CN108573262B - IGR-OMP-based high-dimensional sparse vector reconstruction method - Google Patents

IGR-OMP-based high-dimensional sparse vector reconstruction method Download PDF

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CN108573262B
CN108573262B CN201810431421.8A CN201810431421A CN108573262B CN 108573262 B CN108573262 B CN 108573262B CN 201810431421 A CN201810431421 A CN 201810431421A CN 108573262 B CN108573262 B CN 108573262B
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赵健
申富饶
董珍君
赵金煕
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Abstract

The invention discloses a high-dimensional sparse vector reconstruction method based on IGR _ OMP. According to the characteristic that least square solution needs to be calculated in each step of the OMP algorithm, an incremental IGR _ OMP algorithm is provided based on the Greville recursion process, a series of useful recursion properties are obtained, and the calculation speed is effectively improved; on the basis of analyzing the IGR _ OMP algorithm, an incremental IQR _ OMP algorithm on the basis of QR decomposition is established by utilizing the relationship between the Greville recursion algorithm and the QR decomposition, the algorithm effectively reduces the calculation workload and improves the effectiveness of the algorithm in solving large-scale problems; high-dimensional sparse vector reconstruction simply means that the most sparse coefficient representation of the high-dimensional vector is given on an overcomplete vector system.

Description

IGR-OMP-based high-dimensional sparse vector reconstruction method
Technical Field
The invention relates to a high-dimensional sparse vector reconstruction method based on IGR _ OMP.
Background
Compressed Sensing (CS) is a new key technology in the field of image or signal processing. The essence of this is to effectively determine the sparsest coefficient representation of this high dimensional vector on An overcomplete dictionary using information much lower than the dimensions of the Signal vector or image vector, the core problem is to solve the sparseness of a sublinear system to reconstruct the Signal or image vector (ref: Yan worship et al, compressed sensing and applications 2015, national defense industry publishers, Beijing, E.J. Cand es, M.Wakin, "girder with estimation" An interactive reduction sampling, IEEE Signal Processing Magazine,2008,25(2), 21-30.).
High-dimensional sparse vector reconstruction is a very important problem in the field of signal processing and image processing. Although the classical OMP (Orthogonal Matching Pursuit) algorithm has a lower complexity than other non-greedy algorithms, it is still a little away from real-time signal processing in practical applications.
Disclosure of Invention
Aiming at the defects of the prior art, the invention discloses a high-dimensional sparse vector reconstruction method based on IGR _ OMP (Incremental Greville Recursion based Orthogonal Matching Pursuit algorithm), which comprises the following steps:
let matrix A ═ a1，a2，…，aK]∈RM×K，K＜＜M，rank(A)＝K(RM×K: set of M x K dimensional real matrices, ai(i ═ 1,2 …, K): the ith real column vector in dictionary matrix a, rank (a): rank of matrix a);
note A1＝[a1]，Ak＝[Ak-1,ak]，k＝2,3,…,K(a1: first column vector of matrix A, A1: iteration matrix A, A of 1 st timek: the kth matrix A, A with k column vectorsk-1: the k-1 st iteration matrix A);
computing A by Greville recursion algorithm+I.e. the Generalized Inverse of matrix A (ref: Ben-Isreal A., Greville T.N.E., Generalized applied instruments Theory and Applications, Wiley, New York, 1974):
preliminary test value
Figure BDA0001653500390000011
Recursion step pair K is 2,3, …, K
Figure BDA0001653500390000021
bk＝ak-Ak-1dk；
Figure BDA0001653500390000022
Wherein d iskIs to solve for
Figure BDA0001653500390000023
Temporarily substituted vectors, bkIs dimension and akIdentity solution
Figure BDA0001653500390000024
A temporarily replaced column vector. The algorithm is characterized in that
Figure BDA0001653500390000025
By
Figure BDA0001653500390000026
Are obtained recursively, i.e. from
Figure BDA0001653500390000027
Successive computations
Figure BDA0001653500390000028
Rather than directly for each iteration
Figure BDA0001653500390000029
For the incremental IGR _ OMP high-dimensional vector sparse recovery in compressed sensing, the steps are as follows:
step A1, inputting observation data vector y epsilon RMAnd the dictionary matrix a ═ a1，a2，…，aN]∈RM×NLet the initial support set index
Figure BDA00016535003900000210
Let the initial residual vector r0Is equal to the observation data vector y, i.e. r0Let k be 1. Wherein R isMIs a set of M x 1 dimensional real column vectors, RM×NIs a set of M × N dimensional real number matrices, aiRepresents the ith real column vector in the dictionary matrix a, i ═ 1,2 …, N;
step A2, solving the k-1 iteration residual vector r in the dictionary matrix Ak-1Of the kth iteration of (a) the most strongly correlated column jk：
jk∈argmaxj|(rk-1，aj)|，Ωk＝Ωk-1∪{jk}，
Wherein omegakFor the kth iteration support set index, Ωk-1For the k-1 iteration support set index, ajIs the jth real column vector in the dictionary matrix A;
step A3, supporting the index omega by the k-1 iterationk-1Corresponding dictionary matrix
Figure BDA00016535003900000211
Generalized inverse matrix of
Figure BDA00016535003900000212
Incremental calculation of kth iteration support index omegakCorresponding dictionary matrix
Figure BDA00016535003900000213
Generalized inverse matrix of
Figure BDA00016535003900000214
Figure BDA00016535003900000215
If k is 1, then jkIs j1Then vector of
Figure BDA00016535003900000216
Is equal to
Figure BDA00016535003900000217
Is the inverse of the broad sense of (1)
Figure BDA00016535003900000218
Ωk＝{j1Turning to step A4;
if k ≠ 1, the following calculations are performed sequentially:
Figure BDA00016535003900000219
Figure BDA00016535003900000220
c＝bT/(bTb)，
Figure BDA0001653500390000031
wherein
Figure BDA0001653500390000032
Is a dictionary matrix
Figure BDA0001653500390000033
Middle jkA vector of columns of real numbers,
Figure BDA0001653500390000034
is a vector
Figure BDA0001653500390000035
D is a vector of the provisional substitution, b is a dimension and a vector of the provisional substitution
Figure BDA0001653500390000036
The same M × 1-dimensional vector, c is a provisionally substituted 1 × M-dimensional vector.
