CN106597415A - Method for improving error detection precision of sparse aperture imaging system under Gaussian noise - Google Patents

Abstract

The invention discloses a method for improving error detection precision of a sparse aperture imaging system under Gaussian noise. The method comprises the steps of: acquiring a Fisher matrix about the sparse aperture imaging system based on a log-likelihood function of two images in a phase difference method, and acquiring evaluation factors through inversion of the Fisher matrix, acquisition of a trace of the inversed matrix and statistical analysis; and calculating corresponding evaluation factors through changing external parameters such as defocusing amount of a focal plane image and the number of photons received by a focal plane and a defocused plane, wherein the smaller the values of the evaluation factors are, the higher the precision of detecting errors of the sparse aperture imaging system by using the parameters when the phase difference method is adopted. The method provided by the invention can provide a basis for selecting the optimal phase difference method parameter configuration, and effectively improves the error detection precision of the sparse aperture imaging system.

Description

A kind of method that sparse aperture imaging system error-detecting precision is improved under Gaussian noise

Technical field

The present invention relates to a kind of method 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 technology

Sparse aperture imaging system be by multiple sub-apertures according to certain rules permutation and combination together, for replace one it is big Aperture area, due to the whole heavy caliber of the relative aperture of each sub-aperture it is much smaller, therefore sparse aperture can not only overcome due to A series of too big the brought difficulties of optical system bore, and can obtain and the suitable spatial discrimination of wide-aperture optical system Rate.Sparse aperture imaging system mostly is two anti-system architectures in practical application, and principal reflection mirror is by each little sub- speculum institute Constitute.For the picture matter for obtaining, the phase error between each sub- speculum must be strictly controlled (typically smaller than λ/ 20rms), this is not an easy thing for current processing, for debuging technique；In addition around sparse aperture system Environment, temperature, the factor such as gravity also results in the generation of the sub-aperture not systematic error such as common phase position.

It is used at present detecting that the method for sparse aperture imaging system error mainly has phase difference method, phase difference method is logical Cross and gather unknown extension object through multiple image formed by optical system, according to these picture construction object functions to optics A kind of wavefront sensing methods that the error and unknown object thing of system is estimated.The detection accuracy of phase difference method is limited by all Multifactor, such as noise, the defocusing amount of focal plane image, the wherein photon ratio between different images, noise are topmost shadow The factor of sound.The photon numbers that the presence of thermal noise and receiver each pixel are received when being worked due to picture receiver are once If 20, the image that receiver is obtained will be affected by Gaussian noise.Therefore it is various outer under analysis Gaussian noise Impact 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) disclose a kind of using phase difference method survey The method of amount sparse aperture imaging system error, the method is dilute to 6 sub-apertures using the phase difference method based on gradient search method The Piston errors in thin aperture are detected that wherein the defocusing amount of focal plane image is 0.5 wavelength, and are analyzed when each What testing result was subject to when width image adds the white Gaussian noise that upside deviation is 1% affects.But document is not considered outside which kind of Under the conditions of portion, when carrying out error-detecting to the system affected with Gaussian noise, the result for drawing is more accurate, does not propose Any method carries out the result precision size of error-detecting evaluating the system；Document " the phase difference method pair based on genetic algorithm The piston errors of synthesis telescope are detected " ([J] astronomical research and technology, 2011,8 (4)：It is 369-373) open A kind of method that utilization phase difference method measures three sub-aperture sparse aperture imaging system errors, each width image equally has in document The defocusing amount for having the Gaussian random white noise that variance is 1%, focal plane image is chosen for 1 wavelength, and document is merely provided Under this condition the result of error-detecting, does not equally propose any method to improve the precision of the error-detecting.

The content of the invention

The present invention detects sparse aperture imaging system error when institute under Gaussian noise for existing using phase difference method A kind of deficiency of presence, there is provided method that can effectively judge and improve testing result precision.

