CN106899305A - A kind of primary signal reconstructing method based on Second Generation Wavelets - Google Patents

Abstract

The invention discloses a kind of primary signal reconstructing method based on Second Generation Wavelets.Its implementation is：The present invention uses the Second Generation Wavelets to carry out rarefaction representation to primary signal x as sparse base, x samplings are carried out to primary signal with computing cluster afterwards, measured value y is obtained by calculation matrix Φ and primary signal x, measured value y is stored or is transmitted, the reconstruct of data is finally carried out using segmentation orthogonal matching pursuit, make it possible to by measured value y, i.e. result after primary signal x compressions recovers primary signal x in the distortion range for allowing.The present invention solves the problems, such as that traditional compressed sensing technology can not well carry out rarefaction representation for destructuring mass network data, propose a kind of primary signal reconstructing method based on Second Generation Wavelets, to solve the problems, such as that conventional compression perceives the unworthiness for destructuring mass network data, compression ratio and distortion as small as possible as big as possible is reached.

Description

A kind of primary signal reconstructing method based on Second Generation Wavelets

Technical field

The invention belongs to field of data compression, more particularly to a kind of primary signal reconstructing method based on Second Generation Wavelets, The reconstruct to destructuring mass network data is perceived suitable for conventional compression.

Background technology

Compressed sensing as an emerging sampling theory, by using the sparse characteristic of data, can far below how Under conditions of Qwest's sample rate, data sample of the data volume much smaller than primary signal is obtained, then by nonlinear algorithm weight Structure primary signal.

Traditional compressed sensing technology is developed for being applied to compression of images field, due to view data available point The form of battle array is indicated, and rarefaction representation can be carried out to it well as sparse base with first generation small echo.And for non-knot Structure mass network data, due to its obvious unstructured nature, it is right well to be difficult to as sparse base using first generation small echo It carries out rarefaction representation, and this will cause compressed sensing to be remarkably decreased the quality of destructuring mass network data.

Second Generation Wavelets method, relative to Traditional Wavelet algorithm, is a kind of more fast and effectively wavelet transformation realization side Method, is independent of Fourier conversion, completes the construction to biorthog-onal wavelet filter in time domain completely.This building method exists Structured Design and self-adaptive construction aspect outstanding advantages compensate for the deficiency of traditional frequency domain building method.In compressed sensing technology In, be used as sparse base by Second Generation Wavelets carries out rarefaction representation to primary signal so that destructuring mass data energy Enough by good rarefaction representation.

The content of the invention

It is an object of the invention to propose a kind of primary signal reconstructing method based on Second Generation Wavelets, to solve traditional pressure Contracting perceives the unworthiness problem for destructuring mass network data, to realize as big as possible compression ratio and as small as possible Distortion.

To achieve the above object, technical scheme includes as follows：

A kind of primary signal reconstructing method based on Second Generation Wavelets, comprises the following steps：

Step 1：Rarefaction representation is carried out to primary signal, sparse coefficient is obtained by sparse base and primary signal；

Step 2：Primary signal is sampled, measured value is obtained by calculation matrix and primary signal；

Step 3：Sparse base, measured value and random number seed are stored；

Step 4：Sparse coefficient is obtained by iteration, primary signal is reconstructed using sparse base.

Further according to the primary signal reconstructing method based on Second Generation Wavelets, to primary signal described in step 1 Rarefaction representation is carried out, sparse coefficient is obtained by sparse base and primary signal：

For the point set S={ x for giving₁,x₂,…,x_n, there is x₁<x₂<…<x_n, n=k × 2^l, wherein n, k, l is just whole Number；

Matrix V for any one 2k × 2k, then V^LTable takes following k × 2k half parts of matrix V, V^UExpression takes square K × 2k half parts of the top of battle array V, have

Be used as sparse base by Second Generation Wavelets carries out rarefaction representation to primary signal x so that destructuring mass data Rarefaction representation can be carried out to primary signal x by good rarefaction representation, carried out as follows：

