CN104410423B - Formula genetic iteration reconstructing method is recalled in compressed sensing - Google Patents
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Abstract
The invention discloses formula genetic iteration reconstructing method is recalled in a kind of compressed sensing.The method includes: firstly, initializing the supported collections of sparse signal to be asked, and then by genetic manipulations such as duplication, multiple point crossover, selection, Big mutation rate processing, loop iteration approaches the optimal location information of required sparse signal, and carries out the update supported collection of backtracking formula.The amplitude information that each nonzero element of sparse signal to be asked finally is obtained using least square method projection, completes signal reconstruction.The present invention under conditions of degree of rarefication is unknown, can Accurate Reconstruction go out sparse signal to be asked.
Description
Technical field
The present invention relates to a kind of signal reconfiguring methods, belong to signal processing technology field.
Background technique
Nyquist sampling theorem points out that sampling rate must reach twice of signal bandwidth or more could Accurate Reconstruction letter
Number.Then, although the data sampled in this manner can complete representation original signal, there are biggish for sample value data
Redundancy.Therefore this method acquisition data generally require to carry out compression processing to save memory space.Compressed sensing (Compressed
Sensing, CS) it is also referred to as compression sampling or sparse sampling, breach the signal sampling reason of conventional Nyquist sampling thheorem
By.The theory points out, as long as signal or be sparse on some transform domain, then one and sparse basis not phase can be used
The higher-dimension sparse signal is projected on a lower dimensional space and completes sparse signal compression by the calculation matrix of pass.Restructuring procedure are as follows:
Decompression restores higher-dimension sparse signal from the projection value of low-dimensional.Theoretically, as long as projection contains enough letters of reconstruction signal
Breath can high probability, high-precision restore original signal.Therefore, CS can be widely used in Medical Image Processing, remote sensing figure
As processing, wireless sensor network and exploitation of image capture device etc..
Compressed sensing mainly includes the restructing algorithm of the sparse transformation of signal, the design of calculation matrix and signal.Tradition
Restructing algorithm mainly have: greedy algorithm and base tracing algorithm.Greedy algorithm is mainly the x that original signal is obtained by iteration
Supported collection mainly includes match tracing (Matching pursuit, MP), orthogonal matching pursuit (Or-thogonal
Matching pursuit, OMP) etc., greedy algorithm was obtained extensively with rebuilding the advantage that speed is fast, method for reconstructing is simple in engineering
General application.Base tracing algorithm mainly includes subspace tracking (Basis pursuit, SP), gradient tracking (Gradient
Pursuit, GP) etc..MP and OMP algorithm requires condition of the degree of rarefication ruler as Accurate Reconstruction, however believes in practical applications
Number degree of rarefication ruler be usually unknown.In addition, MP and OMP algorithm idea concentrates on match tracing, need effective son empty
Between track and expand, once atom to be selected enters candidate supported collection, then permanent addition not will be deleted, and cause the atom of mistake can not
It rejects, lacks the thought of " backtracking ".SP closes although being added to supported collection each time with new atom when time supported collection
There is no carry out quantitative assessment (the i.e. shadow of increase or deletion atom pair error redundancy to using solution calculated by least square
Ring), while can not provide and guarantee when time iteration residual error is centainly less than the theoretical of previous iteration residual error.Lacking based on these algorithms
Point causes reconstructed error larger, becomes the key technology difficulty of signal reconstruction urgent need to resolve.
Summary of the invention
The present invention needs known degree of rarefication K and subspace tracing algorithm can not be to support to solve conventional greedy restructing algorithm
It concentrates the atom that newly adds to be judged whether preferably problem, proposes in a kind of compressed sensing and recall formula genetic iteration reconstruct side
Method, process and greedy restructing algorithm process contrary.Genetic algorithm is highly effective for solving the np problem in Combinatorial Optimization, therefore
The thought for recalling formula genetic algorithm can be applied to the reconstruction of compressed sensing.Basic process are as follows: constructed by optimization processing
Preferable initial support collection；It is replicated again, multiple point crossover, selection, Big mutation rate genetic manipulation successive ignition are approached wait ask sparse
The optimal location information of signal is finally projected the amplitude information, that is, final reconstruction result for obtaining each position by least square method；Its
In by save eliminate atom carry out multiple point crossover operation, then in supported collection carry out multiple point crossover operation after atom residual error
Compare, recalls the update supported collection of formula, the loss of optimal solution can be prevented in this way.The method of the present invention can be unknown in degree of rarefication K, and
It does not need efficiently reconstruct final sparse result under conditions of the tracking of subspace.