Step A4, solving a minimization problem
Figure BDA0001653500390000037
Solution x ofk：
Figure BDA0001653500390000038
Wherein
Figure BDA0001653500390000039
Representative vector
Figure BDA00016535003900000310
Euclidean norm of;
step A5, updating the k iteration residual vector
Figure BDA00016535003900000311
Step A6, determining whether the stopping criterion is satisfied, i.e. determining whether the Euclidean norm of the k-th iteration residual vector satisfies | rk‖2ε (ε is a very small number, e.g.: 10)-6) If yes, executing step A7, otherwise updating k to k +1, and repeating steps A2 to A5;
step A7, outputting sparse coefficient vector x ∈ RN：
Figure BDA00016535003900000312
Where x (i) is the i-th element in the sparse coefficient vector x, xk(i) Is a vector xkThe i-th element of (1), RNIs an N × 1 dimensional real number vector.
bkAn essential condition of 0 is ak∈R(Ak-1)＝span{a1，a2，…，ak-1}，
Namely akAnd a1，a2，…，ak-1Linear correlation, Ak-1＝[a1，a2，…，ak-1](akRepresentation matrix AkThe kth column vector, R (A)k-1) Is represented by matrix Ak-1K-1 column vectors of (a) and (b) a1，a2，…，ak-1Represents by the vector a1，a2，…，ak-1Generated vector space, Ak-1For the (k-1) th iteration a matrix,
Figure BDA00016535003900000313
is Ak-1The generalized inverse matrix of (d);
Figure BDA00016535003900000314
bk∈R(Ak-1)⊥is akAt R (A)k-1) Orthogonal projection on the orthogonal complement, if b k0 denotes ak∈R(Ak-1) Then a iskAnd R (A)k-1) Linear correlation (I)MIs an identity matrix of M x M,
Figure BDA00016535003900000315
representation matrix Ak-1Generalized inverse matrix of (A)k-1)⊥Representation generation space R (A)k-1) The orthogonal complement of).
Figure BDA0001653500390000041
Denotes bkIs akAt R (A)k-1)⊥The orthogonal projection of (a) isi(i-1, 2, …, k-1) is a matrix ak-1Vector of ith column):
(bk，ai)＝0,i＝1,2,…,k-1 (4.7)
i.e. bkAnd a1，a2，…，ak-1Are orthogonal.
The invention also discloses a high-dimensional sparse vector reconstruction method based on the IQR _ OMP (Incremental QR decomposition based Orthogonal Matching Pursuit algorithm based on QR decomposition), which comprises the following steps:
step B1, inputting observation data vector y epsilon RmAnd the dictionary matrix a ═ a1，a2，…，an]∈Rm×n(ii) a Let the initial support set index
Figure BDA0001653500390000042
Make the initial residual vector equal to the observed data vector, i.e. r0Y, let k be 1, wherein RmRepresenting a real column vector, R, of dimension m x 1m×nRepresenting a matrix of real numbers of dimension m x n, aiRepresents the ith m × 1-dimensional real column vector in the dictionary matrix a, i is 1,2 …, n;
step B2, solving the k-1 iteration residual vector r in the dictionary matrix Ak-1Of the kth iteration of (a) the most strongly correlated column jk：
jk∈argmaxj|(rk-1，aj)|，Ωk＝Ωk-1∪{jk}，
Wherein omegakFor the kth iteration support set index, Ωk-1For the k-1 iteration support set index, ajA j column real number column vector in the dictionary matrix A;
step B3, Q of the k-1 st iterationk-1And Rk-1Recurrently calculating Q of kth iterationkAnd Rk：
If k is 1, then jk＝j1Then, then
Figure BDA0001653500390000043
Q1The first column of the matrix is Q1(1) or q1Is equal to
Figure BDA0001653500390000044
Namely, it is
Figure BDA0001653500390000045
Turning to the step B4, the method,
if k ≠ 1, the following calculations are performed in sequence:
Figure BDA0001653500390000046
Figure BDA0001653500390000047
Figure BDA0001653500390000048
qk＝q1/αk，
Qk＝[Qk-1:qk]，
Figure BDA0001653500390000051
wherein the vector
Figure BDA0001653500390000052
Is the jth of the dictionary matrix AkColumn vector, qkIs the kth column vector, β, of the matrix QkTemporary replacement vectors for the kth iteration, αkA temporary replacement value for the kth iteration;
step B4, updating the k iteration residual error vector
Figure BDA0001653500390000053
Wherein q iskIs the kth iteration matrix QkThe kth column vector of (1);