(1-1) constructs the matrix M of 2k × 2k_1,i,Wherein i=1 ..., n/ 2k, s_i=(i-1) 2k；

Matrix M_1,iOrthogonal basis U_1,i, U_1,i=[Orth (M_1,i)]^T, Orth represents and seeks orthogonal Base computing；By U_1,iObtain U₁：

By U₁Obtain first basic matrix Ψ₁, first basic matrix Ψ₁It is fortune in the middle of necessary when constructing sparse base Ψ Calculate result, Ψ₁=U₁；

(1-2) constructs a matrix M of 2k × 2k_2,i,Wherein i=1 ..., n/2k, by M_2,iObtain U_2,i, U_2,i=[Orth (M_2,i)]^T；Pass through U again_2,iObtain U₂′：

Wherein n₂=n/4k, by U₂' obtain U₂,Wherein I_n/2It is the unit square of (n/2) × (n/2) Battle array；

By U₂And U₁, obtain second basic matrix ψ₂, ψ₂=U₁U₂；

According to structural matrix M_1,iMethod construct M_1,2iAnd M_1,2i-1；

(1-3) obtains j-th basic matrix ψ_j, wherein j=2 ..., log₂(n/k) the matrix M of 2k × 2k, is constructed_j,i,Wherein i=1 ..., n/2k, by M_j,iU can be obtained_j,i, U_j,i=[Orth (M_j,i)]^T, by U_j,iU can be obtained_j′：

By U_j' U can be obtained_j,

For j=2 ..., log₂(n/k) U obtained by_jAnd U₁, j-th basic matrix ψ can be obtained_j, ψ_j=U₁U₂…U_j；

(1-4) obtains maximum log as j₂(n/k) when, j-th basic matrix ψ of gained_jAs sparse base ψ, i.e.,

(1-5) obtains sparse coefficient s, specific s=ψ by sparse base ψ and primary signal x^-1X, wherein ψ^-1Expression asks dilute Dredge the inverse matrix of base ψ；

The sparse coefficient s is exactly the result of primary signal x rarefaction representations on sparse base ψ.

Further according to the primary signal reconstructing method based on Second Generation Wavelets, to primary signal described in step 2 Sampled, measured value is obtained by calculation matrix and primary signal；

One size of construction is the Bernoulli random matrixes of M × N as calculation matrix Φ, and each of which element is only It is vertical to obey Bernoulli distributions, the i-th row, jth row element Φ_i,jRepresent：

Measured value y, y=Φ x=Φ ψ s=As are obtained by calculation matrix Φ and primary signal x.

Further according to the primary signal reconstructing method based on Second Generation Wavelets, to sparse base, survey described in step 3 Value and random number seed are stored：

The random number seed of sparse base ψ, measured value y and Bernoulli random matrixes as calculation matrix Φ is carried out Storage, is called when needing and primary signal is reconstructed；

The random number seed refers to the true random number that computer is used to generate random matrix, each element in matrix All it is to be calculated through algorithm by a true random number from computer system time, the true random number is random number Seed；Certain in seed, in the case that algorithm is certain, the random matrix of acquisition is identical；In storage, it is not necessary to which storage is passed Defeated whole calculation matrix Φ, only stores random number seed；

Further according to the primary signal reconstructing method based on Second Generation Wavelets, obtained by iteration described in step 4 To sparse coefficient, primary signal is reconstructed using sparse base, carried out as follows：

(4-1) obtains measured value y, random number seed and sparse base ψ, generates calculation matrix Φ, and by calculation matrix Φ Sensing matrix A, A=Φ ψ are obtained with sparse base ψ；

(4-2) does not start residual error r during iteration₀=y, does not start index set during iterationIteration time The initial value 1 of number t, wherein t is iterations, and maximum iteration is t_max；

The residual error r that (4-3) is obtained by last iteration_t-1With the line number M of sensing matrix A, thresholding T is obtained_h,||·||₂2 norms of matrix are sought in expression, and wherein ts is threshold parameter ts, when rt represents the t times iteration Residual error；

