A non-redundant sparse aperture array arrangement method with high cut-off frequency

Abstract

The present invention relates to the field of sparse aperture imaging technology, and in particular to a non-redundant sparse aperture array arrangement method with a high cutoff frequency. The method comprises: establishing an equilateral triangle grid, establishing a coordinate system with a grid point as the origin; determining the number of sub-apertures to be optimized, and obtaining the corresponding coordinate value range matrix M in the equilateral triangle grid according to the number of sub-apertures; selecting a coordinate point; removing the coordinate points that cause redundancy in the coordinate value range matrix M; checking whether the termination condition is met, and if the termination condition is not met, repeating the point selection and redundancy removal steps; if it is met, calculating the objective function; the termination condition is that the number of selected points p is equal to the number of sub-apertures; using an optimization solution algorithm for calculation, and obtaining the optimal result of the non-redundant sparse aperture array guided by the objective function. The advantages are: a non-redundant sparse aperture array arrangement can be stably obtained, and the corresponding modulation transfer function is compact, the cutoff frequency is high, and the distribution is uniform in all directions.

Description

Technical Field

[0001] The present invention relates to the technical field of sparse aperture imaging, and in particular to a non-redundant sparse aperture array arrangement method with a high cutoff frequency. Background Art

[0002] According to the Rayleigh criterion, the larger the aperture of an optical system, the higher its angular resolution, and the better its observation capability and imaging effect. The key to the development of traditional large-aperture telescopes is to manufacture large-aperture primary mirrors. The increase in the aperture of the primary mirror will bring difficulties in various links such as processing, testing, support, assembly and transportation. In order to break through the limitations of traditional telescopes in terms of system aperture, European scientists proposed sparse aperture imaging technology based on the principle of interference imaging, that is, using multiple small-aperture optical systems arranged in a certain array shape to be equivalent to a large-aperture optical system to achieve high-resolution imaging. Sparse aperture is to arrange several smaller optical apertures in a certain rule to form a large optical aperture.

[0003] The parameters that affect the sparse aperture imaging performance include the number, position, and diameter of sub-apertures. Among them, the sparse aperture array arrangement includes the number and position of sub-apertures, and is a key part of the design of sparse aperture optical systems. Reasonable sparse aperture array arrangement can enable the optical system to evenly transmit information in all directions of the frequency domain and have a larger cutoff frequency. However, the design of sparse aperture array arrangement is difficult and cannot be obtained by mathematical methods. The current sparse aperture array arrangement design generally proposes a sub-aperture arrangement rule to design a series of array arrangements, which cannot guarantee that the obtained frequency domain modulation transfer function is optimal; or the array arrangement is obtained by exhaustive enumeration, which requires a huge amount of calculation. Summary of the invention

[0004] In order to solve the above problems, the present invention provides a non-redundant sparse aperture array arrangement method with a high cutoff frequency.

[0005] The present invention aims to provide a non-redundant sparse aperture array arrangement method with a high cutoff frequency, comprising the following steps:

[0006] S1. Establishing an equilateral triangle mesh, establishing a coordinate system with a grid point of the equilateral triangle mesh as the origin;

[0007] S2. Determine the number n of sub-apertures to be optimized, and obtain the corresponding coordinate value range matrix M in the regular triangle grid according to the number n of sub-apertures;

[0008] S3. Select a coordinate point in the coordinate value range matrix M;

[0009] S4. Remove the coordinate points that cause redundancy in the coordinate value range matrix M;

[0010] S5. Check whether the termination condition is met. If the termination condition is not met, repeat steps S3-S4; if the termination condition is met, calculate the objective function; the termination condition is that the number of selected points p is equal to the number of sub-apertures n;

[0011] S6. Use the optimization solution algorithm to calculate and obtain the optimal result of the non-redundant sparse aperture array guided by the objective function.

[0012] Preferably, the specific method for establishing the coordinate system in step S1 is as follows: take a grid point of the equilateral triangle grid as the origin (0,0), take the side length of the minimum equilateral triangle as a unit length, take a minimum equilateral triangle with the origin as the vertex, and take the directions of the two sides starting from the origin as the coordinate axis directions.

[0013] Preferably, n in step S2 is a positive integer greater than 2; the coordinate value range is the grid points in and on the regular hexagon with a side length a with the center as the origin in the regular triangular grid, and the matrix composed of the coordinates of all grid points is the coordinate value range matrix M; the side length of the regular hexagon , N+ is a set of positive integers.

