The Algorithm for Recognizing Superposition of Wave Aberrations from Focal Pattern Based on Partial Sums

Abstract

In this paper, we investigate the possibility of recognizing a superposition of wave aberrations from a focal pattern based on a matrix of partial sums. Due to the peculiarities of the focal pattern, some types of the considered superpositions are recognized ambiguously from the intensity pattern in the focal plane by standard error-reduction algorithms. It is numerically shown that when recognizing superpositions of Zernike functions from the intensity pattern in the focal plane, the use of step-by-step optimization in combination with the Levenberg–Marquardt algorithm yields good results only with an initial approximation close to the solution. In some cases, the root mean square reaches 0.3, which is unacceptable for precise detection in optical systems that require prompt correction of aberrations in real time. Therefore, to overcome this drawback, an algorithm was developed that considers partial sums, which made it possible to increase the convergence range and achieve unambiguous recognition results for aberrations (root mean square does not exceed 10⁻⁸) described by superpositions of Zernike functions up to n = 5.

1. Introduction

The problem of measuring and correcting wavefront aberrations is frequently encountered in optics [1,2,3], for example, in astronomical observations [4,5,6,7], in microscopy [8,9], in optical communication and coding [10,11,12], in ophthalmology [13,14,15,16,17,18], and in other imaging and focusing systems [19,20,21].

In modern optical systems, the accuracy of image formation largely depends on minimizing wave aberrations or wavefront distortions that arise due to imperfect optics, problems with system alignment or inhomogeneity of the medium. These aberrations, superimposed on each other, create complex distortions in the focal plane, which significantly reduces the quality of images. Recognizing the superposition of such aberrations by their manifestation in the focal pattern remains one of the key tasks of adaptive optics and computer diagnostics of optical systems.

Many approaches have been proposed to solve this problem, including interferometry, the shadow method, the Hartmann method, and others [22,23,24]. Traditional methods of analysis, such as decomposition of aberrations into Zernike polynomials, although they allow describing individual types of distortions, often face limitations when working with multicomponent superpositions. Zernike functions are an orthonormal basis defined on a unit circle [25,26,27], which allows the description of wavefront aberrations. In [28,29,30], the efficiency of using the Zernike function basis for weak aberrations (up to 0.4 wavelength) was demonstrated. As the aberration increases, it makes sense to use Zernike phase functions [31,32], which are wavefronts matched with certain Zernike functions of a given value.

In adaptive optics, there are several approaches to wavefront correction [33]. In particular, when a Shack–Hartmann device is used, the wavefront correction is made by minimizing the displacements of the focal spots on the sensor [34]. When analyzing a picture in the focal plane, the size of point spread function (PSF) is minimized [35] and/or the Strehl ratio is maximized [36]. Also, for interferograms, it is possible to track the deviation of the ideal distribution from the distorted one [37]. In addition, methods of adaptive wavefront corrections using the modal decomposition of correlation filter method are considered [38]. A similar approach is possible using filters matched to Zernike functions [27,28,30,32]. However, generally, the Zernike polynomials are used only to represent the wavefront, either before or after correction.

Thus, wavefront detection is associated with a phase problem [39,40], which is solved by indirect measurement methods through the intensity pattern. In this case, the problem of unambiguous phase reconstruction arises, since a certain intensity pattern can be formed with different phase distributions.

In this paper, we propose to use the Levenberg–Marquardt algorithm [41,42], adapted to the analysis of partial sum matrices constructed from focal intensity patterns. The method allows for the sequential (step-by-step) allocation of the contribution of individual aberrations using iterative refinement of the model. However, the Levenberg–Marquardt algorithm, applied directly to intensity patterns in the focal plane, gives a good result only with an initial approximation close to the solution. Therefore, to overcome this drawback, an additional calculation of partial sums was introduced, which allowed for an increase in the convergence range and simplified the identification of aberrations even with a superposition consisting of several Zernike functions. This is especially relevant for systems with a high degree of nonlinearity, where classical approaches lose their effectiveness.

