CN112422133B - Binary sparse signal recovery method for subtraction matching pursuit and application thereof - Google Patents
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Abstract
The invention discloses a binary sparse signal recovery method for subtraction matching pursuit and application thereof, wherein the method is used for binary sparse signal recovery based on explicit SMP or implicit SMP; the explicit SMP based concrete steps are: inputting a noise observation vector, a perception matrix and sparsity; initializing data; in each iteration, the column vector with the index i corresponding to the perception matrix is most related to the residual error to obtain an index s k By subtracting the perceptual matrix correspondence index as s k Updating a residual vector by the column vector until iteration stops, and outputting an estimation sparse signal; the implicit SMP based concrete steps are as follows: inputting a noise observation vector, a perception matrix and sparsity; initializing data; in each iteration, selecting an index, acquiring a column index set, then updating a total index estimation support set, updating the correlation, and circulating iteration until the iteration is stopped; and outputting an estimated sparse signal. The invention improves the sparse signal recovery efficiency and achieves the purpose of better recovery performance of the sparse signal.
Description
Technical Field
The invention relates to the technical field of communication and signal processing, in particular to a binary sparse signal recovery method for subtraction matching pursuit and application thereof.
Background
In many applications such as communication and signal processing, it is often necessary to recover a sparse signal from a linear system with noise interference, where an n-dimensional signal x is called K sparse signal if x has at most K non-zero elements. Compressed sensing is applied in electronic engineering, especially in signal processing, for acquiring and reconstructing sparse or compressible signals. The key idea of compressed sensing is to recover sparse signals from few non-adaptive linear measurements by convex optimization. By using an efficient algorithm, high dimensional signals can be recovered from linear measurements that were previously considered to be highly incomplete, provided that such high dimensional signals can be sparsely represented by appropriate basis.
The Orthogonal Matching Pursuit (OMP) algorithm is a compressed sensing on-the-go algorithmOne of the most commonly used sparse recovery algorithms in the application fields of signals, signals and the like, and the batch processing algorithm OMP is the most efficient OMP realization algorithm at present. The OMP algorithm needs to solve a least squares problem in each iteration, if A is not pre-calculated T A and A T y, the complexity of running K iterations of a batch OMP is mn 2 +2mn+K 2 n+3Kn+K 3 But if A is pre-calculated T A and A T y, the complexity of running K iterations of a batch OMP is K 2 n+3Kn+K 3 Therefore, when m, n and K are larger, the complexity is higher, the efficiency of the batch OMP algorithm is also limited, and sparse recovery of signals is an important research direction in the field of communication and signal application, and is also a hot spot of current research.
Disclosure of Invention
In order to overcome the defects and shortcomings in the prior art, the invention provides a binary sparse signal recovery method for subtraction matching pursuit, which is mainly different from an OMP algorithm in that: in each iteration, a least square problem does not need to be solved, so that the sparse signal recovery efficiency is improved, sufficient conditions for recovering sparse signals are set, the problem of low sparse signal recovery quality is solved, and the purpose of better sparse signal recovery performance is achieved.
The second objective of the present invention is to provide a binary sparse signal recovery system for subtraction matching pursuit.
A third object of the present invention is to provide a storage medium.
It is a fourth object of the invention to provide a computing device.
In order to achieve the purpose, the invention adopts the following technical scheme:
a binary sparse signal recovery method of subtraction matching pursuit comprises the following steps:
performing binary sparse signal recovery based on explicit SMP or implicit SMP;
the specific steps of binary sparse signal recovery based on explicit SMP are as follows:
inputting a noise observation vector y, a perception matrix A and sparsity K;
initializing data, wherein the data initialization comprises initializing iteration times, estimating a support set, estimating sparse signals and residual vectors, and the noise observation vectors are used as initial values of the residual vectors;
setting iteration stop conditions, and selecting an index i in each iteration so that a sensing matrix A corresponds to a column vector A with the index i i And residual r k Most relevant, thus obtaining the index s k The corresponding index is s by subtracting the sensing matrix A k Column vector of
Updating the residual vector, and then entering next iteration until an iteration stop condition is met, and stopping the iteration;
outputting an estimated sparse signal;
the specific steps of performing binary sparse signal recovery based on the implicit SMP are as follows:
inputting a noise observation vector y, a perception matrix A and sparsity K;
initializing data, including initializing iteration number, estimating support and estimating sparse signal, and initializing correlation u 0 ＝A T y, wherein A T Represents the transpose of the perceptual matrix a;
setting iteration stop conditions, selecting an index i in each iteration, and acquiring a column index set s k Immediately followed by updating the total index estimate support
Thereby updating the correlation
Then, entering the next iteration until an iteration stop condition is met, and stopping the iteration;
and outputting an estimated sparse signal.
