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Off-axis telescope misalignment correction based on defocus spot moment features

Wei Tang, Yujia Liu, Yiqian Gan, Siheng Tian, Qiong Tu, Yang Li, Yongmei Huang, and Hongyang Guo

Open Access Open Access

Abstract

Optical mirror misalignments, which are caused by assembly mistakes and changes in the surrounding environment (such as gravity, temperature, and atmosphere), degrade the system’s imaging performance. Therefore, active misalignment correction is essential for ensuring the image quality of the off-axis telescope. In this paper, a novel misalignment correction method without wavefront sensors is proposed. The point spread functions (PSFs) of the system are analytically related to the optical mirror misalignments. On this basis, a fully connected neural network (FCNN) is used to establish the mapping relationship between the misalignments and the discrete orthogonal unbiased finite impulse response (UFIR) moment features, which can effectively characterize changes of intensity and geometric of the spot image. The simulation and experimental results in this paper justify the effectiveness and practicality of the proposed method. This approach offers a low-cost and straightforward technical method for achieving high imaging quality throughout the alignment and observation phases. This approach can prevent the accumulation of errors caused by wavefront detection and the high delay of multiple iterations.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In optical systems, reflective systems have been gradually developed over refractive systems due to some superiority, including larger apertures, longer focal lengths, smaller optical routes, no chromatic aberrations, and higher imaging quality. Moreover, off-axis reflective telescopes, compared with on-axis, axisymmetric reflective telescopes, have a series of advantages, such as unobscured pupils, lower optical energy loss rates, and higher diffraction MTFs at low and medium frequencies. Therefore, they are widely used in optical communication, astronomical observation (e.g., the Advanced Technology Solar Telescope (ATST) and Large UV-Optical-Infrared (LUVOIR) Space Telescope [1,2]), and remote sensing.

With the development of optical design performance and manufacturing technology, the imaging quality of optical systems is no longer limited by the figure errors of mirrors [3]. Instead, the alignment accuracy during the integration stage of the telescope, as well as the spatial position and attitude misalignments of mirrors caused by changes during observation (such as the gravity of the system, temperature changes, and atmospheric perturbations in the external environment), have become the primary constraints affecting the image performance of the system. Consequently, correcting these misalignments is a key process for maintaining the imaging accuracy of off-axis telescopes. The misalignment correction technique used to correct wavefront aberrations induced by mirror misalignments includes two parts: alignment technique during assembly and active optical correction technique during observation. The misalignment correction of off-axis telescopes is difficult due to their lack of rotational symmetry and the coupling effect between different degrees of freedom (DOFs) of misalignment. Off-axis telescopes are more susceptible to mirror misalignments than on-axis, axisymmetric ones [4]. Therefore, finding a high-precision and real-time misalignment correction method is extremely important.

The current misalignment correction methods for off-axis reflector telescope systems are can be broadly categorized as follows. First, the analytic model based on nodal aberration theory (NAT) establishes the analytical relationship between the aberration field and misalignment parameters of the misaligned optical system, providing theoretical support for analyzing the aberration characteristics and their coupling relationship between different types of misalignments. On the basis of the two basic properties of the misaligned optical system discovered by R.A. Buchroeder [5], R.V. Shack [6] proposed NAT. K.P. Thompson [7,8], T. Schmid [9,10], and others conducted further research the research to guide the alignment of on-axis two-mirror and three-mirror telescope systems. Considering the continual development of the off-axis reflective telescope and the challenge of aligning the telescope due to its complex construction, NAT has been applied to the theoretical analysis of the aberration of the misaligned off-axis reflective telescope system by introducing the pupil coordinate transformation and aberration field decenter vectors [11,12]. The effects of lateral and axial misalignments on the aberration field and the aberration compensation characteristics of pupil-offset off-axis telescope systems have been analyzed, and the model has been extended to calculate correction values of misaligned off-axis reflective systems containing freedom surfaces [13–20]. To further improve the solution accuracy, the misalignment correction model has been upgraded to fifth-order NAT from the one based on third-order NAT [13,14]. A series of results have been achieved in the theoretical study of NAT, providing a theoretical basis for analyzing the wavefront aberrations and intensity changes of point spread functions (PSFs) caused by mirror misalignments. A class of numerical misalignment correction methods called computer-aided alignment (CAA) technique constructs a simple model to fit the specific relationship between the misalignments and corresponding wavefront aberration coefficients. CAA techniques include the sensitive table method (STM) [21,22], differential wavefront sampling (DWS) [23–25] and artificial neural network (ANN) methods [26,27]. Apart from these, the merit function (MF) [4,28] regression method utilizes optical simulation software to nonlinearly optimize the misaligned state to the nominal state of the optical system until the evaluation function meets the imaging quality requirement. Numerical model-based fitting methods have been widely used for misalignment correction in real telescope systems.

The mainstream computer-aided alignment techniques require utilizing additional wavefront sensors (e.g., Shack-Hartmann wavefront sensors [29], curvature sensors [30], or shearing interferometers [31]) to detect the real-time wavefront aberration of a specific portion of the field of view or even the entire field of view. These instruments require special calibration which increases the complexity of the system. In addition, these methods are sensitive to external environmental interference, affects the measurement accuracy. An image intensity-based wavefront sensing approach called phase retrieval (PR) has emerged to avoid using these instruments. Another type of method using in-focus and defocus plane images is called phase diversity (PD). The former typical algorithm proposed by Gerchberg and Saxton (G-S algorithm) [32] and improved by Yang-Gu [33] requires many iterations to achieve wavefront aberration detection leading to a lack of real-time performance. With the rapid development of deep learning, the convolutional neural network (CNN) has been introduced to most latter algorithms [34–36], whose training process relies on manual experience for parameter selection. Although the emergence of this type of technology reduces the hardware cost of the system, the accuracy of the misalignment calculation greatly depends on the wavefront detection precision. An image-based misalignment correction technique that skips the wavefront detection was proposed [37]; in this technique, an objective function is established based on the image sharpness evaluation function and the mirror misalignments are solved using a nonlinear optimization algorithm. This type of method requires many time-consuming iterations and easily falls into local extremes with poor robustness. Moreover, this kind of iterative method usually yields a compensation solution rather than a recovery solution. The image quality of the system practically meets the actual demand, but the lens eventually deviates from the ideal initial position.

