CN112130318B - Optical free-form surface characterization method based on Gaussian radial basis function - Google Patents
Optical free-form surface characterization method based on Gaussian radial basis function Download PDFInfo
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Abstract
The method comprises the steps of determining a Gaussian radial basis function expression, obtaining a data point set to be fitted of the optical free-form surface to be fitted, normalizing the data to be fitted, calculating a gradient vector according to the normalized data to be fitted, analyzing and processing the gradient vector, dividing sub-apertures, setting the base number of the Gaussian radial basis functions, obtaining the number of the Gaussian radial basis functions in each sub-aperture, obtaining the total number of the Gaussian radial basis functions, uniformly distributing Gaussian radial basis function central points in the sub-apertures, determining the optimal value range of a coefficient A, determining the final fitting effect and determining the final base function total number, so that the complex optical free-form surface can be subjected to high-precision characterization, the requirements of design, processing and detection of a modern optical system can be met, the method is simple to calculate and easy to realize, The surface shape adaptability is strong, the device is suitable for any caliber, and the high-precision characterization of the optical free-form surface can be realized.
Description
Technical Field
The invention relates to the technical field of optical surface shape fitting, in particular to a Gaussian radial basis function-based optical free-form surface characterization method, aiming at characterizing a free-form surface by utilizing a Gaussian radial basis function to form a novel free-form surface characterization method.
Background
The optical free-form surface belongs to an aspheric surface, and refers to an optical surface without rotational symmetry characteristics. The main types of common optical free-form surfaces include continuous polynomial surfaces, off-axis aspheric surfaces, discontinuous surfaces, fine periodic structures, and the like, wherein the continuous polynomial surfaces and the off-axis aspheric surfaces are widely applied. The optical free-form surface has the characteristic of multiple degrees of freedom, so that the performance of an optical system can be improved, the structure of the optical system can be simplified, and the self weight of the optical system can be reduced, so that the optical free-form surface is widely applied to the fields of optical imaging, illumination, beam shaping and the like at present.
The characterization method of the optical free-form surface is the basis of the design, processing, manufacturing and detection of the optical free-form surface. Different from the traditional optical surface, most optical free-form surfaces cannot be represented by a single mathematical formula, so that research and exploration on a surface shape representation method are needed. Advances in optical free-form surface characterization methods have enabled the development of modern optics.
The characterization method of the optical free-form surface commonly used at present mainly comprises a Zernike polynomial, a Q-type polynomial, a two-dimensional Chebyshev polynomial, a two-dimensional Legendre polynomial, an XY polynomial, a B spline function, a NURBS function, a radial basis function and the like. The Zernike polynomials are widest and mature in application range, terms of the polynomials can correspond to the Seidel aberration, and system aberration evaluation is facilitated; but it is a global mode, which is not good for characterizing the local region with higher complexity in the optical free-form surface, and it can only be applied to circular aperture. Like the Zernike polynomials, the Q-type polynomials, the two-dimensional chebyshev polynomials, the two-dimensional legendre polynomials, and the XY polynomials are all global patterns, which are not favorable for characterizing the local characteristics of the optical free-form surface. The use of B-spline functions and NURBS functions is complicated and also not conducive to characterizing optical free-form surfaces.
The Gaussian radial basis function is one of radial basis functions, belongs to a non-global mode, can well represent local characteristics of an optical free-form surface, and is simple in mathematical expression and applicable to any caliber. However, when the optical free curved surface is represented by using the method, the number of the gaussian radial basis functions, the distribution of the center points of the gaussian radial basis functions and the values of the shape factors corresponding to the gaussian radial basis functions have a large influence on the representation effect.
At present, the difficulty of representing the optical free-form surface by using the Gaussian radial basis function is to research a representing method which is simple in calculation, easy to implement, strong in surface shape adaptability, suitable for any caliber and high in representing capability, so that the high-precision representation of the optical free-form surface is realized.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides the optical free-form surface characterization method based on the Gaussian radial basis function, which can be used for performing high-precision characterization on a complex optical free-form surface and can meet the requirements of design, processing and detection of a modern optical system.
The technical scheme of the invention is as follows: the optical free-form surface characterization method based on the Gaussian radial basis function comprises the following steps:
(1) the expression of the explicit gaussian radial basis function is formula (1):
wherein, Z (X)i,Yi) Is the rise, w, of the optical free-form surface to be fittedjIs a coefficient of (X)i,Yi) Is a Cartesian coordinate system in which εjIs the shape factor of the jth Gaussian radial basis function, (x)0j,y0j) Is the central point of the jth Gaussian radial basis function, m is the number of data points to be fitted, and n is the total number of the Gaussian radial basis functions;
when fitting the optical free curve, if Z (X)i,Yi)、εjAnd (x)0j,y0j) It is known to solve the formula (1) to obtain wjThe fitting result is formula (2):
the deviation of fit is formula (3)
Zerror(Xi,Yi)＝Z(Xi,Yi)-Z'(Xi,Yi)，i＝1,2,...,m (3)；
(2) Acquiring a data point set to be fitted of the optical free-form surface to be fitted;
(3) carrying out normalization processing on data to be fitted;
(4) calculating a gradient vector according to the normalized data to be fitted;
(5) analyzing and processing the gradient vector, wherein the area with larger numerical value has more violent surface shape change; the smaller the numerical value is, the more gradual the surface shape change is;
(6) dividing the sub-aperture;
(7) setting the number of Gaussian radial basis function bases, acquiring the number of Gaussian radial basis functions in each sub-aperture, and acquiring the total number of the Gaussian radial basis functions;
(8) uniformly distributing Gaussian radial basis function central points in the sub-apertures;
(9) defining the optimal value range of the coefficient A
The coefficient a is related to the shape factor epsilon of the gaussian radial basis function, and the expression of the coefficient a is formula (10):
wherein d isjThe minimum distance between the center point of the jth Gaussian radial basis function and the center points of other Gaussian radial basis functions; determining the value of the shape factor of each Gaussian radial basis function by determining the value of the coefficient A;
(10) the final fitting effect and the final total number of basis functions are clear.
