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Time-accuracy imaging quality prediction method for an infrared detection system under hypersonic conditions

Lin Ju, Yinglun Liu, Yue Ming, Xiaotian Shi, Xueshen Li, and Zhigang Fan

Open Access Open Access

Abstract

The imaging quality of infrared detection systems over time directly affects their ability to track targets accurately. In this study, a prediction scheme for the image quality of infrared detection system under hypersonic conditions based on time accuracy has been developed. Further, based on the time discretization, a calculation model has been established for the prediction scheme to perform numerical simulation. In particular, for verifying the reliability of this prediction method and the associated numerical calculation model, a comparison has been made between the numerical simulation results and the wind tunnel test results. The maximum error of the comparison result is less than 4.5%, and the reliability of the method proposed in this paper has been proved.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Infrared detection technology is important in information warfare due to the advantages of high guidance accuracy, strong anti-interference capability, and the ability to work in different weather conditions. Infrared detection technology needs to obtain the position information of a target at all times in their working state and continuously modify their own working status to ensure that they can accurately strike the target [1]. The infrared detection system is used to obtain the original infrared signal of the target to track and identify it in the infrared detection technology. It is essential to predict image quality of infrared detection systems considering time-varying characteristics, so that the precision strike capability of infrared detection technology can be ensured. In addition, with the advancement of modern warfare, infrared detection technology is increasingly used in hypersonic conditions [1,2]. Under hypersonic conditions with time variation, the image quality degradation of infrared detection systems will be more severe [3–5]. Moreover, the fluid and structural physics parameters are operated at a noticeably smaller time scale than thermal evolution. The influence of fluid evolution and structural evolution on the image quality degradation of infrared detection systems is different at different time scales. Hence, the time-varying characteristics of fluid and structural evolution cannot be ignored. These aspects have inspired the research on the time-accuracy imaging quality prediction method for infrared detection systems under hypersonic conditions.

The infrared detection system consists of an optical window and an internal infrared imaging system. A schematic showing the signal transmission in the infrared detection system is presented in Fig. 1. When the infrared detection system is working under hypersonic conditions, the original infrared signal of the target will be transmitted to the infrared imaging system through its external flow field and optical window. However, under hypersonic conditions with time variation, it is affected by the aero-optical effect. This will make the target image look blurry, have jitter, face energy attenuation, and have background noise interference. The aero-optical effect is the main reason for the deterioration of the imaging quality of an infrared detection system under hypersonic conditions [3,4]. Hence, this study aims to develop an imaging quality prediction method of the infrared detection system that is affected by aero-optical effects based on time-accurate technology.

 figure: Fig. 1.

Fig. 1. Schematic showing the signal transmission in the infrared detection system.

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Recently, researchers have been conducting an increasing amount of research on the prediction of image quality under the influence of aero-optical effects. Wang et al. proposed a numerical model for predicting the aero-optical performance in a supersonic flow field [6]. Yang et al. presented a new ray tracing simulation of aero-optical effect through anisotropic inhomogeneous media as supersonic flow field surrounds a projectile [7]. Xu et al. proposed a backward ray-tracing method for aero-optics simulations of flow field [8]. Zhang et al. used a combination of the principle of ray tracing and numerical simulation methods to calculate the optical path difference (OPD) in a signal that passes through an optical window [9]. Niu et al. established a numerical simulation model for the thermal radiation noise of the optical window [10]. Wang et al. established a comprehensive performance simulation method for the aero-optical transmission effect as well as the aero-thermal radiation effect exerted by the optical window [11]. Xiao et al. established the joint aero-optical transmission effect model of the flow field and optical window [3,12,13]. The full impact of aero-optical effects on imaging would include the joint aero-optical transmission effect of the fluid and optical window and the aero-thermal radiation effect exerted by the optical window. However, it can be understood that the research on the current imaging quality prediction methods focuses only on a certain aspect of the aero-optical performance, and the time-varying characteristics of hypersonic conditions are ignored. There is a lack of comprehensive research on this topic, leading to the imaging quality prediction for the infrared detection system in the real working state cannot be realized. In addition, to the best of our knowledge, no study has been done on predicting the aero-optical performance by enjoying the time-accuracy technique.