[0014] Preferably, step S3 of selecting coordinate points specifically includes:

[0015] Each coordinate point in the coordinate value range matrix M corresponds to an index. When selecting a coordinate point, the index corresponding to the point is input; when selecting a point for the first time, the initial value range matrix M0 is used;

[0016] When the input index is an integer multiple greater than 1 of the length of the coordinate value range matrix M, let the index be equal to 1;

[0017] When the input index is greater than the length of the coordinate value range matrix M and is not an integer multiple thereof, let the index be equal to the remainder of the index divided by the length of the coordinate value range matrix M;

[0018] When the coordinate value range matrix M is empty, a point is randomly selected from the initial value range matrix M0.

[0019] Preferably, step S4 specifically includes:

[0020] S401. The autocorrelation vector group obtained by subtracting the coordinates of all the selected points in pairs:

[0021] S402. Move the center of the autocorrelation vector group to the position of each selected point in turn, and remove the points covered by the autocorrelation vector group in the coordinate value range matrix each time the movement occurs.

[0022] Preferably, in step S401, p points are selected, 1≤p≤n, and the coordinates of the pth point are (x p ,y p ); the autocorrelation vector group is determined by the following formula:

[0023] ;

[0024] In the formula, It is a matrix of p*p rows and 2 columns, i and j are the corresponding indices of the coordinates to be subtracted in pairs in the coordinate value range matrix M; is the coordinate represented by the i*jth row of the matrix;

[0025] The coordinates of the points covered by the autocorrelation vector group in step S402 are determined by the following formula:

[0026] ;

[0027] Where i and j are the indices of the coordinates that are subtracted from each other in the coordinate value range matrix M, and k is the MTF p The coordinates of the point to which the center moves are in the index corresponding to the coordinate value range matrix M. MTF in the coordinate value range matrix M for each move p The coordinates of the points to be covered.

[0028] Preferably, the objective function O in step S5 is F The specific expressions are as follows:

[0029] ;

[0030] In the formula, , ; L M The vector group obtained by subtracting the coordinates of the n points MTF The number of unequal vectors in L a The vector group obtained by subtracting the coordinates of the n points MTF The number of vectors that are not equal to each other in the union of the coordinate vectors of all points in a regular hexagon with its center at the origin and side length a; L a-1 The vector group obtained by subtracting the coordinates of the n points MTF The number of vectors that are not equal to the union of the coordinate vectors of all points in a regular hexagon with center at the origin and side length a-1.

[0031] Compared with the prior art, the present invention can achieve the following beneficial effects:

[0032] By removing redundant points through a non-redundant point selection method, the amount of calculation is greatly reduced, and a non-redundant sparse aperture array arrangement can be stably obtained; the objective function is set reasonably, and the modulation transfer function (MTF) corresponding to the calculated array arrangement has the characteristics of compactness, high cutoff frequency, and uniform distribution in all directions, which has certain guiding significance in the design of sparse aperture optical systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a flow chart of a non-redundant sparse aperture array arrangement method with high cutoff frequency provided according to an embodiment of the present invention.

[0034] Figure 2 is an example diagram of a triangular mesh and a coordinate system provided according to an embodiment of the present invention.

[0035] Figure 3 is an example diagram of a method for selecting coordinate points provided according to an embodiment of the present invention.

[0036] Figure 4 is an example diagram of calculating an objective function through autocorrelation according to an embodiment of the present invention.

[0037] Figure 5 The present invention provides a non-redundant sparse aperture array arrangement coordinate points when the number of sub-apertures is 5, and two existing Golay-type non-redundant sparse aperture array arrangement coordinate points are shown as examples; (A) the non-redundant sparse aperture array arrangement coordinate points of the present invention; (B) the first Golay-type non-redundant sparse aperture array arrangement coordinate points; (C) the second Golay-type non-redundant sparse aperture array arrangement coordinate points.

[0038] Figure 6 The following are examples of the autocorrelation of the non-redundant sparse aperture array arrangement coordinate points provided by the embodiment of the present invention and the two existing Golay type non-redundant sparse aperture array arrangement coordinate points when the number of sub-apertures is 5; (A) the autocorrelation of the non-redundant sparse aperture array arrangement coordinate points of the present invention; (B) the autocorrelation of the first type of Golay type non-redundant sparse aperture array arrangement coordinate points; (C) the autocorrelation of the second type of Golay type non-redundant sparse aperture array arrangement coordinate points. . DETAILED DESCRIPTION

[0039] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, the same modules are represented by the same reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, the detailed description thereof will not be repeated.