As a preferred technical scheme, in explicit SMP and implicit SMP, iteration termination conditions are all | | | r k || 2 E ≦ e, e represents a given positiveAnd (4) counting.
As a preferred technical solution, in explicit SMP, the specific steps of iteration include:
in each iteration, through argmax function from
To obtain a residual error r k-1 And each column of projection coefficients of the sensing matrix A
Maximum value of inner product absolute value, and forming set s by corresponding the values to positions of column elements of sensing matrix A k As a column index of the sensing matrix a, a specific calculation formula is:
after finding out the maximum index value corresponding to the perception matrix A, setting an estimated sparse signal
Corresponding set s k The value of the column index position is 1;
updating index collections
Will find the set of column indices s in each loop k And the column index estimation support which is recorded iteratively
Merging, recording all column index values, updating index estimation support set
Obtaining a complete index estimation support set;
the residual r recorded by last iteration calculation k-1 Corresponding matrix to this index set
And updating the residual vector.
As a preferred technical solution, the estimation sparse signal represents an index estimation support set
The estimated sparse signal with a value of 1 corresponding to the column index position is specifically represented as:
wherein,
representing the pseudo-inverse of the perceptual matrix a.
As a preferred technical scheme, in implicit SMP, a total index estimation support set is updated
The method comprises the following specific steps:
will find the column index set s in each loop k And the column index estimation support which is recorded iteratively
Merging, recording all column index values and updating the total index estimation support set
A complete index set is obtained.
As a preferred technical scheme, in implicit SMP, correlation is updated
The calculation method is as follows:
wherein,
estimating a complement of an ensemble for a perceptual matrix A total index
Corresponding to the transposition of the column vector, r k Is the residual error. .
As a preferable technical scheme, in the explicit SMP, based on the RIP tight and sufficient condition, the sensing matrix A meets the requirement
RIP conditions of (1) and
the preconditions of (a);
based on the condition that the cross correlation is tight and sufficient, the perception matrix A meets the requirement
Cross correlation and
the preconditions of (a);
wherein,
μ denotes the cross-correlation between the A column vectors of the perceptual matrix, A i Representing a column vector indexed by the i-perception matrix A, A j The representation index is a j-column vector.
In order to achieve the second object, the present invention adopts the following technical solutions:
a binary sparse signal recovery system with subtraction matching pursuit is provided with an explicit SMP signal recovery module or an implicit SMP signal recovery module;
the explicit SMP signal recovery module includes: the system comprises an explicit SMP data input unit, an explicit SMP data initialization unit, an explicit SMP iteration stop condition setting unit, a residual vector updating unit and an explicit SMP output module;
the explicit SMP data input unit is used for inputting a noise observation vector y, a perception matrix A and sparsity K;
the explicit SMP data initialization unit is used for initializing data, and comprises initialization iteration times, estimation support sets, estimation sparse signals and residual vectors, and the noise observation vectors are used as initial values of the residual vectors;
the explicit SMP iteration unit is used for loop iteration;
the explicit SMP iteration stop condition setting unit is used for setting an iteration stop condition;
the residual vector updating unit is used for updating residual vectors, and in each iteration, one index i is selected, so that the sensing matrix A corresponds to the column vector A with the index i i And residual r k Most relevant, resulting in an index s k The corresponding index is s by subtracting the sensing matrix A k Column vector of
Updating a residual vector;
the explicit SMP output unit is used for outputting an estimation sparse signal;
the implicit SMP signal recovery module comprises: an implicit SMP data input unit, an implicit SMP data initialization unit, an implicit SMP iteration stop condition setting unit, a correlation updating unit and an implicit SMP output module;
the implicit SMP data input unit is used for inputting a noise observation vector y, a perception matrix A and sparsity K;
the implicit SMP data initialization unit is used for data initialization, and comprises initialization iteration times, estimation support and estimation sparse signals, and initialization correlation u 0 ＝A T y, wherein A T Represents the transpose of the perceptual matrix a;
the implicit SMP iteration unit is used for loop iteration;
the implicit SMP iteration stop condition setting unit is used for setting an iteration stop condition;
the correlation updating unit is used for updating the phaseSex of concern
In each iteration, an index i is selected, and a column index set s is obtained k Immediately followed by updating the total index estimate support
Thereby updating the correlation
The implicit SMP output module is used for outputting an estimated sparse signal.
