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CN106899305B - Original signal reconstruction method based on second-generation wavelet - Google Patents

Original signal reconstruction method based on second-generation wavelet Download PDF

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CN106899305B
CN106899305B CN201710011603.5A CN201710011603A CN106899305B CN 106899305 B CN106899305 B CN 106899305B CN 201710011603 A CN201710011603 A CN 201710011603A CN 106899305 B CN106899305 B CN 106899305B
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CN106899305A (en
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白义东
刘涛
王艳
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Shaanxi Shangpin Information Technology Co ltd
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Abstract

The invention discloses an original signal reconstruction method based on second-generation wavelets. The implementation scheme is as follows: the method comprises the steps of sparsely representing an original signal x by using a second-generation wavelet as a sparse basis, then sampling the original signal x by using a computing cluster, obtaining a measured value y from a measurement matrix phi and the original signal x, storing or transmitting the measured value y, and finally reconstructing data by using a piecewise orthogonal matching pursuit, so that the original signal x can be recovered in an allowed distortion range through the measured value y, namely a result obtained after the original signal x is compressed. The invention solves the problem that the traditional compressed sensing technology can not well carry out sparse representation on unstructured massive network data, and provides an original signal reconstruction method based on second-generation wavelets, so as to solve the problem that the traditional compressed sensing technology is not applicable to unstructured massive network data and achieve the compression ratio as large as possible and the distortion as small as possible.

Description

Original signal reconstruction method based on second-generation wavelet
Technical Field
The invention belongs to the field of data compression, and particularly relates to an original signal reconstruction method based on second-generation wavelets, which is suitable for reconstructing unstructured massive network data by traditional compressed sensing.
Background
Compressed sensing is an emerging sampling theory, and by utilizing the sparsity of data, data samples with data volume much smaller than that of an original signal can be obtained under the condition of being far lower than the nyquist sampling rate, and then the original signal is reconstructed through a nonlinear algorithm.
The traditional compressed sensing technology is developed for being applied to the field of image compression, and since image data can be represented in a lattice form, the first generation wavelet can be well represented sparsely by being used as a sparse basis. For the unstructured massive network data, due to the obvious unstructured characteristic, the first generation wavelet is used as a sparse basis and is difficult to be well sparsely represented, so that the quality of the unstructured massive network data by compressed sensing is remarkably reduced.
Compared with the traditional wavelet algorithm, the second generation wavelet method is a faster and more effective wavelet transform realization method, does not depend on Fourier transform, and completely completes the construction of the biorthogonal wavelet filter in the time domain. The construction method has the outstanding advantages in structural design and self-adaptive construction, and overcomes the defects of the traditional frequency domain construction method. In the application of the compressive sensing technology, the original signal is sparsely represented by taking the second-generation wavelet as a sparse basis, so that unstructured massive data can be well sparsely represented.
Disclosure of Invention
The invention aims to provide an original signal reconstruction method based on second-generation wavelets, which is used for solving the problem of inapplicability of traditional compressed sensing to unstructured massive network data so as to realize the compression ratio as large as possible and the distortion as small as possible.
In order to achieve the purpose, the technical scheme of the invention comprises the following steps:
an original signal reconstruction method based on second generation wavelets comprises the following steps:
step 1: carrying out sparse representation on an original signal, and obtaining a sparse coefficient through a sparse basis and the original signal;
step 2: sampling the original signal, and obtaining a measured value by using the measurement matrix and the original signal;
and step 3: storing the sparse basis, the measured value and the random number seed;
and 4, step 4: and obtaining a sparse coefficient through iteration, and reconstructing the original signal by using a sparse basis.
