CN106597415B - The method of sparse aperture imaging system error-detecting precision is improved under a kind of Gaussian noise - Google Patents
The method of sparse aperture imaging system error-detecting precision is improved under a kind of Gaussian noise Download PDFInfo
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Abstract
本发明公开了一种高斯噪声下提高稀疏孔径成像系统误差检测精度的方法。以相位差异法中两幅图像的对数似然函数为基础，求得关于稀疏孔径成像系统的Fisher矩阵，通过对Fisher矩阵求逆矩阵的迹和统计分析得到评价因子；通过改变外界参数如焦平面图像的离焦量、焦平面与离焦平面接受到的光子数比来计算相应的评价因子，评价因子的值越小表明相位差异法采用这些参数检测稀疏孔径成像系统误差的精度就越高。本发明提供的评价方法可为选择一种最佳的相位差异法参数配置提供依据，有效提高稀疏孔径成像系统误差检测精度。
The invention discloses a method for improving the error detection accuracy of a sparse aperture imaging system under Gaussian noise. Based on the log-likelihood function of the two images in the phase difference method, the Fisher matrix of the sparse aperture imaging system is obtained, and the evaluation factor is obtained through the trace and statistical analysis of the inverse of the Fisher matrix. The defocus amount of the plane image and the ratio of the number of photons received by the focal plane and the defocus plane are used to calculate the corresponding evaluation factor. The smaller the value of the evaluation factor, the higher the accuracy of the phase difference method using these parameters to detect the error of the sparse aperture imaging system. . The evaluation method provided by the invention can provide a basis for selecting an optimal parameter configuration of the phase difference method, and effectively improve the error detection accuracy of the sparse aperture imaging system.
Description
Technical field
The present invention relates to a kind of methods for improving sparse aperture imaging system error-detecting precision, especially in Gaussian noise
The lower method for improving sparse aperture imaging system error-detecting precision.
Background technique
Sparse aperture imaging system is that permutation and combination together, is used to replace one big according to certain rules by multiple sub-apertures
Aperture area, since the entire heavy caliber of the relative aperture of each sub-aperture is much smaller, sparse aperture can not only overcome due to
A series of too big brought difficulties of optical system bore, and can obtain and the comparable spatial discrimination of wide-aperture optical system
Rate.Sparse aperture imaging system mostly two anti-system structure in practical application, principal reflection mirror is by each small sub- reflecting mirror institute
It constitutes.Good image quality in order to obtain, the phase error between each sub- reflecting mirror must be strictly controlled (usually less than λ/
20rms), this is not an easy thing for current processing, adjustment technique；In addition around sparse aperture system
Environment, temperature, the factors such as gravity also result in the generation of sub-aperture not systematic errors such as common phase position.
The method for being used to detect sparse aperture imaging system error at present mainly has phase difference method, and phase difference method is logical
It crosses and acquires unknown extension object by multiple image formed by optical system, according to these picture construction objective functions to optics
A kind of wavefront sensing methods that the error and unknown object object of system are estimated.The detection accuracy of phase difference method is limited by all
Multifactor, such as noise, the defocusing amount of focal plane image, the photon ratio between different images, wherein noise is most important shadow
The factor of sound.The photon numbers that the presence of thermal noise and each pixel of receiver receive when due to picture receiver work are once
If 20, receiver image obtained will will receive the influence of Gaussian noise.Therefore various outer under analysis Gaussian noise
Influence of boundary's parameter to sparse aperture imaging system error-detecting precision has great significance.
Document " Optical misalignment sensing and image reconstruction using
Phase diversity " ([J] J.Opt.Soc.Am.A, 1988,5 (6): 914-923) discloses a kind of utilization phase difference method survey
The method for measuring sparse aperture imaging system error, this method are dilute to 6 sub-apertures using the phase difference method based on gradient search method
The Piston error for dredging aperture is detected, and wherein the defocusing amount of focal plane image is 0.5 wavelength, and is analyzed when each
Testing result is by being influenced when the white Gaussian noise that width image is 1% plus variance.However document does not consider outside which kind of
Under the conditions of portion, when carrying out error-detecting to the system influenced with Gaussian noise, the result obtained is more accurate, does not propose
Any method come evaluate the system carry out error-detecting result precision size；Document " the phase difference method pair based on genetic algorithm
The piston error of synthesis telescope is detected " ([J] astronomical research and technology, 2011,8 (4): 369-373) disclosure
A kind of method measuring three sub-aperture sparse aperture imaging system errors using phase difference method, each width image equally has in document
Having variance is 1% Gaussian random white noise, and the defocusing amount of focal plane image is chosen for 1 wavelength, and document merely provides
Under this condition error-detecting as a result, not proposing any method equally to improve the precision of the error-detecting.