In order to achieve the third object, the present invention adopts the following technical solutions:
a storage medium stores a program that realizes the binary sparse signal recovery method of the above subtraction matching pursuit when executed by a processor.
In order to achieve the fourth object, the present invention adopts the following technical means:
a computing device comprises a processor and a memory for storing a processor executable program, wherein the processor executes the program stored in the memory to realize the binary sparse signal recovery method of the subtraction matching pursuit.
Compared with the prior art, the invention has the following advantages and beneficial effects:
(1) The explicit SMP algorithm adopted by the invention forms the residual vector explicitly during each iteration, and the efficiency of the explicit SMP algorithm is higher than that of the existing batch OMP algorithm
And (4) multiplying.
(2) The implicit SMP algorithm employed in the present invention is similar in complexity to the explicit SMP algorithm, but if A can be pre-computed T A and A T y, the complexity of the former can be significantly reduced, so a is calculated in advance T A and A T y, implicit SMP is K times more efficient than existing batch OMP algorithms.
(3) The sensing matrix A meets the requirement of the RIP based on the SMP algorithm
RIP conditions of (1) and
based on the premise that the sensing matrix A meets the requirement of the tight and sufficient cross correlation
Cross correlation and
the method meets the two sufficient conditions, solves the problem of low recovery quality of the sparse signal x, and achieves the purpose of better recovery performance of the sparse signal x.
Drawings
Fig. 1 is a schematic diagram of an overall flow framework of a binary sparse signal recovery method for subtraction matching pursuit according to this embodiment;
FIG. 2 is a diagram illustrating the comparison result between the average CPU time and the measurement number m in the simulation experiment of the present embodiment;
FIG. 3 is a diagram illustrating a comparison result between an average missed detection probability and a measurement number m in a simulation experiment of the present embodiment;
fig. 4 is a diagram illustrating a comparison result between the average false alarm probability and the measurement number m in the simulation experiment of the present embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Examples
As shown in fig. 1, the present embodiment provides a binary sparse signal recovery method for subtraction matching pursuit, including the following steps:
(1) Explicit SMP algorithm
S1: and inputting a known noise observation vector y, a perception matrix A and sparsity K. K denotes sparsity in each of the following. Wherein y = Ax + v is a known noise observation vector having a size of m × 1, where
Is a real domain, the sparse signal x belongs to R n The magnitude n x 1,v is the known noise vector,
this example considers only l 2 Norm noise, here l 2 The norm is in the form of a norm,
x＝(x 1 ,x 2 ,…,x n )，
i.e. for a given positive number e, there is | | | v | | | grind 2 ≤∈。
(m<<n) which is a perceptual matrix of known size m × n. K sparse, i.e. for length n
Vector of which n element values are only K non-zero, where K is<<n, this vector is said to be K sparse or strictly K sparse.
S2: initializing iteration times k =0, and estimating an support set
Estimating sparse signals
Which is an n-dimensional vector, an initial residual vector r 0 ＝y。
S3: iteration is carried out by utilizing known conditions until iteration stopping conditions are met, and estimation sparsity is obtainedSignal
In the explicit SMP algorithm, some priori information is used to set a stop condition, and the iteration termination condition in this embodiment is | | r k || 2 ≦ e, where r k Denotes the residual, and e denotes a given positive number. In each iteration, the SMP selects an index i (from the sensing matrix a)
Representing a total index estimate set
Complement of) such that A i And residual r k Most relevant, thus obtaining the index s k . Then by subtracting
And updating the residual vector, and then entering the next iteration until the iteration stops. Where A is i ∈R m Is that the sensing matrix a corresponds to a column vector with index i,
representing the sensing matrix A with corresponding index s k The column vector of (2).