The invention proposes formula genetic iteration reconstructing method is recalled in compressed sensing, comprising the following steps:
Step 1: required sparse signal θ, which is equivalent to chromosome, carries out population setting, i.e. population is equivalent to required sparse
The supported collection ψ of signal θ, chromosome are the atom of supported collection ψ, input measurement value y, random Gaussian calculation matrix Φ and sparse change
Change matrixSupported collection ψ, the process that encoding scheme initializes supported collection, specific steps are initialized by encoding scheme again
At least further include:
1) required sparse signal θ is equivalent to chromosome and carries out population setting, is i.e. population is equivalent to required sparse signal θ
Supported collection ψ, chromosome be supported collection ψ atom；
2) assuming that supported collection is that N is arranged, the matrix of M dimension, N indicates that chromosome number, M indicate the gene number in every chromosome,
Input measurement value y, random Gaussian calculation matrix Φ ∈ RN×MWith sparse transformation matrix
3) sparse signalWhereinFor random Gaussian calculation matrix and sparse transformation Matrix Multiplication
Long-pending transposition, x1Supported collection is ψ, and the element being not zero in the atom of supported collection ψ is set 1, obtains initial support collection and is represented by ψM×N
={ θ1, θ2..., θN}；
Step 2: being judged whether by the residual error size and sort ascending that calculate atom in supported collection ψ according to residual error size
Atom in supported collection ψ is eliminated into the storehouse S of setting and is saved, if satisfied, then carry out eliminating operation, it is otherwise, atom is straight
It taps into crossover operation；Judge to be directly entered in the atom of crossover operation whether by atom progress copying and saving operation, if full again
Foot, then carry out duplication operation, otherwise, without other operations；
Step 3: in supported collection ψA atom carries out random pair two-by-two and multiple point crossover operates, and by
Newly generated atom replaces the atom of pairing, in storehouse SA atom also carries out random pair two-by-two and multiple spot is handed over
Fork operation, and by the atom of newly generated atom replacement pairing, the process of panmixia and multiple point crossover, is at least also wrapped two-by-two
It includes:
1) two atoms in supported collection ψ are randomly choosed and are matched, storehouse S also carries out random pair operation；
2) crossover probability p is setc, by crossover probability pcDeterminingA atom carries out random pair and multiple point crossover
Operation；
3) the random binary character string τ that a length is M is generated respectively1And τ2, τ1It is matched with two in supported collection ψ
Atom is corresponding, τ2It is corresponding with the atom of two pairings in storehouse S；
If 4) τ1、τ2Middle corresponding position element is 1, then otherwise the corresponding position commutative element for matching atom does not exchange；
5) atom of first wife couple is replaced by the new atom that pairing generates；
Step 4: expanding supported collection ψ by the atom that duplication operation saves, it is residual to calculate separately atom in supported collection ψ and storehouse S
The size and sort ascending of difference, then compare and judge whether to update supported collection；
Step 5: being to carry out cataclysmic mutation or common mutation operation, then judge whether according to Big mutation rate condition criterion
Reach maximum genetic algebra, if reaching, skip to step 6, otherwise, skips to step 2 and carry out loop iteration；
Step 6: projecting the amplitude information for obtaining nonzero element in sparse result by least square method.
Compared with the prior art, the advantages of the present invention are as follows:
It 1, need not precognition degree of rarefication K.
2, by comparing the method for the size of atom residual error in storehouse and supported collection, reach the update supported collection of backtracking formula.