In this study, we have developed a joint prediction scheme involving the time-discretization-based aero-optical effect. This scheme considers the aero-optical transmission effect of the flow field and the optical window in addition to the aero-thermal radiation effect of the optical window. In particular, the prediction scheme takes into account the progress of time-accuracy imaging. A numerical simulation calculation model and an image quality evaluation index calculation model have been established following the prediction scheme. In order to verify the reliability of the prediction method, numerical simulations have been performed on the imaging processing of the infrared detection system under hypersonic working conditions, and the results thus obtained have been compared with the results of the wind tunnel experiment.

2. Time-accuracy image quality prediction model

2.1 Joint scheme with the benefit of the time-accuracy technique

The imaging quality prediction model of the infrared detection system is affected by aero-optical effect has been established in this paper is mainly divided into two aspects. The first is the aero-optical transmission effect exerted by the external flow field. In contrast, the second aspect is the combined effect of aero-optical transmission and aero-thermal radiation exerted by the optical window. It involves three spatial domains: the fluid domain, the structural domain, and the imaging domain. The fluid domain and the structural domain interact via heat transfer, and the relationship between them is coupled. The aero-optical transmission effect in the fluid domain acts on the imaging domain. At the same time, the joint effect of aero-optical transmission and aero-thermal radiation caused by the structural domain acts on the imaging domain. Based on the generation mechanism of the aero-optical effect, a joint scheme with the benefit of the time-accuracy technique has been proposed. The coupling relationship with time variation between fluid and structure has been considered in the joint scheme neglected in previous studies [11,14,15]. In addition, the joint influence of the flow field and the structure on the aero-optical transmission effect and the joint influence of the aero-optical transmission and aero-thermal radiation effect of the structure have been covered by this joint scheme. The specific joint scheme is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Joint prediction scheme incorporating the aero-optical transmission and aero-thermal radiation effect.

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Under hypersonic conditions, the external flow field of infrared detection system will form a strong turbulence boundary layer, and the density of this layer will exhibit a non-uniform gradient change. This causes an uneven gradient distribution in the refractive index field of the external flow field of the infrared detection system. At the same time, the optical window is located at the forefront of the infrared detection system and is directly exposed to hypersonic conditions. The aerodynamic heat flow will heat the optical window, its temperature will rise rapidly, and the heating will be uneven. The high temperature of the optical window and the non-uniform temperature distribution will cause its non-uniform deformation. This will cause thermal stress and thermal strain in the optical window. Subject to the thermo-optical and elasto-optical effects, the refractive index field of the optical window also exhibits non-uniform gradient distribution. When the infrared light propagates in a non-uniformly distributed refractive index field with time variation, the light will be deflected, resulting in distortion of the wave front. This will make the target image collected by the infrared detection system look blurry, have jitter, and face energy attenuation. This is the aero-optical transmission effect. The uneven gradient refractive index field directly contributes to the aero-optical transmission effect. On the other aspect, the optical window heated to a high temperature will generate strong infrared rays. These rays are received by the detector of the infrared imaging system, causing the background noise of the original target image, which will interfere with the recognition of the target. This is the aero-thermal radiation effect. The radiant energy produced by the high temperature is the direct cause of the aero-thermal radiation effects. In addition, the elevated temperature of the optical window will be transferred back to the fluid. This is the coupling between the fluid domain and the structural domain.

