CN115996060A - Anti-noise reconstruction method for compressed sensing system signal based on resolution reduction constraint - Google Patents
Anti-noise reconstruction method for compressed sensing system signal based on resolution reduction constraint Download PDFInfo
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Abstract
The invention discloses an anti-noise reconstruction method for a compressed sensing system signal based on resolution reduction constraint, which relates to the field of target signal observation by a compressed sensing observation system, solves the problems that the existing compressed sensing underdetermined system is sensitive to noise, effective signal reconstruction cannot be realized under noise interference, and the like.
Description
Technical Field
The invention relates to a compressed sensing observation system for observing a target signal and realizing effective reconstruction under the condition of noise interference.
Background
When the signal acquisition (observation) system acquires the target signal, the signal acquisition frequency or the observation frequency of the system is usually required to be more than or equal to the required target signal data volume; however, due to real factors such as time, space, system integration, etc., the system cannot achieve the required observation times, namely, undersampling the target signal. Aiming at the problem of undersampling a target signal, scientific researchers propose a solution based on a compressed sensing principle. The main idea of compressed sensing is to combine the observation and compression steps in the data acquisition process, typically to acquire signals at sub-nyquist rate (e.g. one fifth of the nyquist rate), and to reconstruct from these samples globally with accuracy and probability. In compressed sensing theory and application, sparsity is a prerequisite for achieving reconstruction, so that a signal with either of the following two conditions can achieve efficient reconstruction:
1. the signal itself is sparse, defined as a vector with most elements being zero, i.e. the number k of non-zero elements in the signal is much smaller than the number of zero elements in the signal, called k sparse signal.
2. The k elements of the signal may represent the overall energy of the system and may also be referred to as k sparse signals.
The compressed sensing system is mainly applied to the situation that an output signal is a target signal aliasing, namely, a clear linear relation exists between the output signal and the target signal, so that the compressed sensing system is characterized by discrete linearity in most cases, and after sparse transformation (dictionary) is introduced and sparsity constraint is increased, sparse vectors are calculated preferentially, and then the sparse vectors are combined with the dictionary to realize signal reconstruction.
The influence of input interference, electronic readout noise and thermal noise is unavoidable in the signal acquisition process. To complete signal calculation and reconstruction, the compressed sensing system needs to be calibrated to obtain the system response, and the calibration result of the system response contains uncertainty. Therefore, many interference factors are inevitably introduced into the reconstruction calculation, and the reconstruction accuracy will be affected. Based on the system type, the resolving stability influenced by noise is sequentially an overdetermined system, a positive definite system and an underdefinite system from high to low. In view of the fact that the compressed sensing system belongs to an underdetermined system, noise interference greatly influences signal reconstruction accuracy, and effective reconstruction of a target signal is more likely to be impossible.
Disclosure of Invention
The invention aims to solve the problems that the existing compressed sensing underdetermined system is sensitive to noise, effective signal reconstruction cannot be realized under noise interference, and the like. A method for noise-resistant reconstruction of a compressed sensing system signal based on a resolution reduction constraint is provided.
The anti-noise reconstruction method for the compressed sensing system signal based on the resolution reduction constraint comprises the following steps:
firstly, constructing a compressed sensing system signal acquisition model, linearly discretizing the model, and determining the data range and resolution of an observed target signal;
step two, performing response calibration of a high-resolution compressed sensing system to obtain a basic data source;
step three, determining the data range and resolution of the observed target signal according to the step one, and acquiring a basic data source to perform response resolution reduction calculation of the compressed sensing system;
fourth, the system response after different orders of resolution reduction is subjected to pathological analysis, and the conditions are combined: the system response is less than or equal to the number of times of observing the compressed sensing system after the resolution is reduced, and the resolution reduction system response is selected; drawing a system response condition number change curve changing along with the resolution;
step five, performing resolution reduction operation on the system response by utilizing the resolution selected in the step four, after obtaining the low-resolution system response, performing low-resolution target signal reconstruction by adopting a regularization method, and analyzing the mathematical relationship between the low-resolution target signal and the high-resolution target signal;
step six, introducing the mathematical relationship between the low-resolution target signal and the high-resolution target signal in the step five into the reconstruction calculation of the high-resolution signal, restricting the sparse reconstruction iteration process, and reducing the reconstruction deviation of the high-resolution signal.
The invention has the beneficial effects that:
the invention solves the problem that effective signal reconstruction cannot be realized because the compressed sensing system is sensitive to noise interference, carries out pathological analysis and selection on the response of the reconstruction system under variable resolution, preferentially solves the low-resolution high-precision target signal, and constrains the compressed sensing-the reconstruction of the high-resolution target signal by the low-resolution reconstruction signal, thereby increasing the anti-interference performance of the compressed sensing signal reconstruction system, and realizing the compressed sensing signal reconstruction of anti-noise interference by the self system without introducing additional environmental conditions.