Therefore, a novel off-axis telescope misalignment correction method that avoids the wavefront detection without multiple iterations is proposed in this paper. Because the algorithm based on a CNN directly takes each pixel of the spot image as input, its large amount of data leads to a large training time. The method proposed in this paper reduces the number of data dimensions by using discrete orthogonal unbiased finite impulse response (UFIR) [38] moments to extract the intensity and geometrical features of the defocus spot image as inputs to the network. A mapping relationship is established between feature vectors of defocus spot images and misalignments via a fully connected neural network (FCNN). Once the network is trained, the feature vectors characterizing the defocus spot images of the misalignment system are fed into the model, and a high-precision solution to the mirror misalignments can be obtained.

The structure of this paper is as follows. In Section 2 we introduce the wave aberration characteristics of the misaligned off-axis optical system based on nodal aberration theory and analyze the relationship between the PSF and misalignments based on Fourier Optics. In Section 3, we utilize UFIR moments to extract spot image characteristics to compress data efficiently. We next explain the reasons behind the PSF and the misalignments one-to-many mapping and we propose a solution. Then, the nonlinear relationship between the moment feature vectors of the defocus spot image and misalignments is established based on the FCNN. In Sections 4 and 5, we perform adequate simulations and actual experimental validation. Finally, in Section 6, we conclude this paper.

2. Mapping the relationship between the spot image and misalignments

2.1 Wave aberration characteristics of the misaligned off-axis optical system based on nodal aberration theory

According to nodal aberration theory, the total aberrations on the image surface of an optical system are the sum of the aberration contributions from each element. The wave aberration of an on-axis reflective optical system is expressed as

$$\begin{aligned} W=\sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty}\sum_{m}^{\infty}{({W}_{klm})}_{j}{(\vec{H}\cdot\vec{H})}^{p}{(\vec{\rho}^{\prime}\cdot\vec{\rho}^{\prime})}^{n}{(\vec{H}\cdot\vec{\rho}^{\prime})}^{m}, \end{aligned}$$
where ${W}_{klm}$ represents various types of wave aberration coefficients, $j$ represents the surface serial number of the optical system, $\vec {H}$ and $\vec {\rho }^{\prime }$ respectively denote the normalized field of view vector of the image plane and the normalized pupil vector, $p$, $n$, $m$ denote the power index of its vector operation, $k=2p+m$, $l=2n+m$.

Aberration analysis for off-axis reflective optical systems requires a pupil coordinate transformation matrix. Pupil scaling factor $B$ and pupil off-axis vector $\vec {d}$ are introduced. $\vec {\rho }^{\prime }$ in Eq. (1) is replaced by $B\vec {\rho }+\vec {d}$. The vector aberration function of an off-axis reflective optical system can be expressed as

$$W=\sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty}\sum_{m}^{\infty}{({W}_{klm})}_{j}{(\vec{H}\cdot\vec{H})}^{p}{[(B\vec{\rho}+\vec{d})\cdot(B\vec{\rho}+\vec{d})]}^{n}{[\vec{H}\cdot(B\vec{\rho}+\vec{d})]}^{m}.$$

When there is a mirror misalignment in the off-axis reflective optical system, the image quality of the system degrades and the aberration changes accordingly. The normalized aberration field decenter vector ${\vec {\sigma }}_{j}$ is introduced to denote the offset of the aberration center of the surface $j$ with respect to the ideal system. The misaligned optical system’s effective field of view is denoted by ${\vec {H}}_{Aj}$, so the corresponding field of view vector relationship between the nominal and misaligned states of the optical system is expressed as ${\vec {H}}_{Aj}=\vec {H}-{\vec {\sigma }}_{j}$. The analytical formula between the aberration and misalignments of a misaligned off-axis reflective optical system [11] is constructed as

$$\begin{aligned} W= & \sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty}\sum_{m}^{\infty} {({W}_{klm})}_{j}{[(\vec{H}- {\vec{\sigma}}_{j})\cdot(\vec{H}-{\vec{\sigma}}_{j})]}^{p}{[(B\vec{\rho}+\vec{d})\cdot(B\vec{\rho}+\vec{d})]}^{n}\\ & {[(\vec{H}-{\vec{\sigma}}_{j})\cdot(B\vec{\rho}+\vec{d})]}^{m}. \end{aligned}$$

The purpose of the misalignment correction technique is to enhance the imaging quality of the system. When the elements are misaligned, the imaging quality (the intensity distribution of the PSF) of an off-axis reflective optical system is mainly affected by the primary aberration, which is composed of astigmatism ${W}_{222}$, coma ${W}_{131}$ and spherical aberration ${W}_{040}$. The remaining terms only introduce changes in the position of the PSFs. The formula (3) can be expanded into

$$\begin{aligned} W= & \sum_{j}{W}_{040j}[(B\vec{\rho}+\vec{d})\cdot(B\vec{\rho}+\vec{d})]^2+\sum_{j}{W}_{131j}[{\vec{H}}_{Aj}\cdot(B\vec{\rho}+\vec{d})][(B\vec{\rho}+\vec{d})\cdot(B\vec{\rho}+\vec{d})]\\ & +\dfrac{1}{2}\sum_{j}{W}_{222j}[{\vec{H}}_{Aj}^2\cdot(B\vec{\rho}+\vec{d})^2]+else. \end{aligned}$$

