## Abstract

The core problem of phase diversity phase retrieval (PDPR) is to find suitable optimization algorithms for wave-front sensing of different scales, especially for large-scale wavefront sensing. When dealing with large-scale wave-front sensing, existing gradient-based local optimization algorithms used in PDPR are easily trapped in local minimums near initial positions, and available global optimization algorithms possess low convergence efficiency. We construct a practicable optimization algorithm used in PDPR for large-scale wave-front sensing. This algorithm, named EPSO-BFGS, is a two-step hybrid global optimization algorithm based on the combination of evolutionary particle swarm optimization (EPSO) and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Firstly, EPSO provides global search and obtains a rough global minimum position in limited search steps. Then, BFGS initialized by the rough global minimum position approaches the global minimum with high accuracy and fast convergence speed. Numerical examples testify to the feasibility and reliability of EPSO-BFGS for wave-front sensing of different scales. Two numerical cases also validate the ability of EPSO-BFGS for large-scale wave-front sensing. The effectiveness of EPSO-BFGS is further affirmed by performing a verification experiment.

© 2016 Optical Society of America

## 1. Introduction

The phase diversity(PD) algorithm is a well-known algorithm to improve image quality, which has many applications in high resolution imaging and adaptive optics imaging [1–4]. To achieve near-diffraction-limited imaging, PD algorithm requires the collection of a simultaneous set of two (or more) focused images and defocused images. The focused images are the target image degraded by unknown wave-front aberrations, while the defocused images are the same blurred images, but with an additional known defocus wave-front. The reconstruction of unknown wave-front aberrations requires numerical processing to solve an optimization problem. The error metric, set as the optimization object, is built by the error intensity distribution between actually obtained image intensity and theoretical image intensity. The reconstruction of the parameterized wave-front aberrations can be obtained by minimizing the optimization objective function.

PD algorithm was first introduced by Gonsalves and Chidlaw [5]. This algorithm is suitable for point target and extended target at the same time. Since the work of Gonsalves and Chidlaw, scholars have used PD technique to estimate the wave-front aberrations induced by atmospheric blur or imperfections of the optical system and improve the quality of detected images [1,2,6,7]. In recent years, PD algorithm has been successfully used in adaptive optics (AO) system [3, 4, 8–10]. Other applications of PD algorithm also include measurement of a laser beam [11, 12] and biological microscopy imaging [13].

However, it is not easy to apply PD algorithm to wave-front estimation due to the complicated optimization processing, especially for large-scale wave-front sensing. The error metric is a multiple extreme values function on a multidimensional parameter space. When dealing with large-scale wave-front sensing, the global minimum search of error metric becomes extremely difficult. Gradient-based optimization algorithms, such as conjugate-gradient algorithm(CG) [14, 15] and quasi-Newton algorithm(BFGS) [16–19], are easily trapped in local minimums near the initial positions. Available global optimization algorithms used in PD, such as the genetic algorithm [20], have not focused on large-scale wave-front sensing. Moreover, general global optimization algorithms possess low convergence efficiency when dealing with large-scale wavefront sensing.

In this paper, we construct a two-step optimization algorithm(EPSO-BFGS) based on the combination of particle swarm optimization (PSO) [21–23] and Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The kind of PSO used here(named EPSO) is a modified form of traditional PSO introduced a evolutionary mechanism, which is helpful to improve convergence efficiency. For the first step, EPSO provides global search and obtains a rough global minimum position in limited search steps. And for the second step, BFGS initialized by the rough global minimum position approaches the global minimum with high accuracy and fast convergence speed. EPSO-BFGS gives a guarantee of believable convergence efficiency and acceptable convergence speed.

The rest of this paper is organized as follows: In Section 2 we review the theory of PDPR. Section 3 describes the hybird global optimization algorithm EPSO-BFGS. In Section 4 we discuss numerical simulations and a verification experiment. Finally, our conclusion and future work are given in Section 5.