Further according to the original signal reconstruction method based on the second generation wavelet, the original signal is sparsely represented in step 1, and a sparse coefficient is obtained through a sparse basis and the original signal:
for a given set of points S ═ x1,x2,…,xnIs x1<x2<…<xn，n＝k×2lWherein n, k and l are positive integers;
for any 2k x 2k matrix V, then VLThe table takes the k × 2k halves of the lower edge of the matrix V, VURepresenting the k x 2k half of the upper side of the matrix V, having
Figure BDA0001204936650000011
The original signal x is sparsely represented by taking the second generation wavelet as a sparse basis, so that unstructured massive data can be well sparsely represented, the original signal x is sparsely represented, and the method comprises the following steps:
(1-1) constructing a 2k matrix M1,i，
Figure BDA0001204936650000021
Where i is 1, …, n/2k, si＝(i-1)2k；
Matrix M1,iOf orthogonal basis U1,i，U1,i＝[Orth(M1,i)]TOrth represents solving orthogonal basis operation; by U1,iTo obtain U1：
Figure BDA0001204936650000022
By U1Obtain the first basis matrix Ψ1First base matrix Ψ1Is the intermediate operation result, Ψ, necessary for constructing sparse basis Ψ1＝U1；
(1-2) constructing a 2k matrix M2,i，
Figure BDA0001204936650000023
Where i is 1, …, n/2k, from M2,iTo obtain U2,i，U2,i＝[Orth(M2,i)]T(ii) a Then passes through U2,iTo obtain U2′：
Figure BDA0001204936650000024
Wherein n is2N/4k, from U2' obtaining U2，
Figure BDA0001204936650000025
Wherein In/2An identity matrix of (n/2) × (n/2);
by U2And U1To obtain a second basis matrix psi2，ψ2＝U1U2；
According to the construction matrix M1,iMethod of constructing M1,2iAnd M1,2i-1；
(1-3) obtaining the jth basis matrix psijWhere j is 2, …, log2(n/k) to construct a 2k × 2k matrix Mj,i，
Figure BDA0001204936650000031
Where i is 1, …, n/2k, from Mj,iCan obtain Uj,i，Uj,i＝[Orth(Mj,i)]TFrom Uj,iCan obtain Uj′：
Figure BDA0001204936650000032
By Uj' can obtain Uj，
Figure BDA0001204936650000033
Log for j 2, …2(n/k) obtained UjAnd U1The jth base matrix psi can be obtainedj，ψj＝U1U2…Uj；
(1-4) taking the maximum value log when j2(n/k), the obtained jth base matrix psijI.e. a sparse basis psi, i.e.
Figure BDA0001204936650000035
(1-5) obtaining a sparse coefficient s by using the sparse basis psi and the original signal x, wherein the specific s is psi-1x, wherein ψ-1Representing the inverse matrix of the sparse basis psi;
the sparse coefficients s are the result of the sparse representation of the original signal x on the sparse basis ψ.
Further according to the original signal reconstruction method based on the second generation wavelet, sampling the original signal in the step 2, and obtaining a measurement value by a measurement matrix and the original signal;
constructing a Bernoulli random matrix with the size of M multiplied by N as a measurement matrix phi, wherein each element independently follows the Bernoulli distribution, and the ith row and the jth column are respectively provided with a plurality of elementsBy phii,jRepresents:
Figure BDA0001204936650000034
the measured values y, y Φ x, Φ ψ s, As are obtained from the measurement matrix Φ and the original signal x.
Further according to the original signal reconstruction method based on the second generation wavelet, the sparse basis, the measured value and the random number seed are stored in step 3:
storing the sparse basis psi, the measured value y and the random number seed of the Bernoulli random matrix used as the measurement matrix phi, and calling when the original signal needs to be reconstructed;
the random number seed is a true random number used for generating a random matrix by a computer, each element in the matrix is obtained by calculating the true random number from the system time of the computer through an algorithm, and the true random number is the random number seed; under the condition that the seeds are fixed and the algorithm is fixed, the obtained random matrixes are the same; during storage, the whole measurement matrix phi is not required to be stored and transmitted, and only the random number seed is stored;
further according to the original signal reconstruction method based on the second generation wavelet, in step 4, the sparse coefficient is obtained through iteration, the original signal is reconstructed by using a sparse basis, and the method comprises the following steps:
(4-1) acquiring a measured value y, a random number seed and a sparse basis psi, generating a measurement matrix phi, and obtaining a sensing matrix A from the measurement matrix phi and the sparse basis psi, wherein A is phi psi;
(4-2) residual error r when iteration is not started0Set of indices when iteration is not started, y
Figure BDA0001204936650000041
An initial value 1 of an iteration number t, wherein t is the iteration number, and the maximum iteration number is tmax；