2.2 Analytical relationship between the PSF and misalignments

According to NAT, we establish the mapping relationship between the element misalignment and aberration. According to Eq. (3), when element misalignment is present in an off-axis reflective optical system, the wave aberration at the pupil surface is a function of the field of view, the pupil, and the misalignment, which can be expressed as

$$W=W(\vec{H},\vec{\rho},\vec{\delta}),$$
where $\vec {\delta }$ represents a vector of each element’s misalignment. To investigate the effect of aberrations induced by element misalignments on the final imaging of the system, a generalized pupil function is defined as [39]
$$\mathcal{P}(\vec{\rho},\vec{\delta},\vec{H})=P(\vec{\rho})\exp[jkW(\vec{H},\vec{\rho},\vec{\delta})],$$
where $P(\vec {\rho })$ represents the pupil function of the system; $k=2\pi /\lambda$ and $\lambda$ is the wavelength. Based on Fourier Optics, an analytical relation between the PSF and the misalignment can be expressed as
$$h_{I}(\vec{H},\vec{\delta})=\left|\mathcal{F}\left\{\mathcal{P}(\vec{\rho},\vec{\delta},\vec{H})\right\} \right|^{2} =\left|\mathcal{F}\left\{P(\vec{\rho})\exp[jkW(\vec{H},\vec{\rho},\vec{\delta})]\right\} \right|^{2},$$
where $h_{I}(\vec {H},\vec {\delta })$ represents the PSF of a noncoherent imaging system, and $\mathcal {F}$ represents Fourier transform. When a telescope system observes a point source, the PSF equals the intensity distribution of the observation (spot image). As a result, Eq. (7) demonstrates that a clear analytical relationship exists between the spot image and misalignments.

3. Image feature-based misalignment correction for an off-axis telescope using the FCNN

3.1 UFIR moment-based feature extraction for spot images

In this paper, unbiased Finite Impulse Response (UFIR) moments are used for effective data compression and feature extraction of spot images. Image moments are widely used as image global feature descriptors in image analysis, pattern recognition, machine vision, and other related fields. To avoid errors caused by the numerical integration of continuous orthogonal moments and spatial domain conversion, the discrete orthogonal UFIR moments without a large number of factorial calculations and no parameters to be determined are selected for this paper.

The Nth-order UFIR discrete orthogonal polynomial is defined as

$$T_{n}(x,N) = \sum_{j=0}^{n}a_{jn}(N)x^{j},$$
existing from 0 to $N-1$ and having the coefficients denoted as
$$a_{jn}(N)=({-}1)^{j}\dfrac{M_{(j+1)1}^{(n)}(N)}{\left|\boldsymbol{{H}}_{n}(N)\right|},$$
where $\left |\textbf {\textit {H}}_{n}(N)\right |$ is the determinant and $M_{(j+1)1}^{(n)}(N)$ is the minor of the Hankel matrix $\textbf {\textit {H}}_{n}(N)\in R^{(n+1)\times (n+1)}$.

It can be proved that $T_{n}(x,N)$ satisfies the following orthogonality [40] on $x\in [0,N-1]$

$$\sum_{x=0}^{N-1}\mu(x,N)T_{k}(x,N)T_{n}(x,N)=d_{n}^{2}(N)\delta_{kn}.$$

The square of the weighted norm $d_{n}^{2}(N)$ is given by

$$d_{n}^{2}(N)=\dfrac{n+1}{N(N-1)}\prod_{i=0}^n\dfrac{N-1-i}{N+1}.$$

The non-negative weight $\mu (x,N)$ is the ramp probability density function

$$\mu(x,N)=\dfrac{2x}{N(N-1)}.$$

Given a spot image, the corresponding $(n,m)$ order UFIR moments are defined as

$$U_{nm}=\sum_{x=0}^{N-1}\sum_{y=0}^{M-1}\overline{T}_n(x,N)\overline{T}_m(y,M)f(x,y).$$

The size of the spot image is $N\times M$, with $0\leq x<N,0\leq y<M$. $x,y$ represents the coordinate number of the spot image in two spatial dimensions and their combination indicates the position of each pixel point. $f(x,y)$ represents the PSF, and each pixel point corresponds to a specific value. The scaled UFIR polynomials $\overline {T}_n(x, N),\overline {T}_m(y, M)$ are defined as

$$\overline{T}_n(x,N)=\sqrt{\dfrac{\mu(x,N)}{d_{n}^{2}(N)}}{T}_n(x,N),$$
$$\overline{T}_m(y,M)=\sqrt{\dfrac{\mu(y,M)}{d_{m}^{2}(N)}}{T}_m(y,M),$$
with $n=0,1,2,\ldots, N-1,m=0,1,2,\ldots, M-1$. From the above equations, it can be seen that the calculation of UFIR moments has a certain complexity. Since the UFIR discrete orthogonal polynomial has a recursive relationship [40], the following recursive formula for calculating the scaled UFIR polynomials also holds.
$$\overline{T}_n(x,N)=\alpha'\overline{T}_{n-1}(x,N)+\beta'\overline{T}_{n-2}(x,N),$$
where the coefficients can be obtained as
$$\alpha'=2\dfrac{n^2(2N-1)-x(4n^2-1)}{2n-1}\sqrt{\dfrac{1}{n(n+1)(N-n-1)(N+n)}},$$
$$\beta'={-}\dfrac{2n+1}{2n-1}\sqrt{\dfrac{(n-1)(N+n+1)(N-n)}{(n+1)(N-n-1)(N+n)}}.$$

The initial values $\overline {T}_0(x,N),\overline {T}_1(x,N)$ are respectively calculated as follows

$$\overline{T}_0(x,N)=\sqrt{\dfrac{2x}{N(N-1)}},$$
$$\overline{T}_1(x,N)=[2(2N-1)-6x]\sqrt{\dfrac{x}{(N-2)(N-1)N(N+1)}}.$$

The abovementioned recurrence relationship enables the traditional drawbacks of discrete orthogonal moment factorial coefficients, such as their wide dynamic range and complex calculation process, to be efficiently eliminated. Consequently, the basic functions can be calculated quickly and accurately. In Section 5, comparative experiments were done with some commonly used image feature extraction methods such as Hu [41], Zernike [42], Tchebichef moments [43], and LBP (Local Binary Pattern) operator [44].