2. Materials and Methods

2.1. Wave Aberrations

Zernike functions are defined as follows [25,26,27]:

$$Z_{n,m}(r,\phi )=A_{n,m}{R}_{n,\left|m\right|}(r){\Phi }_m(\phi ),$$

(1)

where $R_{n,m}(r)$ are the radial Zernike polynomials, $A_{nm}$ is a normalizing coefficient, ${\Phi }_m(\phi )$ is the angular component:

$$R_{n,\left|m\right|}(r)={\displaystyle \sum _{k=0}^{\frac{n-\left|m\right|}{2}}\frac{{\left(-1\right)}^k\left(n-k\right)!}{k!\left(\frac{n+\left|m\right|}{2}-k\right)!\left(\frac{n-\left|m\right|}{2}-k\right)!}{r}^{n-2k}}$$

(2)

$$A_{n,m}=\left\{\begin{array}{l}\sqrt{2n+2},m\ne 0\\ \sqrt{n+1},m=0\end{array}\right.$$

(3)

$${\Phi }_m(\phi )=\left\{\begin{array}{l}1,m=0\\ \cos (m\phi ),m>0\\ \sin (m\phi ),m<0\end{array}\right.$$

(4)

Note that the indices n and m have the same parity.

The input field is given by the following relation:

$$g(r,\phi )=\exp \left[-{\displaystyle \frac{r^2}{{\sigma }^2}}\right]\exp \left[i2\pi \psi (r,\phi )\right]$$

(5)

where σ is the radius of the Gaussian beam and $W(r,\phi )$ is the wavefront described as the superposition of the Zernike functions (1):

$$\psi (r,\phi )={\displaystyle \sum _{n=0}^N{\displaystyle \sum _{m=-n}^n{C}_{n,m}{Z}_{n,m}(r,\phi )}}$$

(6)

The amplitude distributions in the focal plane of the lens can be calculated using the Fourier transform:

$$G(u,v)={\displaystyle \frac{2\pi }{\lambda f}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \int g(x,y)\exp \left[-\frac{i2\pi }{\lambda f}\left(xu+yv\right)\right]}dxdy}$$

(7)

where f is the focal length, and λ is the wavelength of the illuminating beam.

2.2. Solution Optimization

To recognize an aberrated wavefront from the focal pattern (7), it is proposed to use the Levenberg–Marquardt optimization method [25]. This problem can be formalized as the minimization of the following functional:

$${\displaystyle \underset{0}{\overset{R}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\left[I_{\Omega }(\rho ,\theta )-I(\rho ,\theta )\right]}^2\rho \mathrm{d}\rho \mathrm{d}\theta }}\to \min$$

(8)

where the point spread function (PSF) of the field under study corresponds to

$$I(\rho ,\theta )={\left|G(\rho ,\theta )\right|}^2$$

(9)

The PSF of the calculated field at each stage of the method [25] corresponds to the focal distribution of the intensity of the set of wave aberrations:

$$I_{\Omega }(\rho ,\theta )={\displaystyle \frac{4{\pi }^2}{{\lambda }^2{f}^2}}\left|{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\exp \left[-\frac{r^2}{{\sigma }^2}\right]\exp \left[i2\pi {\displaystyle \sum _{n,m\in \Omega }{D}_{n,m}{Z}_{n,m}(r,\phi )}\right]\left[-\frac{i2\pi }{\lambda f}r\rho \cos (\theta -\phi )\right]r\mathrm{d}r\mathrm{d}\phi }}\right|$$

(10)

where Ω is a certain set of indices.

For optimization, it is necessary to specify a limited set Ω of Zernike functions that can be included in the resulting superposition $I_{\Omega }(\rho ,\theta )$. Note that Z_{n = 0,m = 0} is excluded from the set of Zernike functions since its presence or absence does not change the intensity in the focal plane.

A set of simplest superpositions is formed from a limited set of aberrations. There are two types of simplest superpositions. The first type contains one radial aberration with m = 0: Z_n,m=0. Such superpositions $D_{n,m=0}{Z}_{n,m=0}(r,\phi )$ have one optimized parameter. The second type paired aberrations, i.e., aberrations with the same order n and orders m of different signs: $D_{n,m}{Z}_{n,m}(r,\phi )+D_{n,-m}{Z}_{n,-m}(r,\phi )$. Such superpositions contain two optimized parameters.

