CN114088348B - Multi-directional slope and curvature mixed wavefront reconstruction method for high-order truncation error - Google Patents

Multi-directional slope and curvature mixed wavefront reconstruction method for high-order truncation error Download PDF

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CN114088348B
CN114088348B CN202111026969.2A CN202111026969A CN114088348B CN 114088348 B CN114088348 B CN 114088348B CN 202111026969 A CN202111026969 A CN 202111026969A CN 114088348 B CN114088348 B CN 114088348B
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reconstruction
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curvature
slope
wavefront
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刘克
钟慧
李艳秋
秦鹏
何飞
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a multidirectional slope and curvature mixed wavefront reconstruction method for high-order truncation errors, which is characterized in that slope and curvature measurement values in additional directions are added to each grid point of a region method model, besides the conventional vertical and horizontal directions, the relationship among a wavefront value, a slope value and a mixed curvature value in any other direction is considered to construct an integral equation, a Taylor expansion is utilized to solve a coefficient, and a multidirectional mixed reconstruction equation with smaller truncation errors is deduced to reconstruct wavefronts; compared with the existing mixed wavefront reconstruction method, the method solves by using the least square method, has high calculation speed, and can meet the application requirements of real-time operation such as adaptive optics and the like; the truncation error is small, and the reconstruction precision is improved; and has strong anti-noise capability.

Description

Multi-directional slope and curvature mixed wavefront reconstruction method for high-order truncation error
Technical Field
The invention belongs to the technical field of optical measurement, and particularly relates to a multi-directional slope and curvature mixed wavefront reconstruction method for high-order truncation errors.
Background
The slope type wavefront sensing technology calculates wavefront aberration by measuring the slope value of the wavefront (the first derivative of the wavefront) using a reconstruction algorithm. The Hartmann-shack wavefront sensor is a typical example of slope type wavefront sensing technology and has wide application in the fields of adaptive optical systems, ophthalmology, optical elements, system aberration detection and the like. In recent years, a new type of wavefront sensing technology of a slope and curvature mixed type has been proposed. Because the sensing technology simultaneously measures the slope and the curvature value of the wavefront, more wavefront information can be obtained compared with a single slope type or curvature type wavefront sensing technology, so that the wavefront aberration detection with higher precision can be realized, and the method is a development direction of the high-precision aberration detection technology.
The hybrid wavefront reconstruction algorithm can utilize wavefront phase information, including a torsional curvature term, characterized by slopes, curvatures within individual sub-apertures, and hybrid curvature values (i.e., all first and second order information of the wavefront). There are a few studies on hybrid wavefront reconstruction algorithms. In 2011, Barwick (Opt Commun, 2011,284(8): 2099-. In 2017, M.Viegers (Opt.Express, 2017,25(10): 11514-. In 2018, Fan Feng (Opt. Eng., 2018,57(7):074106-1-074106-10) introduced an improved approach to incorporate the defocus into a moment-based approach, where first and second order origin moments can be used to reconstruct the wavefront, but the amplitude limits for medium and high order aberrations are more severe and the higher the order, the smaller the range over which the amplitude can be detected.
Although the above methods all utilize the first-order and second-order derivative information of the wavefront, the methods have the problems of large truncation error or weak noise immunity, and the measurement noise is inevitable in practical situations, so a novel hybrid wavefront reconstruction method is needed to improve the reconstruction accuracy and the noise immunity.
Disclosure of Invention
In view of this, the invention provides a multi-directional slope and curvature mixed wavefront reconstruction method for a high-order truncation error, which can improve the wavefront reconstruction accuracy and the anti-noise capability, thereby realizing the precise measurement of high-order aberration.