Summary of the invention
The present invention detects sparse aperture imaging system error when institute using phase difference method under Gaussian noise for existing
Existing deficiency provides a kind of method that can effectively judge and improve testing result precision.
It realizes that the technical solution of the object of the invention is to provide under a kind of Gaussian noise and improves sparse aperture imaging system error
The method of detection accuracy, including such as following steps:
(1) it is obtained under the influence of Gaussian noise using phase difference method, the logarithm about sparse aperture imaging system error
Likelihood function L:
Wherein, σ2For Gaussian noise variance；For object；It is sparse aperture imaging system to object institute
At picture；For image coordinates；Subscript d=1 indicates that the picture of focal plane, subscript d=2 indicate the picture of focal plane；For
The point spread function of sparse aperture imaging system, when subscript d=1,It is as corresponding sparse aperture is imaged for focal plane
It unites point spread function, when subscript d=2,It is focal plane as corresponding sparse aperture imaging system point spread function；*
For convolution operator；For the error of sparse aperture imaging system, by the error of each sub-apertureComposition,N is the sub-aperture quantity of sparse aperture imaging system；
Point spread functionFor sparse aperture imaging system generalized pupil function Fourier transformationMould
Square,Wherein, FT is Fourier transform operator；For n-th of sub-aperture
Generalized pupil function,For the pupil coordinate of sparse aperture imaging system；Generalized pupil functionIt is expressed from the next:
In formula,For the normalization light pupil function of n-th of sub-aperture of sparse aperture imaging system, phase termIt is indicated by zernike polynomial,For the jth item of zernike polynomial,αnjFor the jth item zernike coefficient of n-th of sub-aperture；Phase termThe defocusing amount for indicating focal plane image, for image focal plane,It is 0；
(2) using log-likelihood function L to the zernike polynomial system of sparse aperture imaging system any two sub-aperture
Several second-order partial differential coefficients, and negative desired value is taken to second-order partial differential coefficient solving result, obtain the sparse aperture imaging system
Fisher submatrix about any two sub-aperture:
Wherein, Fishermn(j, k) is about jth row in the Fisher submatrix of m-th of sub-aperture and n-th of sub-aperture
With kth column element, m=1,2 ... N n=1,2 ... N；KdFor as λdThe number of photons received；Im { } is to take imaginary part operator；On
Mark * is complex conjugate operator；onormFor normalized object；
(3) step (2) are repeated, obtains the Fisher matrix of the sparse aperture imaging system comprising whole sub-apertures
Fisher:
(4) it inverts to the Fisher matrix F isher that step (3) obtains, then to inverse matrix track taking:
ε=Trace (INV (Fisher)),
Wherein, Trace () indicates to take the mark of matrix, and INV () is indicated to matrix inversion；
(5) random using Karhunen-Loeve function expansion method simulation generation zernike polynomial M width under 1~J rank
Phase screen takes the order J value of zernike polynomial to be greater than 10, and random phase screen number M is not less than 50, obtains each width phase screen
The random error of corresponding one group of sparse aperture imaging systemIt repeats step (2)~(4) and calculates separately each groupObtained ε,
Obtain evaluation points
(6) change the defocusing amount of focal plane image respectivelyThe photon ratio that focal plane and focal plane receive
K1/K2, step (1)~(5) are repeated, obtain corresponding to different defocusing amountsAnd different focal planes and focal plane receive
Photon ratio K1/K2Corresponding evaluation points
(7) each evaluation points comparedValue, with evaluation pointsDefocusing amount corresponding to value minimumWith
Photon ratio K1/K2As external parameter, sparse aperture imaging system error is detected.
By adopting the above-described technical solution, the invention has the advantages that the evaluation points for passing through calculating sparse aperture systemThus optimal phase difference method measurement parameter configuration is provided, to improve the sparse aperture imaging system under Gaussian noise
Error-detecting precision.