step B5, determining whether the stopping criterion is satisfied, i.e. determining whether the euclidean norm of the k-th residual vector satisfies | rk‖2ε (ε is a very small number, e.g.: 10)-6) If yes, go to step B6; otherwise, updating k to k +1, and repeating the steps B2 to B4;
step B6, solving the minimization problem
Figure BDA0001653500390000054
Solution x ofk：
Figure BDA0001653500390000055
Wherein
Figure BDA0001653500390000056
Support the indicator omega for the kth iterationkA corresponding dictionary matrix is then generated and stored in the memory,
Figure BDA0001653500390000057
j1，j2，…，jk∈Ωk，
Figure BDA0001653500390000058
representative vector
Figure BDA0001653500390000059
Euclidean norm of;
Figure BDA00016535003900000510
is a matrix RkThe generalized inverse of (1) is,
Figure BDA00016535003900000511
is a matrix QkTransposing;
step B7, outputting sparse coefficient vector x:
Figure BDA00016535003900000512
where x (i) denotes the i-th element in the sparse coefficient vector x, xk(i) Representing a vector xkThe ith element in (1).
Using a Greville recursion relation to obtain:
Figure BDA00016535003900000513
then there are:
Figure BDA00016535003900000514
wherein d iskIs to solve for
Figure BDA00016535003900000515
Temporarily substituted vectors, bkIs dimension and akIdentity solution
Figure BDA00016535003900000516
A temporarily replaced column vector.
According to QR Decomposition (QR Decomposition), if the matrix A executed at the k-1 th time is obtained by calculationk-1：
Ak-1＝Qk-1Rk-1 (4.9)
Wherein A isk-1The k-1 th iteration matrix A, Qk-1＝[q1,q2,…qk-1]∈RM×(k-1)Is a column orthogonal matrix (q)i(i-1, 2, …, k-1) is a matrix Qk-1Ith column vector of), qk-1Is a matrix Qk-1The (k-1) th column vector Rk-1An upper triangular matrix of (k-1) × (k-1), taking into account the calculation of step k:
Figure BDA0001653500390000061
wherein beta isk∈R(k-1)×1，αkFor a scalar, we derive (I: identity matrix):
Figure BDA0001653500390000062
Figure BDA0001653500390000063
Figure BDA0001653500390000064
due to the fact that
Figure BDA0001653500390000065
Thereby the device is provided with
Figure BDA0001653500390000066
Wherein
Figure BDA0001653500390000067
Is a matrix RkThe inverse of the matrix of (a) is,
therefore:
Figure BDA0001653500390000068
also, a residual vector r is calculated every stepk＝s-AkxkOne matrix and vector multiplication (s is the reconstructed vector) needs to be calculated once, and r iskEach step needs to be used in A to select and rkThe column vector having the largest inner product in absolute value is added to AkIn the formation of Ak+1For recursive QR decomposition, this is: r iskIs y at R (A)k)⊥Orthogonal projection of (a) i.e.:
Figure BDA0001653500390000069
has the advantages that:
the IGR _ OMP method and the IQR _ OMP two-increment OMP method for sparse reconstruction of high-dimensional signal vectors have the following characteristics:
1. the calculation speed is high. For matrix Ak＝[a1，a2，…，ak]∈RM×k，y∈RM，rank(Ak) Min corresponding to kx‖y-AkSolution of x |
Figure BDA00016535003900000610
If obtained by QR decomposition
Figure BDA00016535003900000611
For each k conventional OMP method, the amount of computation is about
Figure BDA00016535003900000612
Subflups (ref: L.N.Trefethen, D.Bau, Numerical Linear Algebra, Chinese edition copy 2006 by Posts and Telecom Press). While the incremental algorithm IQR _ OMP is approximately
Figure BDA00016535003900000613
Therefore, the workload of the incremental algorithm is obviously superior to that of the traditional OMP algorithm, especially when high dimensional problems.
2. From the viewpoint of computational stability, since
Figure BDA0001653500390000071
While the incremental IQR _ OMP method is designed to be cond2(Ak) Rather than its square. Thus when A iskWhen the condition number of (2) is slightly larger, the normal equation algorithm is not only large in calculation amount, but also can seriously affect the stability of the algorithm. Therefore, the method is obviously superior to the method of the normal equation in the aspects of calculation precision and calculation speed.