By sensing matrix A and residual error r_t-1A vectorial u of a length of N is obtained, there are u=abs (A^Tr_t-1), abs () is represented Modulus value；

For 1≤j≤N, calculate successively<r_t-1,a_j>,<·,·>Inner product of vectors is sought in expression, the N number of numerical value structure that will be obtained Into vectorial u, thresholding T is more than in selection u_hValue, the row sequence number j of correspondence sensing matrix A is constituted into set J₀, the collection is combined into row Sequence number set, J₀Represent the index that each iteration finds；

(4-4) index set Λ_t=Λ_t-1∪J₀, row set A_t=A_t-1∪a_j, seek y=A_ts_tLeast square solution, obtain s_t's Estimate Update residual errort =t+1；

Wherein a_jThe jth row of representing matrix sensing matrix A, A_tRepresent by index Λ_tThe row set of the sensing matrix A for selecting；

If Λ_t=Λ_t-1, or t>t_max, or r_t=0, then stop iteration；

Reconstruct gainedIn Λ_tThere is nonzero term at place, and its value is last time iteration gained sparse coefficient s_t,It is iteration knot To the final estimate of sparse coefficient s, s during beam_tTo the estimate of sparse coefficient s during for the t times iteration；

At the end of (4-5) iteration, sparse coefficient is obtainedUsing sparse base ψ restructural primary signals, that is, obtain original letter The estimate of number x

The estimateThe primary signal for as reconstructing.

The present invention compared with prior art, has the following advantages that：

1. the present invention to primary signal x using Second Generation Wavelets as sparse base due to carrying out rarefaction representation so that non-knot Structure mass data can solve traditional compressed sensing technology based on first generation small echo to non-by good rarefaction representation The unworthiness problem of structuring mass data, obtains larger compression ratio, and Second Generation Wavelets are multinomials, so having meter Calculate fireballing advantage.

2. the present invention is due to carrying out the reconstruct of data using segmentation orthogonal matching pursuit, enabling by measured value y, That is the result after primary signal x compressions, primary signal x is recovered in the distortion range for allowing, and is provided to improving compression ratio Sound assurance.

Brief description of the drawings

Fig. 1 is that a kind of primary signal reconstructing method based on Second Generation Wavelets of the present invention realizes flow chart.

Specific embodiment

To make the objects, technical solutions and advantages of the present invention become more apparent, below in conjunction with accompanying drawing, to of the present invention Scheme and effect are described in further detail.

As shown in figure 1, the present invention is to a kind of primary signal reconstructing method based on Second Generation Wavelets, specific steps are such as It is lower described.

Step 1：Rarefaction representation is carried out to primary signal, sparse coefficient is obtained by sparse base and primary signal.

For the point set S={ x for giving₁,x₂,…,x_n, wherein x₁<x₂<…<x_n, n=k × 2^l, n, k, l is just whole Number.

Matrix V for any one 2k × 2k, then V^LTable takes following k × 2k half parts of matrix V, V^UExpression takes square K × 2k half parts of the top of battle array V, have

Be used as sparse base by Second Generation Wavelets carries out rarefaction representation to primary signal x so that destructuring mass data Rarefaction representation can be carried out to primary signal x by good rarefaction representation, carried out as follows：

(1-1) constructs the matrix M of 2k × 2k_1,iIt is as follows：

Wherein i=1 ..., n/2k, s_i=(i-1) 2k.

Matrix M_1,iOrthogonal basis U_1,i, U_1,i=[Orth (M_1,i)]^T, Orth is represented and is sought orthogonal Base computing, by U_1,iObtain U₁：

By U₁Obtain first basic matrix Ψ₁, first basic matrix Ψ₁It is fortune in the middle of necessary when constructing sparse base Ψ Calculate result, Ψ₁=U₁。

(1-2) constructs a matrix M of 2k × 2k_2,i,Wherein i=1 ..., n/2k, by M_2,iObtain U_{2, i}, U_{2, i}=[Orth (M_{2, i})]^T, then by U_{2, i}Obtain U₂′：

Wherein n₂=n/4k, by U₂' obtain U₂,Wherein I_n/2It is the unit square of (n/2) × (n/2) Battle array.