The Levenberg–Marquardt algorithm [25,26] minimizes the sum of squares of M nonlinear functions of N arguments. For the algorithm to work, it is necessary to calculate the values of intensity and its Jacobian at each point.

To calculate partial derivatives, we use the formula:

$$\begin{array}{l}{I}_p^{\prime }=(G\cdot G^{*}{)}^{\prime }=G_p^{\prime }\cdot G^{*}+G\cdot {(G^{*})}_p^{\prime }=G_p^{\prime }\cdot G^{*}+G\cdot (G_p^{\prime })*=\\ =G_p^{\prime }\cdot G^{*}+(G_p^{\prime }\cdot G^{*})*=2\mathrm{Re}\left[G_p^{\prime }\cdot G^{*}\right].\end{array}$$

(11)

The derivative of field (7) with respect to the parameter σ has the following form:

$$G_{\sigma }^{\prime }={\displaystyle \frac{2\pi }{\lambda f{\sigma }^3}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \int r^{2}g(x,y)\exp \left[-\frac{i2\pi }{\lambda f}\left(xu+yv\right)\right]}dxdy}$$

(12)

The derivative of field (7) with respect to the parameter C_nm has the following form:

$$G_{C_{nm}}^{\prime }={\displaystyle \frac{i4{\pi }^2}{\lambda f}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \int Z_{n,m}(x,y)g(x,y)\exp \left[-\frac{i2\pi }{\lambda f}\left(xu+yv\right)\right]}dxdy}$$

(13)

Thus, the calculation of partial derivatives (12) and (13) is reduced to multiple calculations of the Fourier transform.

Optimization starts from a given starting point (initial values of the parameters being optimized). As a result of optimization, a certain local minimum of the optimized functional is found (9). The optimization algorithm does not guarantee that this will be a global minimum. To increase the efficiency of the optimization, it is proposed to use its partial sums (13) instead of the field (7).

Each pixel of the matrix of partial sums (i, j) contains a value equal to the sum of the values of the pixels of the original matrix, the indices of which do not exceed the corresponding index of the current pixel—that is, the sum of the values of the pixels, the row index of which is no greater than i and the column index is no greater than j. In this case, the pixel with the smallest indices (0, 0) contains the value of a single pixel of the original matrix. And the pixel with the largest indices contains the sum of the entire matrix. Thus, the resulting matrix contains the partial sums of the original matrix (Figure 1).

The partial sum matrix is constructed as follows:

$$\begin{array}{l}S(x_0,y_0)=a(x_0,y_0),\\ S(x_{i>0},y_0)=S(x_{i-1},y_0)+a(x_i,y_0),\\ S(x_0,y_{j>0})=S(x_0,y_{i-1})+a(x_0,y_j),\\ S(x_{i>0},y_{j>0})=a(x_i,y_j)+S(x_{i-1},y_j)+S(x_i,y_{j-1})-S(x_{i-1},y_{j-1}).\end{array}$$

(14)

Thus, we can talk about the transformation of the image a into S, where S is the matrix of partial sums of the matrix a. Note that the calculation of the Jacobian for the formed matrix S is reduced to applying the same procedure to the Jacobian of the original matrix a. So, now the problem can be formalized as minimization of the following functional:

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\left[S_{\Omega }(x,y)-S(x,y)\right]}^2\mathrm{d}x\mathrm{d}y}}\to \min$$

(15)

where $S_{\Omega }$ is the matrix of partial sums of $I_{\Omega }(\rho ,\theta )$, $S$ is the matrix of partial sums of the $I(\rho ,\theta )$.

Figure 2 shows the block diagram of the step-by-step optimization algorithm. The beginning of the algorithm “1” corresponds to the input field $I(\rho ,\theta )={\left|G(\rho ,\theta )\right|}^2$. Then, in the loop “2” by the number of stages corresponding to the specified number of aberrations from the set Ω, the field $\Delta I=I_{\Omega }(\rho ,\theta )-I(\rho ,\theta )$ is compensated in step “3.1” and corresponding partial sums calculated in step “3.2” are performed in the nested loop “3” by a predetermined $\Delta S=S_{\Omega }(x,y)-S(x,y)$ set of aberrations from the set Ω. After completing the loop “3”, the type of aberration “4” and its coefficient at which the minimum value of Functional (15) is achieved are determined, and the obtained aberration is added to the compensating superposition “5”. After completing the loop “2”, calculation “6” of Functional (15) of the obtained field is performed taking into account the compensating superposition. The exit criterion corresponds to block “7”, the execution of which ends the algorithm “8”.

