## Abstract

The original model-based wavefront sensorless adaptive optics (WFSless AO) for extended objects uses the low spatial frequency content of images as the metric function and employs 2N + 1 images to correct N Lukosz aberration modes. We propose an improved method that uses the same metric but requires only N + 2 images to correct N aberration modes. The N + 2 method can achieve comparable corrective accuracy but requiring much smaller number of images compared with the 2N + 1 method. The N + 2 method is not only more efficient, but also provides the flexibility of choosing arbitrary basis modes by involving the non-orthogonality between modes in a linear least-squares optimization process. It is demonstrated that the deformable mirror’s influence functions modes (IFM), orthogonal mirror modes (OMM) and fitted Zernike modes (FZM) all can be used as modal basis for the N + 2 method to generate biased images. It is also proved that the N + 2 method is robust to image sampling rate and image noise, showing good prospects of applications in various imaging systems.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Adaptive optics (AO) has been successfully implemented to compensate for aberrations and restore image quality of the astronomical telescope, retinal imaging system and biomicroscopy. A conventional AO employs a wavefront sensor (WFS) to measure the aberration and a wavefront corrector to remove aberrations based on the phase conjugation principle. However, using a dedicated WFS like Shack-Hartmann can become difficult because of the unsuitable beacon, scintillation, non-common path error or system constraints [1–3]. Wavefront sensorless (WFSless) AO is an alternative strategy to overcome or to avoid these difficulties. In WFSless AO, a distinct WFS is absent and a wavefront corrector is driven in an iterative manner to optimize a metric function and the aberration is eventually minimized. The optimization process is expected to be efficient in searching the metric function’s global extremum, which means a minimum number of wavefront deformations and image measurements. In dynamic aberration correction, an efficient optimization means high control bandwidth [1,4]. In fluorescence biomicroscopy, a minimum number of exposures are desired to reduce the risk of phototoxicity and photobleaching [5].

The optimization algorithms in WFSless AO can be classified into two categories: model-free and model-based [6]. The model-free algorithms like hill climbing [7], genetics [8], simulated annealing [9], stochastic parallel gradient descent [10–11], etc. have been developed for various kinds of imaging systems. Some other mathematical optimization methods are also available as alternatives [12–14]. The model-free algorithms simply use the image-plane measurements to refresh the metric function value and guide the next search direction. The convergence speed of model-free algorithms is dependent on many factors and may vary a lot in practice. A large number of iterations is required to reach convergence. Moreover, the global optimum may not be achieved if the metric function or the control parameters are not chosen properly.

The model-based WFSless AO relies on the derived relationship between the selected aberration modes and a well-suited metric function. Based on this relationship, the aberration can be estimated from a sequence of biased images which are captured in turn after introducing specific aberration modes with known amplitude at the pupil plane using a wavefront corrector like a deformable mirror (DM). The relationship may be derived approximately under certain conditions, but a relatively accurate estimation can still be obtained and will be refined subsequently with only several iterations. Model-based WFSless AO can converge very quickly within a small number of exposures and avoid dropping into local optima. It has been applied to various microscopes to suppress the specimen-induced aberration [15–18].

For point-like source imaging, Booth first proposed a model-based algorithm suitable for small aberrations, using Zernike polynomials as the bias modes and Strehl ratio as the metric function [19]. Later he extended the algorithm to accommodate large aberrations, using Lukosz-Zernike polynomials as the bias modes and root-mean-square (RMS) spot radius as the metric function [20]. Huang & Rao proposed a generalized model-based method which is insensitive to the selection of bias modes [21]. Antonello *et al.* proposed an experimental method to determine the relationship between the metric function and the aberration modes [22]. The above algorithms use N + 1 images to correct N aberration modes simultaneously. Alternatively, a sequential (mode-by-mode) correction scheme can be applied to reduce the interval time between corrections [23].