3, locally optimal solution will not be fallen into, globally optimal solution can be preferably found.
Detailed description of the invention
Fig. 1 is the flow chart that formula genetic iteration reconstructing method is recalled in compressed sensing of the present invention
Specific embodiment
According to an aspect of the present invention, as shown in Figure 1, the specific embodiment of the invention such as following steps:
Step 1: setting population and encoding scheme:
1) sparse signal θ to be asked is equivalent to chromosome and carries out population setting, is i.e. population is equivalent to required sparse signal θ
Supported collection ψ, chromosome be supported collection ψ atom；
2) assuming that supported collection is that N is arranged, the matrix of M dimension, N indicates that chromosome number, M indicate the gene number in every chromosome,
Input measurement value y, random Gaussian calculation matrix Φ ∈ RN×MWith sparse transformation matrix
3) sparse signalWhereinFor random Gaussian calculation matrix and sparse transformation Matrix Multiplication
Long-pending transposition, x1Supported collection is ψ, and the element being not zero in the atom of supported collection ψ is set 1, obtains initial support collection and is represented by ψM×N
{θ1, θ2..., θN}。
Step 2: duplication:
1) by residual error functionCalculate atom residual error F in supported collection ψiAnd sort ascending, wherein i=
1,2 ..., N；
2) generation gap value GAP:GAP ∈ (0.8,1) is set, judges whether atom residual error is greater thanIf more than will then wash in a pan
It eliminatesA biggish atom of residual error is saved into the storehouse S of setting, otherwise, is directly entered multiple point crossover operation；
Whether the residual error of atom, which is less than, is judged to the atom for being directly entered multiple point crossover operationIt is multiple if being less than
SystemA atom is to atom collection K, otherwise, without other operations.
Step 3: multiple point crossover:
1) two atoms in supported collection ψ are randomly choosed and are matched, storehouse S also carries out random pair operation；
2) crossover probability p is setc, by crossover probability pcDeterminingA atom carries out random pair and multiple point crossover
Operation；
3) the random binary character string τ that a length is M is generated respectively1And τ2, τ1It is matched with two in supported collection ψ
Atom is corresponding, τ2It is corresponding with the atom of two pairings in storehouse S；
If 4) τ1、τ2Middle corresponding position element is 1, then otherwise the corresponding position commutative element for matching atom does not exchange；
5) atom of pairing is replaced by the new atom that pairing generates.
Step 4: selection:
1) the atom collection K of duplication is merged into supported collection ψ；
2) the residual error size and sort ascending of atom in supported collection ψ are calculated, atom also carries out residual computations and passs in storehouse S
Increase sorting operation；
3) enabling the smallest atom of residual error in storehouse S is k, and the maximum atom of residual error is m in supported collection ψ, if k is less than m, k
It is exchanged with each other with m, otherwise, does not exchange, this process of repetitive operation is both greater than supported until when the residual error of all atoms in storehouse S
Collect the residual error of the maximum atom of residual error in ψ.
Step 5: cataclysmic mutation:
Cataclysmic mutation can prevent the supported collection of iteration from falling into " precocity " phenomenon (i.e. when algorithm proceeds to certain iteration
When, the residual error of some atom m is far smaller than the residual error of other any atoms in supported collection, is replicated because atom is selected
Probability byFormula determines that many atoms that will result in next supported collection in this way come from the same atom m,
To approximate each other, the limiting case of this phenomenon is exactly all atoms from same previous atom)；Detailed process are as follows:
1) intensive factor a:a ∈ (0.5,1), Big mutation rate Probability p are setbigWith common mutation probability pm, and calculate this time repeatedly
The least residual F of the atom of supported collection ψ in generationminWith mean residual Favg；
If 2) meet condition a × F of Big mutation ratemin< Favg, thenWhereinHave for the t times iteration
Atoms all in this supported collection are set as the shape with least residual atom by the atom of least residual, t=0,1,2 ..., n
Formula, then to each element on atom in supported collection ψ, generate a random chance pkIf pkLess than Big mutation rate Probability pbig
Inversion operation is then carried out, otherwise, is not changed；
If 3) be unsatisfactory for condition a × F of Big mutation ratemin< Favg, then to each element on atom in supported collection ψ,
Generate a random chance pkIf pkLess than common mutation probability pm, then inversion operation is carried out, otherwise, is not changed.