S31: k = k +1, the number of iterations is increased by 1 each time an iteration is performed;
s32: computing
(i.e., calculating)
The absolute value of (a) is,<·>representing the inner product), in each iteration, from the start by the argmax function
To obtain a residual error r k-1 And each column projection coefficient of the perception matrix A
Maximum value of inner product absolute value, and forming set s by the positions of the values corresponding to column elements of sensing matrix A k I.e. as column index of the perceptual matrix a. Here, the number of the first and second electrodes,
representing the corresponding index of the sensing matrix A as the transposition of the i-column vector
And residual r k-1 The absolute value of the inner product of (c). Setting omega to represent a total branch set of the K sparse signals x, setting | omega | to represent a cardinal number of omega, and setting a set
And
and is provided with
Represents the complement of the set omega, so
Representation collection
Complementing;
S33：
in order to clearly mark the estimation signal corresponding to the maximum index value found each time, after the maximum index value corresponding to the perception matrix A is found, the estimation sparse signal is set
Corresponding sets k The value of the column index position is 1. The sparse signals come from communication and signal processing and meet the requirement of acquiring signals with items of 1 or 0;
s34: updating index estimate support sets
Will find the set of column indices s in each loop k And the column index estimation support recorded iteratively
Merging is performed such that all column index values are recorded to update the total index estimate set
Obtaining a complete index estimation support set;
s35: updating residual errors
Here explicit residual vector formation is explicitly embodied.
A sub-matrix representing A, which comprises only
Column of index, x S Is a sub-vector of x, which contains only x
The entries of the index are such that,
is that
And transposing the matrix. For any full rank matrix
Is provided with
And
both represent a matrix separately
And (3) column space up-projection and quadrature complementary projection. The residual r recorded by last iteration calculation k-1 Corresponding matrix to this index set
And updating the residual vector. The iteration termination condition is | | | r k || 2 ≦ e, where the iteration stop condition is ensured by updating the residual in each iteration.
S4: output of
Wherein,
is to estimate the sparse signal. Here, the number of the first and second electrodes,
representing the total index estimate support
Estimated sparse signals having a value of 1 for the column index position, i.e.
Here, the
Representing the pseudo-inverse of the perceptual matrix a, e.g. y = Ax + v, then
Therefore, the first and second electrodes are formed on the substrate,
representing an index estimate support
Is the pseudo-inverse of the perceptual matrix a.
From the above analysis, the present embodiment provides a main difference between the new SMP algorithm and OMP in that the former does not solve the least square problem, thereby improving the efficiency. Thus, if K iterations are performed, the complexity of the explicit SMP algorithm is:
thus, explicit SMP is more efficient than batch OMP algorithms
And (4) doubling.
(2) Implicit SMP algorithm
In some cases, if x is assumed to have K non-zero entries, or the algorithm is run for K iterations, we do not need to explicitly form the residual r k . An implicit SMP algorithm is designed, when A T A and A T y can be pre-computed, which is more efficient compared to explicit SMP algorithms. Where A is T Representing the transpose of a perceptual matrix A T A is a symmetric matrix of size n × n, A T y is a matrix of size n × 1. The method steps of the implicit SMP algorithm are as follows:
p1: inputting known values y, A and sparsity K, wherein y = Ax + v is a noise observation vector with a size of m × 1,A being a sensing matrix with a size of m × n;
p2: initializing iteration number k =0, estimating support set
Estimating sparse signals
u 0 ＝A T y，
The size is n × 1.
P3: iterating by using known conditions until an iteration stop condition | | | r is met k || 2 Is less than or equal to the epsilon, and an estimated sparse signal is obtained
As with the explicit SMP algorithm, the implicit SMP algorithm stops with | | | r k || 2 Is less than or equal to the epsilon. At each iteration, the implicit SMP algorithm selects an index
By calculation of
Obtaining a set of column indices s k Immediately followed by updating the total index estimate support
And pass through
To update
And then entering the next iteration until the iteration stops. Here with the formation of a residual r k In contrast to the explicit SMP algorithm, the implicit SMP algorithm is formed in the kth iteration
Is defined as
Here, ,
estimating a complement of an ensemble for a total index of a perceptual matrix A
Corresponding to the transposition of the column vector, r k Is the residual error.
To represent
And r k The inner product of (d) is the correlation between the two.