For extended objects imaging, Débarre *et al.* developed a model-based algorithm using at least 2N + 1 images to correct N Lukosz modes [24–25]. This algorithm uses the low spatial frequency contents of images as the metric but leads to correction for all spatial frequencies. Yang *et al.* further detailed Huang & Rao’s method mathematically and generalized it to extended objects [26]. However, the performance of this method is dependent on the image contents and a Tukey window has to be applied to alleviate the effect of edge pixels, which limits its application in real scenarios.

In this paper, an improved model-based WFSless AO method is proposed for extended objects imaging. The low spatial frequency content of images is used as the metric function which is the same as Débarre’s method. However, it is optimized in a more efficient manner. Only N + 2 images are required to simultaneously correct N aberration modes in one correction cycle, almost two times faster than the 2N + 1 algorithm in [24]. The influence functions of DM or any other aberration modes can be used to generate biased images. Compared with Yang’s method in [26], our method is insensitive to the image content and edge pixels because the metric is only dependent on the low spatial-frequency components of image. The advantages of proposed N + 2 method is demonstrated both by simulation and experiments.

## 2. Principles of model-based WFSless AO for extended object

#### 2.1 Original algorithm using 2N + 1 images

To better understand the development of our method, we first briefly review the original algorithm presented in [24]. For extended objects and incoherent imaging system, the integral of the low spatial frequency contents of image can be used as the metric function:

*m*is the spatial frequency;

*ξ*is the polar coordinate angle. The region of integration is an annulus of inner radius M

_{1}and outer radius M

_{2}. For small aberrations represented by Lukosz modes and a small M

_{2}, the metric function has a paraboloidal maximum with respect to Lukosz coefficients. For large amplitude aberrations, it is better to use the Lorentzian function to approximate the metric which is proven to be the reciprocal of metric function

*f*. The relationship between the metric function

*g = 1/f*and the aberration

**Φ**is given by Eq. (2) which is also valid for small aberrations.

*P*denotes the pupil area; ${q_2}$ and ${q_3}$ are parameters related to the optical system and the objects to be imaged.

The aberration is expanded by Lukosz modes $\{{{L_i}} \}$ whose gradients are orthogonal, i.e.

where ${\delta _{ij}}$ is the Kronecker delta.Then Eq. (2) can be simplified as

where*l*denotes Lukosz coefficients excluding the first three terms (piston, tip and tilt).

_{i}From Eq. (7), the relationship between the metric function and Lukosz coefficient is quadratic. *l _{i}* can be estimated independently from three measurements of the metric function as

To correct for N Lukosz modes, one can apply the correction simultaneously after all modal coefficients are estimated, requiring 2N + 1 images in total since the unbiased metric function value is the same for different mode’s estimation.

#### 2.2 Improved algorithm using N + 2 images

In the 2N + 1 algorithm, the modal coefficients are estimated sequentially through a quadratic optimization. The modal bases should be Lukosz modes or other modes that have the same orthogonality as defined by Eq. (6). Here we propose a model-based algorithm which is insensitive to the selection of bias modes and using only N + 2 images to correct for N aberration modes. The modal coefficients are estimated simultaneously and the bias modes are not required to be orthogonal. The non-orthogonality (crosstalk) between the bias modes are involved in a linear least-squares optimization process in our algorithm.

The aberration can be expanded by an arbitrary modal basis $\{{{K_i}} \}$ as

It is assumed here that piston and tip/tilt have been removed from $\{{{K_i}} \}$ as they have no effect on the metric function value. Then Eq. (2) can be rewritten as

After introducing a positive bias ${\varphi _{j + }} = \textrm{ + }b{K_j}$, the metric function is given by

Subtracting ${g_0}$ from ${g_{j + }}$, we can obtain

The inner product of $\{{\nabla {K_i}} \}$ is defined as

Then Eq. (12) can be rewritten as

As the parameter ${q_3}$ is usually unknown, an additional negative bias is introduced to eliminate it. Here an arbitrarily chosen mode ${K_s}$ with amplitude *–b* is applied as the negative bias and we can get

After introducing *N* positive biases and one negative bias, we can obtain a set of equations and write them in matrix form as

**G**is obtained from measured metric function values;

**X**,

**Y**and

**Q**are pre-calculated according to Eq. (13) and Eq. (19).

The modal coefficient vector **A** can be solved by simple matrix manipulations as

Assuming the response matrix of DM is **F** whose columns are the influence functions, the control signal of DM for conjugated correction is calculated by

#### 2.3 Wavefront correction procedure of N + 2 algorithm

The flowchart of the N + 2 algorithm is shown in Fig. 1.