Detailed description of the invention
Fig. 1 is the sparse aperture imaging system of Golay3 structure provided in an embodiment of the present invention；
Fig. 2 is the method that sparse aperture imaging system error-detecting precision is improved under a kind of Gaussian noise provided by the invention
Work flow diagram；
Fig. 3 is imageable target object provided in an embodiment of the present invention；
Fig. 4 be it is provided in an embodiment of the present invention by Golay3 sparse aperture imaging system to focal plane formed by object
Picture and focal plane picture；
Fig. 5 is the sparse aperture imaging system evaluation points being calculated under different defocusing amounts in the embodiment of the present invention
Size；
Fig. 6 is that system focal plane image and focal plane image obtain under different photon ratios in the embodiment of the present invention
Sparse aperture imaging system evaluation pointsResult.
Specific embodiment
With reference to the accompanying drawings and examples, technical solution of the present invention is further elaborated.
Embodiment 1
During using phase difference method detection sparse aperture imaging system error, due in the collection process of image
With Gaussian noise, therefore the information that each pixel receives on CCD is a Gaussian random variable, it is assumed that all variables
Between it is mutually indepedent, obeying mean value is 0, variance σ2Gaussian Profile, then CCD every piece image collectedIt is general
Rate density function can be indicated by formula (1) are as follows:
Wherein, σ2S Gaussian noise variance；For object；Subscript d=1 indicates that the picture of focal plane, subscript d=2 indicate
The picture of focal plane；For the point spread function of sparse aperture imaging system, when subscript d=1,For focal plane
As corresponding sparse aperture imaging system point spread function, when subscript d=2,It is focal plane as corresponding sparse hole
Diameter imaging system point spread function；For image coordinates；It * is convolution operator；For the error of sparse aperture imaging system, by each
The error of a sub-apertureComposition,N is the sub-aperture quantity of sparse aperture imaging system.
According to the optical basic principle of information, point spread function is sparse aperture imaging system pupil function Fourier transformationMould square,Wherein, FT is Fourier transform operator；
For the generalized pupil function of n-th of sub-aperture,For the pupil coordinate of sparse aperture imaging system.Generalized pupil functionIt is indicated by following formula (2):
In formula,For the normalization light pupil function of n-th of sub-aperture of sparse aperture imaging system, phase termIt is indicated by zernike polynomial,For the jth item of zernike polynomial,αnjFor the jth item zernike coefficient of n-th of sub-aperture, i.e. phase difference method
Ask for the error of the sparse aperture imaging system of detection；Phase termThe defocusing amount for indicating focal plane image, it is flat for coke
Face image,It is 0；
Above-mentioned probability density function formula (1) is handled using Maximum-likelihood estimation theory, and ignores unrelated constant
, formula (3) log-likelihood function L can be obtained:
By Cram é r-Rao lower bound theory it is found that asking two rank local derviations that can obtain log-likelihood function L
Fisher matrix.Log-likelihood function L is calculated to the zernike polynomial system of sparse aperture imaging system any two sub-aperture
Several second-order partial differential coefficients, and negative desired value is taken to second-order partial differential coefficient solving result, obtain the sparse aperture imaging system
Fisher submatrix about the two any sub-apertures is formula (4):
Wherein, Fishermn(j, k) is about jth row in the Fisher submatrix of m-th of sub-aperture and n-th of sub-aperture
With kth column element, m=1,2 ... N n=1,2 ... N.KdFor as λdThe number of photons received；Im { } is to take imaginary part operator；On
Mark * is complex conjugate operator；onormFor normalized object.
The Fisher matrix form (5) of the sparse aperture imaging system comprising whole sub-apertures can be obtained by repeating the above steps
Fisher:
It inverts again to obtained Fisher matrix F isher, and by formula (6) to inverse matrix track taking:
ε=Trace (INV (Fisher)) (6)
Wherein, Trace () indicates to take the mark of matrix, and INV () is indicated to matrix inversion；
It is simulated using Karhunen-Loeve function expansion method and generates zernike polynomial M width random phase under 1~J rank
Screen, the random error of the corresponding one group of sparse aperture imaging system of each width phase screenRepetitive operation is counted respectively according to the above method
Calculate each groupObtained ε, and statistical average obtains evaluation points
Change the defocusing amount of focal plane image respectivelyThe photon ratio K that focal plane and focal plane receive1/
K2, aforesaid operations step is repeated, obtains corresponding to different defocusing amountsAnd the light that different focal planes and focal plane receive
Subnumber ratio K1/K2Corresponding evaluation pointsEvaluation pointsValue it is smaller, utilize corresponding defocusing amountAnd number of photons
Compare K1/K2It is as a result more accurate as external parameter to sparse aperture error-detecting.