[0040] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation of the present invention.

[0041] Example 1

[0042] like Figure 1 As shown, a non-redundant sparse aperture array arrangement method with a high cutoff frequency is provided, comprising the following steps:

[0043] S1. Establishing an equilateral triangle mesh, establishing a coordinate system with a grid point of the equilateral triangle mesh as the origin; Figure 2 As shown, take point O as the origin in the divided equilateral triangle grid points, with coordinates (0,0); the side length of the smallest triangle in the grid is one unit length; take a smallest triangle with O as the vertex, starting from point O, the directions of the sides of the two triangles are the directions of the coordinate axes (x-axis and y-axis); the coordinates of any grid point in the grid can be expressed as (x,y), where x and y are integers; the coordinates of the upper right corner are (2,1).

[0044] S2. Determine the number n of subapertures to be optimized, where n is a positive integer greater than 2; obtain a coordinate value range matrix M corresponding to the equilateral triangle grid according to the number n of subapertures;

[0045] Specifically, the coordinate value range is the grid points in and on the regular hexagon with a side length a with the center as the origin in the regular triangular grid, and the matrix composed of the coordinates of all grid points is the initial value range matrix M0; the side length of the regular hexagon , where N+ is a set of positive integers;

[0046] If there are two different a's, the larger a is taken as the side length of the regular hexagon. For example, when n=4, n(n-1)=12. In this case, a can be 1 or 2, and 3a(a+1) is 6 and 18 respectively, so a=2 is finally taken. When n=3, Figure 3 As shown, the initial value range matrix M0 of the grid points inside and on the edge of the hexagon with side length a=1 is [0,0; 0,1; 1,0; 0,-1; -1,0; 1,-1; -1,1]; when the number of sub-apertures is between 3 and 10, the side length a of the hexagon is given by the following Table 1:

[0047] Table 1 Value range comparison table

[0048]

[0049] S3. Select a coordinate point in the coordinate value range matrix M;

[0050] The number of selected points is p, where p is a positive integer of 1≤p≤n. The coordinates of the pth point are (x p ,y p ); The value range matrix when taking points for the first time is the initial value range matrix M0, and the subsequent value range matrix is ​​the matrix after removing redundant coordinate points;

[0051] The 3a(a+1)+1 coordinates in the initial value range matrix M0 correspond to coordinate indexes 1 to 3a(a+1)+1 respectively; when selecting the pth point, enter the index o corresponding to the pth point p (1≤o p ≤3a(a+1)+1), the coordinates of the pth point are the oth point in the coordinate value range matrix M p coordinates; (1) when o p is the length of the coordinate range matrix L m When o is greater than an integer multiple of 1, let o p =1; (2) when o p Greater than the coordinate value range matrix length L m And it is not an integer multiple, let o p =mod(o p , L m );(3)When the coordinate value range matrix M is empty, the coordinates of point p are any point in the initial value range matrix M0.

[0052] S4. Remove the coordinate points that cause redundancy in the coordinate value range matrix M;

[0053] S401. The number of points currently selected is p, 1≤p≤n, and the coordinates of the pth point are (x p ,y p ). The MTF of the autocorrelation vector group of p point coordinates p It is a vector group obtained by subtracting p coordinates from each other, determined by the following formula:

[0054] ;

[0055] In the formula, It is a matrix of p*p rows and 2 columns, i and j are the corresponding indices of the coordinates to be subtracted in pairs in the coordinate value range matrix M; is the coordinate represented by the i*jth row of the matrix;

[0056] S402. MTF p The center moves to the positions of p points in sequence, and each time it moves, the MTF is removed from the coordinate value range matrix M. p The point covered, the coordinates of which are determined by the following formula;

[0057] ;

[0058] Where i and j are the indices of the coordinates that are subtracted from each other in the coordinate value range matrix M, and k is the MTF p The coordinates of the point to which the center moves are in the index corresponding to the coordinate value range matrix M. MTF in the coordinate value range matrix M for each move p The coordinates of the covered points; by removing these coordinates, the amount of calculation can be greatly reduced when the number of selected points increases.