In practice, the spot images acquired by the CCD camera for $p$ fields of view are expressed as $f_1,f_2,\ldots,f_p$. The UFIR moments are used to extract the $q$th-order intensity and geometric variation features $U_{nm}$ for each field of view spot image, where $n=0,1,2,\ldots, q,m=0,1,2,\ldots, q$. The spot image moment $U_i$ of one field of view is a $(q+1)\times (q+1)$ feature matrix, which is then arranged in rows as a $1\times [(q+1)\times (q+1)]$ row vector. The spot image moments of $p$ fields of view are expressed as

$$U=[U_{100},U_{101},\ldots,U_{1kk},\ldots,U_{200},\ldots,U_{pqq}].$$

A clear analytical relationship exists between the spot image and the misalignments. As a result, the extracted spot image UFIR moments can be mapped to the misalignments.

3.2 Analysis and solution for one-to-many mapping of in-focus images and misalignments

Misaligned optical elements produce the corresponding aberration in a noncoherent imaging system such as a telescope. The optical system’s PSF is the intensity fraction of the spot on the image plane, which can be expressed as [45]

$$h(x',y')=\left|\mathcal{F}\left\{P(x,y)\exp[jkW(x,y)]\right\} \right|^{2},$$
where $P(x,y)$ is the pupil function and $W(x,y)$ is the wavefront aberration. $h(x',y')$ can also be expressed as
$$h(x',y')=\left|\mathcal{F}\left\{P(x,y)\exp[j\phi(x,y)]\right\} \right|^{2},$$
where $\phi (x,y)$ is the wavefront phase distribution of the pupil surface. Analyzing $\phi (x,y)$ reveals that its odd symmetric transformation $-\phi (-x,-y)$ yields the same PSF. The proof for this is as follows: when the effect of the pupil function $P(x,y)$ is ignored (which can be set to 1 for homogeneous illumination), the PSF corresponding to $\phi (x,y)$ and $-\phi (-x,-y)$ are $h(x,y)$ and $h'(x,y)$, respectively, which can be expressed as:
$$\begin{aligned} h(x',y')= & \left|\iint \exp[j\phi(x,y)]\cdot \exp[{-}j2\pi(x'x+y'y)]dxdy \right|^2 \\ = & \Big|\iint[\cos(\phi(x,y)-2\pi(x'x+y'y))+j\sin(\phi(x,y)-2\pi(x'x+y'y))]dxdy \Big|^2, \end{aligned}$$
$$\begin{aligned} h'(x',y') = & \left|\iint \exp[j(-\phi({-}x,-y))]\cdot \exp[{-}j2\pi(x'x+y'y)]d({-}x)d({-}y)\right|^2 \\ = & \Big|\iint[\cos(-\phi({-}x,-y)-2\pi(x'x+y'y))+j\sin(-\phi({-}x,-y)-2\pi(x'x+y'y))]d({-}x)d({-}y)\Big|^2 \\ = & \Big|\iint[\cos(-\phi(x,y)+2\pi(x'x+y'y))+j\sin(-\phi(x,y)+2\pi(x'x+y'y))]dxdy\Big|^2. \end{aligned}$$

The integral forms in Eqs. (24) and (25) have the same real part and opposite imaginary parts, and they are equal after the square of the mode is taken. Thus, $h(x,y)=h'(x,y)$. It follows that a given random wavefront phase distribution has the same PSF as its odd-symmetric transformed phase, implying that the same PSF at a system’s in-focus point corresponds to more than one wavefront phase distribution. Misalignment causes the wavefront phase to change, resulting in a one-to-many mapping problem in which the set of all possible spot images maps to the set of all possible misalignments. In other words, several different combinations of misalignments correspond to exactly the same focus plane spot image.

The defocus spot image is utilized to solve this multi-solution problem. The defocus phase is denoted as $\Delta \phi (x,y)$ (Changes in phase distribution due to defocus term only). Since $\Delta \phi (x,y)=\Delta \phi (-x,-y)$, the defocus spot intensity distributions of $\phi (x,y)$ and $-\phi (-x,-y)$ corresponding to $I(x',y')$ and $I'(x',y')$ can be expressed as

$$\begin{aligned} I(x',y')= & \Big|\iint[\cos(\phi(x,y)-2\pi(x'x+y'y)+\Delta\phi(x,y)) \\ & +j\sin(\phi(x,y)-2\pi(x'x+y'y)+\Delta\phi(x,y))]dxdy \Big|^2, \end{aligned}$$
$$\begin{aligned} I(x',y')= & \Big|\iint[\cos(-\phi({-}x,-y)-2\pi(x'x+y'y)+\Delta\phi(x,y)) \\ & +j\sin(-\phi({-}x,-y)-2\pi(x'x+y'y)+\Delta\phi(x,y))]d({-}x)d({-}y) \Big|^2 \\ = & \Big|\iint[\cos(-(\phi(x,y))+2\pi(x'x+y'y)+\Delta\phi({-}x,-y)) \\ & +j\sin(-\phi(x,y))+2\pi(x'x+y'y)+\Delta\phi({-}x,-y))]dxdy \Big|^2. \end{aligned}$$

The above equations demonstrate that the real and imaginary parts of the integral differ; thus, $I(x',y')\neq I'(x',y')$. The above method is also applicable if the phase distributions of different misalignment cases with the same PSF do not have an odd symmetric transform relationship. Therefore, the defocus spot can somewhat mitigate the detrimental effect of the multi-solution problem caused by the in-focus spot during the subsequent network training and solving. This theory is experimentally verified in subsection 4.2.

3.3 Misalignment correction method based on moment features of defocus spot images

NAT and Fourier Optics have been used to demonstrate that the spot image is related to mirror misalignments. Furthermore, this relationship is affected by the field of view. Therefore, Misalignment correction is performed for the off-axis reflective telescope illustrated in Fig. 1 by developing a mathematical model between the features of the defocus spot images at arbitrary multiple field positions within a reasonable range and the mirror misalignments via a deep learning algorithm. Firstly, using discrete orthogonal UFIR moments, arbitrary multi-field defocus spot image intensity and geometric change features are extracted. Instead of directly inputting two-dimensional pixel values of the image into the CNN, the one-dimensional moment feature vector is utilized as the input to the FCNN, which reduces the network training time. The FCNN is then used to establish a mapping relationship between the features of defocus spot images and the mirror misalignments. In practice, a well-trained network model can solve for mirror misalignments directly by inputting arbitrary multi-field defocus spot image moment features, eliminating the need for multiple iterations or optimizations. This approach is simple for ordinary field technicians to implement because it requires low hardware demand and relatively fixed network parameters.

 figure: Fig. 1.

