning signal, we can allow errorShared weight is gradually increased, and approaches L₀The rarefaction representation regular terms F of norm_σ(s) Shared weight is opposite to be reduced.Therefore λ can be set intoThe descending series that is positively correlated of value, so as to It can be according to error in iterative processSize adaptation go adjustment weight λ value.

By two and three dimensions relational graph in Fig. 5 it is found that being gradually reduced with σ, rarefaction representation regular terms F_σ(s) increasingly Approach smooth approximate L₀Norm is gradually approached while having gradually decreased the degree of rarefication of reconstruct sparse signal with bigger efficiency The globally optimal solution that we need, to reduce algorithm complexity and improve the precision of image reconstruction, to be more favorable for Reconstruction of the compressed sensing to image.

In order to make the minimization of object function in (5) formula acquire sparse solution s, s is resolved into positive portion and negative portion by us Point.We introduce vector u and v and are expressed as follows as a result:

S=u-v, u >=0, v >=0 (7)

Wherein, u_i=(s_i)₊, v_i=(- s_i)+, i=1,2 ..., n, n are the length of vector s.(x)₊=max { 0, x }

, indicate that positive number is taken to operate.To the conversion of (5) formula are as follows:

Wherein, u >=0, v >=0.By (8) formula at the bound constrained quadratic programming problem of standard.

Wherein,B=Θ^TY,With

In order in an iterative process can be according to errorSize adaptation go to adjust sparse Xiang Quan The value of weight λ, λ is arranged for we are as follows:

λ=0.1 | | Θ s-y | |_∞ (10)

Observation is it is found that the dimension of matrix is twice: s ∈ R in formula (5) in formula (9)ⁿ, and z ∈ R²ⁿ.Matrix dimension Increase have very faint influence to experimental result, for the size of matrix, to the operation of matrix A show more added with Efficiency, for z^T=[u^Tv^T], it is easy to it obtains:

This needs the product calculation of a Θ, it means that for objective functionAlso the product calculation by a Θ is only needed, so the increase of matrix dimension is to calculation The influence of method complexity is little.

In order to solve the objective function in (9), we are by z^(k)To z^(k+1)Successive iteration is carried out, scalar parameter μ is introduced^(k)> 0, thus:

For the vector z of each iteration^(k)Value, we are along negative gradient directionIt scans for, while it Nonnegative quadrant is projected to, recalled a linear search is until J (z^(k)) value have one adequately decline until.Along This gradient direction, if no longer encountering a new restrict, we pass through a guess value μ of an initialization^(k)Make J(z^(k)) minimize as far as possible.Define a new vector g^(k):

Wherein i=1,2 ..., 2n, 2n are z vector element number, we initialize μ by following equation.

Obtain μ₀Specific calculated result are as follows:

μ in order to prevent₀Value it is too large or too small, we are usually μ₀It is limited in section [μ_min, μ_maxIn, wherein 0 < μ_min< μ_max.Be equivalent to make one take in operation mid (μ_min, μ_max, μ₀), as (g^(k))A^Tg^(k)When=0, μ₀=μ_max。 μ is chosen in this way₀Mode be a very novel idea, the maximal and minmal value boundary edge that has ignored μ relative to front Negative gradientDirection J (z^(k)) for the mode that reduces, this new mode be more easier to generate one it is acceptable more Add appropriate μ₀Value.

Approach L₀The gradient project algorithms of norm are shown in the specific implementation steps are as follows:

1) it initializes.As indicated above, s⁽⁰⁾=Θ^⊥Y, Θ^⊥For the pseudo inverse matrix of Θ, m⁽⁰⁾=(s⁽⁰⁾)₊, n⁽⁰⁾=(- s⁽⁰⁾ ₊,μ₀Attenuation factor, α ∈ (0,1)；Invariant γ,Sequence of iterations k, k=0.

2) μ is calculated by formula (13)₀, by mid (μ_min, μ_max, μ₀) determine μ₀Value.

3) recall linear search: in sequence μ₀, α μ₀, α²μ₀... middle search first meets the value conduct of inequality (14) μ^(k)Value.

Then it selects

4) algorithm operation restrains and stops iteration, z at this time after meeting termination condition^(k+1)The as estimated value of gained z；It is no Person, k=k+1,2), 3) process continues iteration for repetition, until meeting termination condition.

