CN106533451B - A kind of stopping criterion for iteration setting method that block-sparse signal restores - Google Patents
A kind of stopping criterion for iteration setting method that block-sparse signal restores Download PDFInfo
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Abstract
The invention discloses the stopping criterion for iteration setting methods that a kind of block-sparse signal restores, it is characterized in that including: 1, measurement block-sparse signal；2, block-sparse signal recovery is carried out using iterative algorithm, determines the residual signals energy probability distribution of each iteration；3, a suitable threshold value is determined for front probability distribution, by the true residual signals energy of current iteration compared with threshold value, court verdict is that non-zero signal block is all detected or there are also non-zero signal blocks not to be detected；4, situation about being detected entirely for front judgement for non-zero signal block, stops iteration, exports the signal of recovery；Otherwise, continue iteration, carry out residual energy next time and adjudicate.The signal recovery problems unknown for degree of rarefication, the present invention can accurately estimate degree of rarefication, reduce unnecessary computing cost, improve block-sparse signal recovery accuracy.
Description
Technical field
The invention belongs to compressed sensing technology field, specifically a kind of block-sparse signal based on residual signals energy
The stopping criterion for iteration setting method of recovery.
Background technique
Compressed sensing technology has obtained extensive concern in past several years, in the processing of such as image video signal, communication
The every field such as signal processing, compressed sensing technology all become strong research and with tools.Compressive sensing theory expression,
It is zero when signal vector has sparsity i.e. many elements, when measurement can use the sample frequency lower than Nyquist, lead to
Cross certain method also can determine to recover in sampling system from deficient.What the work of compressed sensing early stage considered is general sparse
Signal, i.e. nonzero element it is random be distributed in all possible position of vector.It is sparse with block structure with the progress of research
Signal is also paid close attention to.Block-sparse signal be widely present in practical applications, as multi-wave signal, condition of sparse channel gain to
Amount, radar pulse signal, small data packets access etc..Block-sparse signal indicates, only certain when signal sequence is divided into multiple pieces
Block is non-zero.Existing research demonstrates, and considers that the sparse signal of block structure often has than general sparse signal
There is better signal recovery performance.
Wherein, when the sparse recovery of block, since the position of non-zero signal block and quantity are not aware that in advance, algorithm is needed to institute
There is signal to be estimated.It is extensive with block orthogonal matching pursuit (block orthogonal matching pursuit, BOMP) iteration
For double calculation method, BOMP, which needs to pass through relevant calculation in each iterative step and compare, detects most possible non-zero signal block
Position, then carries out signal update again and residual signals update, and when non-zero signal block is all detected, iteration terminates.Cause
This, it may be said that timely stop iteration and directly affects block-sparse signal restorability.
Secondly, the degree of rarefication of block-sparse signal i.e. the number of non-zero signal block, are one important in signal recovery
Parameter, especially for iterative algorithm.Existing work is generally acknowledged that degree of rarefication is known, to can control time of iteration
Number, stops iteration in time.And in practical applications, signal reset terminal does not know the information of degree of rarefication, and the estimation of degree of rarefication is with regard to lattice
It is outer important.If degree of rarefication estimation is too low, some important non-zero signal blocks can be missed；And if that estimates is too high, no
Necessary iteration will affect the performance of signal recovery, while increase the expense of calculating.In the communications field, small data packets access scene
The maximum value possible for considering known degree of rarefication, so that the number of iteration is often greater than true degree of rarefication.It is led in signal processing
Domain, existing (automatic double overrelaxation) ADORE thresholding algorithm can estimate degree of rarefication and progress simultaneously
Signal restores, however is limited to general sparse signal, considers then to be difficult to be applicable in when block structure and bigger signal dimension；Other
A few thing is then to stop iteration by setting termination condition, but these termination conditions generally both depend on empirical value, is lacked
The support of weary theory, is often only applicable to special scenes.
Summary of the invention
The present invention is to overcome the shortcomings of the prior art, proposes a kind of iteration ends that block-sparse signal restores
Condition setting method, to can solve the problem of iterative algorithm cannot stop in time so that signal to be restored degree of rarefication not
In the case where knowing, still accurate can estimate degree of rarefication and terminate iteration in time, thus reduce unnecessary computing cost,
It improves block-sparse signal and restores accuracy.