The technical scheme for realizing the invention is as follows:
the method for reconstructing the multi-directional slope and curvature mixed wavefront of the high-order truncation error comprises the following steps of:
adding slope and curvature measurement values in any horizontal and vertical directions between 0 DEG and 180 DEG at each grid point based on a regional method Southwell model;
constructing an integral equation by using the relation among a horizontal and vertical wave front value, a slope value and a mixed curvature value in any direction between 0 and 180 degrees;
thirdly, obtaining a solution about the integral equation coefficient by using Taylor expansion;
step four, substituting the coefficient solved in the step three into the integral equation in the step two to obtain a reconstruction equation, and expressing the reconstruction equation as AW (equal to H) by using a matrix;
wherein, A is 2(N-1) 2 ×N 2 W is the wavefront value to be measured, and the size is N 2 X 1 column vector, H slope curvature value of magnitude 2(N-1) 2 A column vector of x 1;
step five, calculating the values of A and H, namely recovering the wave front value to be measured as W ═ A T A) + A T H；
Wherein A is T Is the transpose of A, (A) T A) + Is A T A generalized inverse matrix.
Further, the slope and curvature measurements in any horizontal and vertical direction between 0 ° and 180 ° are added at each grid point in step one as:
Figure GDA0003476124700000031
Figure GDA0003476124700000032
the reconstruction region is square and is divided into N × N sub-apertures with a width h, the central positions of the sub-apertures in the t-th row and the k-th column in the reconstruction region are represented by (t, k), t is 1,2, …, N, k is 1,2, …, N, S t,k Is the slope value at an arbitrary directional point (t, k), C t,k Is a curvature value at an arbitrary directional point (t, k),
Figure GDA0003476124700000033
and
Figure GDA0003476124700000034
representing slope values at points (t, k) in the x and y directions, respectively,
Figure GDA0003476124700000035
and
Figure GDA0003476124700000036
representing curvature values at points (t, k) in the x and y directions, respectively,
Figure GDA0003476124700000037
is the torsional curvature term at point (t, k), a, b, c, d, e are non-zero constants.
Further, in the second step, the relationship among the wavefront value, the slope value and the mixed curvature value in any horizontal and vertical direction between 0 ° and 180 ° is expressed as follows:
Af 1 +Bf 2 ＝h(Ef 1 ′+Ff 2 ′)+h 2 (Gf 1 ″+Hf 2 ″)
wherein, f 1 And f 2 The function values at points 1 and 2, f 1 ' and f 2 ' first derivative values at points 1 and 2, respectively, f 1 "and f 2 "are the second derivative values at points 1 and 2, A, B, E, F, G and H are coefficients, and points 1 and 2 are the locations of two reconstruction points adjacent to each other in any direction within the reconstruction region.
Further, in step four, the reconstruction equations in the 45 ° and 135 ° directions are expressed as:
Figure GDA0003476124700000038
Figure GDA0003476124700000039
wherein, the central position of the i-th row and j-th column sub-aperture in the reconstruction region is represented by (i, j), i is 1,2, …, (N-1), j is 1,2, …, (N-1), W i,j Representing the wavefront value, W, at point (i, j) i+1,j+1 Representing the wavefront value at point (i +1, j +1),
Figure GDA00034761247000000310
and
Figure GDA00034761247000000311
respectively represent a slope value and a curvature value at a 45 DEG direction point (i +1, j),
Figure GDA00034761247000000312
and
Figure GDA00034761247000000313
respectively, the slope value and the curvature value at the 135 ° direction point (i, j).
And further, solving a simultaneous equation set of the reconstruction equations in the horizontal direction and the vertical direction and the step four to construct a multidirectional mixed reconstruction equation.
Further, the system of simultaneous reconstruction equations for the horizontal, vertical, 45 ° and 135 ° directions is expressed as:
Figure GDA0003476124700000041
Figure GDA0003476124700000042
Figure GDA0003476124700000043
Figure GDA0003476124700000044
wherein, the first and the second end of the pipe are connected with each other,
Figure GDA0003476124700000045
and
Figure GDA0003476124700000046
representing the slope values at points (i, j) in the x and y directions, respectively,
Figure GDA0003476124700000047
and
Figure GDA0003476124700000048
respectively representing the curvature values at points (i, j) in the x and y directions, obtaining a multidirectional mixed reconstruction equation, representing by using a matrix, andand solving a wave front value to be measured.