3. At present, theoretical results about sparse reconstruction of signal vectors are difficult to be checked in the process of actual calculation. While the incremental algorithms IQR _ OMP and IGR _ OMP given herein are rk,kAnd bkCan be used as test akA relative to the front face1,a2,…,ak-1A measure of the degree of linear independence (ref: Zhao Jinxi, Huang's method of compatible linear equations and its generalization, the mathematics academic journal of higher schools, 01 th, pp 8-17,1981/4/2.). It is important to realize the check of the linear independence degree between vector systems in the calculation process. The incremental algorithm provided by the invention has the characteristics that the calculation expense is obviously reduced, the algorithm has strong robustness, and the linear independence degree of the currently selected atom relative to the previous atom can be given in the calculation process. Numerical results show that the algorithm presented herein has significant advantages.
Drawings
The foregoing and other advantages of the invention will become more apparent from the following detailed description of the invention when taken in conjunction with the accompanying drawings.
FIG. 1 is a flowchart of the IGR _ OMP algorithm.
FIG. 2 is a flow chart of the IQR _ OMP algorithm.
Detailed Description
The invention is further explained below with reference to the drawings and the embodiments.
Among sparse reconstruction algorithms for high-dimensional signal vectors for compressed sensing, Greedy algorithm has been emphasized because of its low computational complexity and fast convergence speed, while Orthogonal Matching Pursuit (OMP) is the most important one (refer to: journal a. Tropp and Anna C. Gilbert. Signal Recovery From measuring instruments Via organic Matching Pursuit [ J ]. IEEE Transactions on Information Theory, VOL.53, NO.12, DECEMBER 2007.).
The classical Orthogonal Matching Pursuit algorithm (OMP, Orthogonal Matching Pursuit) is as follows:
inputting: observed data vector y ∈ RM(RM: m-dimensional real column vector set) and dictionary matrix a ═ a1，a2，…，aN]∈RM×N(ai(i-1, 2, …, N) is the ith column vector of matrix a, RM×N: m × N dimensional real matrix).
And (3) outputting: sparse signal vector x ∈ RN(RN: a set of N-dimensional real column vectors).
Step1. initialization: let the initial support set index
Figure BDA0001653500390000081
Let the initial residual vector equal the observed data vector, i.e. r0Let k be 1.
Step2. identification: solving the k-1 iteration residual error vector r in the matrix Ak-1Of the kth iteration of (a) the most strongly correlated column jk：
Figure BDA0001653500390000082
Wherein omegakFor the kth iteration support set index, Ωk-1For the k-1 iteration support set index, aj(j ═ 1,2, … k) the jth column vector in matrix a.
Step3. calculation
Figure BDA0001653500390000083
Solution of (2)
Figure BDA0001653500390000084
Wherein
Figure BDA0001653500390000085
Is the support index omegakA corresponding dictionary matrix is then generated and stored in the memory,
Figure BDA0001653500390000086
ai(i＝j1，j2，…jk∈Ωk) Is a matrix
Figure BDA0001653500390000087
The ith column vector.
Step4. update the k iteration residual vector
Figure BDA0001653500390000088
Step5. if some stop criterion is met, the euclidean norm of the k-th iteration residual vector meets | rk‖2ε (ε is a very small number, e.g.: 10)-6) Then stop iteration to Step 6;
otherwise let k ← k +1, and repeat Step2 through Step4.
Step6. output coefficient vector x (i): i-th element of output coefficient vector x, xk(i) The method comprises the following steps Representing a vector xkIth element of (2):
Figure BDA0001653500390000089
since the number of non-zero elements K of the solution vector is less than M and less than N, step3 in the algorithm is actually solving an M x K overdetermined linear least square problem. That is, in general, to solve the sparseness of a sublinear system, the OMP algorithm is actually implemented by successively solving an overdetermined linear least squares problem of M × K (K1, 2.., K). That is, it is important to recognize that the greedy algorithm is used to solve the sparse reconstruction problem of the high-dimensional signal vector, which is essentially to solve the least square solution of the indeterminate linear system composed of the column vectors (atoms) of a corresponding to the non-zero elements of x. The OMP algorithm has two key steps:
1. for a matrix (dictionary) A ∈ RM×NColumn (atomic) indices that match non-zero elements in x are determined. According to the principle of greedy algorithm, at step k, the AND r is found in the N columns of Ak-1＝y-Axk-1The vector with the largest absolute value of inner product (reference: S.Mallat and Z.Zhang, Matching pursuits with time-frequency dictionary, IEEE Trans. Signal Process,1993,41, pp.3397-3415.), constitutes a new support set, which is step 2;
2. for large problems, the workload required to solve the least squares solution in the OMP algorithm using the usual normal equations method is often nearly 80% of the total workload. Finding a more efficient algorithm to solve the least squares solution becomes a core problem in the OMP algorithm.