The technical scheme for realizing the object of the invention is to provide raising sparse aperture imaging system error under a kind of Gaussian noise The method of accuracy of detection, including such as following steps：

(1) obtained under the influence of Gaussian noise using phase difference method, with regard to the logarithm of sparse aperture imaging system error Likelihood function L：

Wherein, σ²For Gaussian noise variance；For object；For sparse aperture imaging system to object institute into Picture；For image coordinates；Subscript d=1 represents the picture of focal plane, and subscript d=2 represents the picture of focal plane；For dilute The point spread function of thin aperture imaging system, during subscript d=1,It is focal plane as corresponding sparse aperture imaging system Point spread function, during subscript d=2,It is focal plane as corresponding sparse aperture imaging system point spread function；* it is 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 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 sub-aperture of sparse aperture imaging system, phase termRepresented by zernike polynomial,For the jth item of zernike polynomial,α_njFor the jth item zernike coefficient of n-th sub-aperture；Phase term The defocusing amount of focal plane image is represented, for image focal plane,For 0；

(2) the zernike polynomial system using log-likelihood function L to 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 With regard to the Fisher submatrixs of any two sub-aperture：

Wherein, Fisher_mn(j, k) is jth row in Fisher submatrixs with regard to m-th sub-aperture and n-th sub-aperture With kth column element, m=1,2 ... N n=1,2 ... N；K_dIt is as λ_dThe number of photons for receiving；Im { } is to take imaginary part operator；On Mark * is complex conjugate operator；o_normFor normalized object；

(3) repeat step (2), obtain the Fisher matrixes of the sparse aperture imaging system comprising whole sub-apertures Fisher：

(4) Fisher matrix Fs isher that step (3) is obtained are inverted, then to inverse matrix track taking：

ε=Trace (INV (Fisher)),

Wherein, Trace () represents the mark for taking matrix, and INV () is represented to matrix inversion；

(5) zernike polynomial M width under 1～J ranks is produced using the simulation of Karhunen-Loeve function expansion methods random Phase screen, takes the exponent number J values of zernike polynomial more than 10, and random phase screen number M is not less than 50, obtains each width phase screen The random error of one group of sparse aperture imaging system of correspondenceRepeat step (2)～(4) calculate respectively each groupThe ε for obtaining, Obtain evaluation points

(6) defocusing amount of focal plane image is changed respectivelyThe photon ratio that focal plane and focal plane are received K₁/K₂, repeat step (1)～(5) obtain correspondence difference defocusing amountAnd different focal planes and focal plane are received Photon ratio K₁/K₂Corresponding evaluation points

(7) each evaluation points for obtaining are comparedValue, with evaluation pointsDefocusing amount corresponding to value minimumWith Photon ratio K₁/K₂As external parameter, sparse aperture imaging system error is detected.

As a result of above-mentioned technical proposal, it is an advantage of the invention that：By the evaluation points for calculating sparse aperture systemThus optimal phase difference method measurement parameter configuration is provided, is missed so as to improve the sparse aperture imaging system under Gaussian noise Difference accuracy of detection.

Description of the drawings

Fig. 1 is the sparse aperture imaging system of Golay3 structures 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 that the present invention is provided Workflow diagram；

Fig. 3 is imageable target thing provided in an embodiment of the present invention；

Fig. 4 be it is provided in an embodiment of the present invention by Golay3 sparse apertures imaging system to focal plane formed by object Picture and focal plane picture；

Fig. 5 be in the embodiment of the present invention under different defocusing amounts calculated sparse aperture imaging system evaluation points Size；

Fig. 6 is that system focal plane image and focal plane image are obtained 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 gatherer process of image With Gaussian noise, therefore the information that each pixel is received on CCD is a Gaussian random variable, it is assumed that all of variable Between it is separate, obey average be 0, variance is σ²Gaussian Profile, then every piece image that CCD is gatheredIts is general Rate density function can be expressed as by formula (1)：

Wherein, σ²S Gaussian noise variances；For object；Subscript d=1 represents the picture of focal plane, and subscript d=2 is represented The picture of focal plane；For the point spread function of sparse aperture imaging system, during subscript d=1,For focal plane As corresponding sparse aperture imaging system point spread function, during subscript d=2,It is focal plane as corresponding sparse hole Footpath 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 individual sub-apertureComposition,N is the sub-aperture quantity of sparse aperture imaging system.