## 2. Phase diversity phase retrieval

In this section, we briefly review the phase diversity technique. We will only consider two images separated by a certain defocus distance. Let us suppose that the object is illuminated with non-coherent quasi-monochromatic light, and the imaging system is a linear shift-invariant system [24]. The intensity distribution of the image plane with Gaussian noise can be modeled by the following equation:

where ⊗ stands for the convolution operation,**r**is a two-dimensional position vector in the image plane,

*o*(

**r**) is the object to be found,

*h*(

**r**) is the point-spread function (PSF) of the optical system, and

*n*(

**r**) is Gaussian noise. With the condition of near-field [25], the point-spread function associated with the focused image is given by where

*FT*

^{−1}denotes the inverse Fourier transform,

**v**is a two-dimensional position vector in the pupil plane,

*P*(

**v**) is the binary aperture function with values of 1 inside the pupil and 0 outside, and

*ϕ*(

**v**) is the unknown wave-front aberration. Similarly, the defocused image is given by and

*d*indicates that a known phase diversity

*ϕ*(

_{d}**v**) has been added to the optical system. The common used

*ϕ*(

_{d}**v**) is defocus aberration.

*ϕ*(

_{d}**v**) can be exactly determined by the location of the defocused image plane relative to the focused plane.

The following error metric

**u**is a two-dimensional spatial frequency coordinate,

*FT*denotes the Fourier transform,

*H*(

**u**) is the optical transfer function (OTF) of the focused system, and

*H*(

_{d}**u**) is the OTF of the defocused system.

*E* is minimized by choice of *O*(**u**) when

Substitution of *O*(**u**) into *E* yields

It is notable that Eq. (9) is independence of any object estimate.

In this paper, the unknown wave-front aberration is expanded on a finite set of Zernike polynomials [26]:

The coefficients *a*_{1} − *a*_{3} stand for piston and tilt of the wave-front aberration, which have no effect on the resolution of image. The error metric *E* is therefore defined on a multidimensional parameter space:

For a given parameters **a**, the error metric *E*(**a**) can be calculated.

The goal is to find the optimal parameter set **a** for which the error metric is globally minimum. Many standard gradient-based nonlinear optimization algorithms have been used by scholars, such as steepest decent algorithm(SD), conjugate-gradient algorithm(CG) and quasi-Newton algorithm(BFGS), etc. All gradient-based nonlinear optimization algorithms need an initial estimate for **a**. Then the error metric and its gradient can be obtained at the initial estimate **a**. The gradient of error metric can be calculated by finite difference method or by analytic expression [14]. The gradient provides a search direction, and a new estimate for **a** is made along the search direction to find a smaller error metric. The entire procedure is repeated iteratively until a local minimum is found. In other words, the search of the optimal parameter set **a** with gradient-based optimization algorithms is easily trapped in a local minimum that is not the true solution.

## 3. EPSO-BFGS optimization algorithm

To cope with the disadvantage that the existing gradient-based optimization algorithms used in PDPR are easily trapped in local minimums, we construct a hybrid EPSO-BFGS global optimization algorithm. This algorithm is based on the combination of Particle Swarm Optimization (PSO) and Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. EPSO is an improved form of traditional PSO, which a evolutionary mechanism has been introduced to improve convergence efficiency. Firstly, EPSO provides global search and obtains a rough global minimum position in finite search steps. Then, BFGS initialised by the rough minimum position approaches the global minimum with high accuracy and fast convergence speed.

Particle Swarm Optimization (PSO), belonging to the Evolutionary Algorithm family, is a population-based optimisation algorithm introduced by Kennedy and Eberhart [21]. Since PSO requires no gradient features, it is relatively flexible to be applied to PDPR. PSO employs a number of particles and each particle stands for a potential solution of the whole optimization problem. The initial position and velocity of each particle are randomly generated:

*N*is the number of particles and

*K*is the maximum number of design variables.