It has been pointed out previously that sparse reconstruction of signal vectors is the most sparse solution to determine a large high-order sublinear system from its scientific point of view. The OMP algorithm actually introduces a support set column by column and solves the least square solution of an overdetermined linear system as follows:
Figure BDA0001653500390000091
whereas the usual OMP algorithm is a normal equation method, which calculates at the k-th iteration:
Figure BDA0001653500390000092
for the least squares solution (3.3) to solve the over-determined system of linear equations, since the number of iterations K < M < N. Thus, it is considered that the method of the equation (3.4) is suitably effective as compared with other direct methods. However, there is a fundamental problem in the coefficient matrix of (3.3)
Figure BDA0001653500390000093
Each iteration is added with a column, and each iteration needs to be solved (3.3), and the accumulated calculation amount is very large. For general middle and small sizeThe problem, the effect of which is not obvious, is a big problem for large high-order problems. How to effectively utilize the successive iteration information of the OMP algorithm is the starting point for constructing the incremental new algorithm.
In order to be able to explain the problem more intuitively, the overdetermined linearity problem (3.3) to be solved by the invention is set
Figure BDA0001653500390000094
That is, in the iteration 2000 step, the calculation time of the equation method accumulation is 1667.6 seconds, while the new algorithm accumulation calculation constructed by using the information of the previous iteration only takes 137.36 seconds, and the speed of the same problem is improved by 12 times. At the 2000 th iteration, the normal equation method took 4.5273 seconds, whereas the method of the present invention took only 0.2427 seconds.
It is stated above that the concept of spark or cross-correlation value of A can be utilized to discuss the existence of large sparse solutions. It must be seen that neither of these conditions is very convenient to verify. In fact, since the solution of the present invention is an M × k overdetermined linear system, there are:
theorem 3.1 sets the sublinear system Ax ═ y, where A ∈ RM×N，x∈RN，y∈RM，rank(A)＝M，‖x‖0K and K < M < N, if A ═ a1，a2，…，aN]The vector system formed by any K rows is not related linearly, so that a solution vector x meeting the sparse condition exists, and the solution vector is enabled to be II0＝K。
This condition seems to be difficult to verify, but since the coefficient matrix of the overdetermined linear system is introduced column by column, it is true only to ensure that the vector system composed of the column-by-column introduced vectors is linearly independent. It is important how the new vector can be verified in the calculation process to be linearly independent of the vector system of the advanced support set, which is not done by the conventional OMP algorithm. How to take advantage of this column-wise introduced feature to consider and design an algorithm is the motivation for the present invention to consider new algorithms.
As shown in fig. 1 and 2, the present invention discloses a high-dimensional sparse vector reconstruction method based on IGR _ OMP and a high-dimensional sparse vector reconstruction method based on IQR _ OMP.
Principle of incremental OMP method:
under certain conditions, the OMP algorithm is essentially a super-definite linear system, and the super-definite linear system is characterized in that the column vector with the maximum energy is selected successively to enter a support set. That is, in the k iteration, the column vector with the maximum energy and the originally selected k-1 column vector are selected to form a new Mxk over-definite system
Figure BDA0001653500390000101
Wherein
Figure BDA0001653500390000102
y∈RMThen, a least squares solution is found. From this viewpoint, understanding and analyzing can grasp the essence of OMP method, and can see the defects of traditional OMP method. The OMP algorithm computes the generalized inverse of the submatrix from scratch for each iteration. Whether directly calling standard functions or directly calculating by formulas
Figure BDA0001653500390000103
Many extra computations are added, especially for large problems, which is more prominent as the number of iterations increases. The k-th iteration of the incremental algorithm utilizes information obtained by the previous iteration, so that the effectiveness of the algorithm is greatly improved. Two incremental OMP algorithms are given below.
IGR _ OMP algorithm:
let A ═ a1，a2，…，aK]∈RM×KK < M, rank (A) ═ K (wherein ai(i ═ 1,2, …, K) is the ith column vector of matrix a, RM×K: a set of M × K dimensional real matrices);
note A1＝[a1]，Ak＝[Ak-1,a1]，k＝2,3,…,K(a1: first column vector of matrix A, A1: iteration matrix A, A of 1 st timek: the kth iteration matrix A, Ak-1: the k-1 st iteration matrix A);
computing A by Greville recursion algorithm+(ref: Ben-Isreal A., Greville T.N.E., Generalized invest Theory and Applications, Wiley, New York, 1974.):
preliminary test value
Figure BDA0001653500390000104
Recursion step pair K is 2,3, …, K
Figure BDA0001653500390000111
bk＝ak-Ak-1dk；
Figure BDA0001653500390000112
The algorithm is characterized in that
Figure BDA0001653500390000113
By
Figure BDA0001653500390000114
Are recurrently obtained in which dkIs to solve for
Figure BDA0001653500390000115
Temporarily substituted vectors, bkIs dimension and akIdentity solution
Figure BDA0001653500390000116
A temporarily replaced column vector. So there is the following IGR _ OMP algorithm:
step A1, inputting observation data vector y epsilon RM(RM: m × 1-dimensional real column vector) and dictionary matrix a ═ a1，a2，…，aN]∈RM×N(ai(i-1, 2, …, N) is the ith column vector of matrix a, RM×N: mxn dimensional real matrix), index the initial support set
Figure BDA0001653500390000117
Make the initial residual vector equal to the input observed data vector, i.e. r0Y, let k be 1;
step A2, solving the k-1 iteration residual vector r in the dictionary matrix Ak-1Of the kth iteration of (a) the most strongly correlated column jk(Ωk: support set index, Ω, for the kth iterationk-1: the k-1 th iteration support set index, aj: jth column vector in matrix a):
Figure BDA0001653500390000118
step A3, supporting set index omega by the k-1 iterationk-1Corresponding matrix
Figure BDA0001653500390000119
General inverse of
Figure BDA00016535003900001110
Incremental calculation of kth iteration support set index omegakCorresponding matrix
Figure BDA00016535003900001111
General inverse of
Figure BDA00016535003900001112
(
Figure BDA00016535003900001113
Representative matrix
Figure BDA00016535003900001114
J (d) ofkColumn (ii):
if k is 1, then jk＝j1Then vector of
Figure BDA00016535003900001115
Is a generalized inverse equal to the vector
Figure BDA00016535003900001116
Is the inverse of the broad sense of (1)
Figure BDA00016535003900001117
Ωk＝{j1The flow is turned to the step A4,
if k ≠ 1, the following steps are performed in sequence:
Figure BDA00016535003900001118
Figure BDA00016535003900001119
c＝bT/(bTb)，
Figure BDA00016535003900001120
where d is the temporary replacement vector, b is the temporary replacement vector
Figure BDA00016535003900001121
Column vectors of the same dimension, c is a row vector that is temporarily replaced.
