CN118152709A - Residence time algorithm based on wavelet base fitting - Google Patents
Residence time algorithm based on wavelet base fitting Download PDFInfo
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Abstract
The invention relates to the technical field of optical processing residence time calculation, in particular to a residence time algorithm based on wavelet base fitting. Comprising the following steps: measuring the surface shape of a workpiece to be processed through an interferometer to obtain a surface shape residual error; selecting a processing technology and performing fixed-point processing measurement to obtain a removal function; setting a polishing track type and polishing track parameters according to the surface shape sampling size of a workpiece to be processed; expressing the surface shape of the workpiece to be processed by using wavelet to obtain a surface shape spectrum; expressing a removal function by using a wavelet to obtain a removal function feature matrix; and constructing a matrix equation by using the surface shape spectrum and the removal function feature matrix, and solving the residence time spectrum by using an iteration method so as to calculate the residence time. The advantages are that: the advantage of local fitting of wavelets is utilized, so that the residence time solution is more accurate in a small scale range, thereby realizing accurate removal, and being beneficial to the accurate control of the residence time of elements with extremely high roughness requirements, such as a strong laser reflector or a photoetching objective lens; has repeatability.
Description
Technical Field
The invention relates to the technical field of optical processing residence time calculation, in particular to a residence time algorithm based on wavelet base fitting.
Background
The multi-resolution analysis method is a method for performing image analysis with different resolutions in the field of image science, but the theory can be applied to the characterization of surface shape residual errors. Wherein the expression for solving the residence time problem is as follows:
；
Wherein, Representing convolution,/>Representing track coverage.
Currently, there are mainly the following five methods in terms of residence time calculation: first, bayesian methods, paper Algorithm for ion beam figuring of low-gradient mirrors, mainly describes this method; second, the Fourier transform method, paper ITERATIVE BLIND DECONVOLUTION METHOD FOR DWELL-time adjustment, mainly describes this method; thirdly, a matrix method is mainly introduced in paper 'solving algorithm of magneto-rheological processing residence time of large-caliber optical elements'; fourth, the direct convolution method, which is mainly described in the paper Dwell-time algorithm for polishing large optics.
Each of the above four methods has its own advantages, but has a problem that the residence time is obtained by iterative correction of the solution, which results in the solution not being smooth, so that the dynamic performance of the machine tool is examined.
Fifth, polynomial fitting methods, which fit solutions by smooth analytical polynomials, result in solutions that avoid the machine tool dynamic performance problems described above. The paper Zernike mapping of optimum DWELL TIME IN DETERMINISTIC fabrication of freeform optics mainly describes this method. However, this approach is a global fit and cannot be modified locally to the solution so that the dwell time solution will deviate from the solution of the original problem. Therefore, the solution of the existing residence time algorithm still cannot well balance the dynamic performance of the machine tool with the local accurate residence time.
Disclosure of Invention
The invention provides a residence time algorithm based on wavelet base fitting to solve the problems.