P31: k = k +1, the number of iterations is increased by 1 for each iteration;
P32：
find the index column set, here
Representation collection
The complement of (a) is to be added,
to represent
In each iteration, by the argmax function
In (a) to obtain
Transpose of column vector representing perception matrix A corresponding to index i
And residual r k-1 Which is the correlation of the two), the positions of these values corresponding to the column elements of the perceptual matrix a constitute a set s k I.e. as column index of the perceptual matrix a. Different from explicit SMP algorithmsS3, an implicit SMP algorithm finds a column index set
So that
And calculated in k iterations
The residual error is not directly displayed in the step, and the calculation of the implicit SMP algorithm is embodied;
P33：
like explicit SMP algorithm, estimation sparse signal is also set
Corresponding set s k The value of the column index position is 1;
p34: the index estimation support is updated and,
will find the column index set s in each loop k And the column index estimation support which is recorded iteratively
Merging is performed, so that the total index estimation support is updated by recording all the column index values
Obtaining a complete index estimation support set;
P35：
updating
Unlike explicit SMPs, implicit SMPs form one in k iterations
Is shown as
By calculating
To update
Here, ,
that is to say
Which is the complement of the total index estimate support in the sensing matrix a
Transpose of the corresponding column vector
And residual r k I.e., the correlation of the two.
Wherein,
here, the notation: = definition, A T r k Transpose matrix A representing a perceptual matrix A T And residual vector r k The correlation of (a) with (b) is,
transposed matrix A as perceptual matrix A T And the transpose matrix A of the sensing matrix A T And the sensing matrix A corresponds to an index set s k Column vector of
Correlation; the iteration termination condition is | | | r k || 2 ≤∈Here by updating the correlation in each iteration in an implicit way
And then update | | r k || 2 To ensure that the iteration stop condition is reached.
P4: output of
Wherein,
is to estimate the sparse signal. Here, ,
representing the total index estimate support
The estimated sparse signal having a value of 1 corresponding to the column index position, that is,
here, the
Representing the pseudo-inverse of the perceptual matrix a, e.g. y = Ax + v, then
Therefore, the first and second electrodes are formed on the substrate,
representing an index estimate support
Is the pseudo-inverse of the perceptual matrix a.
Although the complexity of the implicit SMP algorithm is similar to the explicit SMP algorithm, if A can be pre-computed T A and A T y, the complexity of the former can be significantly reduced. Thus, if K iterations are performed, the complexity of the implicit SMP algorithm is:
therefore, in the process of executing K times of iterations, A is calculated in advance T A and A T y, implicit SMP is more than K times more efficient than batch OMP algorithms.
To reach the iteration termination condition | | | r k || 2 And setting a sufficient condition for recovering the sparse signal according to the embodiment, and providing a sufficient condition for stably recovering the sparse signal x by using the constrained isometry and the mutual coherence of the sensing matrix A. The explicit and implicit SMP algorithms have high recovery efficiency for recovering the K sparse signal x, so that sufficient conditions for recovering the K sparse signal x by the SMP algorithms are established. Although the explicit SMP algorithm and the implicit SMP algorithm are two different implementation directions of the SMP algorithm, their sufficient conditions are the same, so the present embodiment only describes the sufficient conditions for the explicit SMP algorithm to recover the signal x.
The SMP algorithm deals with sparse signal structures containing noise signals, i.e. the sparse signal x is recovered from the linear model y = Ax + v. Here the noise vector
Satisfy | | v | the zero calculation 2 Is less than or equal to E. Due to the implementation of the SMP algorithm, the sensing matrix A of the SMP algorithm meets the requirements based on RIP under the condition of tight and sufficient RIP
RIP conditions of (1) and
based on the premise that the sensing matrix A meets the requirement of the tight and sufficient cross correlation
Cross correlation and
meets the two tight and sufficient conditions, solves the problemsThe problem of low recovery quality of the sparse signal x is solved, so that the stopping criterion is | | | r k || 2 The SMP algorithm less than or equal to the epsilon accurately recovers the branch set omega of x, and the purpose of better recovery performance of the sparse signal x is achieved. Wherein for the matrix
And 1. Ltoreq. K.ltoreq.n, k-limiting equidistant constant (RIC). Delta k E (0,1) is the smallest constant, thus
So that here is delta K+1 E (0,1) is the RIC minimum constant for this example, which satisfies the inequality above, and K is the sparsity.