Termination condition are as follows:

Wherein,For kth time iteration J (z^(k)) in z^kThe gradient at place, z^(k)For the obtained z value of kth time, tolA is One constant lower limit, default value 0.01.

Because New Algorithm is to approach L by way of gradient projection (Gradient Projection)₀Norm, therefore This New Algorithm is named as L0GP algorithm by we.In order to verify the validity of L0GP algorithm reconstruction signal, we below will Further emulation experiment is done to the reconstruction of non-sparse two dimensional image signal.In the reconstruction experiment to picture signal, in order to The universality of L0GP algorithm performance is embodied, we will make comparisons with the algorithms most in use of some compressed sensing signal reconstructions, these ratios Compared with algorithm have Subspace Pursuit algorithm (SP), Iterative Reweighted Least Squares algorithm (IRLS), Orthogonal Matching Pursuit algorithm (OMP), Regularized Orthogonal Matching Pursuit algorithm (ROMP), Generalized Orthogonal Matching Pursuit algorithm (GOMP), L1-Basis Pursuit algorithm (L1_BP) and L1-Regularized Least Squares algorithm (L1_LS).

It is 512 × 512 that we, which are used to do the image size that emulates, if two dimensional image signal is arranged in single-row one-dimensional The dimension of signal, signal has exceeded computer capacity, will spend a large amount of calculating time.Therefore, the method that we take is pair The column signal (512 × 1) and row signal (1 × 512) of image carry out projection measurement, i.e., projection pattern shown in Fig. 4, are equivalent to pair The column signal and row signal of image are rebuild, and the reconstruction time of such entire image will also react the reconstruction efficiency of algorithm.By It is non-sparse signal in two dimensional image signal, therefore before image carries out compressed sensing signal reconstruction, needs to carry out image dilute Transformation is dredged, that take herein is discrete wavelet sparse transformation (DWT), i.e. Ψ in x=Ψ s is wavelet basis.The survey that we use Moment matrix Φ is calculation matrix shown in Fig. 3.

For size be 512 × 512 (n=512 × 512) Mandrill image, we to image measurement m=170 × 170 (i.e. sample rate is about) it is secondary restoration and reconstruction are carried out to image, experimental result is as shown in Figure 6.

Then the Y-PSNR (PSNR), structural similarity (SSIM) to eight kinds of algorithm reconstruction images in upper figure and again The structure time has carried out data statistics, obtains result shown in table one.

The experimental result of algorithms of different in the case of one 1/9 sample rate of table

From Fig. 6 and table one as can be seen that under 1/9 sampled measurements rate, compared to the suboptimal algorithm of minimum norm class With the greedy algorithm of match tracing class, the PSNR and SSIM of New Algorithm reconstruction image have a degree of raising, possess Higher image reconstruction accuracy.The eyes of Mandrill image are amplified, it is seen that New Algorithm reconstruction image it is thin Section is apparent from, and image quality is more preferable.

Since the image reconstruction time of the greedy algorithm of match tracing class depends on the number of iterations of setting or terminates item Part, although ROMP algorithm image reconstruction time is very short, its image reconstruction accuracy is poor very compared to for L0GP algorithm It is more.And when the number of iterations takes certain number to, the image reconstruction accuracy of ROMP algorithm can't be with time and iteration time Several increases are obviously improved, and can be consumed a large amount of time cost instead and be calculated cost.Novel L0 is approached due to this The gradient project algorithms of norm approach global optimum's solution as sparse as possible with higher efficiency and speed, so compared to traditional For iterative least square algorithm (IRLS) and L1 norm algorithm, image reconstruction is improved while reconstruction time is considerably reduced Precision, this is also the sharpest edges place of this calculation.

It is the Fingerprint image of 512 × 512 (n=512 × 512) for size, we increase the sampling to image Pendulous frequency, carrying out emulation experiment again, (i.e. sample rate is about to image measurement m=256 × 256) it is secondary come to figure As carrying out restoration and reconstruction, the reconstruction image experimental result of Fig. 7 and table two has been obtained.

The experimental result of algorithms of different in the case of 2 1/4 sample rate of table

By Fig. 7 and table two it is found that the texture and detail section of Fingerprint image are amplified, the texture of image It becomes more fully apparent, noise becomes lower.