The invention aims to provide a residence time algorithm based on wavelet base fitting, which specifically comprises the following steps:
S1, measuring the surface shape of a workpiece to be processed through an interferometer to obtain a discretized surface shape residual error of the workpiece to be processed ；
S2, selecting a processing technology of the workpiece to be processed, and measuring a removal function；
S3, setting polishing track types and polishing track parameters according to the surface shape sampling size of the workpiece to be processed, wherein resident points on the tracks are as follows；
S4, expressing the surface shape of the workpiece to be processed by using wavelet: selecting a Gaussian scale functionAnd/>Constructing a one-dimensional scale function family and a one-dimensional wavelet function family:
；
In the method, in the process of the invention, Representing scale factors,/>Representing a displacement coefficient;
constructing a two-dimensional wavelet function family according to a tensor construction method in multi-resolution analysis by utilizing a one-dimensional scale function family and a one-dimensional wavelet function family 、/>、/>、/>；
Residual error of the surface shape of the workpiece to be processedExpressed as:
；
In the method, in the process of the invention, Is the shape of face/>And/>The number of directional discretized points. Shift wavelet amount/>Is limited in the surface shapeWithin,/>Is the scale factor,/>Is a two-dimensional vertical detail coefficient,/>Is a two-dimensional horizontal detail coefficient,/>Is a two-dimensional oblique detail coefficient. Two-dimensional scale function/>The two-dimensional wavelet functions are divided into three classes, the first being vertical wavelets/>The second type is horizontal waveletThe third class is the oblique wavelet/>。/>Corresponding to horizontal, vertical and inclined numbers respectively; /(I)Is the maximum detail scale sequence number,/>Is the minimum detail scale number;
The rewritten column vector form is as follows:
；
Wherein:
The wavelet basis vectors are:
The surface spectrum is:
；
S5, using wavelet to express a removal function, and converting the convolution kernel into the convolution kernel according to two-dimensional Taylor expansion:
；
In the method, in the process of the invention, Is a removal function/>Is a wavelet feature matrix of (a);
S6, solving a fitting coefficient matrix equation of the construction equation: ; in the/> Is the residence time spectrum;
For a pair of The calculation accuracy of (2) is estimated, and a residual error is set as follows:
；
The error equation vector is:
；
Wherein, Is/>Standard inner product of/>Is an error metric vector for a two-dimensional approximation scale space,/>Is an error metric vector for two-dimensional vertical, horizontal and diagonal wavelet spaces,/>Is the total error metric vector of the calculation method;
Opposite type Regularizing and solving the C by an iteration method;
S7, calculating final residence time ：/>。
Preferably, step S2 is specifically as follows: selecting an experimental piece made of the same material as the workpiece to be processed, measuring the initial surface shape, measuring the surface shape again after polishing time is 10s, subtracting the surface shapes measured before and after processing, and dividing the surface shape by the polishing time for 10s to obtain a removal function within 1s。
Preferably, the iteration target set in the iteration method of step S6 is:
；
Wherein, Due to the minimum residence time of the machine tool dynamic performance limit,/>Is the minimum step size on the track,Is the maximum speed limit of machine tool operation; /(I)Is a regularization factor. /(I)Representing a 2-norm of the matrix or vector;
and solving the iteration target to obtain a solution C.
Preferably, the method for solving the iteration target in step S6 is newton method, gradient descent method, conjugate gradient method, krylov subspace approximation method, biorthogonal method or SBB method.
Compared with the prior art, the invention has the following beneficial effects:
The solving method of the invention utilizes the advantages of local fitting of wavelets, so that the residence time solution is more accurate in a small scale range, and is beneficial to the accurate control of the residence time of elements with extremely high requirements on roughness, such as a strong laser reflector or a photoetching objective lens. The method has repeatability, and can calculate accurate residence time aiming at local part, thereby realizing accurate removal and guiding the processing practice of elements such as a strong laser reflector or a photoetching objective lens.
Drawings
Fig. 1 is a flow chart of a residence time algorithm based on wavelet base fitting provided in accordance with the present invention.
FIG. 2 is a diagram of a core workflow program framework provided in accordance with the present invention.
Detailed Description
Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, like modules are denoted by like reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, a detailed description thereof will not be repeated.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not to be construed as limiting the invention.
The residence time algorithm based on wavelet base fitting specifically comprises the following steps:
S1, measuring the surface shape of a workpiece to be processed through an interferometer to obtain a discretized surface shape residual error of the workpiece to be processed ；
S2, selecting a processing technology of the workpiece to be processed, and measuring a removal function；
S3, setting polishing track types and polishing track parameters according to the surface shape sampling size of the workpiece to be processed, wherein resident points on the tracks are as follows；
S4, expressing the surface shape of the workpiece to be processed by using wavelet: selecting a Gaussian scale functionAnd/>Constructing a one-dimensional scale function family and a one-dimensional wavelet function family:
(1)
constructing a two-dimensional wavelet according to a tensor construction method in multi-resolution analysis; tensor product construction two-dimensional scale function The two-dimensional wavelet functions are divided into three classes, the first being vertical waveletsThe second class is horizontal wavelet/>The third class is the oblique wavelet/>。
The tensor construction method expression is as follows:
(2)
Wherein, Representing the tensor product. The construction rule of (2) is as follows: /(I)(3)
According to a multi-resolution analysis theoretical formulaWherein/>Represented as a straight sum. Knowing the surface shape residual error of the workpiece to be processed/>Can be expressed as:
(4)
Some limits are made on the expression, so that the calculation is convenient; Is the maximum detail scale sequence number,/> Is the minimum detail scale sequence number; for example, the Gaussian wavelet support length selected is/>While the surface shape is of a caliberThe field size of view of the observed roughness is/>；
Then；
；
Where ceil is rounded up and floor is rounded down.Representing a support set of functions. In the/>Is the shape of face/>And/>The number of directional discretized points. Shift wavelet amount/>Define in-plane shape size/>Within,/>Is the scale factor,/>Is a two-dimensional vertical detail coefficient,/>Is a two-dimensional horizontal detail coefficient,/>Is a two-dimensional oblique detail coefficient. /(I)Corresponding to horizontal, vertical and inclined numbers respectively.