Has the beneficial effects that:
1. the multidirectional slope and curvature mixed wavefront reconstruction method for the high-order truncation error breaks through the defect that the traditional technology carries out reconstruction only by using information in the x direction and the y direction, and achieves the aim of high-order aberration precision measurement by constructing an equation with smaller truncation error by using the relation among a wavefront value in any horizontal direction and a vertical direction, a slope value and a mixed curvature value between 0-180 degrees.
2. According to the method, multi-direction reconstruction is adopted, so that a loop comprises more grid points, the reconstruction precision is improved, and the method has strong anti-noise capability;
3. the wavefront reconstruction process of the invention is solved by using a least square method, and compared with iterative computation processes of other wavefront reconstruction technologies, the wavefront reconstruction process of the invention has the advantages of small computation amount and high computation speed, and can be applied to application fields with higher real-time requirements, such as adaptive optics and the like.
Drawings
FIG. 1 is a schematic diagram of the 45 and 135 orientation slope and curvature measurements.
FIG. 2 is a flow chart of the method of the present invention.
Fig. 3 is a schematic structural diagram of reconstruction based on a region-based Southwell model.
FIG. 4 is a comparison graph of the reconstruction results of wavefront reconstruction in the method of the present invention and the conventional method without considering noise; (a) the Barwick algorithm, HB1, HB2 algorithm; (b) pathak algorithm and HM1, HM2 algorithm; (c) HB1, HB2 algorithm and HM1, HM2 algorithm.
FIG. 5 is a diagram comparing the reconstruction results of the wavefront reconstruction with the conventional method when the SNR is 10 in consideration of noise; (a) the Barwick algorithm, HB1, HB2 algorithm; (b) the Pathak algorithm and HM1, HM2 algorithms; (c) HB1, HB2 algorithm and HM1, HM2 algorithm.
FIG. 6 is a comparison graph of the reconstruction result of the wavefront reconstruction of the method of the present invention and the conventional method when the SNR is 30 under the noise consideration; (a) the Barwick algorithm and HB1, HB2 algorithms; (b) the Pathak algorithm and HM1, HM2 algorithms; (c) HB1, HB2 algorithm and HM1, HM2 algorithm.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
FIG. 1 schematically shows a schematic diagram of the slopes and curvatures in the 45 ° and 135 ° directions, based on a regional Southwell model, in addition to the conventional horizontal and vertical slope and curvature values, the invention considers the slopes and curvature values in the 45 ° and 135 ° directions, and adds additional grid points (i-1, j-1), (i +1, j-1), (i +1, j +1), and (i-1, j +1) to the loop to improve the reconstruction accuracy.