By U₂And U₁, obtain second basic matrix ψ₂, ψ₂=U₁U₂。

According to structural matrix M_{1, i}Method construct M_1,2iAnd M_1,2i-1。

(1-3) obtains j-th basic matrix ψ_j, wherein j=2 ..., log₂(n/k) the matrix M of 2k × 2k, is constructed_{J, i},Wherein i=1 ..., n/2k, by M_{J, i}U can be obtained_{J, i}, U_{J, i}=[Orth (M_{J, i})]^T, by U_{J, i}U can be obtained_j′：

By U_j' U can be obtained_j,

For j=2 ..., log₂(n/k) U obtained by_jAnd U₁, j-th basic matrix Ψ can be obtained_j, ψ_j=U₁U₂…U_j。

(1-4) obtains maximum log as j₂(n/k) when, j-th basic matrix ψ of gained_jAs sparse base ψ, i.e.,

(1-5) obtains sparse coefficient s, specific s=ψ by sparse base ψ and primary signal x^-1x.Wherein ψ^-1Expression asks dilute Dredge the inverse matrix of base ψ.

The sparse coefficient s is exactly the result of primary signal x rarefaction representations on sparse base ψ.

Step 2：Primary signal is sampled, measured value is obtained by calculation matrix and primary signal.

One size of construction is the Bernoulli random matrixes of M × N as calculation matrix Φ, and each of which element is only It is vertical to obey Bernoulli distributions, the i-th row, jth row element Φ_i,jRepresent：

Measured value y, y=Φ x=Φ ψ s=As are obtained by calculation matrix Φ and primary signal x.

Step 3：Sparse base, measured value and random number seed are stored.

The random number seed of sparse base ψ, measured value y and Bernoulli random matrixes as calculation matrix Φ is carried out Storage, is called when needing and primary signal is reconstructed.

The random number seed refers to the true random number that computer is used to generate random matrix, each element in matrix All it is to be calculated through algorithm by a true random number from computer system time, the true random number is random number Seed.Certain in seed, in the case that algorithm is certain, the random matrix of acquisition is identical.In storage, it is not necessary to which storage is passed Defeated whole calculation matrix Φ, only stores random number seed.

Step 4：Sparse coefficient is obtained by iteration, primary signal is reconstructed using sparse base.

Carry out as follows：

(4-1) obtains measured value y, random number seed and sparse base ψ, generates calculation matrix Φ, and by calculation matrix Φ Sensing matrix A, A=Φ ψ are obtained with sparse base ψ；

(4-2) does not start residual error r during iteration₀=y, does not start index set during iterationIteration time The initial value 1 of number t, wherein t is iterations, and maximum iteration is t_max；

The residual error r that (4-3) is obtained by last iteration_t-1With the line number M of sensing matrix A, thresholding T is obtained_h,||·||₂2 norms of matrix, wherein t are asked in expression_sIt is threshold parameter t_s, r_tWhen representing the t times iteration Residual error；

By sensing matrix A and residual error r_t-1Obtain a vectorial u of a length of N, u=abs (A^Tr_t-1), abs () is represented and asked Modulus value；

For 1≤j≤N, calculate successively<r_t-1,a_j>,<,·>Inner product of vectors is sought in expression, and the N number of numerical value that will be obtained is constituted Vectorial u, thresholding T is more than in selection u_hValue, the row sequence number j of correspondence sensing matrix A is constituted into set J₀, the collection is combined into row sequence Number set, J₀Represent the index that each iteration finds；

(4-4) index set Λ_t=Λ_t-1∪J₀, row set A_t=A_t-1∪a_j, seek y=A_ts_tLeast square solution, obtain s_t's Estimate Update residual errort =t+1；

Wherein a_jThe jth row of representing matrix sensing matrix A, A_tRepresent by index Λ_tThe row set of the sensing matrix A for selecting；

If Λ_Λ=Λ_t-1, or t>t_max, or r_t=0, then stop iteration；

Reconstruct gainedIn Λ_tThere is nonzero term at place, and its value is last time iteration gained sparse coefficient s_t,For iteration terminates When to the final estimate of sparse coefficient s, s_tTo the estimate of sparse coefficient s during for the t times iteration；

At the end of (4-5) iteration, sparse coefficient is obtainedUsing sparse base ψ restructural primary signals, that is, obtain original letter The estimate of number x

The estimateThe primary signal for as reconstructing.