At each stage, we form a set of analyzed superpositions ${\displaystyle \sum _{n,m\in \Omega }{D}_{n,m}{Z}_{n,m}(r,\phi )}$. Each analyzed superposition of the set contains the current resulting superposition and one simplest superposition that is not included in the current resulting superposition. For each of the analyzed superpositions of the set, we perform optimization. As a result, for each analyzed superposition, we obtain a set of optimized parameters $D_{n,m}$ and the value of Functionals (8) and (15). The superposition from the set with the minimum value of the functional becomes the current superposition. At each stage, the number of added simplest superpositions is reduced by one and the number of simplest superpositions in the resulting superposition is increased by one.

3. Results

Below are illustrations of normalized intensity distributions (from 0 to 1) with the following calculation parameters: f = 100 mm, σ = 100 mm, λ = 0.000633 mm, the size of the images in the focal plane is 2 mm × 2 mm.

3.1. One Wave Aberration

Let us consider an aberrated wavefront $g(r,\phi )=\exp \left[-{\displaystyle \frac{r^2}{{\sigma }^2}}\right]\exp \left[i2\pi \psi (r,\phi )\right]$ (5) defined by one odd wave aberration $\psi (r,\phi )={\displaystyle \sum _{n=0}^N{\displaystyle \sum _{m=-n}^n{C}_{n,m}{Z}_{n,m}(r,\phi )}}$ (6), where the coefficient C_3.1 = 1; the remaining coefficients are equal to 0. Figure 3 shows the wavefront phase corresponding to the PSF along with the partial sum matrix obtained from the PSF.

Let us perform the minimization of Functional (8) according to the PSF pictures. Figure 4a shows the values of Functional (8) for the tabulated selection of values of the coefficient D_3,1. The PSF of the calculated field (10) corresponds to the presence of one Zernike function in the wavefront Z_3,1. It is evident that it is possible to find a solution (C_3,1 = 1) even in such a simple case, if negative values are not considered, and also if the initial point is not greater than 1.3.

We will perform the minimization of Functional (15) by the matrices of partial sums obtained from the PSF. Figure 4b shows the values of Functional (15) for the tabulated selection of the values of the coefficient D_3,1. The presence of two convergence regions for odd aberrations is an indicator that it is necessary to check the value of the coefficient with the opposite sign. It is evident that the solution (C_3,1 = 1.3) allows achieving the minimum RMS.

Let us consider an aberrated wavefront (5) defined by one even wave aberration (6), where the coefficient C_2,2 = 1; the remaining coefficients are equal to 0. Figure 5 shows the wavefront phase corresponding to the PSF and the partial sum matrix obtained from the PSF.

We will perform the minimization of Functional (8) by the patterns of the PSF and Functional (15) by the matrices of partial sums obtained from the PSF. Figure 6 shows the values of the corresponding functionals. It is clearly visible that the graph in Figure 6a has two regions and has a wavy character in neighborhoods. This means that when optimizing one parameter, the required value can only be found from a limited region. With a random choice of the initial value, some local extremum can be found far from the global minimum. As for the graph in Figure 6b, the presence of two convergence regions is not critical for even aberrations, since these are two equivalent solutions, and the global extremum is achieved regardless of the choice of the initial value.

Thus, minimization of Functional (15) based on the partial sum matrix (14) instead of the PSF (10) allows us to achieve a global extremum regardless of the initial value, which allows us to accurately determine the type and weight of the aberration in the studied wavefront, specified by one wave aberration.