With reference to fig. 2, the method for reconstructing the multi-directional slope and curvature mixed wavefront of the high-order truncation error of the present invention specifically comprises the following steps:
step one, based on a regional method Southwell model, additionally adding slope and curvature measurement values in any horizontal and vertical directions between 0 and 180 degrees at each grid point:
Figure GDA0003476124700000061
Figure GDA0003476124700000062
the reconstruction region is square and is divided into N × N sub-apertures with a width h, the central positions of the sub-apertures in the t-th row and the k-th column in the reconstruction region are represented by (t, k), t is 1,2, …, N, k is 1,2, …, N, S t,k Is the slope value at an arbitrary directional point (t, k), C t,k Is a curvature value at an arbitrary directional point (t, k),
Figure GDA0003476124700000063
and
Figure GDA0003476124700000064
representing slope values at points (t, k) in the x and y directions, respectively,
Figure GDA0003476124700000065
and
Figure GDA0003476124700000066
representing curvature values at points (t, k) in the x and y directions, respectively,
Figure GDA0003476124700000067
is the torsional curvature term at point (t, k), a, b, c, d, e are non-zero constants;
step two, considering the relation among a horizontal wave front value and a vertical wave front value in any direction between 0 and 180 degrees, a slope value and a mixed curvature value to construct an integral equation, wherein the truncation error of the integral equation is O (h) 5 )：
Af 1 +Bf 2 ＝h(Ef 1 ′+Ff 2 ′)+h 2 (Gf 1 ″+Hf 2 ″)
Wherein, f 1 And f 2 The function values at points 1 and 2, f 1 ' and f 2 ' first derivative values at points 1 and 2, respectively, f 1 "and f 2 "second derivative values at point 1 and point 2, A, B, E, F, G and H are coefficients, point 1 and point 2 are two reconstruction point positions adjacent to each other in any direction in the reconstruction region, the higher the truncation error order of the reconstruction equation is, the more accurate the value is, and similarly, the truncation error can be constructed as O (H) (H 7 ) Integral equation of (1), more than two gradient values of the multi-inclusion of the fifth order, truncation error from O (h) 5 ) To O (h) 7 )；
Thirdly, obtaining a solution about an equation coefficient by using Taylor expansion;
taylor expansion of the above expression at point 1.5, where point 1.5 is the middle of points 1 and 2, equalizes the derivative front coefficients on both sides of the equation, thereby solving for the values of A, B, E, F, G and H.
Step four, after solving the coefficient, executing the following steps:
(1) substituting the coefficients solved in step three into the integral equation in step two, taking the directions of 45 ° and 135 ° as examples:
Figure GDA0003476124700000071
(2) the truncation error of O (h) in the directions of 45 degrees and 135 degrees can be obtained 5 ) The reconstruction equation of (1):
Figure GDA0003476124700000072
Figure GDA0003476124700000073
wherein, the central position of the i-th row and j-th column sub-aperture in the reconstruction region is represented by (i, j), i is 1,2, …, (N-1), j is 1,2, …, (N-1), W i,j Representing the wavefront value, W, at point (i, j) i+1,j+1 Represents the wavefront value at point (i +1, j +1),
Figure GDA0003476124700000074
and
Figure GDA0003476124700000075
respectively represent a slope value and a curvature value at a 45 DEG direction point (i +1, j),
Figure GDA0003476124700000076
and
Figure GDA0003476124700000077
respectively, the slope value and the curvature value at the 135 ° direction point (i, j). The two equations above can be expressed in a matrix as:
AW＝H
wherein, A is 2(N-1) 2 ×N 2 W is the wavefront value to be measured, and the size is N 2 X 1 column vector, H slope curvature value of magnitude 2(N-1) 2 X 1 column vector, similarly, can be constructed with a truncation error of O (h) 7 ) Reconstruction equations in the 45 ° and 135 ° directions.
Step five, calculating A and H values, and solving the wavefront value to be measured by using a least square method as follows:
W＝(A T A) + A T H；
wherein A is T Is the transpose of A, (A) T A) + Is A T A generalized inverse matrix of.
Preferably, the method can combine reconstruction equations in the x and y directions and a simultaneous equation set of the reconstruction equations in any horizontal and vertical directions between 0 and 180 degrees obtained in the fourth step to construct a multidirectional mixed reconstruction equation:
take x, y, 45 ° (diagonal) and 135 ° directions as examples:
Figure GDA0003476124700000081
Figure GDA0003476124700000082
Figure GDA0003476124700000083
Figure GDA0003476124700000084
wherein the content of the first and second substances,
Figure GDA0003476124700000085
and
Figure GDA0003476124700000086
representing the slope values at points (i, j) in the x and y directions, respectively,
Figure GDA0003476124700000087
and
Figure GDA0003476124700000088
the curvature values at points (i, j) in the x and y directions are respectively expressed, and the truncation error is O (h) 5 ) The multi-direction mixed reconstruction equation can be expressed by a matrix and solved by a least square methodObtaining the wavefront value to be measured, solving the wavefront value in the process similar to the fourth step to the fifth step, obtaining the truncation error O (h) by applying the method similar to the method 7 ) The multi-directional hybrid reconstruction equation of (1).