Reconstruct for big data quantity as image class and high sampling rate signal, the greedy algorithm of OMP class is without very Good theoretical guarantee, by data above statistics it is found that the precision of reconstructed image and time can not often reach ideal simultaneously, Exact Reconstruction signal has certain randomness.Because reconstructed image precision can't when the number of iterations reaches a certain level It is obviously improved with the increase of the number of iterations, can accordingly pay very big time cost instead, so for the greedy of OMP class Greedy algorithm often considers time cost on the basis of meeting reconstruction accuracy, and compromise is taken in reconstitution time and reconstruction accuracy Value.

Comprehensive analysis comprehensively considers, L0GP New Algorithm from signal reconstruction time and reconstruction accuracy the two evaluation indexes Performance be classic.To the validity of two kinds of image reconstruction results of Mandrill and Fingerprint, also demonstrates and newly mention Universality of the algorithm out to different type image reconstruction.Compared to minimum norm class algorithm and tracking class algorithm, L0GP is novel Algorithm performance is more outstanding, also demonstrates that New Algorithm is with higher efficiency in principle analysis part and bigger probabilistic approximation is complete The hypothetic deduction of the optimal solution as sparse as possible of office.

Part of that present invention that are not described in detail belong to the well-known technology of those skilled in the art.

Those of ordinary skill in the art it should be appreciated that more than embodiment be intended merely to illustrate the present invention, And be not used as limitation of the invention, if in spirit of the invention, to embodiment described above variation, Modification will all be fallen in the range of claims of the present invention.

Claims (5)

1. one kind approaches L based on compressed sensing₀The gradient projection image rebuilding method of norm, it is characterized in that: this method includes as follows Step:

Step 1 defines an approximating function first, makes the functional value of approximating function from approximation by the parameter value in adjustment function L₂Norm is to L₀Norm is approached, rarefaction representation solution of this approximating function model for the sparse signal of approximate substitution reconstruct；

Step 2, then using the sparse representation model of this approximating function as regular terms, and with the mode of gradient projection approach use up May sparse Wavelet representation for transient coefficient, so that the error of reconstructed image Cephalometry and actual measured value is approached item minimum, in turn Reconstruct sparse signal；

Step 3, during constantly iterative solution reconstruction signal, parameter value in approximating function according to reconstructed error carry out from Adjustment is adapted to, to realize sparse table aspect from L₂To L₀Constantly approaching for norm, finally reaches the mesh of more accurate reconstruction signal 's.

2. according to claim 1 approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, feature It is: due to solving L₂The convex optimized algorithm objective function of norm be it is convex, be easily found globally optimal solution, algorithm complexity also compared with It is low, but the degree of rarefication for reconstructing sparse signal is not high enough, causes the reconstruction accuracy of sparse signal inadequate, to make the matter of image reconstruction It measures relatively low.

3. according to claim 1 approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, feature It is: due to seeking minimum L₀Norm non-convex optimization algorithm objective function be it is non-convex, locally optimal solution is easily trapped into, so being not easy Globally optimal solution is found, algorithm complexity is higher, but required pendulous frequency is less, and reconstruction signal has most sparse solution, therefore Possess the reconstruction accuracy of better sparse signal, to improve image reconstruction quality.

4. according to claim 2 or 3 approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, it is special Sign is: L₂Norm convex optimized algorithm is more suitable for rebuilding the structure and profile information of the low frequency of image, and minimum L₀Norm Non-convex optimization algorithm is more conducive to rebuild the details and texture image of image high frequency, and by L₂Norm is to L₀Norm is forced Near procedure is then the process of a continuous iterative approach optimal solution.

5. according to claim 4 approach L based on compressed sensing₀The gradient projection image rebuilding method of norm, feature It is: in L₂Norm approaches L₀During the continuous iterative approach optimal solution of norm, falls into local optimum and conciliate instable risk It has been be lowered that, and a possibility that solution falls into the globally optimal solution environs of needs is greatly increasing, and forces in this way Nearly minimum L₀The obtained solution of the algorithm of norm can preferably approach global optimum, so that the precision of reconstruction signal be enable to mention It is high.