(4-3) residual r from last iterationt-1And the number M of the rows of the sensing matrix A to obtain a threshold Th，
Figure BDA0001204936650000042
||·||2Representing 2 norm of matrix calculation, wherein ts is a threshold parameter ts, and rt represents residual error in the t iteration;
from the sensing matrix A and the residual rt-1A vector u of length N is obtained with u ═ abs (a)Trt-1) Abs (·) represents the modulo value;
for j is more than or equal to 1 and less than or equal to N, calculating in sequence<rt-1,aj>，<·,·>Expressing to obtain vector inner product, forming obtained N numbers into vector u, selecting u with value greater than threshold ThThe value of (A) is set J by the column number J of the corresponding sensor matrix A0Said set being a set of column sequence numbers, J0Representing the index found for each iteration;
(4-4) index set Λt＝Λt-1∪J0Column set At＝At-1∪ajCalculating y as AtstLeast squares solution of (1) to obtain stIs estimated value of
Figure BDA0001204936650000043
Figure BDA0001204936650000044
Updating residual errors
Figure BDA0001204936650000045
t＝t+1；
Wherein a isjRepresents the jth column, A, of the matrix sensor matrix AtRepresentation by index ΛtThe selected column set of the sensing matrix A;
if Λt＝Λt-1Or t is>tmaxOr rtIf the value is 0, stopping iteration;
reconstructing the resultant
Figure BDA0001204936650000046
At ΛtA non-zero term is positioned, and the value of the non-zero term is a sparse coefficient s obtained in the last iterationt，
Figure BDA0001204936650000047
Is the final estimate of the sparse coefficient s at the end of the iteration, stThe estimated value of the sparse coefficient s in the t iteration is obtained;
(4-5) obtaining sparse coefficients at the end of the iteration
Figure BDA0001204936650000048
Reconstructing the original signal by sparse basis psi to obtain the estimated value of the original signal x
Figure BDA0001204936650000049
Figure BDA00012049366500000410
The estimated value
Figure BDA00012049366500000411
I.e. the reconstructed original signal.
Compared with the prior art, the invention has the following advantages:
1. according to the invention, the second-generation wavelet is used as a sparse basis to carry out sparse representation on the original signal x, so that unstructured massive data can be well sparsely represented, the problem of inapplicability of the traditional compression sensing technology based on the first-generation wavelet to the unstructured massive data is solved, a larger compression ratio is obtained, and the second-generation wavelet is a polynomial, so that the method has the advantage of high calculation speed.
2. The invention uses the segment orthogonal matching pursuit to reconstruct the data, so that the original signal x can be recovered in an allowed distortion range through the measured value y, namely the result after the original signal x is compressed, and the invention provides a powerful guarantee for improving the compression ratio.
Drawings
Fig. 1 is a flowchart of an implementation of an original signal reconstruction method based on second-generation wavelets according to the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the following describes the solutions and effects of the present invention in detail with reference to the accompanying drawings.
As shown in fig. 1, the present invention specifically performs the following steps on the original signal reconstruction method based on the second-generation wavelet.
Step 1: and carrying out sparse representation on the original signal, and obtaining a sparse coefficient through a sparse basis and the original signal.
For a given set of points S ═ x1,x2,…,xnIn which x1<x2<…<xn，n＝k×2lN, k and l are positive integers.
For any 2k x 2k matrix V, then VLThe table takes the k × 2k halves of the lower edge of the matrix V, VURepresenting the k x 2k half of the upper side of the matrix V, having
Figure BDA0001204936650000051
The original signal x is sparsely represented by taking the second generation wavelet as a sparse basis, so that unstructured massive data can be well sparsely represented, the original signal x is sparsely represented, and the method comprises the following steps:
(1-1) constructing a 2k matrix M1,iThe following were used:
Figure BDA0001204936650000052
where i is 1, …, n/2k, si＝(i-1)2k。
Matrix M1,iOf orthogonal basis U1,i，U1,i＝[Orth(M1,i)]TOrth denotes the orthogonal basis calculation, by U1,iTo obtain U1：
Figure BDA0001204936650000053
By U1Obtain the first basis matrix Ψ1First base matrix Ψ1Is an intermediate operation node necessary for constructing sparse basis ΨFruit, fruit juice1＝U1。
(1-2) constructing a 2k matrix M2,i，
Figure BDA0001204936650000054
Where i is 1, …, n/2k, from M2,iTo obtain U2，i，U2，i＝[Orth(M2，i)]TThen through U2，iTo obtain U2′：
Figure BDA0001204936650000061
Wherein n is2N/4k, from U2' obtaining U2，
Figure BDA0001204936650000062
Wherein In/2Is an identity matrix of (n/2) × (n/2).