The invention defines the Gaussian radial basis function expression, obtains a data point set to be fitted of the optical free-form surface to be fitted, normalizes the data to be fitted, calculates a gradient vector according to the normalized data to be fitted, analyzes and processes the gradient vector, divides sub-apertures, sets the base number of the Gaussian radial basis function, obtains the number of the Gaussian radial basis functions in each sub-aperture, obtains the total number of the Gaussian radial basis functions, uniformly distributes the central points of the Gaussian radial basis functions in the sub-apertures, defines the optimal value range of the coefficient A, defines the final fitting effect and defines the final total number of the basis functions, the method has the advantages of simple calculation, easy realization, strong surface shape adaptability, suitability for any caliber and realization of high-precision characterization of the optical free-form surface.
Drawings
FIG. 1 is a flow chart of a method for characterizing an optical free-form surface based on Gaussian radial basis functions according to the present invention.
FIG. 2 is a diagram of data to be fitted after normalization of an optical free-form surface to be fitted.
Fig. 3 is a diagram illustrating the sub-aperture division and the distribution of the center points of the basis functions.
Fig. 4 is a fitting deviation distribution diagram corresponding to the final fitting effect.
Detailed Description
As shown in fig. 1, the method for characterizing an optical free-form surface based on a gaussian radial basis function includes the following steps:
(1) the expression of the explicit gaussian radial basis function is formula (1):
wherein, Z (X)i,Yi) Is the rise, w, of the optical free-form surface to be fittedjIs a coefficient of (X)i,Yi) Is a Cartesian coordinate system in which εjIs the shape factor of the jth Gaussian radial basis function, (x)0j,y0j) Is the central point of the jth Gaussian radial basis function, m is the number of data points to be fitted, and n is the total number of the Gaussian radial basis functions;
when fitting the optical free curve, if Z (X)i,Yi)、εjAnd (x)0j,y0j) It is known to solve the formula (1) to obtain wjThe fitting result is formula (2):
the deviation of fit is formula (3)
Zerror(Xi,Yi)＝Z(Xi,Yi)-Z'(Xi,Yi)，i＝1,2,...,m (3)；
(2) Acquiring a data point set to be fitted of the optical free-form surface to be fitted;
(3) carrying out normalization processing on data to be fitted;
(4) calculating a gradient vector according to the normalized data to be fitted;
(5) analyzing and processing the gradient vector, wherein the area with larger numerical value has more violent surface shape change; the smaller the numerical value is, the more gradual the surface shape change is;
(6) dividing the sub-aperture;
(7) setting the number of Gaussian radial basis function bases, acquiring the number of Gaussian radial basis functions in each sub-aperture, and acquiring the total number of the Gaussian radial basis functions;
(8) uniformly distributing Gaussian radial basis function central points in the sub-apertures;
(9) defining the optimal value range of the coefficient A
The coefficient a is related to the shape factor epsilon of the gaussian radial basis function, and the expression of the coefficient a is formula (10):
wherein d isjThe minimum distance between the center point of the jth Gaussian radial basis function and the center points of other Gaussian radial basis functions; determining the value of the shape factor of each Gaussian radial basis function by determining the value of the coefficient A;
(10) the final fitting effect and the final total number of basis functions are clear.
The invention defines the Gaussian radial basis function expression, obtains a data point set to be fitted of the optical free-form surface to be fitted, normalizes the data to be fitted, calculates a gradient vector according to the normalized data to be fitted, analyzes and processes the gradient vector, divides sub-apertures, sets the base number of the Gaussian radial basis function, obtains the number of the Gaussian radial basis functions in each sub-aperture, obtains the total number of the Gaussian radial basis functions, uniformly distributes the central points of the Gaussian radial basis functions in the sub-apertures, defines the optimal value range of the coefficient A, defines the final fitting effect and defines the final total number of the basis functions, the method has the advantages of simple calculation, easy realization, strong surface shape adaptability, suitability for any caliber and realization of high-precision characterization of the optical free-form surface.
Preferably, in the step (2), the data points to be fitted are points on the optical free-form surface to be fitted, and are coordinates (X) of the optical free-form surface to be fitted in a cartesian coordinate systemi,Yi) Corresponding rise Z (X)i,Yi) The number of the elements in the point set is m, and the acquisition mode of the data point set to be fitted comprises the following steps: and artificially generating and measuring the related data obtained by measuring the optical free-form surface real object to be fitted by using a contourgraph, a three-coordinate measuring machine or an interferometer.