Step A4, minimizing problems
Figure BDA0001653500390000121
Solution of (2)
Figure BDA0001653500390000122
Wherein
Figure BDA0001653500390000123
Step A5, updating the k iteration residual vector
Figure BDA0001653500390000124
Step A6, determining whether the stopping criterion is satisfied, i.e. the Euclidean norm of the k-th iteration residual vector is fullFoot | rk‖2ε (ε is a very small number, e.g.: 10)-6) If yes, executing step A7, otherwise updating k to k +1, and repeating steps A2 to A5;
step A7, outputting sparse coefficient vector x ∈ RN(RN: n × 1-dimensional real column vector, x (i): output the ith element, x, of a sparse coefficient vector xk(i) The method comprises the following steps Step A4 minimizing the problem solution vector xkIth element of (2):
Figure BDA0001653500390000125
the nature of the column Greville recursion:
properties 4.1bkAn essential condition of 0 is ak∈R(Ak-1)＝span{a1，a2，…，ak-1I.e. akAnd a1，a2，…，ak-1The correlation is linear.
This is because:
Figure BDA0001653500390000126
this description bk∈R(Ak-1)⊥Is akAt R (A)k-1) Orthogonal projection on the orthogonal complement, if b k0 means ak∈R(Ak-1) Therefore a iskAnd R (A)k-1) The linear correlation and vice versa.
II b in the actual calculationk‖2Is an important index which reflects akAnd R (A)k-1) Degree of linear correlation of.
In the compressed sensing problem, b does not occur from the assumption of the sensing matrixkCase 0. But | bk‖2This index is very important and reflects akAnd R (A)k-1) Degree of linear correlation of. If | bk‖2Not equal to 0, but very small indicates akAnd a1，a2，…，ak-1Close to linear correlation, i.e. a very weak degree of linear independence, is the case of so-called numerical instability. This is important, and it solves the problem of checking a in the course of calculationkWith its front vector system a1，a2，…，ak-1Is a big problem in numerical calculation.
Properties 4.2
Figure BDA0001653500390000131
That is to say bkIs akAt R (A)k-1)⊥The orthogonal projection of (a) is as follows:
(bk，ai)＝0,i＝1,2,…,k-1 (4.7)
i.e. bkAnd a1，a2，…，ak-1Are orthogonal.
IQR _ OMP algorithm:
if already calculated:
Ak-1＝Qk-1Rk-1 (4.9)
wherein Qk-1＝[q1,q2,…qk-1]∈RM×(k-1)Of a column orthogonal matrix, Rk-1An upper triangular matrix of (k-1) × (k-1). The calculation of step k is considered below:
Figure BDA0001653500390000132
wherein beta isk∈R(k-1)×1，αkIs a pure quantity. And (3) obtaining:
Figure BDA0001653500390000133
Figure BDA0001653500390000134
Figure BDA0001653500390000135
due to the fact that
Figure BDA0001653500390000136
Thereby the device is provided with
Figure BDA0001653500390000137
Therefore:
Figure BDA0001653500390000138
similarly, the residue r is calculated every stepk＝s-AkxkA matrix is computed once multiplied by the vector (s: the vector to be reconstructed), and rkEach step needs to be used in A to select and rkThe column vector with the largest inner product is supplemented to AkIn the formation of Ak+1. In this case, for recursive QR decomposition
Property 4.4 in the QR _ OMP algorithm, rkIt is true that y is at R (A)k)⊥Orthogonal projection of (a) i.e.:
Figure BDA0001653500390000139
k＝1,2,…,K
computing a residual vector r using the equation (4.9)kBrings great convenience, avoids direct multiplication of a matrix and a vector, and can calculate one inner product in each step because the calculation of an intermediate quantity { x ] is avoided in an inner loopkThis can reduce the computational effort for large problems. Therefore, there are:
the IQR-OMP algorithm:
step B1, inputting observation data vector y epsilon Rm(Rm: m × 1 dimensional real column vector set) dictionary matrix a ═ a1，a2，…，an]∈Rm×n(ai(i-1, 2, … n) is the ith column vector of matrix a, Rm×n: a set of m × n dimensional real number matrices); let the initial support set index
Figure BDA0001653500390000141
Make the initial residual vector equal to the observed data vector, i.e. r0Y, let k be 1;
step B2, solving the k-1 iteration residual vector r in the dictionary matrix Ak-1The most strongly correlated column j ofk：
jk∈arg maxj|(rk-1，aj)|，Ωk＝Ωk-1∪{jk}；
Wherein omegakIs the kth iteration support set index, Ωk-1Is the k-1 th iteration support set index, ajIs the jth column vector in matrix a.