By U2And U1To obtain a second basis matrix psi2，ψ2＝U1U2。
According to the construction matrix M1，iMethod of constructing M1，2iAnd M1，2i-1。
(1-3) obtaining the jth basis matrix psijWhere j is 2, …, log2(n/k) to construct a 2k × 2k matrix Mj，i，
Figure BDA0001204936650000063
Where i is 1, …, n/2k, from Mj，iCan obtain Uj，i，Uj，i＝[Orth(Mj，i)]TFrom Uj，iCan obtain Uj′：
Figure BDA0001204936650000064
By Uj' can obtain Uj，
Figure BDA0001204936650000065
Log for j 2, …2(n/k) obtained UjAnd U1The jth basis matrix Ψ can be obtainedj，ψj＝U1U2…Uj。
(1-4) taking the maximum value log when j2(n/k), the obtained jth base matrix psijI.e. a sparse basis psi, i.e.
Figure BDA0001204936650000066
(1-5) obtaining a sparse coefficient s by using the sparse basis psi and the original signal x, wherein the specific s is psi-1x. Wherein psi-1Representing the inverse matrix of the sparse basis psi.
The sparse coefficients s are the result of the sparse representation of the original signal x on the sparse basis ψ.
Step 2: the original signal is sampled and a measurement value is obtained from the measurement matrix and the original signal.
Constructing a Bernoulli random matrix with the size of M multiplied by N as a measurement matrix phi, wherein each element independently follows the Bernoulli distribution, and the ith row and jth column elements use phii,jRepresents:
Figure BDA0001204936650000071
the measured values y, y Φ x, Φ ψ s, As are obtained from the measurement matrix Φ and the original signal x.
And step 3: the sparse basis, the measured value and the random number seed are stored.
And storing the sparse basis psi, the measured value y and the random number seed of the Bernoulli random matrix used as the measurement matrix phi, and calling when the original signal needs to be reconstructed.
The random number seed is a true random number used for generating a random matrix by a computer, each element in the matrix is obtained by calculating the true random number from the system time of the computer through an algorithm, and the true random number is the random number seed. Under the condition that the seeds are fixed and the algorithm is fixed, the obtained random matrixes are the same. During storage, the whole measurement matrix phi is not required to be stored and transmitted, and only the random number seed is stored.
And 4, step 4: and obtaining a sparse coefficient through iteration, and reconstructing the original signal by using a sparse basis.
The method comprises the following steps:
(4-1) acquiring a measured value y, a random number seed and a sparse basis psi, generating a measurement matrix phi, and obtaining a sensing matrix A from the measurement matrix phi and the sparse basis psi, wherein A is phi psi;
(4-2) residual error r when iteration is not started0Set of indices when iteration is not started, y
Figure BDA0001204936650000072
An initial value 1 of an iteration number t, wherein t is the iteration number, and the maximum iteration number is tmax；
(4-3) residual r from last iterationt-1And the number M of the rows of the sensing matrix A to obtain a threshold Th，
Figure BDA0001204936650000078
||·||2Representing a 2 norm of the matrix, where tsIs a threshold parameter ts，rtRepresenting the residual error at the t-th iteration;
from the sensing matrix A and the residual rt-1A vector u, u-abs (a) of length N is obtainedTrt-1) Abs (·) represents the modulo value;
for j is more than or equal to 1 and less than or equal to N, calculating in sequence<rt-1,aj>，<·,·>Expressing to obtain vector inner product, forming obtained N numbers into vector u, selecting u with value greater than threshold ThThe value of (A) is set J by the column number J of the corresponding sensor matrix A0Said set being a set of column sequence numbers, J0Representing the index found for each iteration;
(4-4) index set Λt＝Λt-1∪J0Column set At＝At-1∪ajCalculating y as AtstLeast squares solution of (1) to obtain stIs estimated value of
Figure BDA0001204936650000073
Figure BDA0001204936650000074
Updating residual errors
Figure BDA0001204936650000075
t＝t+1；
Wherein a isjRepresents the jth column, A, of the matrix sensor matrix AtRepresentation by index ΛtThe selected column set of the sensing matrix A;
if ΛΛ＝Λt-1Or t is>tmaxOr rtIf the value is 0, stopping iteration;
reconstructing the resultant
Figure BDA0001204936650000076
At ΛtA non-zero term is positioned, and the value of the non-zero term is a sparse coefficient s obtained in the last iterationt，
Figure BDA0001204936650000077
Is the final estimate of the sparse coefficient s at the end of the iteration, stThe estimated value of the sparse coefficient s in the t iteration is obtained;
(4-5) obtaining sparse coefficients at the end of the iteration
Figure BDA0001204936650000081
Reconstructing the original signal by sparse basis psi to obtain the estimated value of the original signal x
Figure BDA0001204936650000082
Figure BDA0001204936650000083
The estimated value
Figure BDA0001204936650000084
I.e. the reconstructed original signal.
