Machine learning for improved image-based wavefront sensing

---restore content starts---

Image-based Wavefront sensing is a method to measure wavefront error by using parametric physical model and nonlinear optimization to calculate point diffusion function (PSF). When performing image-based wavefront sensing, a PSF is captured on the detector, and the physical model creates a wave front that generates an analog PSF that matches the optimized data. A good strategy is to parameterize the wavefront with a polynomial (such as the zernike polynomial), thereby reducing the number of dimensions and forcing the physical fit to a smooth-changing wavefront. In order to determine the coefficients of these polynomial to reconstruct the wave front, a gradient-based nonlinear optimization method can be used to adjust the coefficient values. In order to determine the coefficients of these polynomial to reconstruct the wave front, a gradient-based nonlinear optimization method can be used to adjust the coefficient values. The gradient-based optimizer uses the search direction computed from the gradient of the cost function to minimize the cost function relative to the value of the unknown parameter set. In this paper, the cost function is a normalized mean square error (NMSE) metric with constant gain and deviation. The gradient-based optimizer adjusts the coefficients of the polynomial base until the variation of the error measure or gradient value is small enough.

The nonlinear optimization algorithm can stall in local minima and the gradient is zero. In the case of local minima, the calculated PSF is inconsistent with the data, except for a very small number of fuzzy cases, it is locally minimal. In order to avoid local minimum stagnation, when the wavefront error is large, additional information, such as the defocus plane or smoothness knowledge, or a good initial estimation of the polynomial coefficients are needed. When the initial guess is not close enough to the real solution, the likelihood of stagnation at the local minimum is called the "capture range problem", and the distance from the initial estimate to the true solution is close enough to the convergence range called the "Capture Scope". Global optimization can be performed by estimating both Fanggenpo pre-error (RMS Wfe) and using multiple random start guesses with the same number of RMS Wfe to randomly select a starting estimate that is close enough to the true solution within the capture range.

To generate initial estimates within a given PSF capture range, we turn to machine learning and neural networks. Neural networks have previously been applied to phase recovery, attempting to restore the zernike coefficients. The attempt to use a network, each pixel of the PSF as part of the input vector, this is the matrix multiplied by a single "hidden" vector, and then by the nonlinear B-function and the matrix is multiplied by the Zenit polynomial coefficient corresponding to the output vector.

Since we have used a physical model to describe the propagation and detection of light on the intensity plane, we can simply create a simulated PSFS based on the zernike coefficient and input it into CNN. This method assumes that there are known values in our model, such as pupil amplitude, f number, and pixel spacing. For our case, we consider a uniform illumination of the JWST aperture. It is a 0 fill in the array, twice times the aperture width, resulting in a nquist sampled psf (image). We produce PSFS based on the second-to five-order global Zernike polynomial and do not include any error in each segment. All PSFs are normalized to a maximum value of 1 before entering CNN. We use Minibatch training with PSFS minibatch size. CNN's Learning parameters are updated by using a reverse propagation gradient for each CNN operation. The small batch of training updates the CNN parameters based on a gradient from a small batch of inputs, rather than just a single input, which increases the convergence speed [18]. Machine learning relies on the random gradient descent algorithm, where updates are based on gradients and a parameter called learning rate. The learning rate controls the step size of the update, and the smaller learning rate represents the smaller movement in the parameter space. To minimize this, we use Adam, a gradient-based stochastic optimization algorithm that has an adaptive learning rate, which means that it is initialized to a user-defined value and then updated based on user-defined values.

We initially trained 5,000 cycles of PSFS models with only 2.3RMS like aberration (quite a large aberration), with an initial learning rate of 2x10?2 and a reduction of half the 1000 cycles. Then we allow the company to have anywhere from 1 to 4 RMS wave aberration RMS wave and training 20000 era, start learning rate in 1x10?3 and lower 0.5x10?3 after the first 10,000th period. Finally, we include noise, background noise, and bad points in our company, including Poisson noise and optionally including detectors. A Minibatch An example of these input methods, as can be seen in Figure 3. The peak photon and any additional noise parameters are selected from a uniformly distributed random distribution, and table 1 shows the low and high values.

These noise choices make our CNN robust to all kinds of noise that can be found in the experiment, and we trained 50,000 times on these noise PSFS, starting with the 2.5x10?3 learning rate and dropping to 1.0x10?3,0.75x10 after every 10,000 times. ? 3,0.5x10?3 and 0.3x10?3. After the training, our verification loss is the 0.373-wave RMS difference between the predicted and true zernike coefficients, and the remaining rmswfe grows monotonically in the training area.

To determine the effectiveness of the CNN predictions, we used Monte Carlo analysis. The PSFS we simulate is only made up of the zernike coefficients predicted by CNN. The noise in these simulated PSFS is generated in the same way as the training PSFs. We used RMS WFE values ranging from 0.25 waves to 4.0 waves, in increments of 0.25 waves. For each number of RMS WFE, we simulated 250 different PSFS. As a result, some of the matches in our simulated detector window are correct, but unlike the real PSF outside the above window. We increase the size of the image array to 512x512 pixels, because the detector on JWST is at least 1024x1024 pixels (23,24). To prevent errors, we have increased the sampling of the pupil domain by one-fold. We know this is reasonable, because the detector on JWST is at least 1024x1024 pixels [23,24]. Also reduce the PSF to 256x256 pixels and then enter CNN. This means that we do not need to train CNN, we can use a larger array for optimization. These steps improve the convergence of our analysis.

Machine learning for improved image-based wavefront sensing