And carrying out normalization processing on the data to be fitted, and carrying out analysis, fitting and other processing on the normalized data to be fitted in subsequent steps. And through normalization processing, the data to be fitted in the step two are placed in a uniform range for fitting, so that the condition that the data to be fitted are adapted to the dynamic range according to the data to be fitted in the fitting process is avoided. Preferably, in the step (3), the rise is normalized to the coordinate [0,1 ] in a cartesian coordinate system]Internal; normalizing the caliber to the unit circle, and the coordinate (X)i,Yi) Carrying out normalization processing; normalized, coordinate (X)i,Yi) Become (x)i,yi) Rise becomes z (x)i,yi)。
Preferably, in the step (4), a gradient vector of each data point is calculated according to the normalized data to be fitted:
preferably, in the step (5), according to the solution result of the gradient vector of each point in the formula (4), the amplitude of the gradient vector of each point is solved by using the formula (5):
and (5) analyzing the complexity of the surface shape change of the optical free-form surface to be fitted according to the solving result of the formula (5). G (x)i,yi) The area with larger numerical value has more severe surface shape change; g (x)i,yi) The smaller the value of the area, the more gradual the surface shape change.
The region with the most severe surface shape change occupies G (x)i,yi) The vast majority of the medium-large-valued points, resulting in annihilation of regions that are less intense than the region but need to be highlighted as well, are pairs G (x)i,yi) And (3) carrying out root cutting treatment:
at this time, each region having a relatively sharp surface shape changes occupies G' (x)i,yi) More large value points in.
Preferably, in the step (6), the number N of sub-apertures is set, and N is a positive integer after being squared; and dividing the aperture of the optical free-form surface by using the rectangular aperture according to the number of the sub-apertures.
Preferably, in the step (7), the base number of the Gaussian radial basis function is nbaseNumber n of Gaussian radial basis functions within each subaperturejComprises the following steps:
nj＝ceil(k·PVj),j＝1,2,...,N (7)
wherein j is the jth sub-aperture, njIs the number of Gaussian radial basis functions in the jth sub-aperture, PVjIs G' (x) in the jth sub-aperturei,yi) Is greater than the difference between the maximum value and the minimum value of (c),
k is related to the base number of the Gaussian radial basis function, ceil () is an integer operation; the expression for k is:
the total number n of gaussian radial basis functions is:
and (3) artificially setting a Gaussian radial basis function base number, acquiring the number of Gaussian radial basis functions in each sub-aperture according to formulas (7) and (8), and acquiring the total number of the Gaussian radial basis functions according to a formula (9).
Preferably, in the step (8), the number n of Gaussian radial basis functions in each sub-aperture is determinedjN, uniformly distributing the gaussian radial basis function center point in each sub-aperture, thereby obtaining the coordinate (x) of the gaussian radial basis function center point0j,y0j)，j＝1,2,...,n；
The specific distribution mode is as follows:
(I) when n isjWhen the value is 1, the central points of the Gaussian radial basis functions are distributed in the center of the sub-aperture;
(II) when n isjWhen the value is 2, the central points of the Gaussian radial basis functions are uniformly distributed on the central line of the sub-aperture;
(III) when njNot equal to 1 and njIf not equal to 2, if
The central point of the Gaussian radial basis function is uniformly distributed in the subaperture in a square matrix shape, and the number of rows and columns is l1(ii) a If it is
Is a non-integer, then l'1＝ceil(l1) At this time, if l2＝nj/l'1Is an integer, then the center point of the Gaussian radial basis function is according to l2×l'1The rectangular shape of the porous material is uniformly distributed in the sub-aperture; if l2＝nj/l'1Is a non-integer of l'2＝floor(l2) L to l'2×l'1Is distributed in a rectangular shape njA part of the center points of the Gaussian radial basis functions in (1), and the remaining nj-l'2×l'1The center points of the Gaussian radial basis functions are uniformly distributed in the first row.
After the coefficients A are clarified, the shape factor ε of each Gaussian radial basis functionjThe coordinates (x) of the center point of the Gaussian radial basis function obtained in the step eight can be determined0j,y0j) That is, the coefficient w can be solved according to the formula (1)jThereby completing the fitting of the optical free-form surface.
For the same optical free-form surface to be fitted, under the condition that the total number n of Gaussian radial basis functions is different, the fitting effect shows the trend that the fitting effect is firstly improved and then reduced along with the increase of the coefficient A, namely the PV of the fitting deviation is firstly reduced and then increased along with the increase of the coefficient A. Meanwhile, the values of the coefficient A corresponding to the best fitting effect which can be achieved by the total number n of the Gaussian radial basis functions are all within a certain value range of the coefficient A, and the range is the optimal value range of the coefficient A.