Fig. 3 is a schematic structural diagram of reconstruction based on a regional Southwell model, in which black dots represent positions of wavefront reconstruction points, which coincide with reference positions of slope and curvature value measurement points represented by crosses, and circular rings represent intermediate positions of two adjacent wavefront reconstruction points.
As shown in fig. 4, comparing the wavefront reconstruction results of the method of the present invention with the conventional methods without considering noise, (a) Barwick algorithm and HB1, HB2 algorithm (b) Pathak algorithm and HM1, HM2 algorithm (c) HB1, HB2 algorithm and HM1, HM2 algorithm, wherein HB1 and HB2 are mixed reconstruction methods in any direction (taking 45 ° and 135 ° as examples) of the present invention, and HB1 truncation error is O (h 1) (h) of the present invention 5 ) The truncation error of HB2 is O (h) 7 ) HM1 and HM2 are multidirectional hybrid reconstruction methods of the present invention, and HM1 truncation error is O (h) 5 ) HM2 truncation error is O (h) 7 ) Barwick is the existing mixed type reconstruction method, Pathak is the multi-direction slope type reconstruction method, and as can be seen from the simulation result of FIG. 4, compared with the existing mixed type reconstruction method, the relative reconstruction error value of the multi-direction slope and curvature mixed wavefront reconstruction method of the high-order truncation error of the invention is reduced by about four orders of magnitude, and through multi-direction reconstruction, more grid points are combined in a given loop, so that the reconstruction accuracy can be improved to a certain extent.
As shown in fig. 5, when the signal-to-noise ratio difference is 10 under the noise consideration, the method of the present invention compares (a) the Barwick algorithm with HB1 and HB2 algorithm (b) the Pathak algorithm with HM1 and HM2 algorithm (c) HB1 and HB2 algorithm and HM1 and HM2 algorithm with the conventional methods for reconstructing the wavefront reconstruction result, and as can be seen from the simulation result of fig. 5, the method of the present invention has the advantages of smaller relative reconstruction error for higher order aberration and stronger noise immunity compared with the existing algorithms under the condition of the measurement noise, and the multidirectional hybrid reconstruction methods HM1 and HM2 of the present invention have smaller relative reconstruction error than the hybrid reconstruction methods HB1 and HB 2.
As shown in fig. 6, when the signal-to-noise ratio is preferably 30 under the noise consideration, compared with the wavefront reconstruction result of the conventional method, the method of the present invention (a) uses Barwick algorithm and HB1, HB2 algorithm (b) uses Pathak algorithm and HM1, HM2 algorithm (c) uses HB1, HB2 algorithm and HM1, HM2 algorithm, and from the simulation result of fig. 6, it can be seen that the relative reconstruction errors of Barwick algorithm and Pathak algorithm do not change significantly as the signal-to-noise ratio increases from 10 to 30, while the relative reconstruction errors of HB1, HB2, HM1, and HM2 of the present invention are all significantly reduced, and have strong anti-noise capability.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (3)

1. The method for reconstructing the multi-directional slope and curvature mixed wavefront of the high-order truncation error is characterized by comprising the following steps of:
step one, adding a horizontal and vertical slope value in any direction between 0 and 180 degrees at each grid point based on a regional Southwell model
Figure FDA0003749661520000011
And mixed curvature value
Figure FDA0003749661520000012
The reconstruction region is square and is divided into N × N sub-apertures with a width h, the central positions of the sub-apertures in the t-th row and the k-th column in the reconstruction region are represented by (t, k), t is 1,2, …, N, k is 1,2, …, N, S t,k Is the slope value at an arbitrary directional point (t, k), C t,k Is a mixed curvature value at an arbitrary directional point (t, k),