Step 6: the amplitude of each nonzero element determines in sparse result:
1) maximum genetic algebra is set as MAXGEN, if reaching MAXGEN loop iteration, can converge to optimal atom,
That is otherwise the optimal solution of sparse result skips to step 2；Wherein optimal atom is still made of " 1 " and " 0 ", wherein " 1 " indicates former
Beginning sparse signal is nonzero element, and " 0 " indicates that original sparse signal is neutral element；
2) on the basis of each nonzero element position information has determined in sparse result, using least square method each
Position is projected to determine its amplitude information；Assuming that there is a nonzero element in sparse result on the position q, then the non-zero entry
The amplitude p of element are as follows:Wherein TqIndicate the q column of T,<>indicates inner product operation；Restore matrix
Claims (5)
1. recalling formula genetic iteration reconstructing method in compressed sensing, it is characterised in that the supported collection in compressed sensing to be equivalent to lose
Population in propagation algorithm makes the optimal support of initial support collection Step wise approximation by duplication, intersection, selection and cataclysmic mutation
Collection is final reconstruction result, and the method at least includes the following steps:
Step 1: required sparse signal θ, which is equivalent to chromosome, carries out population setting, i.e. population is equivalent to required sparse signal
The supported collection ψ of θ, chromosome are the atom of supported collection ψ, input measurement value y, random Gaussian calculation matrix Φ and sparse transformation square
Battle arraySupported collection ψ, the process initialized by encoding scheme to supported collection, specific steps are initialized by encoding scheme again
At least further include:
1) required sparse signal θ is equivalent to chromosome and carries out population setting, is i.e. the population branch that is equivalent to required sparse signal θ
Support collection ψ, chromosome are the atom of supported collection ψ；
2) assume that supported collection is that N is arranged, the matrix of M dimension, N indicates that chromosome number, M indicate the gene number in every chromosome, inputs
Measured value y, random Gaussian calculation matrix Φ ∈ RN×MWith sparse transformation matrix
3) sparse signalWhereinFor random Gaussian calculation matrix and sparse transformation matrix product
Transposition, x1Supported collection is ψ, and the element being not zero in the atom of supported collection ψ is set 1, obtains initial support collection and is represented by ψM×N=
{θ1, θ2..., θN}；
Step 2: being judged whether according to residual error size to branch by the residual error size and sort ascending that calculate atom in supported collection ψ
Atom in support collection ψ is eliminated into the storehouse S of setting and is saved, if satisfied, then carry out eliminating operation, otherwise, by atom directly into
Enter crossover operation；Judge to be directly entered in the atom of crossover operation whether by atom progress copying and saving operation, if satisfied, then again
Duplication operation is carried out, otherwise, without other operations；
Step 3: in supported collection ψA atom carries out random pair two-by-two and multiple point crossover operation, and by newly producing
Raw atom replaces the atom of pairing, in storehouse SA atom also carries out random pair two-by-two and multiple point crossover behaviour
Make, and replace the atom of pairing by newly generated atom, two-by-two the process of panmixia and multiple point crossover, at least further include:
1) two atoms in supported collection ψ are randomly choosed and are matched, storehouse S also carries out random pair operation；
2) crossover probability p is setc, by crossover probability pcDeterminingA atom carries out random pair and multiple point crossover behaviour
Make；
3) the random binary character string τ that a length is M is generated respectively1And τ2, τ1With the atom of two pairings in supported collection ψ
It is corresponding, τ2It is corresponding with the atom of two pairings in storehouse S；
If 4) τ1、τ2Middle corresponding position element is 1, then otherwise the corresponding position commutative element for matching atom does not exchange；
5) atom of first wife couple is replaced by the new atom that pairing generates；
Step 4: expanding supported collection ψ by the atom that duplication operation saves, atom residual error in supported collection ψ and heap Zhan S is calculated separately
Size and sort ascending, then compare and judge whether to update supported collection；
Step 5: being to carry out cataclysmic mutation or common mutation operation, then judge whether to reach according to Big mutation rate condition criterion
Maximum genetic algebra skips to step 6 if reaching, and otherwise, skips to step 2 and carries out loop iteration；
Step 6: projecting the amplitude information for obtaining nonzero element in sparse result by least square method.