Preferably, in the step (9), an optimization method is used to obtain an optimal value range of the coefficient a, and the optimization method includes the following sub-steps:
(9.1) setting the initial value and the step value of the coefficient A. The initial value of the coefficient a should be set to a small value greater than zero and the step value to a small value greater than zero;
(9.2) setting the value of the current coefficient A according to the initial value and the step value of the coefficient A, and combining the coordinate (x) of the central point of the Gaussian radial basis function obtained in the step eight0j,y0j) Calculating the corresponding fitting result according to the formula (1) and the formula (10);
(9.3) calculating the fitting deviation corresponding to the current coefficient A according to the formula (3), and calculating the PV of the fitting deviation;
(9.4) judging whether the fitting effect corresponding to the current coefficient A is in a descending trend or not according to the PV variation of the fitting deviation, and if not, repeating the second step to the fourth step; if the current value is in the descending trend, stopping updating the value of the coefficient A;
(9.5) analyzing a range with a good fitting effect according to the PV of the fitting deviation corresponding to each coefficient A, wherein the range of the coefficient A corresponding to the range is the optimal value range of the coefficient A; and simultaneously, obtaining the best fitting effect achieved by the current basis function base number, namely the minimum value in PV of fitting deviation corresponding to the coefficient A in the best value range.
The fitting effect of the optical free-form surface can be improved along with the increase of the total number n of the Gaussian radial basis functions, but after n reaches a certain number, the improvement effect of the increase of n on the fitting effect is very limited. According to this rule, preferably, in the step (10), the final fitting effect and the final total number n of gaussian radial basis functions are determined by an optimization method, and the optimization method comprises the following sub-steps:
(10.1) manually setting the fitting precision requirement; analyzing and judging whether the current fitting effect meets the fitting precision requirement or not according to the best fitting effect corresponding to the current Gaussian radial basis function base number obtained in the step (9); if the fitting precision requirement is not met, continuing to execute the next step; if the fitting precision requirement is met, the best fitting effect corresponding to the Gaussian radial basis function base number at the moment is the final fitting effect, the total number of the Gaussian radial basis functions at the moment is the final Gaussian radial basis function total number, the final Gaussian radial basis function total number is calculated according to the formula (9), and the subsequent steps are not executed;
(10.2) the number of bases n of the Gaussian radial basis function artificially set in step (7)baseOn the basis of the numerical value, setting a base number stepping value of the basis function;
(10.3) increasing the number of bases n of the Gaussian radial basis function according to the step valuebaseAfter updating the Gaussian radial basis function base number, sequentially executing the step (7) and the step (8) to obtain the Gaussian radial basis function number in each sub-aperture and the Gaussian radial basis function number in each sub-aperture under the current Gaussian radial basis functionDistribution of the heart points and the total number of Gaussian radial basis functions;
(10.4) setting the value of the coefficient A according to the optimal value range of the coefficient A obtained in the step (9), fitting according to the formula (1) and the formula (10), and determining the optimal fitting effect corresponding to the current Gaussian radial basis function base number, namely the minimum value in the PV of the fitting deviation corresponding to the coefficient A in the optimal value range;
(10.5) analyzing and judging whether the current fitting effect meets the fitting precision requirement or not according to the best fitting effect corresponding to the current Gaussian radial basis function base number; if the fitting precision requirement is not met, repeating the steps (3) - (4); and if the fitting precision requirement is met, stopping updating the Gaussian radial basis function base number, wherein the best fitting effect corresponding to the Gaussian radial basis function base number at the moment is the final fitting effect, the total number of the Gaussian radial basis functions at the moment is the final Gaussian radial basis function total number, and the final Gaussian radial basis function total number is calculated according to the formula (9).
One embodiment of the present invention is described in detail below. The optical free-form surface characterization method based on the Gaussian radial basis function is realized in the following mode:
the process of establishing the optical free-form surface characterization method based on the Gaussian radial basis function is shown as the attached figure 1, and the specific implementation steps are as follows:
the method comprises the following steps: and (4) defining a Gaussian radial basis function expression.
The expression of the Gaussian radial basis function for representing the optical free-form surface is as follows:
wherein, Z (X)i,Yi) Is the rise, w, of the optical free-form surface to be fittedjIs a coefficient of (X)i,Yi) Is a Cartesian coordinate system in which εjIs the shape factor of the jth Gaussian radial basis function, (x)0j,y0j) Is the central point of the jth Gaussian radial basis function, m is the number of data points to be fitted, and n is the total number of the Gaussian radial basis functions.
Step two: and acquiring a data point set to be fitted of the optical free-form surface to be fitted.
An optical free-form surface to be fitted is artificially constructed, the caliber is a circular caliber, the diameter of the caliber is 20mm, and the expression is as follows:
and acquiring a data point set to be fitted of the optical free-form surface to be fitted in a manner of artificial generation. Artificially generating coordinates (X) corresponding to uniform grids in the circular caliber in a Cartesian coordinate system according to the expression and caliber sizei,Yi) And its corresponding rise Z (X)i,Yi). The number m of elements in the point set is 7825.
Step three: and normalizing the data to be fitted.
And (4) carrying out normalization processing on the data to be fitted acquired in the step two, and carrying out analysis, fitting and other processing on the normalized data to be fitted in the subsequent steps. In a Cartesian coordinate system, the rise Z (X)i,Yi) Normalized to the coordinates [0,1 ]]Internal; normalizing the calibre to within the unit circle, i.e. the coordinate (X)i,Yi) Normalization processing is performed to within the unit circle. Normalized, coordinate (X)i,Yi) Become (x)i,yi) Rise Z (X)i,Yi) Becomes z (x)i,yi). The normalized data to be fitted is shown in fig. 2.