The above description is only for the preferred embodiment of the present invention, and the technical solution of the present invention is not limited thereto, and any known modifications made by those skilled in the art based on the main technical idea of the present invention belong to the technical scope of the present invention, and the specific protection scope of the present invention is subject to the description of the claims.

Claims (4)

1. An original signal reconstruction method based on second generation wavelets is characterized by comprising the following steps:
step 1: carrying out sparse representation on an original signal, and obtaining a sparse coefficient through a sparse basis and the original signal:
for any 2k x 2k matrix V, then VLRepresenting k x 2k halves, V, taking the lower side of the matrix VURepresenting the k x 2k half of the upper side of the matrix V, having
Figure FDA0002418612680000011
The original signal x is sparsely represented by taking the second generation wavelet as a sparse basis, so that unstructured massive data can be well sparsely represented, the original signal x is sparsely represented, and the method comprises the following steps:
(1-1) construction of a 2k × 2k matrix M1, i'
Figure FDA0002418612680000012
Where i is 1, …, n/2k, si＝(i-1)2k；
Matrix M1,iOf orthogonal basis U1,i，U1,i＝[Orth(M1,i)]TOrth represents solving orthogonal basis operation; by U1,iTo obtain U1：
Figure FDA0002418612680000013
By U1Obtain the first basis matrix Ψ1First base matrix Ψ1Is the intermediate operation result, Ψ, necessary for constructing sparse basis Ψ1＝U1；
(1-2) constructing a 2k matrix M2,i，
Figure FDA0002418612680000014
Where i is 1, …, n/2k, from M2,iTo obtain U2,i，U2,i＝[Orth(M2,i)]T(ii) a Then passes through U2,iTo obtain U2′：
Figure FDA0002418612680000021
Wherein n is2N/4k, from U2' obtaining U2，
Figure FDA0002418612680000022
Wherein In/2An identity matrix of (n/2) × (n/2);
by U2And U1To obtain a second basis matrix psi2，ψ2＝U1U2；
According to the construction matrix M1,iMethod of constructing M1,2iAnd M1,2i-1；
(1-3) obtaining the jth basis matrix psijWhere j is 2, …, log2(n/k) to construct a 2k × 2k matrix Mj,i，
Figure FDA0002418612680000023
Where i is 1, …, n/2k, from Mj,iCan obtain Uj,i，Uj,i＝[Orth(Mj,i)]TFrom Uj,iCan obtain Uj′：
Figure FDA0002418612680000024
By Uj' can obtain Uj，
Figure FDA0002418612680000025
Log for j 2, …2(n/k) obtained UjAnd U1The jth base matrix psi can be obtainedj，ψj＝U1U2…Uj；
(1-4) taking the maximum value log when j2(n/k), the obtained jth base matrix psijI.e. a sparse basis psi, i.e.
Figure FDA0002418612680000026
(1-5) obtaining a sparse coefficient s by using the sparse basis psi and the original signal x, wherein the specific s is psi-1x, wherein ψ-1Representing the inverse matrix of the sparse basis psi;
the sparse coefficient s is the result of sparse representation of the original signal x on a sparse basis ψ;
step 2: sampling the original signal, and obtaining a measured value by using the measurement matrix and the original signal;
and step 3: storing the sparse basis, the measured value and the random number seed;
and 4, step 4: obtaining a final estimation value of a sparse coefficient s through iteration
Figure FDA0002418612680000031
And reconstructing the original signal by using the sparse basis.