Fig. 1. Schematic of an off-axis reflective telescope misalignment correction method based on defocus spot image moment features via the FCNN.

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The processing for implementing the off-axis two-mirror reflective telescope as an example of a misalignment correction method is presented as follows. The technology is divided into two parts: neural network training and misalignment correction.

  • (1) A dataset is constructed for a neural network model of a misaligned system. Specifically, the ideal spatial position of the secondary mirror relative to the primary mirror should be recorded when the system is well-aligned, as well as the multi-field spot images in this state. The secondary mirror is then added to the known random misalignments of each degree of freedom, and defocus spot images of the corresponding multiple fields are captured using a CCD camera. Before features are extracted, the image needs to be appropriately preprocessed, which includes using morphological operations to extract the center of the spot to crop the image and normalizing the gray values of the image. The discrete orthogonal UFIR moments are then utilized to extract features from defocus spot images. The one-dimensional UFIR moment feature vectors are used as inputs to the neural network, and the corresponding misalignments in each degree of freedom are used as outputs of the network, which are taken as a set of samples. The preceding steps are repeated several times to create a complete dataset.
  • (2) The appropriate network model structure and parameters are selected. The neural network model is trained to establish the nonlinear mapping relationship between the arbitrary multi-field defocus spot image features and the misalignments.
  • (3) The CCD camera captures multi-field defocus spot images while correcting the telescope’s misalignment. These images are then used to generate a one-dimensional UFIR moment feature vector, which is fed into a well-trained neural network model to calculate misalignments. The inverse of the data is fed into the Stewart platform, which controls the spatial position of the secondary mirrors. The spot images are taken again to assess the system’s image quality. The process of aligning the primary and secondary mirrors is completed when the image quality meets the requirements. If the image quality does not meet the desired standards, the previous steps are repeated.

4. Simulations

In this section, the feasibility of this method is verified by the detailed misalignment correction simulation flow for the off-axis telescope design case of the Unobscured Gregorian telescope and the self-established off-axis two-mirror telescope model. In this study, the impact of different parameters, including the number of fields of view, feature order, and camera defocus distance, on the accuracy of the method is investigated.

Since the primary mirror is much larger and heavier than the other mirrors, it serves as a reference point in the alignment process. The experiments in this paper are conducted on the premise of an unmoved primary mirror. The rotation of the secondary mirror around the Z-axis does not significantly impact the imaging quality of the system, and thus is neglected. However, the translation of the secondary mirror along the Z-axis primarily results in defocusing, which is easier to correct than other aberrations. The experiments described in this paper are conducted to correct the lateral misalignments of the secondary mirror, including decenter (XDE, YDE) and tilt (XTE, YTE) along the X-axis and Y-axis using the effective correction of the defocus produced by the axial misalignment as a basis.

The FCNN is selected to relate defocus spot image features and misalignments due to their nonlinear relationship. The inputs of the network are the multi-order UFIR moment features of multi-field defocus spot images, and the outputs are the misalignments of the secondary mirror’s four degrees of freedom on the X-axis and the Y-axis. The initial parameter settings of the network structure are shown in Table 1 and are slightly adjusted according to the changes in the input and output parameters in different experiments.

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Table 1. Parameters of the FCNN

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The detailed simulation flow for correcting misalignments is as follows:

  • (1) An off-axis reflective telescope model is designed in the simulation software and optimized to a nominal state. Multiple noncollinear fields of view and the image plane defocus position are constructed, and the system’s wavefront PV and RMS values are recorded.
  • (2) 2000 sets of X-axis and Y-axis decenter and tilt obeying the uniform distribution are generated to change the position of the secondary mirror and the corresponding multi-field defocus spot image data are preserved as well as the wavefront information.
  • (3) The discrete orthogonal UFIR moments are used to extract features from defocus spot images. The normalized one-dimensional UFIR moment feature vector is used as the neural network’s input, and the corresponding four-degree-of-freedom misalignments are used as the network’s outputs, yielding a dataset of 2000 sample sets.
  • (4) The effectiveness of the proposed method is demonstrated from two perspectives. First, the trained neural network model takes the features of the defocus spot image used for testing to calculate the misalignments. These values are then compared with the real values to measure the accuracy of the algorithm. Evaluation indices such as the mean absolute error (MAE) and root mean square error (RMSE) are used to determine the accuracy. The residual value is calculated by subtracting the output value of the network from the actual value. This value is then applied to the controlled surface. The system’s wavefront information is then recorded and compared to the ideal value, which demonstrates the calibration effect of the present algorithm.

4.1 Effect of the field of view and feature order on the accuracy

The three-dimensional layout of the Unobscured Gregorian off-axis telescope model used for the experiments in this subsection is shown in Fig. 2. The optical parameters of each surface are listed in Table 2. The off-axis value of the pupil is $100 mm$, and the operating wavelength is $10 \mu m$. The ideal nominal of the model has a wavefront PV value of $0.2146\lambda$ and an RMS value of $0.0478\lambda$. A decenter range of $\pm 0.5 mm$ and a tilt range of $\pm 0.4^{\circ}$ following uniform distributions are set for the X-axis and Y-axis.

 figure: Fig. 2.

Fig. 2. The three-dimensional layout of the Unobscured Gregorian off-axis telescope.

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Table 2. Optical parameters of the Unobscured Gregorian off-axis telescope

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To investigate the effect of different numbers of fields of view on the accuracy of the above method, the defocus spot images were selected to be acquired at five non-collinear fields of view, three non-collinear fields of view and one central field of view for the experimental study, with a fixed camera defocus position and the same number of feature orders of a spot image(extracting the 6th order UFIR moments). The corresponding misalignment solution accuracy for 200 test samples is shown in Table 3. This accuracy is obtained using the deviations between the estimated and actual values of the decenter and the tilt on the X-axis and Y-axis.