Referring to attached drawing 1, it is the sparse aperture imaging system of Golay3 structure provided in this embodiment.The broad sense light of system
Pupil functionA writeable accepted way of doing sth (7):
Wherein in the range of sub-aperture n,ForA is the area of three sub-apertures of Golay3 sparse aperture
The sum of；Outside the range of sub-aperture n,It is 0；
The aberration of n-th of sub-apertureIt is indicated with the 1st of zernike polynomial to the 11st:
Referring to attached drawing 2, it is that the inspection of sparse aperture imaging system error is improved under a kind of Gaussian noise provided in this embodiment
Survey the work flow diagram of the method for precision.
Step S101 according to fig. 2, the design parameter that Golay3 sparse aperture imaging system is arranged are as shown in table 1:
Table 1.Golay3 sparse aperture imaging system parameters
Referring to attached drawing 3, it is imageable target object provided in this embodiment；The present embodiment chooses sparse aperture imaging system simultaneously
It unites two error-detectings that width is carried out at image to object, referring to attached drawing 4, it is provided in this embodiment to pass through Golay3
Sparse aperture imaging system is to focal plane picture formed by object and focal plane picture；Wherein width (a) figure is focal plane picture, separately
An outer width (b) figure is defocused image, it is assumed that imaging system noise be variance be 1% Gauss additive white noise and.
Step S102 according to fig. 2, obtaining formula (9) using the data of this two width figure is log-likelihood function L
Wherein σ2It is 0.01, λ1And λ2Respectively correspond Fig. 4 (a) and Fig. 4 (b), and h1And h2Respectively correspond formula (10) and formula
(11) as follows:
Step S103 according to fig. 2 is obtained by Cram é r-Rao lower bound theory about m-th of sub-aperture and n-th
The Fisher submatrix Fisher of a sub-aperturemn, wherein m=1,2,3 n=1,2,3, submatrix FishermnMiddle jth row and
Kth column element Fishermn(j, k) is represented by formula (12):
KdFor as λdThe number of photons received, in the present embodiment total number of light photons, i.e. K1And K2The sum of be 10000.
Step S104 according to fig. 2 obtains the Fisher matrix of the sparse aperture imaging system comprising whole sub-apertures
Formula (13):
Step S105 according to fig. 2, the sparse aperture imaging system comprising whole sub-apertures that (13) formula is obtained
Fisher matrix inversion, then to inverse matrix track taking, can obtain
ε=Trace (INV (Fisher)),
It is simulated using Karhunen-Loeve function expansion method and generates zernike polynomial random phase of 50 width under 1~11 rank
Position screen, calculates each groupObtained ε carries out statistical average and obtains evaluation points
Step S106 according to fig. 2, in the present embodiment, by the main defocusing amount that focal plane image is discussed and focal plane
Influence of the photon ratio received respectively with focal plane image to systematic error detection accuracy.Assuming that two images receive altogether
The number of photons arrived is 10000.
Referring to attached drawing 5, it is the sparse aperture imaging system evaluation being calculated under different defocusing amounts in the present embodiment
The factorResult；It illustrate two width imaging subnumber ratio of sparse aperture imaging system be 1: 1, using image focal plane and from
Image focal plane calculates separately to obtain evaluation points in the case that focal plane image has different defocusing amountsResult.
When the defocusing amount of focal plane image is in 1.63 wavelength it can be seen from Fig. 5 result, evaluation pointsValue it is minimum, this
Show to carry out error-detecting under the defocusing amount, detection accuracy is higher than the error measurement under other defocusing amounts.