Then rewrite (4) to column vector inner product form:
(5)
Wherein:
The wavelet basis vectors are:
(6)
The surface spectrum is:
(7)
S5, using wavelet to express a removal function, and converting the convolution kernel into the convolution kernel according to two-dimensional Taylor expansion:
(8)
is a removal function/> Is a wavelet feature matrix of (a); this step can be referred to the paper "Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet"
S6, solving a fitting coefficient matrix equation of the construction equation:
；
Wherein, Is the residence time spectrum;
This equation is based on the following considerations:
It is known that:
Then:
due to the orthogonality of the wavelet basis:
Thereby obtaining the following steps:
thereby obtaining an equation 。
Estimating the calculation accuracy of the above method, and setting up residual errors as follows:
；
The error equation vector is:
；
Wherein, Is/>Standard inner product of/>Is an error metric vector for a two-dimensional approximation scale space,/>Is an error metric vector for two-dimensional vertical, horizontal and diagonal wavelet spaces,/>Is the total error metric vector of the calculation method;
the problem represented is still a pathological problem, so Tikhonov regularization is needed to improve stability, and an iterative method is used for solving. Thereby setting up an iteration target as follows:
(16)
Wherein, Due to the minimum residence time of the machine tool dynamic performance limit,/>Is the minimum step size on the track,Is the maximum speed limit for machine tool operation. /(I)Is a regularization factor. /(I)Representing the 2-norm of the matrix or vector.
For a pair ofThe formula is regularized and the solution C is carried out by an iteration method.
The method for solving the iteration target is Newton method, gradient descent method, conjugate gradient method, krylov subspace approximation method, double orthogonalization method or SBB method.
S7, calculating final residence time：/>; And (5) finishing the solving.
The residence time solution flow chart proposed by the present patent is shown in fig. 1, the code framework is shown in fig. 2,Representing acquisition of corresponding data,/>Corresponding data are set, ceil is rounded up, floor is rounded down,Representing a support set of functions. /(I)Is an iteration initial value,/>Representation of the function/>Independent variable at minimumIs a value of (2).
The core of the invention is that: and selecting a Gaussian scale function and a wavelet function for data preprocessing. The solution is smoother through a fitting method, and the dynamic performance of the machine tool is met. The advantage of local fitting of wavelets is utilized, so that the residence time solution is more accurate in a small scale range, and the residence time accurate control of elements with extremely high requirements on roughness, such as a strong laser reflector or a photoetching objective lens, is facilitated. The method has repeatability, and can calculate accurate residence time aiming at local part, thereby realizing accurate removal and guiding the processing practice of elements such as a strong laser reflector or a photoetching objective lens.
The method provided by the invention belongs to a fifth polynomial fitting method in the background technology, but can realize local correction of solutions and analyze high-frequency information by relying on the theoretical background of multi-resolution analysis, so that the roughness of a machined element is accurately controlled. The solution is more accurate while the solution meets the dynamic performance problem of the machine tool.
In summary, the invention has the advantage that the influence of the algorithm on the roughness can be effectively solved in the calculation problem of the component residence time with higher requirements on the local roughness. The core idea is to apply the powerful ability of wavelet modification on a small scale to solve the dwell time problem. While ensuring the smoothness of the solution, the accurate residence time can be obtained, and an effective means can be provided for controlling the local roughness.