[0059] like Figure 3 As shown in the figure, when a=3, the index value range is 1-7. When o1, o2, and o3 are 1, 1, and 6 respectively, the coordinates of the first point are the first point (0,0) in M0, located at the origin. At this time, the autocorrelation of the first point is (0,0). After its center is moved to the coordinates of the first point, the superimposed redundant point is (0,0); when the second point is taken, the coordinate value range matrix M=[0,1;1,0;0,-1;-1,0;1,-1;-1,1], and the coordinates of the second point are the first point (0,1) in M. At this time, the autocorrelation of the first two points is , after its center moves to the coordinates of the first two points respectively, the superimposed redundant points are [0,0; 0,-1; 0,1; 0,2]; when the third point is taken, the coordinate value range matrix M=[1,0; -1,0;1,-1;-1,1], and the length of M is L m =4, less than o3; let o3=mod(o3, L m )=2, the coordinates of the third point are the first point in M ​​(-1,0).

[0060] S5. Check whether the termination condition is met. If the termination condition is not met, repeat steps S3-S4; if the termination condition is met, calculate the objective function; the termination condition is that the number of selected points p is equal to the number of sub-apertures n;

[0061] If the termination condition is met, calculate the objective function O F , the objective function O F It can be specifically expressed as follows:

[0062] ;

[0063] In the formula, , ; L M The vector group obtained by subtracting the coordinates of the n points The number of unequal vectors in L a The vector group obtained by subtracting the coordinates of the n points The number of vectors that are not equal to each other in the union of the coordinate vectors of all points in a regular hexagon with its center at the origin and side length a; L a-1The vector group obtained by subtracting the coordinates of the n points The number of vectors that are not equal to each other in the union of the coordinate vectors of all points in a regular hexagon with center at the origin and side length a-1;

[0064] When n=3, o1, o2, and o3 are 1, 1, and 6 respectively, the final autocorrelation of the n points is as follows Figure 4 As shown; L M =7,L a =9,L a-1 =7.

[0065] The setting of the above objective function can make the obtained sparse aperture array arrangement closer to a regular hexagon in the frequency domain and more evenly distributed; by increasing α, a non-redundant sparse aperture array arrangement can be ensured; by increasing γ, a large cutoff frequency can be ensured for the obtained sparse aperture array arrangement.

[0066] S6. Calculate using an optimization algorithm to obtain the optimal result of the non-redundant sparse aperture array guided by the objective function;

[0067] The input of the optimization algorithm is the index of the coordinates of the n points in step S5 in the coordinate value range matrix M, and the range of each index is 1≤o p ≤3a(a+1)+1(1≤p≤n).

[0068] Example 2

[0069] According to the non-redundant sparse aperture array arrangement method with high cutoff frequency of the present invention, the following is designed: Figure 5 (A) Sparse aperture array, the corresponding autocorrelation array is as follows Figure 6 As shown in (A), the number of autocorrelation points is n(n-1)+1, so the obtained Figure 5 The sparse aperture array in (A) is non-redundant, demonstrating the effectiveness of the de-redundancy method in step S4. Figure 5 The sparse aperture array of (A) is obtained by the objective function of step S5, which aims to obtain a more compact, high cutoff frequency, and isotropically uniform sparse aperture array. Figure 6 As shown, taking the distance between the two nearest autocorrelation points as the unit, the sum of the squares of the distances from each autocorrelation point to the center is the moment of inertia of the autocorrelation array, which is an indicator to measure the compactness of the non-redundant sparse aperture system. The smaller the moment of inertia, the more compact the array and the less the loss of mid- and low-frequency information. Figure 6 The moments of inertia of (A), (B), and (C) are 72, 74, and 86, respectively. Therefore, the non-redundant sparse aperture array of the present invention is more compact. Figure 6(A) (B) (C) have the same cutoff frequency, but a circle is drawn with the autocorrelation center as the center and the cutoff frequency as the radius. Obviously Figure 6 (A) has a minimum cutoff frequency in only two directions, while Figure 6 The minimum cutoff frequency of (B) (C) has four directions, Figure 6 (A) More uniform.

[0070] The present invention has a non-redundant sparse aperture array arrangement method with a high cut-off frequency, realizes non-redundant array arrangement through a non-redundant point selection method, realizes optimization calculation through an optimization solution algorithm, realizes convergence of the optimal array arrangement by setting an objective function, and improves the effect of sparse aperture array arrangement optimization.

[0071] It should be understood that the various forms of processes shown above can be used to reorder, add or delete steps. For example, the steps described in the disclosure of the present invention can be performed in parallel, sequentially or in different orders, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and this document does not limit this.