The present invention to achieve the above object of the invention, adopts the following technical scheme that
The characteristics of stopping criterion for iteration setting method that a kind of block-sparse signal of the present invention restores is to carry out as follows:
Step 1, signal sending end generate the block-sparse signal as composed by the block that N number of dimension is d × 1 I-th of dimension for indicating the block-sparse signal s is the transposition of the block of d × 1；By institute
The element for stating all non-zero signal blocks in block-sparse signal s is normalized to 0 mean value unit energy, 1≤i≤N；
Signal reset terminal carries out linear measurement to the block-sparse signal s with using formula (1), obtains measurement vector y:
Y=Bs+z (1)
In formula (1), B indicate a line number be M, the calculation matrix that columns is N × d=Nd, and have: B=[B1,B2,…,
Bi,…,BN], BiI-th of dimension for indicating the calculation matrix B is the submatrix of M × d, and the element in the calculation matrix B is equal
Obedience mean value is 0, variance isMultiple Gauss distribution；Z is the noise vector that dimension is M × 1, every in the noise vector z
A element obey mean value be 0, variance σ2Multiple Gauss distribution；
Step 2 carries out signal recovery to the measurement vector y using iterative algorithm:
Step 2.1, definition k are the number of iterations, define ΛkFor the indexed set institute for the block that preceding k iterative detection goes out
The indexed set for the block that the kth time iteration of composition needs to update；
Step 2.2, initialization k=1；
Step 2.3 carries out kth time detection to not detected non-zero signal blocks all in the block-sparse signal s, obtains
The indexed set of the block of kth time detection；- 1 iteration of the indexed set of the block of kth time detection and kth is needed more
The indexed set Λ of new blockk-1The indexed set Λ for the block for needing to update when composition kth time iterationk；
Step 2.4, using formula (2) to the indexed set ΛkIn the block-sparse signal s corresponding block into
Row least square updates, and obtains the updated block of kth time iteration
In formula (2),Indicate the indexed set ΛkThe corresponding submatrix in the calculation matrix B；It indicates
The indexed set ΛkThe conjugate transposition of corresponding submatrix in the calculation matrix B；
Step 2.5, the residual signals r that kth time iteration is obtained using formula (3)k:
Step 2.6, the residual signals r that the kth time iteration is obtained using formula (4)kENERGY Ek:
Step 2.7, the residual energy E that the kth time iteration is respectively obtained using formula (5) and formula (6)kMean μkWith side
Difference
In formula (5) and (6), nkIndicate the number for the non-zero signal block not detected also after kth time iteration；It is described to obtain
The residual signals r of kth time iterationkENERGY EkApproximate Gaussian distributed
Step 2.8 utilizes the thresholding η for controlling false dismissal probability of formula (7) setting kth time iterationk,1:
In formula (7), pmIt is maximum false dismissal probability allowed, Φ-1(pm) it is cumulative distribution letter in standardized normal distribution
Several independents variable is pmInverse function；
Step 2.9 utilizes the thresholding η for controlling probability of false detection of formula (8) setting kth time iterationk,0:
In formula (8), pfIt is maximum probability of false detection allowed, φ-1(pf) it is cumulative distribution letter in standardized normal distribution
Several independents variable is pfInverse function；
Step 2.10, the threshold value η that kth time iteration is finally set using formula (9)k:
ηk=min (ηk,1,ηk,0) (9)
Step 2.11, by the residual energy E of the kth time iterationkWith the threshold value η of the kth time iterationkCompare, if Ek
< ηk, then it represents that the non-zero signal block is all detected, and stops iteration, under the block that output kth time iteration updates
Mark set ΛkWith updated blockTo recover the signal of signal sending end；Otherwise, indicate that there is also be not detected
The non-zero signal block of survey, and after enabling k+1 be assigned to k, return step 2.3 sequentially executes.