## 3. Simulation

A model-based WFSless AO simulation system, as shown in Fig. 2(a), is built with an incoherent imaging system coupled with a 37-channel DM which is consistent with the DM used in our experiments (see Section 4). The actuator layout of the DM is depicted in Fig. 2(b) where the red inner-circle denotes the clear aperture of DM (∼70% in diameter). 70% is an optimal value as suggested by Ref. [27] to get better correction results. The actuator’s influence function is modeled by a Gaussian function as

*c*is the inter-actuator coupling factor and is set to 0.15;

*d*is the actuator spacing. The influence functions of central actuator and edge actuator are illustrated in Fig. 2(c).

The extended object is a USAF resolution test chart sampled by 256×256 pixels. The pupil aberration is sampled by a 128 × 128 grid and then embedded into a 256 × 256 matrix with zeros padding outside the pupil. The detected image is generated by two-dimensional discrete convolution of the object and the point spread function (PSF).

#### 3.1 Modal basis

A prerequisite of using the original 2N + 1 algorithm to estimate the aberration is a precisely calibrated DM that can reproduce Lukosz modes. However, analytic Lukosz modes cannot be fully generated by a low-order DM [28]. In practice, the most convenient modal basis is the influence function modes (IFM) that can be easily reproduced by DM in situ.

In optical design, manufacture and test, Zernike polynomials is commonly used to describe aberrations. The analytical Zernike modes should be fitted by the DM in practice. DM fitted Zernike modes (FZM) can be calculated by Eq. (23). The FZM obtained from the 37-channel DM in Fig. 2(b) are shown in Fig. 3(a).

where**Z**denotes analytical Zernike modes; $\tilde{{{\textbf Z}}}$ denotes FZM;

**F**denotes IFM.

Another useful basis is the orthogonal mirror modes (OMM) that make up DM’s deformation space [29,30]. OMM can be derived from IFM by a singular value decomposition (SVD).

where columns of**U**constitute OMM;

**S**is a diagonal matrix that contains singular values. A small singular value indicates that large control signals are required to produce a unit amplitude of the corresponding mirror mode. The OMM as shown in Fig. 3(b) are arranged in a descending order of singular values (from left to right, top to bottom).

We investigate the performance of N + 2 algorithm using three different modal bases as mentioned above, i.e. IFM, FZM and OMM. The input aberrations are generated by applying random driving signals to actuators of DM. One hundred aberration samples were generated and then their RMS values were normalized to 1 rad, 3rad and 5 rad for correction. The mean RMS value of residual aberration varying with the correction cycle is plotted in Fig. 4. To hold the low frequency approximation, the spatial frequency range for integration is from M_{1 }= 1/128 to M_{2 }= 6/128. The spatial frequency value is normalized to the cutoff frequency of the imaging system. The optimal values of M_{1} and M_{2} have been investigated in Section 7.3 of Ref. 24. From Ref. 24, a smaller M_{2} leads to better correction accuracy. The value of M_{1} should also be small and excluding the central peak value in the spectrum. The bias amplitude was set to 0.01 rad RMS for the three modal bases.

From Fig. 4, the aberrations can be effectively corrected and converge to a quite small value after several iterations, no matter what kind of modal basis is used. Using IFM or OMM leads to the same results that are slightly better than using FZM. This is because the DM generated aberrations can be fully described either by IFM or OMM, but not by the truncated FZM. In practice, if the low spatial frequency aberration is dominant, it is more efficient to use only low-order OMM or FZM as modal basis. The IFM may not be a good choice especially when a DM having a large number of actuators is used as it requires a lot of exposures.