2. recalling formula genetic iteration reconstructing method in compressed sensing according to claim 1, it is characterised in that initial branch
The process that atom in support collection ψ is handled, specific steps are at least further include:
1) by residual error functionCalculate atom residual error F in supported collection ψiAnd sort ascending, wherein i=1,
2 ..., N；
2) generation gap value GAP:GAP ∈ (0.8,1) is set, judges whether atom residual error is greater thanIf more than will then eliminateA biggish atom of residual error is saved into the storehouse S of setting, otherwise, is directly entered multiple point crossover operation；
3) whether the residual error of atom, which is less than, is judged to the atom for being directly entered multiple point crossover operationIt is replicated if being less thanA atom is to atom collection K, otherwise, without other operations.
3. recalling formula genetic iteration reconstructing method in compressed sensing according to claim 1, it is characterised in that supported collection ψ
The process supplemented and updated, specific steps are at least further include:
1) the atom collection K of duplication is merged into supported collection ψ；
2) the residual error size and sort ascending of atom in supported collection ψ are calculated, atom also carries out residual computations and is incremented by row in storehouse S
Sequence operation；
3) enabling the smallest atom of residual error in storehouse S is k, and the maximum atom of residual error is m in supported collection ψ, if k is less than m, k and m phase
It is interchangeable, it otherwise, does not exchange, this process of repetitive operation, until the residual error of all atoms in Dang Dui Zhan S is both greater than in supported collection ψ
The residual error of the maximum atom of residual error.
4. recalling formula genetic iteration reconstructing method in compressed sensing according to claim 1, it is characterised in that judge whether
Meet Big mutation rate condition, then definitive variation action type；Specific steps are at least further include:
1) intensive factor a:a ∈ (0.5,1), Big mutation rate Probability p are setbigWith common mutation probability pm, and calculate in this iteration
The least residual F of the atom of supported collection ψminWith mean residual Favg；
If 2) meet condition a × F of Big mutation ratemin< Favg, thenWhereinThere is Minimum Residual for the t times iteration
Atoms all in this supported collection are set as the form with least residual atom by the atom of difference, t=0,1,2 ..., n, then
To each element on atom in supported collection ψ, a random chance p is generatedkIf pkLess than Big mutation rate Probability pbigThen carry out
Otherwise inversion operation does not change；
If 3) be unsatisfactory for condition a × F of Big mutation ratemin< Favg, then to each element on atom in supported collection ψ, generate
One random chance pkIf pkLess than common mutation probability pm, then inversion operation is carried out, otherwise, is not changed.
5. recalling formula genetic iteration reconstructing method in compressed sensing according to claim 1, it is characterised in that pass through heredity
Heredity processing in algorithm obtains the position of nonzero element and amplitude information in supported collection, and specific steps are at least further include:
1) it sets maximum genetic algebra as MAXGEN, if reaching MAXGEN loop iteration, optimal atom can be converged to, i.e., it is dilute
Otherwise the optimal solution for dredging result skips to step 2；Wherein optimal atom is still made of " 1 " and " 0 ", wherein " 1 " indicates original dilute
Thin signal is nonzero element, and " 0 " indicates that original sparse signal is neutral element；
2) in sparse result each nonzero element position information oneself through determination on the basis of, at various locations using least square method
It projects to determine its amplitude information；Assuming that there is a nonzero element in sparse result on the position q, then the nonzero element
Amplitude p are as follows:Wherein TqIndicate the q column of T,<>indicates inner product operation；Restore matrix
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