Step four: and calculating a gradient vector according to the normalized data to be fitted.
Calculating the gradient vector of each rise data point according to the normalized data to be fitted:
step five: the gradient vectors are processed analytically.
And solving the amplitude of the gradient vector of each point according to the gradient vector result of each point:
for G (x)i,yi) And (3) carrying out root cutting treatment:
step six: and dividing the sub-aperture.
The number N of the sub apertures is set to 144, and the aperture of the optical free-form surface is divided by using the rectangular aperture according to the number of the sub apertures, as shown in fig. 3.
Step seven: and setting the Gaussian radial basis function base number, acquiring the Gaussian radial basis function number in each sub-aperture, and acquiring the total Gaussian radial basis function number.
Setting the base number n of Gaussian radial basis functionbase200, k is calculated to be approximately equal to 18, and the number n of Gaussian radial basis functions in each sub-aperturejRespectively as follows: n is1＝0，n2＝0，n3＝0，n4＝0，n5＝4，n6＝4，n7＝4，n8＝4，n9＝0，n10＝0，n11＝0，n12＝0，n13＝0，n14＝0，n15＝4，n16＝4，n17＝4，n18＝1，n19＝1，n20＝1，n21＝4，n22＝4，n23＝0，n24＝0，n25＝0，n26＝4，n27＝4，n28＝1，n29＝1，n30＝1，n31＝1，n32＝1，n33＝1，n34＝4，n35＝4，n36＝0，n37＝0，n38＝4，n39＝1，n40＝1，n41＝1，n42＝1，n43＝1，n44＝1，n45＝1，n46＝1，n47＝4，n48＝4，n49＝4，n50＝4，n51＝1，n52＝1，n53＝1，n54＝1，n55＝1，n56＝1，n57＝1，n58＝1，n59＝1，n60＝4，n61＝4，n62＝1，n63＝1，n64＝1，n65＝1，n66＝2，n67＝2，n68＝1，n69＝1，n70＝1，n71＝1，n72＝4，n73＝4，n74＝1，n75＝1，n76＝2，n77＝3，n78＝4，n79＝2，n80＝1，n81＝1，n82＝1，n83＝1，n84＝4，n85＝4，n86＝1，n87＝1，n88＝3，n89＝3，n90＝3，n91＝1，n92＝1，n93＝1，n94＝1，n95＝1，n96＝4，n97＝0，n98＝4，n99＝1，n100＝3，n101＝2，n102＝2，n103＝1，n104＝1，n105＝1，n106＝1，n107＝4，n108＝4，n109＝0，n110＝4，n111＝4，n112＝1，n113＝1，n114＝1，n115＝1，n116＝1，n117＝1，n118＝1，n119＝4，n120＝0，n121＝0，n122＝0，n123＝4，n124＝4，n125＝1，n126＝1，n127＝1，n128＝1，n129＝4，n130＝4，n131＝0，n132＝0，n133＝0，n134＝0，n135＝0，n136＝4，n137＝4，n138＝4，n139＝4，n140＝4，n141＝4，n142＝0，n143＝0，n1440. The total number n of gaussian radial basis functions is 258.
Step eight: and uniformly distributing Gaussian radial basis function center points in the sub-apertures.
According to the number n of Gaussian radial basis functions in each sub-aperturejJ 1, 2.. 144. gaussian radial basis function center points are uniformly distributed in each sub-aperture, thereby obtaining coordinates (x) of the gaussian radial basis function center points0j,y0j) J 1, 2.., 258. The distribution of the center points of the gaussian radial basis functions and the coordinate positions thereof are shown in fig. 3.
Step nine: and defining the optimal value range of the coefficient A.
Obtaining the optimal value range of the coefficient A by using an optimization algorithm, wherein the optimization algorithm is realized by the following steps:
the first step is as follows: the initial value of the coefficient a is set to 0.5, and the step value of the coefficient a is set to 0.5.
The second step is that: and (4) integrating the coordinates of the central point of the Gaussian radial basis function obtained in the step eight when the current coefficient A is 0.5, and calculating the corresponding fitting result according to the expressions (1) and (10).
The third step: the fitting deviation when the current coefficient a is 0.5 is calculated from equation (3), and PV thereof is calculated to be 0.7174.
The fourth step: and judging whether the fitting effect corresponding to the current coefficient A is in a descending trend or not through the change of PV of the fitting deviation. And taking the minimum value in PV of fitting deviation corresponding to the value of the coefficient A taken in the past as the best fitting effect corresponding to the current Gaussian radial basis function base number. And if the difference value of the PV of the fitting deviation corresponding to the current coefficient A and the best fitting effect is more than 10 times of the minimum fitting deviation, determining that the fitting deviation corresponding to the current coefficient A is in a descending trend.