Where μ denotes the cross-correlation between the column vectors of the perceptual matrix A, A i Representing a column vector indexed by the i-perception matrix A, A j The representation index is a j column vector.
RIP-based SMP stringent and adequate conditions: it is first demonstrated if the sensing matrix A satisfies
Given the constraint of positive e, then satisfy any i e Ω and x i Any K-sparse signal x of =1, its support Ω, can be recovered exactly in K iterations. Here, Ω is set to represent the support of the K sparse signal x, and | Ω | represents the cardinality of Ω. Such sparse signals come from communications and signal processing where a signal with an entry of 1 or 0 needs to be acquired, and then proved about δ K+1 The sufficient condition is a tight sufficient condition, and the [ epsilon ] sufficient condition is an approximately tight sufficient condition.
Tightly sufficient conditions based on mutually dry SMPs: if the perceptual matrix A is column normalized and the mutual coherence of A satisfies
Then under a certain condition of being within the constraint, any K sparse signal x support omega can be obtainedTo recover exactly in K iterations, and furthermore it is demonstrated that the sufficient condition for μ is a tight sufficient condition and the condition for e is an approximately tight sufficient condition.
In this embodiment, the performance of the SMP algorithm is tested, and the operation efficiency and recovery effect of the SMP, OMP, batch OMP, coSaMP, and SP algorithms are compared through simulation experiments in multiple tests.
As shown in fig. 2, which shows the average CPU time of the five algorithms described above compared to the number of measurements m, K =100, n =1024, where the SMP algorithm is most efficient and much more efficient than both OMP and batch OMP.
As shown in fig. 3 and fig. 4, the average values of the missing detection probability and the false alarm probability of the five algorithms for the measurement number m are respectively shown, K =100, n =1024, wherein the average missing detection probability and the average false alarm probability of the SMP algorithm are both smaller than those of the OMP and batch OMP algorithms, indicating that the former has better recovery performance. The average miss probability and average false positive probability of OMP and batch OMP are the same, because batch OMP is a fast implementation method of OMP. Although fig. 3 and 4 show that the average probability of missed detection and the average probability of false alarm are less for CoSaMP and SP algorithms than for SMP algorithms when m is larger, they are less efficient than SMP algorithms as shown in connection with fig. 2.
The binary sparse signal recovery method based on subtraction matching pursuit in the embodiment takes a batch processing OMP algorithm as an improved comparison algorithm, and is mainly different from the OMP algorithm in that a least square problem does not need to be solved in each iteration, so that the efficiency is improved.
Compressed sensing is a key technique for stably reconstructing a K sparse signal x from a linear model y = Ax + v. In the sparse recovery process of the SMP through signals, the last output result ensures the stable and efficient recovery of the sparse signal x, and the Subtraction Matching Pursuit (SMP) designed in the embodiment gives two SMP implementation methods which are respectively called as an explicit SMP algorithm and an implicit SMP algorithm according to whether a residual vector is explicitly formed during each iteration, wherein the explicit SMP algorithm has higher efficiency than the batch processing OMP algorithm
And the efficiency of the implicit SMP algorithm is faster than that of the batch OMP algorithm by more than K times, and the final SMP algorithm obtained by experimental comparison through different algorithms is higher than that of the batch OMP algorithm and has better recovery performance.