#### 3.2 Algorithm comparison

The performance of proposed N + 2 algorithm is compared with the 2N + 1 algorithm. In the N + 2 algorithm, OMM are employed as the bias modes. In the 2N + 1 algorithm, DM fitted Lukosz modes (FLM) are adopted in [24] where a 140-actuator DM is used and the modal fitting error is quite small. However, the fitting error is non-negligible for a 37-actuator DM herein, so it is better to use mirror modes whose gradients are orthogonal to avoid fitting error. The gradient orthogonal mirror modes (GOMM) can be obtained by applying SVD on the gradients of response matrix **F** [28,6]. GOMM have the same orthogonality as Lukosz modes and can be fully reproduced by a DM.

Columns of ${\textbf {U}^{\prime}}$ are the GOMM. ${\textbf {U}^{\prime}}$ can be calculated by

From Eq. (26), ${\textbf {U}^{\prime}}$ is a linear transformation of the response matrix **F**, so GOMM can also be fully generated by the DM.

For one hundred aberration samples whose RMS values are normalized to 1 rad and 5rad, the RMS of residual aberration varies with sampling number is shown in Fig. 5. The sampling number is actually the number of captured images in order to estimate the aberration. Both the N + 2 algorithm and the 2N + 1 algorithm are implemented iteratively for four correction cycles. As N is equal to 37 here, 39 images are required in the N + 2 algorithm in one correction cycle and 75 images in the 2N + 1 algorithm. The RMS of the residual aberration is evaluated immediately after each correction is finished. The sampling number represents the number of exposures. It is clear that, for the 2N + 1 algorithm, the accuracy of using GOMM is much better than using FLM because the DM fitting error is totally avoided. The N + 2 algorithm has comparable accuracy to the 2N + 1 algorithm after undergoing the same number of correction cycles. However, the N + 2 algorithm is more efficient since it requires much less images in a correction cycle.

#### 3.3 Image sampling rate

We are trying to apply the proposed WFSless AO method to a general imaging system where the image sampling rate may not satisfy the Nyquist criterion which is required by traditional focal-plane wavefront sensing methods like phase retrieval. To investigate the effect of image sampling rate on the wavefront estimation accuracy of N + 2 algorithm, a Nyquist sampling image with 256×256 pixels is generated. Then the sampling rate is reduced by repeating pixel binning of 4×4. In this way, we can get under-sampled images with 64×64 and 16×16 pixels.

An initial aberration is generated with 3 rad RMS. After three correction cycles, the results of N + 2 algorithm using different image sampling rates are shown in Fig. 6. From the RMS value of residual aberrations, the corrective accuracy of using Nyquist-sampled images is better than using under-sampled images. However, the correction is still remarkable even using the 16×16 images that lose most of the object details. This is understandable because the algorithm only needs the low spatial frequency content of images to estimate the aberration. Most of the low spatial frequency content are still retained in under-sampled images. The image spatial frequency range (i.e. M_{1} and M_{2}) used to calculated the metric function is kept constant for different image sampling rates. Please note that we use a normalized spatial frequency. The maximum spatial frequency normalized to 1 is the imaging system’s cutoff frequency which is a constant in our simulations.

#### 3.4 Image noise

The bias amplitude should be optimized according to the image noise level. If there is no image noise, a smaller bias amplitude leads to better correction accuracy. For actual images with certain signal to noise ratio (SNR), the bias amplitude should be large enough to generate a biased image surpassing the influence of image noise.

For 100 random aberrations with initial RMS of 3 rad, after three times correction of the N + 2 algorithm, the RMS of the residual aberration varies with the bias amplitude and image SNR is shown in Fig. 7. From Fig. 7, there exists an optimal bias amplitude for a given SNR. The correction accuracy decreases for a worse SNR, but is still acceptable even for 40 dB SNR. In practice, the optimal bias amplitude can be given empirically according to the image noise level. The bias modes in Fig. 7 are OMM. When using other modal bases like FZM or IFM, the results have the same trend.