2. The method for reconstructing an original signal based on second-generation wavelets according to claim 1, wherein the step 2 is performed by sampling the original signal, obtaining a measurement value from the measurement matrix and the original signal, and performing the following steps:
constructing a Bernoulli random matrix with the size of M multiplied by N as a measurement matrix phi, wherein each element independently follows the Bernoulli distribution, and the ith row and jth column elements use phii,jRepresents:
Figure FDA0002418612680000032
and obtaining a measured value y from the measurement matrix phi and the original signal x, wherein y is phi x and phi ψ s is As, and A is a sensing matrix.
3. The method for reconstructing an original signal based on second-generation wavelets according to claim 1, wherein the step 3 of storing the sparse basis, the measured value and the random number seed is performed by the following steps:
storing the sparse basis psi, the measured value y and the random number seed of the Bernoulli random matrix used as the measurement matrix phi, and calling when the original signal needs to be reconstructed;
the random number seed is a true random number used for generating a random matrix by a computer, each element in the matrix is obtained by calculating the true random number from the system time of the computer through an algorithm, and the true random number is the random number seed; under the condition that the seeds are fixed and the algorithm is fixed, the obtained random matrixes are the same; during storage, the whole measurement matrix phi is not required to be stored and transmitted, and only the random number seed is stored.
4. Method for reconstructing an original signal based on second-generation wavelets according to claim 1, wherein the final estimation value of the sparse coefficient s is obtained through iteration in step 4
Figure FDA0002418612680000033
Reconstructing an original signal by using a sparse basis, and performing the following steps:
(4-1) acquiring a measured value y, a random number seed and a sparse basis psi, generating a measurement matrix phi, and obtaining a sensing matrix A from the measurement matrix phi and the sparse basis psi, wherein A is phi psi;
(4-2) residual error r when iteration is not started0Set of indices when iteration is not started, y
Figure FDA0002418612680000034
An initial value 1 of iteration times t, wherein t is the iteration times, and the maximum iteration times is tmax;
(4-3) residual r from last iterationt-1And passThe number of rows M of the sensing matrix a, to obtain a threshold Th,
Figure FDA0002418612680000041
||·||2representing a 2 norm of the matrix, where tsIs a threshold parameter ts，rtRepresenting the residual error at the t-th iteration;
from the sensing matrix A and the residual rt-1A vector u, u-abs (a) of length N is obtainedTrt-1) Abs (·) represents the modulo value;
for j is more than or equal to 1 and less than or equal to N, calculating in sequence<rt-1,aj>，<·,·>Expressing to obtain vector inner product, forming obtained N numbers into vector u, selecting vector u with value greater than threshold ThThe value of (A) is set J by the column number J of the corresponding sensor matrix A0Said set being a set of column sequence numbers, J0Representing the index found for each iteration;
(4-4) index set Λt＝Λt-1∪J0Column set At＝At-1∪ajCalculating y as AtstLeast squares solution of (1) to obtain stIs estimated value of
Figure FDA0002418612680000042
Figure FDA0002418612680000043
Updating residual errors
Figure FDA0002418612680000044
Wherein a isjRepresents the jth column, A, of the matrix sensor matrix AtRepresentation by index ΛtThe selected column set of the sensing matrix A;
if Λt＝Λt-1Or t is>tmaxOr rtIf the value is 0, stopping iteration;
reconstructing the resultant
Figure FDA0002418612680000045
At ΛtA non-zero term is positioned, and the value of the non-zero term is a sparse coefficient s obtained in the last iterationt，
Figure FDA0002418612680000046
Is the final estimate of the sparse coefficient s at the end of the iteration, stThe estimated value of the sparse coefficient s in the t iteration is obtained;
(4-5) when the iteration is finished, obtaining the final estimated value of the sparse coefficient s
Figure FDA0002418612680000047
Reconstructing the original signal by sparse basis psi to obtain the estimated value of the original signal x
Figure FDA0002418612680000048
Figure FDA0002418612680000049
The estimated value
Figure FDA00024186126800000410
I.e. the reconstructed original signal.
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