There are many kinds of modified PSO algorithms and we select the model of Clerc [27], which a constriction factor ϕ is introduced to improve convergence efficiency. The position and velocity of each particle are updated by the following equations:

In this work, we set *c*_{1} = *c*_{2} = 2.05. *r*_{1} and *r*_{2} are uniformly distributed random numbers between 0 and 1; *p _{i,l}* is the best local position of particle i itself;

*p*is the best global position of particle i itself;

_{i,g}*p*and

_{i,l}*p*share information among all particles.

_{i,g}*p*and

_{i,l}*p*are updated as follows:

_{i,g}We introduce a evolutionary mechanism to the PSO used here, namely, EPSO. Ascending sort of all error metrics reveals the performance of each particle. After the end of each search step, indexes of all particles are updated according to ascending sort. All particles are divided into two parts: the front part and the rear part. The front part is possessed of more possibility than the rear part for the global minimum search of error metric *E*. The positions and velocities of the rear part are reset as the front part:

The mechanism can enhance the survival of advantageous particles and improve convergence efficiency.

A rough global minimum position ${p}_{i,g}^{(M)}$ can be obtained in limited search steps M by EPSO. The rough global minimum position provides a proper initial position for gradient-based optimization algorithms. We select a widely used quasi-Newton algorithm(BFGS) as the second step to approach the global minimum with high accuracy. BFGS algorithm is also suitable for multi-variable optimization problems.

In fact, for gradient-based optimization algorithms used in PDPR, a proper initial position is unknown and a zero position **a** = 0 is usually initialized. Improper initial position is the very reason that gradient-based optimization algorithms are easily trapped in local minimums. EPSO-BFGS can avoid being trapped in a local minimum. The flow chart and sketch map for EPSO-BFGS are shown in Fig. 1 and Fig. 2.

## 4. Numerical simulations and experimental results

#### 4.1. Numerical simulations

In this section we describe and discuss the simulation results. We assume all images are Nyquist sampled. The size of each data frame is 200 × 200 pixels and the intensity levels are quantized to 16 bits. The diameter of the pupil is set to 100 pixels. The amount of defocus used for the phase diversity is set to be 1.0*λ* peak to valley (PV), which is commonly used [28–30]. Zernike polynomials are used here up to the fifteenth term, so the computed parameter space is

The simulated wave-front aberration *ϕ _{true}* is Kolmogorov aberrated and generated in the pupil plane by the work of Noll and Roddier [26, 31], which lets us easily adjust the turbulence strength

*D*/

*r*

_{0}, where

*D*is a simulated telescope diameter and

*r*

_{0}is Fried’s parameter [32]. The rms value of simulated wavefront is defined as follows:

*ϕ*is the average wave-front of

_{aver}*ϕ*,

_{true}*N*is the total number of pixels within the pupil aperture. The accuracy of the reconstructed wavefront

_{T}*ϕ*is quantified with the rms phase error between the true wave-front aberration and the reconstructed wave-front aberration. The definition of rms phase error is

We will firstly show the different performances between three widely used gradient-based optimization algorithms(SD,CG and BFGS) and our hybrid algorithm EPSO-BFGS. In the case of different turbulence strength *D*/*r*_{0}, Table 1 lists the RMS errors achieved by SD, CG, BFGS and EPSO-BFGS algorithms. Each RMS error is the average value of three same calculation cases. As indicated before, the initial condition of SD, CG and BFGS algorithms is **a** = 0. When the turbulence strength *D*/*r*_{0} = 1, the three gradient-based optimization algorithms work equally well as EPSO-BFGS algorithm. However, when dealing with large turbulence strength, the three gradient-based optimization algorithms obviously lost much accuracy of calculation. Comparatively, EPSO-BFGS algorithm can accurately recover the simulated wave-front aberrations of several large turbulence strength *D*/*r*_{0}.