Figure FDA0003749661520000013
and
Figure FDA0003749661520000014
representing slope values at points (t, k) in the x and y directions, respectively,
Figure FDA0003749661520000015
and
Figure FDA0003749661520000016
representing curvature values at points (t, k) in the x and y directions, respectively,
Figure FDA0003749661520000017
is the torsional curvature value at point (t, k), a, b, c, d, e are non-zero constants;
step two, constructing an integral equation by utilizing the relation among a horizontal wave front value, a vertical wave front value and a mixed curvature value in any direction between 0 and 180 degrees, wherein the relation is expressed as:
Af 1 +Bf 2 ＝h(Ef 1 ′+Ff 2 ′)+h 2 (Gf 1 ″+Hf 2 ″)
wherein, f 1 And f 2 The function values at points 1 and 2, f 1 ' and f 2 ' is the first derivative value at points 1 and 2, respectively, f 1 "and f 2 "is the second derivative value of point 1 and point 2, A, B, E, F, G and H are coefficients, point 1 and point 2 are the positions of two adjacent reconstruction points in any direction in the reconstruction region, and H is the width of the sub-aperture obtained after the reconstruction region is divided;
thirdly, obtaining a solution of a coefficient related to the integral equation by using Taylor expansion;
step four, substituting the coefficient solved in the step three into the integral equation in the step two to obtain a reconstruction equation, and expressing the reconstruction equation as AW (equal to H) by using a matrix;
wherein, A is 2(N-1) 2 ×N 2 W is the wavefront value to be measured and N is the magnitude 2 X 1 column vector, H slope curvature value, magnitude 2(N-1) 2 A column vector of x 1;
the reconstruction equations in the 45 ° and 135 ° directions are expressed as:
Figure FDA0003749661520000021
Figure FDA0003749661520000022
wherein, the central position of the i-th row and j-th column sub-aperture in the reconstruction region is represented by (i, j), i is 1,2, …, (N-1), j is 1,2, …, (N-1), W i,j Representing the wavefront value, W, at point (i, j) i+1,j+1 Represents the wavefront value at point (i +1, j +1),
Figure FDA0003749661520000023
and
Figure FDA0003749661520000024
respectively represent a slope value and a mixed curvature value at a 45 DEG direction point (i +1, j),
Figure FDA0003749661520000025
and
Figure FDA0003749661520000026
respectively representing a slope value and a mixed curvature value at a 135 ° directional point (i, j);
step five, calculating the values of A and H, namely recovering the wavefront value to be measured as W ═ A T A) + A T H；
Wherein A is T Is the transpose of A, (A) T A) + Is A T A generalized inverse matrix.
2. The method for multidirectional slope and curvature mixed wavefront reconstruction of higher-order truncation errors of claim 1, wherein a multidirectional mixed reconstruction equation is constructed by solving simultaneous equations of reconstruction equations in horizontal and vertical directions and in step four.
3. The method for multidirectional slope and curvature hybrid wavefront reconstruction of higher order truncation errors of claim 2, wherein the simultaneous system of reconstruction equations for horizontal, vertical, 45 ° and 135 ° directions is expressed as:
Figure FDA0003749661520000027
Figure FDA0003749661520000028
Figure FDA0003749661520000029
Figure FDA00037496615200000210
wherein the content of the first and second substances,
Figure FDA00037496615200000211
and
Figure FDA00037496615200000212
representing the slope values at points (i, j) in the x and y directions, respectively,
Figure FDA00037496615200000213
and
Figure FDA00037496615200000214
and respectively representing curvature values at points (i, j) in the x direction and the y direction, obtaining a multidirectional mixed reconstruction equation, representing by using a matrix, and solving a wavefront value to be measured.
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