Referring to attached drawing 6, it be in the present embodiment system focal plane image and focal plane image under different photon ratios
Obtained sparse aperture imaging system evaluation pointsResult.Its expression sparse aperture imaging system is Jiao Ping in piece image
Face image, in addition piece image is focal plane image and under the premise of defocusing amount is 0.5 wavelength, and two images have difference
Photon ratio, thus the evaluation points being calculatedSize.As seen from Figure 6, with image focal plane and defocus
The reduction of flat image photon ratio between the two, evaluation pointsValue also reduce therewith, this show by reduce focal plane
Number of photons and increase the number of photons of focal plane, recycle this two images to detect sparse aperture imaging system error, inspection
The precision of survey is available significantly to be improved.
Claims (1)
1. under a kind of Gaussian noise improve sparse aperture imaging system error-detecting precision method, it is characterised in that including such as with
Lower step:
(1) it is obtained under the influence of Gaussian noise using phase difference method, the log-likelihood about sparse aperture imaging system error
Function L:
Wherein, σ2For Gaussian noise variance；For object；It is sparse aperture imaging system to formed by object
Picture；For image coordinates；Subscript d=1 indicates that the picture of focal plane, subscript d=2 indicate the picture of focal plane；It is sparse
The point spread function of aperture imaging system, when subscript d=1,It is focal plane as corresponding sparse aperture imaging system point
Spread function, when subscript d=2,It is focal plane as corresponding sparse aperture imaging system point spread function；It * is volume
Integrating symbol；For the error of sparse aperture imaging system, by the error of each sub-apertureComposition,N is the sub-aperture quantity of sparse aperture imaging system；
Point spread functionFor sparse aperture imaging system generalized pupil function Fourier transformationMould it is flat
Side,Wherein, FT is Fourier transform operator；For the wide of n-th sub-aperture
Adopted pupil function,For the pupil coordinate of sparse aperture imaging system；Generalized pupil functionIt is expressed from the next:
In formula,For the normalization light pupil function of n-th of sub-aperture of sparse aperture imaging system, phase termBy
Zernike polynomial expression, For the jth item of zernike polynomial,N=1,2 ... N, αnjFor the jth item zernike coefficient of n-th of sub-aperture；Phase term
The defocusing amount for indicating focal plane image, for image focal plane,It is 0；
(2) using log-likelihood function L to the zernike polynomial coefficient of sparse aperture imaging system any two sub-aperture
Second-order partial differential coefficient, and negative desired value is taken to second-order partial differential coefficient solving result, obtain the sparse aperture imaging system about
The Fisher submatrix of any two sub-aperture:
Wherein, Fishermn(j, k) is about jth row and kth in the Fisher submatrix of m-th of sub-aperture and n-th of sub-aperture
Column element, m=1,2 ... N n=1,2 ... N；KdFor as λdThe number of photons received；Im { } is to take imaginary part operator；Subscript * is multiple
Conjugate operator；onormFor normalized object；
(3) step (2) are repeated, obtain the Fisher matrix F isher of the sparse aperture imaging system comprising whole sub-apertures:
(4) it inverts to the Fisher matrix F isher that step (3) obtains, then to inverse matrix track taking:
ε=Trace (INV (Fisher)),
Wherein, Trace () indicates to take the mark of matrix, and INV () is indicated to matrix inversion；
(5) zernike polynomial M width random phase under 1~J rank is generated using the simulation of Karhunen-Loeve function expansion method
Screen takes the order J value of zernike polynomial to be greater than 10, and random phase screen number M is not less than 50, and it is corresponding to obtain each width phase screen
The random error of one group of sparse aperture imaging systemIt repeats step (2)~(4) and calculates separately each groupObtained ε, obtains
Evaluation points
(6) change the defocusing amount of focal plane image respectivelyThe photon ratio K that focal plane and focal plane receive1/
K2, step (1)~(5) are repeated, obtain corresponding to different defocusing amountsAnd the light that different focal planes and focal plane receive
Subnumber ratio K1/K2Corresponding evaluation points ε；
(7) each evaluation points comparedValue, with evaluation pointsDefocusing amount corresponding to value minimumAnd number of photons
Compare K1/K2As external parameter, sparse aperture imaging system error is detected.
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| CN108050937B (en) * | 2018-01-08 | 2019-07-16 | 浙江大学 | Method and device for detecting intermediate frequency error of large aperture optical element |
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