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Table 3. Comparison between the misalignment solution accuracy for different numbers of fields of view

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We choose 6th-order (49 features corresponding to a 2D spot image), 4th-order (25 features), and 3rd-order (16 features) moment features to investigate the impacts of different feature orders on the accuracy. The camera defocus position, position and number of fields of view(5 noncollinear fields of view) are fixed. Table 4 shows the corresponding misalignment solution accuracy for the 200 test samples, which are the same as those of the above experiment.

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Table 4. Comparison between the misalignment solution accuracy for different feature orders

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Table 3 and Table 4 illustrate that the proposed method achieves its highest accuracy when the 6th-order moment features are extracted from five noncollinear fields of view of the defocus spot images for calculating the misalignment. Figure 3 shows that after 200 misaligned test samples are corrected using our proposed method, approximately $92{\% }$ or more of the samples are corrected to a near-ideal state. The remaining samples meet the requirements for systematic observation. Based on the data presented in Fig. 4(a), approximately $85{\% }$ of the residuals of the decenter error are within $10\mu m$ and only $1{\% }$ in the range of $0.02mm$ to $0.03mm$. As illustrated in Fig. 4(b), approximately $85{\% }$ of the residual tilt errors are within $0.02 ^{\circ}$, and the maximum deviation of the very few samples is within $0.05 ^{\circ}$. Overall, the residual error follows a normal distribution. The method proposed in this paper can meet the project’s accuracy requirements and is highly competent in correcting decenter and tilt.

 figure: Fig. 3.

Fig. 3. UG telescope 200-group samples wavefront aberration statistics. (a) Wavefront PV statistics and (b) wavefront RMS statistics for the misaligned system. (c) Wavefront PV statistics and (d) wavefront RMS statistics for the corrected system.

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 figure: Fig. 4.

Fig. 4. The residual decenter and tilt error distribution on estimated value and actual value of UG telescope. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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4.2 Influence of different camera defocus distances on the accuracy

The experiments in this section were conducted using a self-designed off-axis two-mirror telescope. The three-dimensional layout of the model is depicted in Fig. 5, and the optical parameters of the surfaces, as shown in Table 5, include an entrance pupil diameter of $148 mm$, and a working wavelength of $532 nm$.

 figure: Fig. 5.

Fig. 5. The three-dimensional layout of the Unobscured Gregorian off-axis telescope.

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Table 5. Optical parameters of off-axis two-mirror telescope

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The three cases involving CCD camera defocus distances of $-15 mm, -10 mm, and -5 mm$ are analyzed in Table 6. First, the nominal state defocus term coefficients of the Zernike polynomial coefficients of the three cases are recorded. For all three cases, uniformly distributed decenter ($\pm 0.6mm$) and tilt ($\pm 0.2^{\circ}$) perturbations are artificially added to the X- and Y-axes. The accuracy of the algorithm is evaluated on 2,000 sets to investigate the effect of camera defocus distance.

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Table 6. Comparison between the misalignment solution accuracy for different choices of defocus position of the camera

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According to the above experimental results, the larger the defocus distance is, the better the neural network learns the correct mapping relationship between the spot image and misalignments, and the higher the misalignment solution accuracy. The actual experimental platform is validated by placing the CCD camera at a defocus of $-15mm$ to obtain more accurate misalignment solution.

5. Experimental validation

We built an experimental platform to verify the feasibility and accuracy of our off-axis two-mirror telescope system model. The experimental platform is displayed in Fig. 6. The wavelength of the light source is $532 nm$, and a collimating beam expander is used to simulate an infinitely far point target. The Stewart platform precisely controls the spatial position of the secondary mirror relative to the primary mirror. The axial guide rail is used to quickly switch the camera between in-focus and defocus states, and various defocus distances can be set. The lateral displacement platform is used to switch the optical path for camera image acquisition and Hartmann wavefront detection states. In this experiment, the CCD camera is placed at the front defocus of $15mm$, with the pixel size of $5.5\mu m$ and a pixel number of $1024\times 1024$. Importantly, the sensor is used only to evaluate misalignment correction effects and is not involved in assisting misalignment correction.

 figure: Fig. 6.

Fig. 6. (a) Overall optical path of the experimental platform. (b) Detailed physical map of the off-axis two-mirror telescope used for misalignment correction experiments.

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Comparative experiments were done with some commonly used image feature extraction methods such as Hu, Zernike, Tchebichef moments, and LBP (Local Binary Pattern) operator. Efficiency is measured by the length of time it takes to extract the features of an image of size 256*256. The experimental results in Table 7 show that UFIR moments have high computational accuracy and efficiency, making them suitable for the intensity and geometric change of spot images for feature extraction. The deviation of the misalignment estimated values from the actual values for the four degrees of freedom is summarized in Table 8. Figure 7 shows the distribution of the residual error between the estimated and actual values of the misalignment for the 200 test samples. Approximately $85{\% }$ of the decenter errors are within $0.03 mm$, and only a small percentage are within $0.05$ to $0.1 mm$. Approximately $86{\% }$ of the tilt errors are within $0.02^{\circ}$, and only a very small portion are within $0.04$ to $0.05^{\circ}$. Both results meet the accuracy requirements of the actual system.

 figure: Fig. 7.

Fig. 7. The residual decenter and tilt error distribution on estimated value and actual value of the experimental telescope without noise. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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Table 7. Solution accuracy and efficiency of the proposed approach based on different image features

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Table 8. Error analysis between the estimated value and actual value of the experimental platform without noise

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Five sets of samples are randomly selected to demonstrate the deviation of the misalignment estimated values of this method used for the experimental platform in Table 9. These results show that the output value of the FCNN is close to the actual value. Figure 8 illustrates that all five sets of misalignments are effectively corrected, and the corrected PV and RMS values are significantly lower than those in the misaligned state and are similar to those in the nominal state. The spot image is also restored from the scattered state to a round spot.

 figure: Fig. 8.

Fig. 8. Misalignment correction effect of the experimental platform without noise. Each column shows wavefront data and in-focus spot images for each of the three states.