The above is only and the preferred embodiment of the present invention is described, technical scheme is not limited to This, any known deformation that those skilled in the art are made on the basis of major technique of the invention design belongs to the present invention Claimed technology category, the specific protection domain of the present invention is defined by the record of claims.

Claims (5)

1. a kind of primary signal reconstructing method based on Second Generation Wavelets, it is characterised in that comprise the following steps：

Step 1：Rarefaction representation is carried out to primary signal, sparse coefficient is obtained by sparse base and primary signal；

Step 2：Primary signal is sampled, measured value is obtained by calculation matrix and primary signal；

Step 3：Sparse base, measured value and random number seed are stored；

Step 4：Sparse coefficient is obtained by iteration, primary signal is reconstructed using sparse base.

2. the primary signal reconstructing method of Second Generation Wavelets is based on according to claim 1, it is characterised in that institute in step 1 State carries out rarefaction representation to primary signal, and sparse coefficient is obtained by sparse base and primary signal：

For the point set S={ x for giving₁,x₂,…,x_n, there is x₁<x₂<…<x_n, n=k × 2^l, wherein n, k, l is positive integer；

Matrix V for any one 2k × 2k, then V^LTable takes following k × 2k half parts of matrix V, V^UExpression takes matrix V K × 2k half parts of top, have

Be used as sparse base by Second Generation Wavelets carries out rarefaction representation to primary signal x so that destructuring mass data can By good rarefaction representation, rarefaction representation is carried out to primary signal x, carried out as follows：

(1-1) constructs the matrix M of 2k × 2k_1,i’Wherein i=1 ..., n/2k, s_i=(i-1) 2k；

Matrix M_1,iOrthogonal basis U_1,i, U_1,i=[Orth (M_1,i)]^T, Orth represents and seeks orthogonal Base computing；By U_1,iObtain U₁：

By U₁Obtain first basic matrix Ψ₁, first basic matrix Ψ₁It is necessary intermediate operations knot when constructing sparse base Ψ Really, Ψ₁=U₁；

(1-2) constructs a matrix M of 2k × 2k_2,i,Wherein i=1 ..., n/2k, by M_2,iObtain U_2,i, U_2,i=[Orth (M_2,i)]^T；Pass through U again_2,iObtain U₂′：

Wherein n₂=n/4k, by U₂' obtain U₂,Wherein I_n/2It is the unit matrix of (n/2) × (n/2)；

By U₂And U₁, obtain second basic matrix ψ₂, ψ₂=U₁U₂；

According to structural matrix M_1,iMethod construct M_1,2iAnd M_1,2i-1；

(1-3) obtains j-th basic matrix ψ_j, wherein j=2 ..., log₂(n/k) the matrix M of 2k × 2k, is constructed_j,i,Wherein i=1 ..., n/2k, by M_j,iU can be obtained_j,i, U_j,i=[Orth (M_j,i)]^T, by U_j,iU can be obtained_j′：

By U_j' U can be obtained_j,

For j=2 ..., log₂(n/k) U obtained by_jAnd U₁, j-th basic matrix ψ can be obtained_j, ψ_j=U₁U₂…U_j；

(1-4) obtains maximum log as j₂(n/k) when, j-th basic matrix ψ of gained_jAs sparse base ψ, i.e.,

(1-5) obtains sparse coefficient s, specific s=ψ by sparse base ψ and primary signal x^-1X, wherein ψ^-1Sparse base ψ is sought in expression Inverse matrix；

The sparse coefficient s is exactly the result of primary signal x rarefaction representations on sparse base ψ.