[0072] The above specific implementations do not constitute a limitation on the protection scope of the present invention. It should be understood by those skilled in the art that various modifications, combinations, sub-combinations and substitutions can be made according to design requirements and other factors. Any modification, equivalent substitution and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A non-redundant sparse aperture array arrangement method with high cutoff frequency, characterized in that: The following steps are involved: S1. Establish an equilateral triangle mesh, and establish a coordinate system with a grid point of the equilateral triangle mesh as the origin; take a point O as the origin among the divided equilateral triangle grid points, and the coordinates are (0,0); the side length of the smallest triangle in the mesh is one unit length; take a smallest triangle with O as the vertex, and starting from point O, the directions of the sides of the two triangles are the directions of the coordinate axes; S2. Determine the number n of subapertures to be optimized, and obtain the corresponding coordinate value range matrix M in the regular triangular mesh according to the number n of subapertures; the coordinate value range is the grid points in and on the regular hexagon with a side length a with the center as the origin in the regular triangular mesh, and the matrix composed of the coordinates of all grid points is the coordinate value range matrix M; S3. Select a coordinate point in the coordinate value range matrix M; S4. Remove the coordinate points that cause redundancy in the coordinate value range matrix M; S401. The autocorrelation vector group obtained by subtracting the coordinates of all the selected points in pairs: S402. Move the center of the autocorrelation vector group to the position of each selected point in turn, and remove the points covered by the autocorrelation vector group in the coordinate value range matrix each time; S5. Check whether the termination condition is met. If the termination condition is not met, repeat steps S3-S4; if the termination condition is met, calculate the objective function; the termination condition is that the number of selected points p is equal to the number of subapertures n; objective function O F The specific expressions are as follows: The F =αL M -βL a -γL a-1 ; In the formula, α>β+γ, β<γ; L M The number of mutually unequal vectors in the vector group MTFn obtained by subtracting the coordinates of the n points in pairs; L a The number of mutually unequal vectors in the union of the vector group MTFn obtained by subtracting the coordinates of the n points in pairs and the coordinate vectors of all points in a regular hexagon with its center at the origin and a side length of a; L a-1 The number of unequal vectors in the union of the vector group MTFn obtained by subtracting the coordinates of the n points in pairs and the coordinate vectors of all points in a regular hexagon with its center at the origin and a side length of a-1; S6. Use the optimization solution algorithm to calculate and obtain the optimal result of the non-redundant sparse aperture array guided by the objective function.

2. The method for arranging a non-redundant sparse aperture array with a high cutoff frequency according to claim 1, characterized in that: The specific method for establishing the coordinate system in step S1 is as follows: take a grid point of the equilateral triangle grid as the origin (0,0), take the side length of the minimum equilateral triangle as a unit length, take a minimum equilateral triangle with the origin as the vertex, and take the directions of the two sides starting from the origin as the coordinate axis directions.

3. The method for arranging a non-redundant sparse aperture array with a high cutoff frequency according to claim 2, characterized in that: The n described in step S2 is a positive integer greater than 2; the side length of the regular hexagon a=argmin(3a(a+1)-n(n-1)), a∈N+, N+ is a positive integer set.

4. The method for arranging a non-redundant sparse aperture array with a high cutoff frequency according to claim 3, characterized in that: The step S3 of selecting coordinate points specifically includes: Each coordinate point in the coordinate value range matrix M corresponds to an index. When selecting a coordinate point, the index corresponding to the point is input; when selecting a point for the first time, the initial value range matrix M0 is used; When the input index is an integer multiple greater than 1 of the length of the coordinate value range matrix M, let the index be equal to 1; When the input index is greater than the length of the coordinate value range matrix M and is not an integer multiple thereof, let the index be equal to the remainder of the index divided by the length of the coordinate value range matrix M; When the coordinate value range matrix M is empty, a point is randomly selected from the initial value range matrix M0.

5. The method for arranging a non-redundant sparse aperture array with a high cutoff frequency according to claim 4, characterized in that: In step S401, p points are selected, 1≤p≤n, and the coordinates of the pth point are (x p ,y p ); the autocorrelation vector group is determined by the following formula: Where, MTF p It is a matrix with p*p rows and 2 columns, i and j are the corresponding indices of the coordinates to be subtracted in pairs in the coordinate value range matrix M; MTF p (i*j) is the coordinate represented by the i*jth row of the matrix; The coordinates of the points covered by the autocorrelation vector group in step S402 are determined by the following formula: Where i and j are the indices of the coordinates that are subtracted from each other in the coordinate value range matrix M, and k is the MTF p The coordinates of the point to which the center moves are in the coordinate value range matrix M corresponding to the index, MTF p (i,j)+(x k ,y k ) is the MTF in the coordinate value range matrix M at each move p The coordinates of the points to be covered.

Images