The current best fit effect is 0.7174, and the fit effect corresponding to the coefficient a being 0.5 is not in a downward trend; taking A to 1 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.1497, wherein the best fitting effect is 0.7174 at the moment and the current fitting deviation is not in a descending trend; taking A to be 1.5 according to the step value, and calculating the PV of the current fitting deviation to be 0.0208, wherein the best fitting effect is 0.1497 and the current fitting deviation is not in a descending trend; taking A to 2 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0118, wherein the best fitting effect is 0.0208 at the moment, and the current fitting deviation is not in a descending trend; taking A to be 2.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0211, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 3 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0234, wherein the best fitting effect is 0.0118 at the moment and is not in a descending trend; taking A to be 3.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0254, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 4 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0268, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 4.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0281, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0364, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to be 5.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0385, wherein the best fitting effect is 0.0118 at the moment, and the current fitting deviation is not in a descending trend; taking A to 6 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0443, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to be 6.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0549, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 7 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0676, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to be 7.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0775, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 8 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0871, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 8.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.0962, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 9 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.1092, wherein the best fitting effect is 0.0118 at the moment and is not in a descending trend; taking A to be 9.5 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.1208, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking A to 10 according to the stepping value, and calculating the PV of the current fitting deviation to be 0.1294, wherein the best fitting effect is 0.0118 at the moment and the current fitting deviation is not in a descending trend; taking a to 10.5 from the step value, the PV of the current fitting deviation is calculated to be 0.1411, the best fitting effect is 0.0118 at the moment, the updating of the coefficient a is stopped in a descending trend.
The fifth step: and setting the range of the corresponding coefficient A as the optimal value range of the coefficient A when the PV of the fitting deviation is less than 5 times of the optimal fitting effect. Number of basis of Gaussian radial basis function nbaseWhen 200, the corresponding best fit effect is 0.0118. When PV of the fitting deviation is less than 0.0118 × 5, 0.0590, the corresponding range of the coefficient a, that is, the optimal value range of the coefficient a is [1.5, 6.5 ]]。
Step ten: the final fitting effect and the final total number of basis functions are clear.
And (4) determining the final fitting effect and the final total number of Gaussian radial basis functions through an optimization algorithm. The optimization algorithm is realized by the following steps:
the first step is as follows: and manually setting the requirement of the fitting accuracy, wherein the requirement of the fitting accuracy is that the PV of the fitting deviation is less than or equal to 10E-05. And the best fitting effect corresponding to the current Gaussian radial basis function base number calculated in the ninth step is 0.0118, the fitting precision requirement is not met, and the next step is continuously executed.
The second step is that: the base number n of the Gaussian radial basis function artificially set in the step sevenbaseOn the basis of 200, a base number step value of the basis function is set to 100.
The third step: increasing the base number n of Gaussian radial basis functions according to the step valuebaseThe updated base number n of the Gaussian radial basis functionbaseAnd (300) sequentially executing the seventh step and the eighth step, and acquiring the number of Gaussian radial basis functions in each sub-aperture, the distribution of central points of the Gaussian radial basis functions in each sub-aperture and the total number of the Gaussian radial basis functions under the current Gaussian radial basis function basis number.
And setting the values of the coefficients A to be 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6 and 6.5 respectively according to the optimal value range of the coefficients A obtained in the step nine. When the coefficient a is 2, the best fit effect is obtained, and PV of the fitting deviation is 0.0028.
The fourth step: when n isbaseWhen the fitting deviation is 300, the PV of the fitting deviation corresponding to the optimal fitting effect is 0.0028, and the fitting precision requirement is not met; taking n according to the base number step value of the basis functionbaseWhen the coefficient A is 2.5, the best fitting effect is obtained, PV of the corresponding fitting deviation is 9.8664E-04, and the fitting precision requirement is not met; taking n according to the base number step value of the basis functionbaseWhen the coefficient A is 2.5, the best fitting effect is obtained, PV of the corresponding fitting deviation is 2.7938E-04, and the fitting precision requirement is not met; taking n according to the base number step value of the basis functionbaseWhen the coefficient A is equal to 3, the best fitting effect is obtained, PV of the corresponding fitting deviation is 1.8691E-04, and the fitting precision requirement is not met; taking n according to the base number step value of the basis functionbaseWhen the coefficient A is equal to 3, the best fitting effect is obtained, PV of the corresponding fitting deviation is 3.0446E-05, and the fitting precision requirement is met. The updating of the basis function base number is stopped.
The PV of the fitting deviation corresponding to the final fitting effect is 3.0446E-05, and the distribution of the fitting deviation is shown in FIG. 4. The final total number of gaussian radial basis functions n is 767.
The invention has the following beneficial effects:
1. the method has simple calculation, can obtain better fitting effect through a simple optimization algorithm, does not need to perform a complex iterative calculation process, and simplifies the calculation.
2. The method is easy to implement, relevant key parameters are determined through an optimization algorithm, and experience is not needed, so that the method is easy to implement.
3. The surface shape adaptability is strong, and for optical free-form surfaces with different complexity, the related setting is determined through a related algorithm, so that the adaptability is strong.
4. The method is suitable for any caliber, not only for common calibers such as round, rectangle and the like, but also for other arbitrary calibers such as ellipse, hexagon and the like.