## 4. Experimental demonstration

The experimental system layout is depicted in Fig. 8. The extended object illuminated by a LED (λ_{0 }= 625nm) is a resolution test target. This target consists of 25 groups of fringes in four directions and the stripe width is from 40µm to 10µm (spatial frequency ranges from 25 lp/mm to 100 lp/mm). The cutoff frequency of the imaging system is about 32 lp/mm. DM_{1} (Thorlabs, DMP40) is a 40-actuator piezoelectric DM used to generate aberrations. DM_{2} which is conjugated to DM_{1} is a 37-channel membrane DM (OKO Tech) used for correction. The conjugation quality between DM_{1} and DM_{2} is not critical since DM_{1} is only used to produce aberrations. The focal plane camera is a 12-bit CMOS sensor with 5.86 µm pixel size (Point Grey, GS3-U3-23S6M-C). With the illumination of a plug-in fiber-coupled laser, the influence functions of DM_{2} were measured by a commercial Shack-Hartmann (S-H) WFS (Imagine Optic, HASO3-76GE) using 59×59 subapertures. The S-H WFS is also used to measure the residual wavefront aberrations after correction.

The measured IFM of DM_{2} is illustrated in Fig. 9(a). Please note that the tip/tilt components have already been removed from the IFM. To evaluate the performance of N + 2 algorithm under different bias modes, FZM and OMM are derived from the measured IFM. GOMM is also derived and will be used in 2N + 1 algorithm for comparison. FZM, OMM and GOMM are shown in Figs. 9(b)–(d) respectively.

The initial aberration as shown in Figs. 10(a_{1}) and (a_{2}) is produced by DM_{1}. This aberration can be corrected by DM_{2} using the measurement of the S-H WFS, which is actually the traditional AO correction. The corrected image and corresponding phase map are shown in Figs. 10(b_{1}) and (b_{2}). This traditional correction result is used to compare with the results of WFSless AO and evaluate the convergence accuracy.

The correction results of N + 2 algorithm with different bias modes and 2N + 1 algorithm are compared in Fig. 11. The metric function was calculated immediately after each time correction. For N + 2 algorithm with IFM, all 37 modes have to be used for correction. For N + 2 algorithm with OMM or FZM, and 2N + 1 algorithm with GOMM, only the first 15 terms are used for correction. High order modes are discarded to avoid actuator clipping [29,30]. Although IFM is easy to calibrate and reproduce, it requires more images, thus taking more time to converge. The convergence rate of N + 2 algorithm with OMM is slightly better than using FZM, and both of them are faster than 2N + 1 algorithm with GOMM. We tested 20 different aberrations and the conclusions are the same. Although inferior in convergence rate, the correction result of the 2N + 1 algorithm is noticeably better than the N + 2 algorithm after the first correction cycle. The 2N + 1 algorithm also has a smallest residual error after 4 correction cycles, although their difference is marginal.

The correction results of N + 2 algorithm with different image sampling rate are shown in Fig. 12. The experimental imaging system is designed as a 2×Nyquist sampling system. The under-sampled images of 44×44 pixels are obtained by pixel binning of 8×8 from the original 352×352 images. The correction accuracy of using under-sampled images is close to that of using 2×Nyquist sampling images, which is consistent with the simulation results in Sec. 3.3.

## 5. Conclusion

An improved model-based WFSless AO method using N + 2 images is proposed for extended objects imaging system. Compared with the original 2N + 1 algorithm, this method has faster convergence when using the same number of bias modes (i.e. same N). IFM, OMM and FZM all can be used as the modal basis for N + 2 method to generate biased images. In practice, if we have some prior knowledge about the error source, the modal basis can be chosen as the dominant modes in aberrations to further accelerate the correction. However, if IFM is adopted as modal basis, the number of bias modes have to be equal to the number of actuators. This is a drawback of using IFM although it is easy to obtain in practice. We recommend to use OMM as the modal basis as it has better correction accuracy in our experiments. As mentioned in Ref. 24, our method is applicable for most objects except it has noticeable periodicity in a predominant direction like a one-dimensional grid pattern.

## Funding

National Natural Science Foundation of China (11874087, 61505008).

## Disclosures

The authors declare no conflicts of interest.

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