Step B3, iterating Q from the k-1 st timek-1And Rk-1Recursion calculating the kth iteration QkAnd Rk：
If k is 1, then jk＝j1Then, then
Figure BDA0001653500390000142
Matrix Q1Is first column of (1) as Q1(1) or q1Is equal to
Figure BDA0001653500390000143
Namely, it is
Figure BDA0001653500390000144
Turning to the step B4, the method,
if k ≠ 1, then the following is performed in order:
Figure BDA0001653500390000145
Figure BDA0001653500390000146
Figure BDA0001653500390000147
qk＝q1/αk，
Qk＝[Qk-1:qk]，
Figure BDA0001653500390000148
wherein
Figure BDA0001653500390000149
Is the j-th in the matrix AkColumn vector, qkIs the kth column vector, β, of the matrix Qk、αkA temporary substitute element for the kth execution;
step B4, updating the k iteration residual error vector
Figure BDA00016535003900001410
Step B5, determine if the stop criterion is met, i.e. the euclidean norm of the k-th iteration residual vector meets | rk‖2ε (ε is a very small number, e.g.: 10)-6) If yes, go to step B6;
otherwise, updating k to k +1, and repeating the steps B2 to B4;
step B6, minimizing problems
Figure BDA0001653500390000151
The solution of (a):
Figure BDA0001653500390000152
give a
Figure BDA0001653500390000153
Is the support index omegakA corresponding dictionary matrix is then generated and stored in the memory,
Figure BDA0001653500390000154
ai(i＝j1，j2，…jk∈Ωk) Is a matrix
Figure BDA0001653500390000155
The ith column vector.
Step B7, output sparse coefficient vector x (i): output the ith element of sparse coefficient vector x, xk(i) The method comprises the following steps Step B6 finding solution xkIth element of (2):
Figure BDA0001653500390000156
the relationship is recurred by using Greville:
Figure BDA0001653500390000157
therefore, there are:
the process of calculating the M-P generalized inverse by 4.5Grville recursion in theorem is consistent with QR recursion. In fact, under the condition of theorem 3, there are
Figure BDA0001653500390000158
Examples
The computing environment for designing the numerical experiment is DELL-PC WORK GROUP Intel (R) core (TM) i 7-47903.6 Ghz, the internal memory 8.0GB, the Windows7 flagship edition, and the computing software is MATLAB R2010 b.
A e R in Case12000×5000，K＝600；
A e R in Case23000×6000，K＝1000；
A e R in Case310000×15000，K＝3000；
The calculated time comparisons for incremental OMP algorithms 1-D are shown in Table 1 below:
TABLE 1
Type of situation IQR_OMP IGR_OMP OMP
Case1 7.6331 9.112 30.8978
Case2 29.2647 34.8170 164.7083
Case3 810.4921 962.1627 10386.0
The above three algorithms are illustrated as follows:
an algorithm based on QR decomposition and a new residue recurrence formula given herein for the IQR _ OMP algorithm;
the OMP algorithm which is established on the basis of a Greville recurrence formula and a new residual recurrence formula and is given in the specification during the IGR _ OMP algorithm;
the OMP algorithm is the classical OMP algorithm, in which the generalized inverse uses the form of an equation. Namely, it is
Figure BDA0001653500390000161
It should be noted that the same type of raw data used by different algorithms is usedA∈RM×NAnd y ∈ RMAre the same.
Results of the calculations the psnr of the three calculation methods for the second case was 118.7734, the psnr of the three methods of the third type was 144.3529, and the MSE was 2.38611e-010. It can be seen that the algorithms GR _ OMP and QR _ OMP presented herein have significant advantages, particularly with respect to large problems, in a fast and efficient manner. For example, in the third case above, the algorithm presented herein has a computation time of less than one tenth of that of the OMP algorithm, as shown in Table 2(2-D Lena image (256X 256)) and Table 3(2-D case (1024X 1024)):
TABLE 2
IGR_OMP OMP IRLS SP GBP IHT CoSaMP
CPUtime(s) 0.18 0.31 8.37 1.40 4.18 0.23 2.30
Psnr 22.06 22.08 23.63 21.04 22.77 15.66 19.77
TABLE 3
IQR_OMP OMP IRLS SP GBP IHT CoSaMP
CPUtime(s) 11.30 31.86 5809 243 2325 14.03 533
Psnr 45.90 45.90 48.70 46.34 47.83 46.79 45.27
(1) In the Lena image of 256 × 256 in table 2, the measurement matrix is a random matrix, and the sparse transform DCT transform is calculated column by column. Wherein the method comprises the following steps:
IGR _ OMP: is an incremental OMP algorithm set up on the basis of Greville recursion as given herein;
IQR _ OMP: is the incremental OMP algorithm set forth herein based on QR decomposition;
OMP: is a classical OMP method established on a normal equation;
IRLS: iterative weighted Least squares (iterative weighted Least Square);
SP: a subspace tracking method;
GBP: greedy Basis Pursuit (Greedy Basis Pursuit);
IHT: an iterative hard threshold method;
and (3) CoSaMP: a compressive sampling matching pursuit method.