It should be appreciated that various forms of the flows shown above may be used to reorder, add, or delete steps. For example, the steps described in the present disclosure may be performed in parallel, sequentially, or in a different order, provided that the desired results of the technical solutions of the present disclosure are achieved, and are not limited herein.
The above embodiments do not limit the scope of the present invention. It will be apparent to those skilled in the art that various modifications, combinations, sub-combinations and alternatives are possible, depending on design requirements and other factors. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the scope of the present invention.
Claims (4)
1. The residence time algorithm based on wavelet base fitting is characterized by comprising the following steps:
S1, measuring the surface shape of a workpiece to be processed through an interferometer to obtain a discretized surface shape residual error of the workpiece to be processed ；
S2, selecting a processing technology of the workpiece to be processed, and measuring a removal function；
S3, setting polishing track types and polishing track parameters according to the surface shape sampling size of the workpiece to be processed, wherein resident points on the tracks are as follows；
S4, expressing the surface shape of the workpiece to be processed by using wavelet: selecting a Gaussian scale functionAnd/>Constructing a one-dimensional scale function family and a one-dimensional wavelet function family:
；
In the method, in the process of the invention, Representing scale factors,/>Representing a displacement coefficient;
constructing a two-dimensional wavelet function family according to a tensor construction method in multi-resolution analysis by utilizing a one-dimensional scale function family and a one-dimensional wavelet function family 、/>、/>、/>；
Residual error of the surface shape of the workpiece to be processedExpressed as:
；
In the method, in the process of the invention, Is the shape of face/>And/>The number of directional discretized points. Shift wavelet amount/>Define in-plane shape size/>Within,/>Is the scale factor,/>Is a two-dimensional vertical detail coefficient,/>Is a two-dimensional horizontal detail coefficient,/>Is a two-dimensional oblique detail coefficient. Two-dimensional scale function/>The two-dimensional wavelet functions are divided into three classes, the first being vertical wavelets/>The second class is horizontal wavelet/>The third class is the oblique wavelet/>。/>Corresponding to horizontal, vertical and inclined numbers respectively; Is the maximum detail scale sequence number,/> Is the minimum detail scale number;
The rewritten column vector form is as follows:
；
Wherein:
The wavelet basis vectors are:
The surface spectrum is:
；
S5, using wavelet to express a removal function, and converting the convolution kernel into the convolution kernel according to two-dimensional Taylor expansion:
；
In the method, in the process of the invention, Is a removal function/>Is a wavelet feature matrix of (a);
S6, solving a fitting coefficient matrix equation of the construction equation: ; in the/> Is the residence time spectrum;
For a pair of The calculation accuracy of (2) is estimated, and a residual error is set as follows:
；
The error equation vector is:
；
Wherein, Is/>Standard inner product of/>Is an error metric vector for a two-dimensional approximation scale space,Is an error metric vector for two-dimensional vertical, horizontal and diagonal wavelet spaces,/>Is the total error metric vector of the calculation method;
Opposite type Regularizing and solving the C by an iteration method;
S7, calculating final residence time ：/>。
2. The residence time algorithm based on wavelet base fitting according to claim 1, wherein said step S2 is specifically as follows: selecting an experimental piece made of the same material as the workpiece to be processed, measuring the initial surface shape, measuring the surface shape again after polishing time is 10s, subtracting the surface shapes measured before and after processing, and dividing the surface shape by the polishing time for 10s to obtain a removal function within 1s。
3. The residence time algorithm based on wavelet base fitting according to claim 2, wherein the iteration targets established in the step S6 iteration method are:
；
Wherein, Due to the minimum residence time of the machine tool dynamic performance limit,/>Is the minimum step distance on the track,/>Is the maximum speed limit of machine tool operation; /(I)Is a regularization factor. /(I)Representing a 2-norm of the matrix or vector;
and solving the iteration target to obtain a solution C.
4. A dwell time algorithm based on wavelet base fitting according to claim 3, wherein the method of solving the iterative objective in step S6 is newton' S method, gradient descent method, conjugate gradient method, krylov subspace approximation method, biorthogonal method or SBB method.
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