According to the optical general 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 sub-aperture,For the pupil coordinate of sparse aperture imaging system.Generalized pupil functionRepresented by following formula (2)：

In formula,For the normalization light pupil function of n-th sub-aperture of sparse aperture imaging system, phase termRepresented by zernike polynomial,For the jth item of zernike polynomial,α_njFor the jth item zernike coefficient of n-th sub-aperture, i.e. phase difference method Ask for the error of the sparse aperture imaging system of detection；Phase termThe defocusing amount of focal plane image is represented, it is flat for burnt Face image,For 0；

Above-mentioned probability density function formula (1) is processed using Maximum-likelihood estimation theory, and ignores unrelated constant , formula (3) log-likelihood function L can be obtained：

It is theoretical from Cram é r-Rao lower bound, obtain by seeking log-likelihood function L two rank local derviations Fisher matrixes.Calculate zernike polynomial systems of the log-likelihood function L to 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 It is formula (4) with regard to the Fisher submatrixs of the two any sub-apertures：

Wherein, Fisher_mn(j, k) is jth row in Fisher submatrixs with regard to m-th sub-aperture and n-th sub-aperture With kth column element, m=1,2 ... N n=1,2 ... N.K_dIt is as λ_dThe number of photons for receiving；Im { } is to take imaginary part operator；On Mark * is complex conjugate operator；o_normFor normalized object.

Repeat the above steps can obtain the Fisher matrix forms (5) of the sparse aperture imaging system comprising whole sub-apertures Fisher：

Again Fisher matrix Fs isher to obtaining are inverted, and by formula (6) to inverse matrix track taking：

ε=Trace (INV (Fisher)) (6)

Wherein, Trace () represents the mark for taking matrix, and INV () is represented to matrix inversion；

Zernike polynomial M width random phase under 1～J ranks is produced using the simulation of Karhunen-Loeve function expansion methods Screen, the random error of one group of sparse aperture imaging system of each width phase screen correspondenceRepeat as stated above and count respectively Calculate each groupThe ε for obtaining, and statistical average draws evaluation points

Change the defocusing amount of focal plane image respectivelyPhoton ratio K that focal plane and focal plane are received₁/ K₂, repeat aforesaid operations step, obtain correspondence difference defocusing amountAnd the light that different focal planes and focal plane are received Subnumber compares K₁/K₂Corresponding evaluation pointsEvaluation pointsValue it is less, using 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 accompanying drawing 1, it is the sparse aperture imaging system of the Golay3 structures that the present embodiment is provided.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 apertures Sum；Outside the scope of sub-aperture n,For 0；

The aberration of n-th sub-apertureRepresented to the 11st with the 1st of zernike polynomial：

Referring to accompanying drawing 2, it is that the inspection of sparse aperture imaging system error is improved under a kind of Gaussian noise that the present embodiment is provided Survey the workflow diagram of the method for precision.

According to Fig. 2 steps S101, the design parameter for arranging Golay3 sparse aperture imaging systems is as shown in table 1：

Table 1.Golay3 sparse aperture imaging system parameters

Referring to accompanying drawing 3, it is the imageable target thing that the present embodiment is provided；The present embodiment chooses sparse aperture imaging system simultaneously Two width to object of uniting carry out error-detecting into image, and referring to accompanying drawing 4, it is the present embodiment offer by 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 Gauss additive white noise that variance is 1% and.