Similarly, in Table 2 and Fig. 3, we show the performances of SD, CG, BFGS and EPSO-BFGS algorithms dealing with wave-front sensing of different RMS. Each RMS error is the average value of three same calculation cases. From the initial position **a** = 0, SD, CG and BFGS algorithms carry on the search for the global minimum position. When the RMS value of unknown wave-front is 0.1*λ*, SD, CG and BFGS algorithms are able to obtain satisfactory RMS errors. That means the PDPR optimization problem is probable convex optimization for small-scale wave-front sensing and SD, CG and BFGS algorithms are competent.

However, when dealing with large-scale wave-front aberrations, Table 2 and Fig. 3 reveal the fact that the search of SD, CG and BFGS algorithms only go ahead a short distance. PDPR optimization problem is non-convex and complicated for large-scale wave-front sensing and SD, CG and BFGS algorithms are easily trapped in a local minimum. The initial position and velocity of EPSO-BFGS algorithm are randomly initialized. And the global search mechanism of EPSO-BFGS algorithm ensures a more optimal position approach than all local minimum positions, that is, a rough global minimum position. The rough global minimum position provides a proper initializing position for BFGS algorithm. RMS errors obtained by EPSO-BFGS algorithm are remarkably smaller than other algorithms. The excellent performance of EPSO-BFGS algorithm confirms the validity and robustness of our mechanism of EPSO-BFGS algorithm.

It is notable that BFGS algorithm, as the second step of EPSO-BFGS algorithm, plays an important role for the second approach after initialized by the rough global minimum position obtained by EPSO. Table 3 is a list of RMS errors achieved by EPSO and EPSO-BFGS algorithms for different particle number N and iteration number M. The coefficients **a** used here(RMS=0.1619*λ*, PV=1.0310*λ*) is case1 listed in Table 4. We can see that BFGS algorithm can obviously improve the performance of EPSO when particle number N and iteration number M are small. A proper initial position can be obtained by EPSO algorithm when particle number N and iteration number M are big enough, which ensures that BFGS algorithm searches the global minimum position towards the unknown wave-front(RMS errors fluctuate near 0.017*λ*). For the cases that RMS errors obtained by EPSO-BFGS fluctuate near 0.017*λ* confirm the validity of EPSO-BFGS. We would like to point out that the complementary role of BFGS is weak when particle number N and iteration number M are big enough. Even so, we think that BFGS algorithm is indispensable because better performance is always desired.

Finally, we will show the ability of EPSO-BFGS algorithm for large-scale wave-front sensing. The coefficients **a** used here for simulations are listed in Table 4, of which case2(RMS=1.0*λ*, PV=5.8325*λ*) and case3(RMS=1.2*λ*, PV=8.6304*λ*) are large-scale wave-front aberrations. Reconstructed coefficients **a** for case2 and case3 are also listed in Table 4. And in Fig. 4 and Fig. 5, phase retrieval and reconstructed object are given. It can be seen that high accuracy can be achieved by EPSO-BFGS algorithm for the two large-scale wave-front aberrations, where the RMS phase errors are 0.01092*λ* and 0.00096*λ*, respectively. The simulation results confirm the capability of our EPSO-BFGS algorithm for large-scale wave-front sensing.

#### 4.2. Experimental results

A preliminary experiment has been made to test the practical performance of EPSO-BFGS for PDPR. The simplified block diagram of the optical system used here is shown in Fig. 6. The equivalent focal length *F* is 300*mm*, the pupil diameter *D* is 10*mm*, and the CCD(Andor zyla 4.2) pixel size is 6.5*μm*. A common halogen lamp is selected as the light source. A fiber bundle is used as the extended object to be measured. To obtain quasi-monochromatic image, a filter(633*nm*) is located at the collimated beam. Lens L1 is used to collimate the light from the fiber. The collimated beam proceeds through the fiber, a beam splitter, a artificial deformed mirror and reflected by the beam splitter. Some unknown aberrations are introduced to the optical system by adjusting the deformation of the mirror. After passing through the lens L2, the focal images with aberrations can be obtained by the CCD located at the focal plane. Then, we move the CCD by the distance *d* to get the the defocused images. The corresponding peak-to-valley optical path difference Δ [30] is equal to