3.2. Superposition of Wave Aberrations

Let us consider an aberrated wavefront $g(r,\phi )=\exp \left[-{\displaystyle \frac{r^2}{{\sigma }^2}}\right]\exp \left[i2\pi \psi (r,\phi )\right]$ defined by the superposition of two wave aberrations $\psi (r,\phi )={\displaystyle \sum _{n=0}^N{\displaystyle \sum _{m=-n}^n{C}_{n,m}{Z}_{n,m}(r,\phi )}}$, where the coefficients C_3,1 = 0.7, C_3,–1 = 0.7; the remaining coefficients are equal to 0.

In

Table 1. The action of the step-by-step algorithm for determining the superposition in the wavefront (5), where $\psi (r,\phi )=0.7Z_{3,1}(r,\phi )+0.7Z_{3,-1}(r,\phi )$.

Stage Number, k		Stage Number, k
0	1	0	1
Field intensity		Partial sums
$I$	$I_{\Omega }$	$S$	$S_{\Omega }$
-	D3,1  = 0.7 D3,–1  = 0.7	-	D3,1  = 0.7 D3,–1  = 0.7
RMS		RMS
-	9.9 × 10−9	-	9.9 × 10−9
Recovered phase		Recovered phase

, the first column (k = 0) presents the PSF of the field under study $I(\rho ,\theta )$; the second column presents the PSF of the calculated field (10) $I_{\Omega }(\rho ,\theta )$ with the weighting coefficients D_3,1 = 0.7, D_3,–1 = 0.7 found using the Levenberg–Marquardt method when minimizing Functional (8) and the corresponding root mean square deviation (RMS) $I(\rho ,\theta )$ from $I_{\Omega }(\rho ,\theta )$ at the k-th stage. At the first stage, the set of analyzed superpositions consists of the simplest superpositions with “zero” initial values of the optimized coefficients D_n±m. It should be noted that the derivatives according to the aberration coefficient at zero are equal to 0. Therefore, a value close to 0 should be taken, for example, a value of 0.01. The superposition with the minimum value of Functional (8) will become the current superposition. Already at the first stage, the RMS was reduced to 9.9 × 10⁻⁹ and the initial aberration coefficients were found.

The second block of Table 1 (columns 3 and 4) presents the results of using the Levenberg–Marquardt method to minimize Functional (15) and the corresponding root mean square deviation (RMS) $S$ from $S_{\Omega }$ at the k-th stage.

As can be seen from Table 1, with a simple superposition, minimization of two different functionals leads to the same result. However, as indicated in Section 2.1, when using Functional (8), the correct solution is not guaranteed when choosing a random starting point for optimization. As for Functional (15), the correct solution is guaranteed regardless of the starting point of optimization, i.e., a global extremum is achieved (not a local one), up to the sign of the weighting coefficient. It is worth noting that the uncertainty of the sign of the aberration weighting coefficient is preserved, regardless of the chosen functional.

Let us consider an aberrated wavefront (5), defined by the superposition of three wave aberrations (6), where the coefficients C_3,1 = 0.7, C_{3,– 1} = 0.7, C _2,2 = 2.0; the remaining coefficients are equal to 0.

Table 2. The action of the step-by-step algorithm for determining the superposition in the wavefront (5), where $\psi (r,\phi )=0.7Z_{3,1}(r,\phi )+0.7Z_{3,-1}(r,\phi )+2.0Z_{2,2}(r,\phi )$.

Stage Number, k			Stage Number, k
0	1	2	0	1	2
Field intensity			Partial sums
$I$	$I_{\Omega }$	$I_{\Omega }$	$S$	$S_{\Omega }$	$S_{\Omega }$
-	D3,1  = 1.00976 D3,–1  = 1.01018	D3,1 = 0.7 D3,–1 = 0.7 D2,2 = 2 D2,–2 = 7.9 × 10−9	-	D3,1  = 1.00976 D3,–1  = 1.01018	D3,1 = 0.7 D3,– 1 = 0.7 D2,2 = 2 D2,–2 = 7.9 × 10−9
RMS			RMS
-	0.835498	9.443586 × 10−8	-	0.077372	2.818248 × 10−9
Recovered phase			Recovered phase