According to Fig. 2 steps S102, formula (9) is obtained for log-likelihood function L using the data of this two width figure

Wherein σ²For 0.01, λ₁And λ₂Correspond to Fig. 4 (a) and Fig. 4 (b) respectively, and h₁And h₂Difference corresponding (10) and formula (11) it is as follows：

According to Fig. 2 steps S103, obtained with regard to m-th sub-aperture and n-th by Cram é r-Rao lower bound theories The Fisher submatrix Fisher of individual sub-aperture_mn, wherein, m=1,2,3 n=1,2,3, submatrix Fisher_mnMiddle jth row and Kth column element Fisher_mn(j, k) is represented by formula (12)：

K_dIt is as λ_dThe number of photons for receiving, in the present embodiment total number of light photons, i.e. K₁And K₂Sum is 10000.

According to Fig. 2 steps S104, the Fisher matrixes of the sparse aperture imaging system comprising whole sub-apertures are obtained Formula (13)：

According to Fig. 2 steps S105, the sparse aperture imaging system comprising whole sub-apertures that (13) formula is obtained Fisher matrix inversions, then to inverse matrix track taking, can obtain

ε=Trace (INV (Fisher)),

Zernike polynomial random phase of 50 width under 1～11 rank is produced using the simulation of Karhunen-Loeve function expansion methods Position screen, calculates each groupThe ε for obtaining, carries out statistical average and draws evaluation points

According to Fig. 2 steps S106, in the present embodiment, by the main defocusing amount that focal plane image is discussed and focal plane Impact of the photon ratio received respectively with focal plane image to systematic error accuracy of detection.Assume that two width images receive altogether The number of photons for arriving is 10000.

Referring to accompanying drawing 5, it is the calculated sparse aperture imaging system evaluation under different defocusing amounts in the present embodiment The factorResult；It illustrates the width imaging subnumber ratio of sparse aperture imaging system two for 1: 1, using image focal plane and from Image focal plane, in the case that focal plane image has different defocusing amounts, is calculated respectively evaluation pointsResult. Can be seen that when the defocusing amount of focal plane image is in 1.63 wavelength by Fig. 5 results, evaluation pointsValue it is minimum, this Show to carry out error-detecting under the defocusing amount, accuracy of detection is higher than the error measurement under other defocusing amounts.

Referring to accompanying drawing 6, it be in the present embodiment system focal plane image and focal plane image under different photon ratios The sparse aperture imaging system evaluation points for obtainingResult.It represents that sparse aperture imaging system is burnt flat in piece image Face image, in addition piece image is that two width images have difference on the premise of focal plane image and defocusing amount are 0.5 wavelength Photon ratio, so as to calculated evaluation pointsSize.As seen from Figure 6, with image focal plane and out of focus The reduction of plane picture photon ratio therebetween, evaluation pointsValue also reduce therewith, this show by reduce focal plane Number of photons and increase focal plane number of photons, recycle this two width image to detect sparse aperture imaging system error, examine The precision of survey can be improved significantly.

Claims (1)

1. under a kind of Gaussian noise improve sparse aperture imaging system error-detecting precision method, it is characterised in that include such as with Lower step：

(1) obtained under the influence of Gaussian noise using phase difference method, with regard to the log-likelihood of sparse aperture imaging system error Function L：

$L=\frac{1}{2{\mathrm{\σ}}^2}\underset{d}{\Σ}\underset{x}{\Σ}{\[{\mathrm{\λ}}_d\left(\stackrel{\→}{x}\right)-o\left(\stackrel{\→}{x}\right)*h_d(\stackrel{\→}{x},\stackrel{\→}{\mathrm{\α}})\]}^2,$