5. The characterization capability is high, and the complex optical free-form surface can be characterized with high precision.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the present invention in any way, and all simple modifications, equivalent variations and modifications made to the above embodiment according to the technical spirit of the present invention still belong to the protection scope of the technical solution of the present invention.
Claims (10)
1. The optical free-form surface characterization method based on the Gaussian radial basis function is characterized by comprising the following steps of: which comprises the following steps:
(1) the expression of the explicit gaussian radial basis function is formula (1):
wherein, Z (X)i,Yi) Is the rise, w, of the optical free-form surface to be fittedjIs a coefficient of (X)i,Yi) Is a Cartesian coordinate system in which εjIs the shape factor of the jth Gaussian radial basis function, (x)0j,y0j) Is the central point of the jth Gaussian radial basis function, m is the number of data points to be fitted, and n is the total number of the Gaussian radial basis functions;
when fitting the optical free curve, if Z (X)i,Yi)、εjAnd (x)0j,y0j) It is known to solve the formula (1) to obtain wjThe fitting result is formula (2):
the deviation of fit is formula (3)
Zerror(Xi,Yi)＝Z(Xi,Yi)-Z'(Xi,Yi)，i＝1,2,...,m (3)；
(2) Acquiring a data point set to be fitted of the optical free-form surface to be fitted;
(3) carrying out normalization processing on data to be fitted;
(4) calculating a gradient vector according to the normalized data to be fitted;
(5) analyzing and processing the gradient vector, wherein the area with larger numerical value has more violent surface shape change; the smaller the numerical value is, the more gradual the surface shape change is;
(6) dividing the sub-apertures, wherein the number of the sub-apertures is N;
(7) setting the base number of the Gaussian radial basis function to obtain the number n of the Gaussian radial basis function in each sub-aperturejJ is 1,2, …, N, and obtaining the total number N of gaussian radial basis functions;
(8) uniformly distributing Gaussian radial basis function central points in the sub-apertures;
(9) defining the optimal value range of the coefficient A
The coefficient a is related to the shape factor epsilon of the gaussian radial basis function, and the expression of the coefficient a is formula (10):
wherein d isjThe minimum distance between the center point of the jth Gaussian radial basis function and the center points of other Gaussian radial basis functions; determining the value of the shape factor of each Gaussian radial basis function by determining the value of the coefficient A;
(10) defining the final fitting effect and the final total number of the basis functions;
wherein the total number n of Gaussian radial basis functions is:
2. the optical free-form surface based on gaussian radial basis function of claim 1A characterization method, characterized by: in the step (2), the data points to be fitted are points on the optical free-form surface to be fitted, and are coordinates (X) of the optical free-form surface to be fitted in a Cartesian coordinate systemi,Yi) Corresponding rise Z (X)i,Yi) The number of the elements in the point set is m, and the acquisition mode of the data point set to be fitted comprises the following steps: and artificially generating and measuring the related data obtained by measuring the optical free-form surface real object to be fitted by using a contourgraph, a three-coordinate measuring machine or an interferometer.
3. The method for characterizing an optical free-form surface based on gaussian radial basis functions of claim 2, wherein: in the step (3), the rise is normalized to the coordinate [0,1 ] under a Cartesian coordinate system]Internal; normalizing the caliber to the unit circle, and the coordinate (X)i,Yi) Carrying out normalization processing; normalized, coordinate (X)i,Yi) Become (x)i,yi) Rise becomes z (x)i,yi)。
4. The method for characterizing an optical free-form surface based on Gaussian radial basis functions as claimed in claim 3, wherein: in the step (4), a gradient vector of each data point is calculated according to the normalized data to be fitted:
5. the method according to claim 4, wherein the method comprises the following steps: in the step (5), according to the gradient vector solving result of each point in the formula (4), the amplitude of each gradient vector is solved by using the formula (5):
the region with the most severe surface shape change occupies G (x)i,yi) The vast majority of the medium-large-valued points, resulting in annihilation of regions that are less intense than the region but need to be highlighted as well, are pairs G (x)i,yi) And (3) carrying out root cutting treatment:
at this time, each region with a drastically changed surface shape occupies G' (x)i,yi) And (4) medium and large value points.
6. The method for characterizing an optical free-form surface based on Gaussian radial basis functions as claimed in claim 5, wherein: in the step (6), the number N of sub-apertures is set, and the number N is a positive integer after being squared; and dividing the aperture of the optical free-form surface by using the rectangular aperture according to the number of the sub-apertures.
7. The method according to claim 6, wherein the method comprises the following steps: in the step (7), the base number of the Gaussian radial basis function is nbaseNumber n of Gaussian radial basis functions within each subaperturejComprises the following steps:
nj＝ceil(k·PVj),j＝1,2,...,N (7)
wherein j is the jth sub-aperture, njIs the number of Gaussian radial basis functions in the jth sub-aperture, PVjIs G' (x) in the jth sub-aperturei,yi) K is related to the base number of the Gaussian radial basis function, ceil () is an integer operation; the expression for k is:
the total number n of gaussian radial basis functions is:
and (3) artificially setting a Gaussian radial basis function base number, acquiring the number of Gaussian radial basis functions in each sub-aperture according to formulas (7) and (8), and acquiring the total number of the Gaussian radial basis functions according to a formula (9).