Among these methods, the IQR _ OMP and IGR _ OMP algorithms are the method of the present invention, and the other methods are the software given by doctor Chengfu Huo, university of hong kong science and technology.
The calculations show that the incremental algorithms IGR _ OMP and IQR _ OMP presented and discussed herein have significant advantages.
The present invention provides a method for incremental orthogonal matching pursuit for high-dimensional sparse vector reconstruction, and a plurality of methods and approaches for implementing the technical solution are provided, the above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, a plurality of improvements and modifications may be made without departing from the principle of the present invention, and these improvements and modifications should also be regarded as the protection scope of the present invention. All the components not specified in the present embodiment can be realized by the prior art.

Claims (1)

1. A method suitable for fast image compression based on IGR _ OMP is characterized in that compressed sensing is used in the image processing field, information far lower than the signal vector or image vector dimension is utilized to determine the sparsest coefficient representation of the high-dimension vector on an overcomplete dictionary, and sparse solution of a sublinear system is solved to reconstruct an image vector, and the method specifically comprises the following steps:
step A1, inputting observation data vector y epsilon RMAnd the dictionary matrix a ═ a1，a2，…，aN]∈RM×NLet the initial support set index
Figure FDA0002680454320000011
Let the initial residual vector r0Is equal to the observation data vector y, i.e. r0Y, let k be 1, wherein RMIs a set of M x 1 dimensional real column vectors, RM×NIs a set of M × N dimensional real number matrices, aiRepresents the ith real column vector in the dictionary matrix a, i ═ 1,2 …, N;
step A2, solving the k-1 iteration residual vector r in the dictionary matrix Ak-1Of the kth iteration of (a) the most strongly correlated column jk：
jk∈arg maxj|(rk-1，aj)|，Ωk＝Ωk-1∪{jk}，
Wherein omegakFor the kth iteration support set index, Ωk-1For the k-1 iteration support set index, ajIs the jth real column vector in the dictionary matrix A;
step A3, supporting the index omega by the k-1 iterationk-1Corresponding dictionary matrix
Figure FDA0002680454320000012
Generalized inverse matrix of
Figure FDA0002680454320000013
Incremental calculation of kth iteration support index omegakCorresponding dictionary matrix
Figure FDA0002680454320000014
Generalized inverse matrix of
Figure FDA0002680454320000015
Step A4, solving a minimization problem
Figure FDA0002680454320000016
Solution x ofk：
Figure FDA0002680454320000017
Wherein the content of the first and second substances,
Figure FDA0002680454320000018
step A5, updating the k iteration residual vector
Figure FDA0002680454320000019
Step A6, determining whether the stopping criterion is satisfied, i.e. determining whether the Euclidean norm of the k-th iteration residual vector satisfies | | | rk||2≦ ε, if satisfied, perform step A7, otherwise update k to k +1, and repeat step A2Go to step a 5;
step A7, outputting sparse coefficient vector as x ∈ RN：
Figure FDA00026804543200000110
Where x (i) is the i-th element in the sparse coefficient vector x, xk(i) Is a vector xkThe i-th element of (1), RNIs a real number column vector set with dimension Nx 1;
step a3 includes: if k is 1, then jkIs j1I.e. by
Figure FDA0002680454320000021
Ωk＝{j1Turning to step A4;
if k ≠ 1, the following calculations are performed sequentially:
Figure FDA0002680454320000022
Figure FDA0002680454320000023
c＝bT/(bTb)，
Figure FDA0002680454320000024
wherein
Figure FDA0002680454320000025
Is a dictionary matrix
Figure FDA0002680454320000026
Middle jkA vector of columns of real numbers,
Figure FDA0002680454320000027
is a vector
Figure FDA0002680454320000028
D is a vector of the provisional substitution, b is a dimension and a vector of the provisional substitution
Figure FDA0002680454320000029
The same M × 1-dimensional vector, c is a temporally substituted 1 × M-dimensional vector;
in the step a4, the method comprises the steps of,
Figure FDA00026804543200000210
representative vector
Figure FDA00026804543200000211
Euclidean norm of;
the 2-D Lena images with the formats of 256 × 256 and 1024 × 1024 respectively are compressed by using the methods of the steps A1-A7, the measurement matrix is a random matrix, and the sparse transform DCT transform is calculated column by column.
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