The residual energy of iteration each in the sparse recovery of block has been carried out portraying for probability distribution by the present invention, and is based on this point
Cloth proposes a kind of stopping criterion for iteration that can control missing inspection and probability of false detection.Compared with the prior art, beneficial skill of the invention
Art effect is embodied in:
1, the present invention can determine one by the probability distribution of calculating residual energy within the scope of the false dismissal probability of permission
Highest threshold value only terminates iteration when residual energy is less than the thresholding, makes it possible to preferably control missing inspection non-zero signal
Block number, to ensure that the higher detection probability of non-zero signal block.
2, the present invention can determine one by the probability distribution of calculating residual energy within the scope of the probability of false detection of permission
Minimum threshold value, and when being used for stopping criterion for iteration, unnecessary the number of iterations is reduced, so that it is complicated to reduce calculating
Degree.
3, the present invention is not needed using block sparsity as priori knowledge, but the meter of theoretical property is carried out for residual energy
It calculates, then derives suitable termination thresholding, therefore the setting method of the termination condition is suitable for various pieces of sparse recovery iteration and calculates
Method.
Detailed description of the invention
Fig. 1 is to restore an analogous diagram in the number of iterations in block-sparse signal using the present invention；
Fig. 2 is to restore an analogous diagram in accurate performance in block-sparse signal using the present invention；
Fig. 3 is to restore an analogous diagram in non-zero signal block detection probability in block-sparse signal using the present invention.
Specific embodiment
In the present embodiment, block-sparse signal is widely existed in practical application, as small in image signal process, communication
Data packet accesses scene etc..The block-sparse signal recovery considered in the present embodiment comprises the following processes: the measurement of block-sparse signal,
Signal recovery, the setting and judgement of stopping criterion for iteration are carried out with iterative algorithm.Assuming that for the signal by N number of dimension for d × 1
Block-sparse signal composed by block, wherein only NaA block is the N of non-zeroa< < N, NaIteration is arranged in namely degree of rarefication
The purpose of termination condition is exactly to be correctly found NaThe position of a non-zero signal block and demodulate come after timely stop iteration.Tool
Body, the stopping criterion for iteration setting method that a kind of block-sparse signal restores is to carry out as follows:
Step 1, signal sending end generate the block-sparse signal as composed by the block that N number of dimension is d × 1 I-th of dimension for indicating the block-sparse signal s is the transposition of the block of d × 1；By institute
The element for stating all non-zero signal blocks in block-sparse signal s is normalized to 0 mean value unit energy, 1≤i≤N；
Signal reset terminal carries out linear measurement to the block-sparse signal s with using formula (1), obtains measurement vector y:
Y=Bs+z (1)
In formula (1), B indicate a line number be M, the calculation matrix that columns is N × d=Nd, and have: B=[B1,B2,…,
Bi,…,BN], BiI-th of dimension for indicating the calculation matrix B is the submatrix of M × d, and the element in the calculation matrix B is equal
Obedience mean value is 0, variance isMultiple Gauss distribution, that is, each column are normalized, the measurement square generated at random
Battle array substantially meets (restricted isometry property) RIP condition of compressive sensing theory；Z is that dimension is M × 1
Noise vector, each element in the noise vector z obey mean value be 0, variance σ2Multiple Gauss distribution；
Step 2 carries out signal recovery to the measurement vector y using iterative algorithm:
Step 2.1, definition k are the number of iterations, define ΛkFor the indexed set institute for the block that preceding k iterative detection goes out
The indexed set for the block that the kth time iteration of composition needs to update；
Step 2.2, initialization k=1；
Step 2.3 carries out kth time detection to not detected non-zero signal blocks all in the block-sparse signal s, obtains
The indexed set of the block of kth time detection；- 1 iteration of the indexed set of the block of kth time detection and kth is needed more
The indexed set Λ of new blockk-1The indexed set Λ for the block for needing to update when composition kth time iterationk；
Step 2.4, using formula (2) to the indexed set ΛkIn the block-sparse signal s corresponding block into
Row least square updates, and obtains the updated block of kth time iteration
In formula (2),Indicate the indexed set ΛkThe corresponding submatrix in the calculation matrix B；It indicates
The indexed set ΛkThe conjugate transposition of corresponding submatrix in the calculation matrix B；
Step 2.5, the residual signals r that kth time iteration is obtained using formula (3)k:
Step 2.6, the residual signals r that the kth time iteration is obtained using formula (4)kENERGY Ek:
Step 2.7, the residual energy E that the kth time iteration is respectively obtained using formula (5) and formula (6)kMean μkWith side
Difference
In formula (5) and (6), nkIndicate the number for the non-zero signal block not detected also after kth time iteration；It is described to obtain
The residual signals r of kth time iterationkENERGY EkApproximate Gaussian distributed
Step 2.8, in an iterative process, when the last one non-zero signal block is detected, the energy of residual signals
An apparent downward trend is often had, is based on the trend, it is believed that as residual energy EkWhen less than some thresholding, it can terminate
Iteration.Utilize the thresholding η of the control false dismissal probability of formula (7) setting kth time iterationk,1:
In formula (7), pmIt is maximum false dismissal probability allowed, Φ-1(pm) it is cumulative distribution letter in standardized normal distribution
Several independents variable is pmInverse function；Missing inspection refers to that certain non-zero signal blocks are not detected, and is judged to stand growth model
Situation；
Step 2.9 utilizes the thresholding η for controlling probability of false detection of formula (8) setting kth time iterationk,0:
In formula (8), pfIt is maximum probability of false detection allowed, Φ-1(pf) it is cumulative distribution letter in standardized normal distribution
Several independents variable is pfInverse function；Erroneous detection refers to that certain stand growth models are detected, and is judged to the non-zero signal block
The case where.
Step 2.10, the threshold value η that kth time iteration is finally set using formula (9)k:
ηk=min (ηk,1,ηk,0) (9)
The setting of thresholding herein is based on such criterion: the thresholding the low more is conducive to reduce false dismissal probability, and thresholding is too
It is low to will increase probability of false detection to a certain degree.In actual signal recovery, guarantee that non-zero signal block is detected as far as possible
Often more important, a small amount of unnecessary iteration is allowed.So in the case where comprehensively considering omission factor and false detection rate, finally
Thresholding select ηk,1And ηk,0In minimum value.On the other hand, threshold value set by each iteration has with current iteration number k
It closes, so threshold value can also change in different iteration.
Step 2.11, by the residual energy E of the kth time iterationkWith the threshold value η of the kth time iterationkCompare, if Ek
< ηk, then it represents that the non-zero signal block is all detected, and stops iteration, under the block that output kth time iteration updates
Mark set ΛkWith updated blockTo recover the signal of signal sending end；Otherwise, indicate that there is also be not detected
The non-zero signal block of survey, and after enabling k+1 be assigned to k, return step 2.3 sequentially executes.
The effect for restoring stopping criterion for iteration setting method the present invention is based on the block-sparse signal of residual signals energy can be with
It is showed by analogous diagram 1, Fig. 2 and Fig. 3.Wherein about the setting of simulation parameter: d=50, N=640, M=2000 measure square
The element of battle array and non-zero signal block is all generated by multiple Gauss variable, is obeyed respectivelyWith the distribution of CN (0,1).About
The threshold value η of settingkThe missing inspection of permission and probability of false detection are respectively as follows: pm=0.1%, pf=0.5%.Here using most common
BOMP iterative algorithm, one block of each iterative detection, the number of final iteration reflect the degree of rarefication of estimation
Introduce in analogous diagram and in harness comparison: condition one (Condition 1) is indicated when adjacent iteration twice
The iteration stopping when variation of reconstruction signal is less than empirical value；Condition two (Condition 2) is indicated when the energy of residual signals is small
Iteration stopping when noise vector energy.In figure, abscissa is that Signal to Noise Ratio (SNR) (is defined asUnit is dB), consider noise
It in 0~8dB and block sparsity is 12 and 20 two kind of situation than range.In Fig. 1, ordinate is the number of iterations (Iteration
Number), the present invention always can timely stop iteration, as the increase of signal-to-noise ratio the number of iterations of the invention is substantially equal to true
Real degree of rarefication；And condition one has erroneous detection block and a large amount of unnecessary iteration under low signal-to-noise ratio；Condition two
The number of iterations is always less than degree of rarefication, and part non-zero signal block is caused to be missed.In Fig. 2, ordinate is normalized mean square error
Poor (normalized MSE, normalized mean square error), the mean square error of the invention under low signal-to-noise ratio
Condition two can be slightly above, this is because, with the increase of signal-to-noise ratio, the present invention then has most caused by a small amount of erroneous detection block
Low mean square error.In Fig. 3, ordinate is the successful detection probability of non-zero signal block, it can be seen that the present invention can stop in time
Only iteration and on the basis of guaranteeing lower mean square error, also has both higher non-zero signal block detection probability.