Drawings
FIG. 1 is a schematic diagram of a compressed sensing signal acquisition system in a method for noise-proof reconstruction of a compressed sensing system signal based on a resolution reduction constraint according to the present invention;
FIG. 2 is a schematic diagram of a compressed sensing signal reconstruction structure based on a resolution reduction constraint according to the present invention;
FIG. 3 is a diagram of a reduced resolution compressed sensing system response selection method according to the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
As shown in fig. 1, the compressed sensing (underdetermined) system is one of effective methods for achieving high-resolution signal acquisition under limited observation times, but the system itself lacks anti-interference capability, and the positive or overdetermined system has anti-interference capability and cannot perform high-resolution signal solution under limited observation times. The relation between the reconstruction and the compressed sensing reconstruction of the positive or overdetermined system is analyzed, and the relation is introduced into the compressed sensing signal reconstruction, so that the anti-interference capability is improved for the compressed sensing-high resolution signal reconstruction system, namely the compressed sensing signal reconstruction based on the resolution reduction constraint, and the accuracy of the signal reconstruction is further ensured.
Referring to fig. 2, a method for compressing a perception system signal against noise reconstruction based on a resolution reduction constraint includes:
observation value g obtained by mth observation of target signal by compressed sensing system m Can be expressed as:
∫ x Ω m (x)φ(x)dx＝g m
discretizing by adopting rectangular mean integral, the formula is changed into:
wherein Δx represents the signal resolution of the current compressed sensing system, and if the compressed sensing system performs M observations on the target signal, the signal is expressed as a matrix:
Ωφ＝g
wherein the method comprises the steps of
The observation matrix is obtained after the calibration of the compressed sensing system, wherein M is the observation times of the compressed sensing system, N is the data length of the signal after the discretization, and the number of times is +.>
Discretizing the result for the target signal, < >>
Is an observed value output by the target signal after passing through the compressed sensing system. The anti-noise reconstruction method for the compressed sensing system signal based on the resolution reduction constraint specifically comprises the following steps:
1. determining the data range [ x ] in which the target signal phi is located start ，x end ]Resolution Deltax 0 ，x start For starting point of target signal, x end For the end point of the target signal, the resolution of the needed compressed sensing system response is the same as the resolution of the target signal, and the system response is expressed as
And M < N.
2. Performing compressed sensing system calibration to obtain system response, wherein resolution obtained by the system calibration is required to be delta x c Satisfy Deltax 0 ＝Δx c /t，t∈Z + Namely, the system is calibrated with higher resolution to ensure the accuracy of the calculation of the resolution, and the response of the calibrated system is as follows:
3. performing resolution reduction calculation of the compressed sensing system to enable the resolution delta x to be low k ＝kΔx 0 ，k∈Z + Wherein the data amount after constraint and resolution reduction is less than or equal to the compressed sensing observation times, namely (x) end -x start )/Δx k And M is less than or equal to. The current system is changed from an underdetermined system for solving high resolution and large data volume to a positive or overdetermined system for solving low resolution and small data volume. Low resolution of deltax k System response of (2)
The calculation method is that
Wherein x is j And x i Discrete points obtained under different resolutions respectively;
wherein the method comprises the steps of
j is E Z, and x can be obtained by the same way i Is a calculation method of (a).
4. And carrying out resolution-variable system response pathological analysis, carrying out multi-value resolution-reducing calculation based on resolution-reducing calculation, constructing corresponding resolution-reducing system response, calculating the number of system response conditions after resolution reduction, and drawing a system response condition number change curve changing along with resolution.
5. The smaller condition number means the stronger the anti-noise interference capability of the current reconstruction system; thus, referring to FIG. 3, the resolution Δx in the curve at the low value inflection point is selected s Performing resolution-reducing operation to obtain corresponding system response omega s As an environment for implementing constraint, the signal reconstruction under the current low resolution is performed by using a Gihonov regularization method, so that the objective function of the data reconstruction after resolution reduction is as follows:
the low resolution signal reconstruction solution is:
in the method, in the process of the invention,
the observation value is output after the target signal passes through the compressed sensing system; i is an identity matrix, mu is a weight coefficient, and the solution is carried out by an L-cut or generalized cross validation method.
6. For reconstructing the compressed sensing target signal, because the data quantity of the target signal is larger than the observation times, a sparse transformation (i.e. dictionary) mode is adopted to add constraint to the current underdetermined system, and the dictionary is selected
Let->
ΩDy＝g subject to:Dy＝φ
Wherein, the liquid crystal display device comprises a liquid crystal display device,
for k sparse vectors (k<<L), the dictionary D is used for mapping the original signal data into a new data domain, so that the mapped data y has sparsity, and after the y is obtained by calculation, a target signal phi can be obtained by calculation according to the selected dictionary D.
7. The mathematical relationship between the reconstructed reduced resolution signal and the required high resolution signal is:
this relationship is introduced into the reconstruction calculation of the sparse representation of the target signal phi, at which time the target function of the system is written:
min||y|| 1
wherein ε is 0 For l calculated after systematic error analysis 2 A norm; epsilon s Absolute value of maximum relative error discretized by different resolutions; for the objective function described above, a range of functions including, but not limited to: and solving and calculating based on a convex optimization and constraint matching pursuit method, so as to obtain a more accurate approximate solution.