3. the primary signal reconstructing method based on Second Generation Wavelets according to claim 1 or claim 2, it is characterised in that in step 2 It is described that primary signal is sampled, measured value is obtained by calculation matrix and primary signal；

One size of construction is the Bernoulli random matrixes of M × N as calculation matrix Φ, and each of which element independently takes From Bernoulli distributions, the i-th row, jth row element Φ_i,jRepresent：

${\mathrm{\&Phi;}}_{i,j}=\left\{\begin{array}{cc}\frac{1}{\sqrt{M}}& P=\frac{1}{2}\\ -\frac{1}{\sqrt{M}}& P=\frac{1}{2}\end{array}\right.=\frac{1}{\sqrt{M}}\left\{\begin{array}{cc}+1& P=\frac{1}{2}\\ -1& P=\frac{1}{2}\end{array}\right.$

Measured value y, y=Φ x=Φ ψ s=As are obtained by calculation matrix Φ and primary signal x.

4. the primary signal reconstructing method of Second Generation Wavelets is based on according to claim any one of 1-3, it is characterised in that step Sparse base, measured value and random number seed are stored described in rapid 3：

The random number seed of sparse base ψ, measured value y and Bernoulli random matrixes as calculation matrix Φ is stored, It is called when needing and primary signal is reconstructed；

The random number seed refers to the true random number that computer is used to generate random matrix, and each element in matrix is It is calculated through algorithm by a true random number from computer system time, the true random number is random several Son；Certain in seed, in the case that algorithm is certain, the random matrix of acquisition is identical；In storage, it is not necessary to storage transmission Whole calculation matrix Φ, only stores random number seed.

5. the primary signal reconstructing method of Second Generation Wavelets is based on according to claim any one of 1-4, it is characterised in that step Sparse coefficient is obtained by iteration described in rapid 4, primary signal is reconstructed using sparse base, carried out as follows：

(4-1) obtains measured value y, random number seed and sparse base ψ, generates calculation matrix Φ, and by calculation matrix Φ and dilute Dredge base ψ and obtain sensing matrix A, A=Φ ψ；

(4-2) does not start residual error r during iteration₀=y, does not start index set during iterationIterations t Initial value 1, wherein t be iterations, maximum iteration is t_max；

The residual error r that (4-3) is obtained by last iteration_t-1With the line number M of sensing matrix A, thresholding T is obtained_h, ||·||₂2 norms of matrix, wherein t are asked in expression_sIt is threshold parameter t_s, r_tRepresent residual error during the t times iteration；

By sensing matrix A and residual error r_t-1Obtain a vectorial u of a length of N, u=abs (A^Tr_t-1), abs () represents modulus value；

For 1≤j≤N, calculate successively<r_t-1,a_j>,<·,·>Inner product of vectors is sought in expression, and the N number of numerical value that will be obtained constitutes vector U, thresholding T is more than in selection vector u_hValue, the row sequence number j of correspondence sensing matrix A is constituted into set J₀, the collection is combined into row sequence Number set, J₀Represent the index that each iteration finds；

(4-4) index set Λ_t=Λ_t-1∪J₀, row set Λ_t=Λ_t-1∪a_j, seek y=A_ts_tLeast square solution, obtain s_t's Estimate Update residual errort =t+1；

Wherein a_jThe jth row of representing matrix sensing matrix A, A_tRepresent by index Λ_tThe row set of the sensing matrix A for selecting；

If Λ_t=Λ_t-1, or t>t_max, or r_t=0, then stop iteration；

Reconstruct gainedIn Λ_tThere is nonzero term at place, and its value is last time iteration gained sparse coefficient s_t,It is right at the end of for iteration The final estimate of sparse coefficient s, s_tTo the estimate of sparse coefficient s during for the t times iteration；

At the end of (4-5) iteration, sparse coefficient is obtainedUsing sparse base ψ restructural primary signals, that is, obtain primary signal x's Estimate

The estimateThe primary signal for as reconstructing.