shows the action of the step-by-step algorithm using two different Functionals (8) and (15). As a result, correct weighting coefficients of aberrations were obtained. At the first stage k =1, higher-order aberration coefficients of the coma type were obtained (n = 3, m = ±1), the minimum RMS based on Functional (8) was 0.835498, and based on Functional (15) the RMS was 0.077372. This difference in RMS is explained by the smoother structure of the partial sum matrices compared to the PSF. At the second stage, we form a set of analyzed superpositions based on the current superposition $D_{n,m}{Z}_{n,m}(r,\phi )+D_{n,-m}{Z}_{n,-m}(r,\phi )$ and a set of the simplest superpositions $D_{3,1}{Z}_{3,1}(r,\phi )+D_{3,-1}{Z}_{3,-1}(r,\phi )$ with optimized coefficients from the previous stage. The superposition with the best functional becomes the current superposition. It is worth noting that at the second stage k = 2 values of the higher-order coma-type aberration coefficients (n = 3, m = ±1) were refined and the coefficients of astigmatism-type aberrations (n = 2, m = ±2). The minimum RMS based on Functional (8) was 9.443586 × 10⁻⁸, and based on Functional (15) the RMS was 2.818248 × 10⁻⁹.

Let us consider an aberrated wavefront (5), defined by a superposition of five wave aberrations (6), where the coefficients C_3,1 = 0.7, C_3,–1 = 0.7, C_2,2 = 1.0, C_3,3 = 1.0, C_3,–3 = –1.0; the remaining coefficients are equal to 0.

Table 3. The action of the step-by-step algorithm for determining the superposition in the wavefront (5), where $\psi (r,\phi )=0.7Z_{3,1}(r,\phi )+0.7Z_{3,-1}(r,\phi )+1.0Z_{2,2}(r,\phi )+1.0Z_{3,3}(r,\phi )-1.0Z_{3,-3}(r,\phi )$.

Stage Number, k
0	1	2	3	4
-	D2.2 = 1.01778 D2,–2 = –1.6780	D2.2 = 1.57367 D2,– 2 = –1.6213 D3.3 = –0.5814 D3,–3 = –0.5605	D2,2 = 2.329595 D2,–2 = –0.03393 D3,3 = 0.522402 D3,–3 = –0.56748 D3,1  = 0.319884 D3,–1 = 0.308649	D2,2 = 1 D2,–2 = –8.8 × 10−8 D3,3 = 1 D3,–3 = –1 D3.1 = 0.7 D3,–1 = 0.7 D4,4 = –1.4 × 10−7 D4,–4 = 3.1 × 10−7
Field intensity
$I$	$I_{\Omega }$	$I_{\Omega }$	$I_{\Omega }$	$I_{\Omega }$
RMS
-	0.931765	1.321484	1.009246	5.974689 × 10−7
Partial sums
$S$	$S_{\Omega }$	$S_{\Omega }$	$S_{\Omega }$	$S_{\Omega }$
RMS
-	0.266242	0.348046	0.100404	3.542697 × 10−8
Recovered phase

shows the action of the step-by-step algorithm using two different Functionals (8) and (15). At the first stage k = 1, the coefficients of the astigmatism-type aberration are obtained (n = 2, m = ± 2) at which the minimum RMS is achieved based on Functional (8) and (15). Then, at the second stage, k = 2 values of the coma-type aberration coefficients (n = 3, m = ±1) were refined and the coefficients of trefoil-type aberration (n = 3, m = ±3) at which the minimum RMS is achieved. At the third stage k = 3, the already found values of the aberration coefficients were also refined and new coma-type aberration coefficients were obtained (n = 3, m = ±1) at which the minimum RMS is achieved. At the last stage k = 4, all coefficients of the initial superposition of aberrations are determined.

Thus, it is shown that recognition of superposition of wave aberrations by focal pattern based on partial sums allows one to determine weight coefficients in the studied superposition of aberration. The given examples show that for recognition of superposition of K aberrations not less than K stages are required. Moreover, in contrast to standard recognition of superposition of wave aberrations by focal pattern, the proposed method allows one to obtain global minimum in the problem of functional minimization.

4. Discussions

The main advantage of partial sum matrices is the increase in the convergence region in the functional optimization problem. As a result, this ensures more stable convergence to the global minimum. That is, the number of local minima decreases.