Example 2
The embodiment provides a binary sparse signal recovery system for subtraction matching pursuit, which is provided with an explicit SMP signal recovery module or an implicit SMP signal recovery module;
in this embodiment, the explicit SMP signal recovery module includes: the system comprises an explicit SMP data input unit, an explicit SMP data initialization unit, an explicit SMP iteration stop condition setting unit, a residual vector updating unit and an explicit SMP output module;
in the embodiment, the explicit SMP data input unit is configured to input a noise observation vector y, a sensing matrix a, and a sparsity K;
in this embodiment, the explicit SMP data initialization unit is configured to initialize data, and includes initializing iteration times, estimating a branch set, estimating a sparse signal, and a residual vector, where the noise observation vector is used as an initial value of the residual vector;
in this embodiment, the explicit SMP iteration unit is used for loop iteration;
in this embodiment, the explicit SMP iteration stop condition setting unit is configured to set an iteration stop condition;
in this embodiment, the residual vector updating unit is configured to update the residual vector, and in each iteration, select an index i such that the sensing matrix a corresponds to the column vector a with the index i i And residual r k Most relevant, thus obtaining the index s k The corresponding index is s by subtracting the perception matrix A k Column vector of
Updating a residual vector;
in the embodiment, the explicit SMP output unit is configured to output an estimation sparse signal;
in this embodiment, the implicit SMP signal recovery module includes: an implicit SMP data input unit, an implicit SMP data initialization unit, an implicit SMP iteration stop condition setting unit, a correlation updating unit and an implicit SMP output module;
in this embodiment, the implicit SMP data input unit is configured to input a noise observation vector y, a perceptual matrix a, and a sparsity K;
in this embodiment, the implicit SMP data initialization unit is configured to initialize data, and includes initializing iteration number, estimating a support set, and estimating a sparse signal, and initializing correlation u 0 ＝A T y, wherein A T Represents the transpose of the perceptual matrix a;
in this embodiment, an implicit SMP iteration unit is used for loop iteration;
in this embodiment, an implicit SMP iteration stop condition setting unit is configured to set an iteration stop condition;
in the present embodiment, the correlation update unit is used to update the correlation
In each iteration, an index i is selected, and a column index set s is obtained k Immediately followed by updating the total index estimate support
Updating dependencies
In this embodiment, the implicit SMP output module is configured to output an estimation sparse signal.
Example 3
The present embodiment provides a storage medium, which may be a storage medium such as a ROM, a RAM, a magnetic disk, an optical disk, or the like, and the storage medium stores one or more programs, and when the programs are executed by a processor, the binary sparse signal recovery method of the subtraction matching pursuit of embodiment 1 is implemented.
Example 4
The embodiment provides a computing device, which may be a desktop computer, a notebook computer, a smart phone, a PDA handheld terminal, a tablet computer, or other terminal devices with a display function, and the computing device includes a processor and a memory, where the memory stores one or more programs, and when the processor executes the programs stored in the memory, the binary sparse signal recovery method of subtraction matching pursuit according to embodiment 1 is implemented.
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.
Claims (10)
1. A binary sparse signal recovery method of subtraction matching pursuit is characterized by comprising the following steps:
performing binary sparse signal recovery based on explicit SMP or implicit SMP;
the specific steps of binary sparse signal recovery based on explicit SMP are as follows:
inputting a noise observation vector y, a perception matrix A and sparsity K;
initializing data, wherein the data initialization comprises initializing iteration times, estimating a support set, estimating sparse signals and residual vectors, and the noise observation vectors are used as initial values of the residual vectors;
setting iteration stop conditions, and selecting an index i in each iteration so that the sensing matrix A corresponds to a column vector A with the index i i And residual r k Most relevant, resulting in a set of column indices s k By subtracting the set of corresponding column indices of the perceptual matrix A as s k Column vector of
Updating the residual vector, and then entering next iteration until an iteration stop condition is met, and stopping the iteration;
outputting an estimated sparse signal;
the specific steps of performing binary sparse signal recovery based on the implicit SMP are as follows:
inputting a noise observation vector y, a perception matrix A and sparsity K;
initializing data, including initializing iteration number, estimating support and estimating sparse signal, and initializing correlation u 0 ＝A T y, wherein A T Represents the transpose of the perceptual matrix a;
setting iteration stop conditions, selecting an index i in each iteration, and acquiring a column index set s k Immediately followed by updating the total index estimate support
Updating dependencies
Then, entering the next iteration until an iteration stop condition is met, and stopping the iteration;
and outputting an estimated sparse signal.
2. The binary sparse signal recovery method for subtraction matching pursuit according to claim 1, wherein iteration termination conditions are all | | | r in explicit SMP and implicit SMP k || 2 ≦ e, e represents a given positive number.
3. The binary sparse signal recovery method for subtractive match tracking according to claim 1 wherein in explicit SMP, the specific steps of iteration comprise:
in each iteration, by the argmax function
To obtain a residual error r k-1 And each column projection coefficient of the perception matrix A
Maximum value of inner product absolute value, and forming column index set s by using the positions of the values corresponding to column elements of sensing matrix A k As a column index of the sensing matrix a, a specific calculation formula is:
after finding out the maximum index value corresponding to the perception matrix A, setting an estimated sparse signal
Corresponding set of column indices s k The value of the column index position is 1;
updating a total index estimate support
Will find the column index set s in each loop k And the column index estimation support which is recorded iteratively
Merging, recording all column index values and updating the total index estimation support set
Obtaining a complete index estimation support set;
residual r recorded by last iteration calculation k-1 Corresponding matrix to this index set
And updating the residual vector.