8. The method according to claim 7, wherein the method comprises the following steps: in the step (8), the number n of Gaussian radial basis functions in each sub-aperture is determinedjN, uniformly distributing the gaussian radial basis function center point in each sub-aperture, thereby obtaining the coordinate (x) of the gaussian radial basis function center point0j,y0j) J is 1,2,. ang, n; the specific distribution mode is as follows:
(I) when n isjWhen the value is 1, the central points of the Gaussian radial basis functions are distributed in the center of the sub-aperture;
(II) when n isjWhen the value is 2, the central points of the Gaussian radial basis functions are uniformly distributed on the central line of the sub-aperture;
(III) when njNot equal to 1 and njIf not equal to 2, if
The central point of the Gaussian radial basis function is uniformly distributed in the subaperture in a square matrix shape, and the number of rows and columns is l1(ii) a If it is
Is a non-integer, then l'1＝ceil(l1) At this time, if l2＝nj/l′1Is an integer, then the center point of the Gaussian radial basis function is according to l2×l′1The rectangular shape of the porous material is uniformly distributed in the sub-aperture; if l2＝nj/l′1Is a non-integer of l'2＝floor(l2) L to l'2×l′1Is distributed in a rectangular shape njIn a part of the Gaussian radial basis functionsHeart point, remaining nj-l′2×l′1The center points of the Gaussian radial basis functions are uniformly distributed in the first row.
9. The method for characterizing an optical free-form surface based on gaussian radial basis functions of claim 8, wherein: in the step (9), an optimization method is used to obtain the optimal value range of the coefficient a, and the optimization method comprises the following sub-steps:
(9.1) setting an initial value and a step value of the coefficient A, wherein the initial value of the coefficient A is set to be a smaller value larger than zero, and the step value is set to be a smaller value larger than zero;
(9.2) setting the value of the current coefficient A according to the initial value and the step value of the coefficient A, and combining the coordinate (x) of the central point of the Gaussian radial basis function obtained in the step (8)0j,y0j) Calculating the corresponding fitting result according to the formula (1) and the formula (10);
(9.3) calculating the fitting deviation corresponding to the current coefficient A according to the formula (3), and calculating the PV of the fitting deviation;
(9.4) judging whether the fitting effect corresponding to the current coefficient A is in a descending trend or not according to the PV change of the fitting deviation, and repeating (9.2) to (9.4) if the fitting effect corresponding to the current coefficient A is not in the descending trend; if the current value is in the descending trend, stopping updating the value of the coefficient A;
(9.5) analyzing a range with a good fitting effect according to the PV of the fitting deviation corresponding to each coefficient A, wherein the range of the coefficient A corresponding to the range is the optimal value range of the coefficient A; and simultaneously, obtaining the best fitting effect achieved by the current basis function number as the minimum value of the fitting deviation PV corresponding to the coefficient A in the best value range.
10. The method for characterizing an optical free-form surface based on gaussian radial basis functions of claim 9, wherein: in the step (10), the final fitting effect and the final total number n of gaussian radial basis functions are determined by an optimization method, and the optimization method comprises the following sub-steps:
(10.1) manually setting the fitting precision requirement; analyzing and judging whether the current fitting effect meets the fitting precision requirement or not according to the best fitting effect corresponding to the current Gaussian radial basis function base number obtained in the step (9); if the fitting precision requirement is not met, continuing to execute the next step; if the fitting precision requirement is met, the best fitting effect corresponding to the Gaussian radial basis function base number at the moment is the final fitting effect, the total number of the Gaussian radial basis functions at the moment is the final Gaussian radial basis function total number, the final Gaussian radial basis function total number is calculated according to the formula (9), and the subsequent steps are not executed;
(10.2) the number of bases n of the Gaussian radial basis function artificially set in step (7)baseOn the basis of the numerical value, setting a base number stepping value of the basis function;
(10.3) increasing the number of bases n of the Gaussian radial basis function according to the step valuebaseAfter updating the base number of the Gaussian radial basis functions, sequentially executing the step (7) and the step (8) to obtain the number of the Gaussian radial basis functions in each sub-aperture, the distribution of the center points of the Gaussian radial basis functions in each sub-aperture and the total number of the Gaussian radial basis functions under the current Gaussian radial basis functions;
(10.4) setting the value of the coefficient A according to the optimal value range of the coefficient A obtained in the step (9), fitting according to the formulas (1) and (10), and determining the optimal fitting effect corresponding to the base number of the current Gaussian radial basis function as the minimum value in the PV of the fitting deviation corresponding to the coefficient A in the optimal value range;
(10.5) analyzing and judging whether the current fitting effect meets the fitting precision requirement or not according to the best fitting effect corresponding to the current Gaussian radial basis function base number; if the fitting precision requirement is not met, repeating the steps (3) - (4); and if the fitting precision requirement is met, stopping updating the Gaussian radial basis function base number, wherein the best fitting effect corresponding to the Gaussian radial basis function base number at the moment is the final fitting effect, the total number of the Gaussian radial basis functions at the moment is the final Gaussian radial basis function total number, and the final Gaussian radial basis function total number is calculated according to the formula (9).
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