It is easy to see that the number of optimization stages is limited. However, the solution is often found before all stages are completed. A sufficiently small value of the functional (less than some given value) serves as a sign. On the other hand, if the number of simplest superpositions in the desired superposition is equal to K, then it can be stated that no less than K stages will be required.

Another consequence is the simplification of step-by-step optimization, since when using partial sum matrices, the convergence region increases, then it is possible to select “zero” initial values of the coefficients of the added simplest superpositions at all stages.

Among the problems requiring additional research, the following can be highlighted:

(1)

Even simplest superpositions are determined up to the sign at the first stage. If the initial superposition contains only even simplest superpositions, then the solution can be found up to the sign. It is worth noting that if the initial superposition contains at least one odd simplest superposition, then the sign of even simplest superpositions can be checked (specified) only when adding the first odd simplest superposition.

(2)

With an increase in the aberration level, the size of the formed PSF pattern expands, so the peripheral part of the analyzed pattern may not fall into the recorded/observed area, which will lead to incorrect results. Therefore, one more analyzed integral characteristic, corresponding to the total intensity value, can be used. Taking into account the standard deviation, it will be of decisive importance if such a characteristic does not match in the reference and current images.

(3)

It should be noted that at the k-th stage, the simplest aberration corresponding to the minimum value of the functional may be absent from the desired superposition. In this case, more stages will need to be performed for clarification. As the number of simplest superpositions in the initial set increases, the number of false (not included in the initial superposition) aberrations increases.

Despite the existing problems requiring additional research, the algorithm for recognizing the superposition of wave aberrations by the focal pattern based on partial sums removes a number of limitations and allows achieving the minimum value of the functional in the Levenberg–Marquardt optimization method, in contrast to recognizing the superposition of wave aberrations by the focal pattern.

Table 4. The action of the step-by-step algorithm for determining the superposition in the wavefront (5), where $\psi (r,\phi )=0.5Z_{4,0}(r,\phi )+0.5Z_{4,-2}(r,\phi )$ (example of false recognition by focal pattern).

Stage Number, k
0	1	.…	8
-	D2,0 = 0.9 D2,2 = 1.08, D2,–2 = 0.78 D3,3 = –2.73, D3,–3 = –2.46 D3,1 = 0.7, D3,–1 = 0.7 D4,4 = 3.84, D4,–4 = 3.84 D4,2 = 0.46, D4,–2 = 0.39 D4,0 = 0.34	…	…
Field intensity
$I$	$I_{\Omega }$	…	$I_{\Omega }$
		…
RMS
-	0.833421	…	0.338210
Recovered phase
		…

shows an example of the action of a step-by-step algorithm for determining the superposition, in which false recognition is performed by the focal pattern.

5. Conclusions

In this paper, an algorithm for recognizing the superposition of wave aberrations from a focal pattern based on a partial sum matrix has been developed. It is shown numerically that when recognizing superpositions of Zernike functions from the focal pattern based on the matrix of partial sums, step-by-step optimization in combination with the Levenberg–Marquardt algorithm gives a guaranteed result for any initial approximation, i.e., the global minimum is achieved. In some cases, the RMS reaches 0.3.

Using the example of even one wave aberration, it is shown that minimization of the functional based on the PSF does not guarantee reaching a global extremum, which leads to the weight coefficient-recognition accuracy depending on the initial values of the optimized parameters. As a result of optimization, a certain local minimum of the optimized functional is found. To increase the efficiency of the optimization method and obtain a guaranteed result, it is proposed to use its partial sums instead of the focal picture, i.e., the matrix of partial sums.

Using the example of one aberration and superposition of up to five aberrations, it is shown that recognition of superposition of wave aberrations by focal pattern based on partial sums allows one to uniquely determine weight coefficients in the studied superposition of aberration. The results of recognition of aberrations allow RMS values not exceeding 10⁻⁸ in less than eight stages of the algorithm.

The developed algorithm combines the mathematical rigor of inverse problem methods with practical applicability, providing a balance between accuracy and computational speed. Its implementation can become an important step in the development of optical systems for wavefront measurements.