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Table 9. Difference between actual value and estimated value by using the proposed approach without noise

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In practical applications, detection noise significantly impacts on the quality of the spot image. The existence of image noise reduces the alignment accuracy of the optical system. Different levels of noise evaluated by signal-to-noise ratio (SNR) are thus added to 200 sets of test defocus spot images to test the proposed approach’s robustness. Figure 9 demonstrates that the solution accuracy of the proposed approach is actually related to the noise level. As the signal-to-noise ratio increases, the solution accuracy gradually improves. When the SNR is higher than 30 dB, solution accuracy is close to a noiseless state and the influence of noise on the alignment accuracy of the proposed approach can be ignored. We have specifically analyzed the solution accuracy of the proposed approach at a 20 dB noise level, which is recorded in Table 10. According to Fig. 10, approximately $86.5{\% }$ of the decenter errors are within $0.04 mm$, and only a small portion of them are within $0.07$ to $0.1 mm$. Approximately $85{\% }$ of the tilt errors are within $0.03^{\circ}$, and only a small percentage are within $0.6$ to $0.1^{\circ}$. Both of these results demonstrate the robustness of the current algorithm. Figure 11 illustrates that, although the corrected results are not as good as those of the noiseless case, they do meet the system’s observation needs, which reflects the noise immunity of this method.

 figure: Fig. 9.

Fig. 9. The relationship between solution accuracy of the proposed approach and signal-to-noise ratio (SNR). It is shown that when the SNR is higher than 30 dB, the influence of noise on the solution accuracy of the proposed approach can be ignored.

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 figure: Fig. 10.

Fig. 10. The residual decenter and tilt error distribution on estimated value and actual value of the experimental telescope with 20 dB noise. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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 figure: Fig. 11.

Fig. 11. Misalignment correction effect of the experimental platform with 20dB noise. Each column shows wavefront data and in-focus spot image for each of the three states.

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Table 10. Error analysis between the estimated value and actual value of the experimental platform with 20 dB noise

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6. Conclusions

In this paper, the feasibility of extracting features from defocus spot images to correct misaligned off-axis reflective telescopes is demonstrated using a real experimental platform. The off-axis telescope has a more complex structure than the on-axis telescope, and the spot image and the misalignments have a one-to-many mapping relationship due to the coupling effect between different misaligned degrees of freedom. A precise and comprehensive derivation of the equations that explain why the defocus image can avoid the misaligned multi-solution problem is provided. Discrete orthogonal UFIR moments are utilized to extract the intensity and geometrical change features of the defocus spot image with high computational accuracy and efficiency. Subsequently, the complex nonlinear mathematical relationship between the feature vectors and the misalignments is fit by the FCNN. The well-trained network can compute the misalignments quickly. The misalignments can then be input into the displacement control platform to achieve misaligned system correction, allowing the relative position deviation of the mirrors to be corrected in real time. The misalignment correction method in this paper is convenient for engineering applications. Additionally, it avoids wavefront detection, incurs a lower hardware cost, and does not require multiple iterations, making it efficient and robust (free from stagnation issues). Because of its accuracy and efficiency, it can also be applied to on-axis reflective telescope systems, complex multi-mirror systems, and optical systems with freedom surfaces for misalignment correction.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time based on data protection policy but may be obtained from the authors via email upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time based on data protection policy but may be obtained from the authors via email upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. Schematic of an off-axis reflective telescope misalignment correction method based on defocus spot image moment features via the FCNN.

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Fig. 2.
Fig. 2. The three-dimensional layout of the Unobscured Gregorian off-axis telescope.

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Fig. 3.
Fig. 3. UG telescope 200-group samples wavefront aberration statistics. (a) Wavefront PV statistics and (b) wavefront RMS statistics for the misaligned system. (c) Wavefront PV statistics and (d) wavefront RMS statistics for the corrected system.

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Fig. 4.
Fig. 4. The residual decenter and tilt error distribution on estimated value and actual value of UG telescope. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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Fig. 5.
Fig. 5. The three-dimensional layout of the Unobscured Gregorian off-axis telescope.

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Fig. 6.
Fig. 6. (a) Overall optical path of the experimental platform. (b) Detailed physical map of the off-axis two-mirror telescope used for misalignment correction experiments.

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Fig. 7.
Fig. 7. The residual decenter and tilt error distribution on estimated value and actual value of the experimental telescope without noise. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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Fig. 8.
Fig. 8. Misalignment correction effect of the experimental platform without noise. Each column shows wavefront data and in-focus spot images for each of the three states.

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Fig. 9.
Fig. 9. The relationship between solution accuracy of the proposed approach and signal-to-noise ratio (SNR). It is shown that when the SNR is higher than 30 dB, the influence of noise on the solution accuracy of the proposed approach can be ignored.

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Fig. 10.
Fig. 10. The residual decenter and tilt error distribution on estimated value and actual value of the experimental telescope with 20 dB noise. (a) Distribution of the residual decenter error. (b) Distribution of the residual tilt error.

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Fig. 11.
Fig. 11. Misalignment correction effect of the experimental platform with 20dB noise. Each column shows wavefront data and in-focus spot image for each of the three states.

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Tables (10)

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Table 1. Parameters of the FCNN

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Table 2. Optical parameters of the Unobscured Gregorian off-axis telescope

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Table 3. Comparison between the misalignment solution accuracy for different numbers of fields of view

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Table 4. Comparison between the misalignment solution accuracy for different feature orders

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Table 5. Optical parameters of off-axis two-mirror telescope

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Table 6. Comparison between the misalignment solution accuracy for different choices of defocus position of the camera

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Table 7. Solution accuracy and efficiency of the proposed approach based on different image features

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Table 8. Error analysis between the estimated value and actual value of the experimental platform without noise

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Table 9. Difference between actual value and estimated value by using the proposed approach without noise

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Table 10. Error analysis between the estimated value and actual value of the experimental platform with 20 dB noise

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Equations (27)

Equations on this page are rendered with MathJax. Learn more.