The technical features of the above-described embodiments may be arbitrarily combined, and all possible combinations of the technical features in the above-described embodiments are not described for brevity of description, however, as long as there is no contradiction between the combinations of the technical features, they should be considered as the scope of the description.
The above examples illustrate only a few embodiments of the invention, which are described in detail and are not to be construed as limiting the scope of the invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention. Accordingly, the scope of protection of the present invention is to be determined by the appended claims.
Claims (8)
1. The anti-noise reconstruction method for the compressed sensing system signal based on the resolution constraint is characterized by comprising the following steps of: the method is realized by the following steps:
firstly, constructing a compressed sensing system signal acquisition model, linearly discretizing the model, and determining the data range and resolution of an observed target signal;
step two, performing response calibration of a high-resolution compressed sensing system to obtain a basic data source;
step three, determining the data range and resolution of the observed target signal according to the step one, and acquiring a basic data source to perform response resolution reduction calculation of the compressed sensing system;
fourth, the system response after different orders of resolution reduction is subjected to pathological analysis, and the conditions are combined: the system response is less than or equal to the number of times of observing the compressed sensing system after the resolution is reduced, and the resolution reduction system response is selected; drawing a system response condition number change curve changing along with the resolution;
step five, performing resolution reduction operation on the system response by utilizing the resolution selected in the step four, after obtaining the low-resolution system response, performing low-resolution target signal reconstruction by adopting a regularization method, and analyzing the mathematical relationship between the low-resolution target signal and the high-resolution target signal;
step six, introducing the mathematical relationship between the low-resolution target signal and the high-resolution target signal in the step five into the reconstruction calculation of the high-resolution signal, restricting the sparse reconstruction iteration process, and reducing the reconstruction deviation of the high-resolution signal.
2. The reduced resolution constraint-based compressed sensing system signal anti-noise reconstruction method of claim 1, wherein: in step one, the data range [ x ] of the observed target signal phi is determined start ，x end ]Resolution Deltax 0 The resolution of the needed compressed sensing system response is the same as the resolution of the observed target signal, and the system response is that
M is less than N; m is the number of observations of the compressed sensing system, and N is the data length of the signal after the dispersion.
3. The reduced resolution constraint-based compressed sensing system signal anti-noise reconstruction method of claim 2, wherein: in the second step, obtaining the basic data source includes obtaining a resolution of Δx required to calibrate the response of the compressed sensing system c Satisfy Deltax 0 ＝Δx c /t，t∈Z + And the system response obtained through calibration is as follows:
4. a compressed sensing system signal anti-noise reconstruction method based on a resolution reduction constraint according to claim 3, wherein: the specific process of the third step is as follows:
setting a low resolution Deltax k ＝kΔx 0 ，k∈Z + Wherein the data amount after constraint and resolution reduction is less than or equal to the compressed sensing observation times M, namely (x) end -x start )/Δx k ≤M；
Low resolution of deltax k System response of (2)
The calculation method comprises the following steps:
wherein x is j And x i Discrete points obtained at different resolutions, respectively.
5. The method for noise-resistant reconstruction of a compressed sensing system signal based on a resolution reduction constraint of claim 4, wherein: in the fourth step, a system response condition number change curve with the change of the resolution is drawn, and the data volume after the resolution reduction operation is required to be smaller than or equal to the compressed sensing observation times, so that the resolution deltax at the inflection point of a low value in the curve is selected s Corresponding system responseShould omega s 。
6. The reduced resolution constraint-based compressed sensing system signal anti-noise reconstruction method of claim 5, wherein: using resolution Deltax s Performing resolution reduction calculation on the system response to obtain a corresponding system response omega s As a low-resolution target signal reconstruction condition, performing low-resolution target signal reconstruction by using a Gihonov regularization method, and reconstructing a target signal after resolution reduction, wherein the target function is as follows:
the low resolution signal reconstruction solution is:
in the method, in the process of the invention,
the observation value is output after the target signal passes through the compressed sensing system; i is an identity matrix; mu is a weight coefficient, and the solution is carried out by an L-cut or generalized cross validation method.
7. The reduced resolution constraint-based compressed sensing system signal anti-noise reconstruction method of claim 1, wherein: in the sixth step, the mathematical relationship between the low resolution target signal obtained by reconstruction after resolution reduction and the target signal with the required resolution is used as constraint, and is introduced into the reconstruction of the target signal with the required resolution, and the obtained constraint formula is as follows:
min||y|| 1
wherein ε 0 For l calculated after systematic error analysis 2 A norm; epsilon s Absolute value of maximum relative error discretized by different resolutions; taking the constraint as an objective function to realize the reconstruction of the compressed sensing high-resolution target signal with anti-noise interference;
in the method, in the process of the invention,
and k is a sparse vector, and D is a dictionary.
8. The reduced resolution constraint-based compressed sensing system signal anti-noise reconstruction method of claim 7, wherein: the objective function is solved and calculated by a method based on convex optimization or constraint matching pursuit.
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