Wherein, σ²For Gaussian noise variance；For object；It is sparse aperture imaging system to formed by object Picture；For image coordinates；Subscript d=1 represents the picture of focal plane, and subscript d=2 represents the picture of focal plane；For sparse The point spread function of aperture imaging system, during subscript d=1,For focal plane as corresponding sparse aperture imaging system point expands Scattered function, during subscript d=2,It is focal plane as corresponding sparse aperture imaging system point spread function；* it is convolution Operator；For the error of sparse aperture imaging system, by the error of each sub-apertureComposition,N For 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 sub-aperture of sparse aperture imaging system, phase termBy Ze Nike Polynomial repressentation, For the jth item of zernike polynomial, N=1,2 ... N, α_njFor the jth item zernike coefficient of n-th sub-aperture；Phase termRepresent the out of focus of focal plane image Amount, for image focal plane,For 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 with regard to The Fisher submatrixs of any two sub-aperture：

$\begin{array}{c}{\mathrm{Fisher}}_{mn}(j,k)=\underset{d}{\Σ}\underset{x}{\Σ}\frac{4{K_d}^2}{{\mathrm{\σ}}^2}\{\[o_{norm}*\mathrm{Im}\{FT\left\{g_{dm}(\stackrel{\→}{u},{\stackrel{\→}{\mathrm{\α}}}_m)Z_j\right\}{G}_d^{*}(\stackrel{\→}{x},\stackrel{\→}{\mathrm{\α}})\}\]\\ \×\[o_{norm}*\mathrm{Im}\left\{FT\right\{g_{dn}(\stackrel{\→}{u},{\stackrel{\→}{\mathrm{\α}}}_n)Z_k\left\}{G}_d^{*}(\stackrel{\→}{x},\stackrel{\→}{\mathrm{\α}})\right\}\]\end{array},$

Wherein, Fisher_mn(j, k) is jth row and kth in Fisher submatrixs with regard to m-th sub-aperture and n-th sub-aperture Column element, m=1,2 ... N n=1,2 ... N；K_dIt is as λ_dThe number of photons for receiving；Im { } is to take imaginary part operator；Subscript * is multiple Conjugate operator；o_normFor normalized object；

(3) repeat step (2), obtain Fisher matrix Fs isher of the sparse aperture imaging system comprising whole sub-apertures：

$Fisher=\left[\begin{array}{cccc}{\mathrm{Fisher}}_{11}& {\mathrm{Fisher}}_{12}& \mathrm{...}& {\mathrm{Fisher}}_{1n}\\ {\mathrm{Fisher}}_{21}& {\mathrm{Fisher}}_{22}& \mathrm{...}& {\mathrm{Fisher}}_{2n}\\ .& .& & .\\ .& .& & .\\ {\mathrm{Fisher}}_{n1}& {\mathrm{Fisher}}_{n2}& \mathrm{...}& {\mathrm{Fisher}}_{nn}\end{array}\right],$

(4) Fisher matrix Fs isher that step (3) is obtained are inverted, then to inverse matrix track taking：

ε=Trace (INV (Fisher)),

Wherein, Trace () represents the mark for taking matrix, and INV () is represented to matrix inversion；

(5) zernike polynomial M width random phase under 1～J ranks is produced using the simulation of Karhunen-Loeve function expansion methods Screen, takes the exponent number J values of zernike polynomial more than 10, and random phase screen number M is not less than 50, obtains each width phase screen correspondence The random error of one group of sparse aperture imaging systemRepeat step (2)～(4) calculate respectively each groupThe ε for obtaining, obtains Evaluation points

$\stackrel{\‾}{\mathrm{\ϵ}}=\mathrm{\ϵ}/M;$

(6) defocusing amount of focal plane image is changed respectivelyPhoton ratio K that focal plane and focal plane are received₁/ K₂, repeat step (1)～(5) obtain correspondence difference defocusing amountAnd the light that different focal planes and focal plane are received Subnumber compares K₁/K₂Corresponding evaluation points ε；

(7) each evaluation points for obtaining are comparedValue, with evaluation pointsDefocusing amount corresponding to value minimumAnd number of photons Compare K₁/K₂As external parameter, sparse aperture imaging system error is detected.