4. The subtractive match tracked binary sparse signal recovery method of claim 3, wherein the estimated sparse signal represents a total index estimate support
The estimated sparse signal with a value of 1 corresponding to the column index position is specifically represented as:
wherein,
representing the pseudo-inverse of the perceptual matrix a.
5. The subtractive match-tracked binary sparse signal recovery method of claim 1, wherein in implicit SMP, the total index estimation support is updated
The method comprises the following specific steps:
will find the column index set s in each loop k And the column index estimation support which is recorded iteratively
Merging, recording all column index values and updating the total index estimation support set
A complete index set is obtained.
6. The binary sparse signal recovery method of subtractive match tracking according to claim 1 wherein in implicit SMP, correlation is updated
The calculation method of (A) is as follows:
wherein,
estimating a complement of an ensemble for a total index of a perceptual matrix A
Corresponding to the transposition of the column vector, r k Is the residual error.
7. The binary sparse signal recovery method for subtraction matching pursuit according to claim 1, wherein in explicit SMP, based on RIP tight sufficiency condition, the sensing matrix A satisfies
RIP conditions of (1) and
the preconditions of (a);
based on the condition that the cross correlation is tight and sufficient, the perception matrix A meets the requirement
Cross correlation and
the preconditions of (a);
wherein,
μ denotes the cross-correlation between the A column vectors of the perceptual matrix, A i Representing a column vector indexed by the i-perception matrix A, A j The representation index is a j-column vector.
8. A binary sparse signal recovery system with subtraction matching pursuit is characterized by being provided with an explicit SMP signal recovery module or an implicit SMP signal recovery module;
the explicit SMP signal recovery module includes: the system comprises an explicit SMP data input unit, an explicit SMP data initialization unit, an explicit SMP iteration stop condition setting unit, a residual vector updating unit and an explicit SMP output module;
the explicit SMP data input unit is used for inputting a noise observation vector y, a perception matrix A and sparsity K;
the explicit SMP data initialization unit is used for initializing data, and comprises initialization iteration times, an estimation support set, an estimation sparse signal and a residual vector, and the noise observation vector is used as an initial value of the residual vector;
the explicit SMP iteration unit is used for loop iteration;
the explicit SMP iteration stop condition setting unit is used for setting an iteration stop condition;
the residual vector updating unit is used for updating the residual vector, and in each iteration, one index i is selected, so that the sensing matrix A corresponds to the column vector A with the index i i And residual r k Most relevant, resulting in a set of column indices s k By subtracting the sensing matrix A corresponding column index set as s k Column vector of
Updating a residual vector;
the explicit SMP output unit is used for outputting an estimation sparse signal;
the implicit SMP signal recovery module includes: an implicit SMP data input unit, an implicit SMP data initialization unit, an implicit SMP iteration stop condition setting unit, a correlation updating unit and an implicit SMP output module;
the implicit SMP data input unit is used for inputting a noise observation vector y, a perception matrix A and sparsity K;
the implicit SMP data initialization unit is used for data initialization, and comprises initialization iteration times, estimation support and estimation sparse signals, and initialization correlation u 0 ＝A T y, wherein A T Represents the transpose of the perceptual matrix a;
the implicit SMP iteration unit is used for loop iteration;
the implicit SMP iteration stop condition setting unit is used for setting an iteration stop condition;
the correlation updating unit is used for updating the correlation
In each iteration, an index i is selected, and a column index set s is obtained k Immediately followed by updating the total index estimate support
Thereby updating the correlation
The implicit SMP output module is used for outputting an estimated sparse signal.
9. A storage medium storing a program, wherein the program when executed by a processor implements the binary sparse signal recovery method of subtractive matching pursuit according to any one of claims 1 to 7.
10. A computing device comprising a processor and a memory for storing a processor-executable program, wherein the processor, when executing the program stored in the memory, implements the subtraction matching pursuit binary sparse signal recovery method of any of claims 1 to 7.
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