Claims (1)
1. the stopping criterion for iteration setting method that a kind of block-sparse signal restores, it is characterized in that carrying out as follows:
Step 1, signal sending end generate the block-sparse signal as composed by the block that N number of dimension is d × 1 I-th of dimension for indicating the block-sparse signal s is the transposition of the block of d × 1；To own in the block-sparse signal s
The element of non-zero signal block is normalized to 0 mean value unit energy, 1≤i≤N；
Signal reset terminal carries out linear measurement to the block-sparse signal s with using formula (1), obtains measurement vector y:
Y=Bs+z (1)
In formula (1), B indicate a line number be M, the calculation matrix that columns is N × d=Nd, and have: B=[B1,B2,…,Bi,…,
BN], BiI-th of dimension for indicating the calculation matrix B is the submatrix of M × d, and the element in the calculation matrix B is obeyed
Value is that 0, variance isMultiple Gauss distribution；Z is the noise vector that dimension is M × 1, each element in the noise vector z
Obey mean value be 0, variance σ2Multiple Gauss distribution；
Step 2 carries out signal recovery to the measurement vector y using iterative algorithm:
Step 2.1, definition k are the number of iterations, define ΛkIt is made of the indexed set of the block of preceding k iterative detection out
The indexed set for the block that kth time iteration needs to update；
Step 2.2, initialization k=1；
Step 2.3 carries out kth time detection to not detected non-zero signal blocks all in the block-sparse signal s, obtains kth
The indexed set of the block of secondary detection；- 1 iteration of the indexed set of the block of kth time detection and kth is needed to update
The indexed set Λ of blockk-1The indexed set Λ for the block for needing to update when composition kth time iterationk；
Step 2.4, using formula (2) to the indexed set ΛkCorresponding block carries out most in the block-sparse signal s
Small two multiply update, obtain the updated block of kth time iteration
In formula (2),Indicate the indexed set ΛkThe corresponding submatrix in the calculation matrix B；Described in expression
Indexed set ΛkThe conjugate transposition of corresponding submatrix in the calculation matrix B；
Step 2.5, the residual signals r that kth time iteration is obtained using formula (3)k:
Step 2.6, the residual signals r that the kth time iteration is obtained using formula (4)kENERGY Ek:
Step 2.7, the residual energy E that the kth time iteration is respectively obtained using formula (5) and formula (6)kMean μkWith variance
In formula (5) and (6), nkIndicate the number for the non-zero signal block not detected also after kth time iteration；To obtain the kth time
The residual signals r of iterationkENERGY EkApproximate Gaussian distributed
Step 2.8 utilizes the thresholding η for controlling false dismissal probability of formula (7) setting kth time iterationk,1:
In formula (7), pmIt is maximum false dismissal probability allowed, Φ-1(pm) be in standardized normal distribution cumulative distribution function from
Variable is pmInverse function；
Step 2.9 utilizes the thresholding η for controlling probability of false detection of formula (8) setting kth time iterationk,0:
In formula (8), pfIt is maximum probability of false detection allowed, Φ-1(pf) be in standardized normal distribution cumulative distribution function from
Variable is pfInverse function；
Step 2.10, the threshold value η that kth time iteration is finally set using formula (9)k:
ηk=min (ηk,1,ηk,0) (9)
Step 2.11, by the residual energy E of the kth time iterationkWith the threshold value η of the kth time iterationkCompare, if Ek< ηk,
It then indicates that the non-zero signal block is all detected, and stops iteration, the indexed set for the block that output kth time iteration updates
ΛkWith updated blockTo recover the signal of signal sending end；Otherwise, indicate non-there is also what is be not detected
Zero-signal block, and after enabling k+1 be assigned to k, return step 2.3 sequentially executes.
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