As we have mentioned in our numerical simulations that choosing *d* such that Δ = *λ* is a relatively better choice to obtain accurate results. So the distance *d* here is 4.5576 *mm*. The measured object takes the area of 300 × 300 pixels and the exposure time is 20*ms*.

The computed parameter space is **a** = [*a*_{4}, *a*_{5}, ⋯, *a*_{15}], the same as our numerical simulations above. The size of measured object is larger than our numerical simulations, which means more computational cost. To get believable results, we set particle number *N* = 500 and iteration number *M* = 500. After the parameter estimates of the unknown phase aberrations, we still need to employ a Wiener-Helstrom noise filter [14] to perform the object restoration:

*O*(

**u**). In fact, It is difficult to get the power spectral density of the noise and that of the object. Usually, a common approach is to assume the radio to be a small constant.

We will firstly compare the performances of different optimization algorithms used in PDPR for a simple large-scale wave-front sensing, that is, a 2*λ*(PV) defocus aberration. We make the deformation of the mirror zero, and the “focal” image is obtained by moving the CCD by the distance 2*d*, which means a 2*λ*(PV) defocus aberration is added to the optical system. Then, the defocused image is obtained by moving the CCD by the distance *d* again. As shown in Fig. 7, we can see from the reconstructed image that the performances of SD, CG and BFGS algorithms are poor. The three gradient-based optimization algorithms provide similar wrong phase sensing. As we have pointed out in our numerical simulations, SD, CG and BFGS algorithms are easily trapped in a local minimum for PDPR. In comparison, EPSO-BFGS algorithm provides a satisfactory performance. Although there are still some small errors, the reconstructed phase by EPSO-BFGS is quite similar with a defocus aberration. We have found the computed parameter *a*_{4}(stands for defocus aberration) is 2.059*λ*(PV).

To testify the effectiveness of EPSO-BFGS algorithm, three different random wave-front aberrations are generated by the deformed mirror. Reconstructed wavefront aberrations and images are shown in Fig. 8. The three measured wavefront aberrations are (a)(RMS=0.155*λ*, PV=0.765*λ*), (b)(RMS=0.261*λ*, PV=1.389*λ*) and (c)(RMS=0.416*λ*, PV=2.136*λ*), respectively. Experimental results are satisfying and practically prove the feasibility of the proposed EPSO-BFGS algorithm for PDPR.

## 5. Conclusion

In this paper, we construct a hybrid EPSO-BFGS global optimization algorithm for PDPR. The two-step mechanism of EPSO-BFGS algorithm overcomes the shortcomings that traditional gradient-based optimization algorithms are easily trapped in local minimums. At the same time, EPSO-BFGS algorithm is stable and reliable. Also, the formulas of EPSO-BFGS algorithm is simple and easy to implement. Computer simulations illustrate that this algorithm can cope with various scales of wave-front sensing. And for large-scale wave-front sensing, EPSO-BFGS can also obtain satisfactory performance. The performed experiment confirms the effectiveness of EPSO-BFGS. Our work provide a practicable algorithm for the complicated nonlinear optimization processing of PDPR, which could enhance the ability of PDPR technique used in several fields such as high resolution imaging and adaptive optics imaging. As part of our future work, we are interested in the physical mode of PDPR. It is desired to find an modified form of the error metric to improve the convergence of PDPR.

## Funding

National Natural Science Foundation of China (NSFC) (6147515) and 61405194); Science Foundation of State Key Laboratory of Applied Optics.

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