W = ∑ j ∑ p ∞ ∑ n ∞ ∑ m ∞ ( W k l m ) j ( H → ⋅ H → ) p ( ρ → ′ ⋅ ρ → ′ ) n ( H → ⋅ ρ → ′ ) m ,
W = ∑ j ∑ p ∞ ∑ n ∞ ∑ m ∞ ( W k l m ) j ( H → ⋅ H → ) p [ ( B ρ → + d → ) ⋅ ( B ρ → + d → ) ] n [ H → ⋅ ( B ρ → + d → ) ] m .
W = ∑ j ∑ p ∞ ∑ n ∞ ∑ m ∞ ( W k l m ) j [ ( H → − σ → j ) ⋅ ( H → − σ → j ) ] p [ ( B ρ → + d → ) ⋅ ( B ρ → + d → ) ] n [ ( H → − σ → j ) ⋅ ( B ρ → + d → ) ] m .
W = ∑ j W 040 j [ ( B ρ → + d → ) ⋅ ( B ρ → + d → ) ] 2 + ∑ j W 131 j [ H → A j ⋅ ( B ρ → + d → ) ] [ ( B ρ → + d → ) ⋅ ( B ρ → + d → ) ] + 1 2 ∑ j W 222 j [ H → A j 2 ⋅ ( B ρ → + d → ) 2 ] + e l s e .
W = W ( H → , ρ → , δ → ) ,
P ( ρ → , δ → , H → ) = P ( ρ → ) exp ⁡ [ j k W ( H → , ρ → , δ → ) ] ,
h I ( H → , δ → ) = | F { P ( ρ → , δ → , H → ) } | 2 = | F { P ( ρ → ) exp ⁡ [ j k W ( H → , ρ → , δ → ) ] } | 2 ,
T n ( x , N ) = ∑ j = 0 n a j n ( N ) x j ,
a j n ( N ) = ( − 1 ) j M ( j + 1 ) 1 ( n ) ( N ) | H n ( N ) | ,
∑ x = 0 N − 1 μ ( x , N ) T k ( x , N ) T n ( x , N ) = d n 2 ( N ) δ k n .
d n 2 ( N ) = n + 1 N ( N − 1 ) ∏ i = 0 n N − 1 − i N + 1 .
μ ( x , N ) = 2 x N ( N − 1 ) .
U n m = ∑ x = 0 N − 1 ∑ y = 0 M − 1 T ¯ n ( x , N ) T ¯ m ( y , M ) f ( x , y ) .
T ¯ n ( x , N ) = μ ( x , N ) d n 2 ( N ) T n ( x , N ) ,
T ¯ m ( y , M ) = μ ( y , M ) d m 2 ( N ) T m ( y , M ) ,
T ¯ n ( x , N ) = α ′ T ¯ n − 1 ( x , N ) + β ′ T ¯ n − 2 ( x , N ) ,
α ′ = 2 n 2 ( 2 N − 1 ) − x ( 4 n 2 − 1 ) 2 n − 1 1 n ( n + 1 ) ( N − n − 1 ) ( N + n ) ,
β ′ = − 2 n + 1 2 n − 1 ( n − 1 ) ( N + n + 1 ) ( N − n ) ( n + 1 ) ( N − n − 1 ) ( N + n ) .
T ¯ 0 ( x , N ) = 2 x N ( N − 1 ) ,
T ¯ 1 ( x , N ) = [ 2 ( 2 N − 1 ) − 6 x ] x ( N − 2 ) ( N − 1 ) N ( N + 1 ) .
U = [ U 100 , U 101 , … , U 1 k k , … , U 200 , … , U p q q ] .
h ( x ′ , y ′ ) = | F { P ( x , y ) exp ⁡ [ j k W ( x , y ) ] } | 2 ,
h ( x ′ , y ′ ) = | F { P ( x , y ) exp ⁡ [ j ϕ ( x , y ) ] } | 2 ,
h ( x ′ , y ′ ) = | ∬ exp ⁡ [ j ϕ ( x , y ) ] ⋅ exp ⁡ [ − j 2 π ( x ′ x + y ′ y ) ] d x d y | 2 = | ∬ [ cos ⁡ ( ϕ ( x , y ) − 2 π ( x ′ x + y ′ y ) ) + j sin ⁡ ( ϕ ( x , y ) − 2 π ( x ′ x + y ′ y ) ) ] d x d y | 2 ,
h ′ ( x ′ , y ′ ) = | ∬ exp ⁡ [ j ( − ϕ ( − x , − y ) ) ] ⋅ exp ⁡ [ − j 2 π ( x ′ x + y ′ y ) ] d ( − x ) d ( − y ) | 2 = | ∬ [ cos ⁡ ( − ϕ ( − x , − y ) − 2 π ( x ′ x + y ′ y ) ) + j sin ⁡ ( − ϕ ( − x , − y ) − 2 π ( x ′ x + y ′ y ) ) ] d ( − x ) d ( − y ) | 2 = | ∬ [ cos ⁡ ( − ϕ ( x , y ) + 2 π ( x ′ x + y ′ y ) ) + j sin ⁡ ( − ϕ ( x , y ) + 2 π ( x ′ x + y ′ y ) ) ] d x d y | 2 .
I ( x ′ , y ′ ) = | ∬ [ cos ⁡ ( ϕ ( x , y ) − 2 π ( x ′ x + y ′ y ) + Δ ϕ ( x , y ) ) + j sin ⁡ ( ϕ ( x , y ) − 2 π ( x ′ x + y ′ y ) + Δ ϕ ( x , y ) ) ] d x d y | 2 ,
I ( x ′ , y ′ ) = | ∬ [ cos ⁡ ( − ϕ ( − x , − y ) − 2 π ( x ′ x + y ′ y ) + Δ ϕ ( x , y ) ) + j sin ⁡ ( − ϕ ( − x , − y ) − 2 π ( x ′ x + y ′ y ) + Δ ϕ ( x , y ) ) ] d ( − x ) d ( − y ) | 2 = | ∬ [ cos ⁡ ( − ( ϕ ( x , y ) ) + 2 π ( x ′ x + y ′ y ) + Δ ϕ ( − x , − y ) ) + j sin ⁡ ( − ϕ ( x , y ) ) + 2 π ( x ′ x + y ′ y ) + Δ ϕ ( − x , − y ) ) ] d x d y | 2 .
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