Fluid domain calculations need to acquire the general density in the external flow field, the temperature, pressure, and the heat flux of the inner surface of the external flow field. The refractive index field is calculated using the external flow field density, whereas the other results are used as the boundary conditions for the heat transfer calculation of the structure. Complete the coupling calculation between the flow field and the structure field. The structural domain calculation obtains the temperature, deformation, stress, strain, and radiant heat flux due to the high temperature of the optical window. Among them, the temperature, stress, and strain results use the thermo-optical effect and the elasto-optical effect to calculate the refractive index field of the optical window. The radiant heat flow results are used for calculating the energy of the infrared radiation rays. In the imaging domain, ray tracing is used to track the target signal to obtain the optical path length(OPL)of the signal to the infrared imaging system. On the other hand, the ray is radiated from the optical window by tracing to obtain the irradiance reaching the detector surface.

2.2 Numerical simulation model based on time-discreteness

Under hypersonic conditions, the fluid and structural physics parameters are operated at a noticeably smaller time scale than thermal evolution. In addition, the influence of fluid evolution and structural evolution on the aero-optical properties is different at different time scales. Hence, time marching is required for capturing the time records at a small time scale, which encapsulate the longer time scale. In this study, a numerical simulation model based on time-discreteness has been developed for predicting the image quality of the infrared detection system. The numerical simulation model considers the coupling between the fluid domain and the structural domain based on time accuracy. In addition, this model considers the different time scales of the aero-optical effects on the imaging quality. The numerical simulation model based on time-discreteness is shown schematically in Fig. 3.

 figure: Fig. 3.

Fig. 3. Schematic showing the coupling scheme base on time-discreteness.

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In Fig. 3, $\Delta {t_F}$ is the result update time step of the fluid domain, $\Delta {t_S}$ is the result update time step of the structural domain, and $\Delta {t_T}$ is the result update time step of the thermal domain. The steps involved in the coupling scheme are as follows:

Firstly, time is updated to $t + \Delta {t_F}$. ① The fluid solver output pressure results are delivered to the structure solver, which can later update the pressure boundary conditions. ④The fluid solver density results are passed to the optical solver, and the refractive index results are updated.

Next, time is updated to $t + \Delta {t_S}$. ② The structural solver updates the deformation results and transfers them and the un-updated thermal results back to the fluid solver. The boundary conditions of the fluid solver are updated, and the calculation is continued. ⑤The structural solver updates the stress, strain, and deformation results and transfers them to the optical solver. The refractive index results are updated.

Finally, time is updated to $t + \Delta {t_T}$. ③The heat flux output produced by the fluid solver will be delivered to the thermal solver. ⑧The structural temperature results are updated and transferred back to the fluid solver. The boundary conditions of the fluid solver are updated, and the calculation continues. ⑥ The updated structure temperature results are passed to the optical solver, and the refractive index results are updated.⑦ The structural radiation heat flux results are updated and passed to the optical solver to update the results of the detector illuminance.

The fluid, structural, and thermal fields are all governed by the equations of continuity of mass, momentum, and energy. The compressible Navier-Stokes (N-S) equations constitute the basic equations for solving the fluid motion. The N-S equations can be expressed as [16–18]:

$$\frac{{\partial \rho }}{{\partial t}} + \vec{\nabla } \cdot ({\rho \vec{v}} )= 0,$$
$$\frac{{\partial ({\rho \vec{v}} )}}{{\partial t}} + \vec{\nabla } \cdot ({\rho \vec{v} \otimes \vec{v} - \sigma } )= 0,$$
$$\frac{{\partial e}}{{\partial t}} + \vec{\nabla } \cdot ({e\vec{v} - \sigma \cdot \vec{v} + \vec{q}} )= 0,$$
where $\rho$ denotes the density, $\vec{v}$ denotes the velocity in the medium associated with the spatial domain, e denotes the specific internal energy, $\sigma$ denotes the stress tensor, and $\vec{q}$ denotes the heat flux.

In calculation for the fluid domain considering the time factor, the spatial discretization of the N-S equations can be expressed as [18–20]

$$\frac{1}{J}\frac{{\partial {\textbf Q}}}{{\partial t}} = R ({\textbf Q} ),$$
$${\textbf Q} \equiv {[{\rho ,\rho \vec{v},e} ]^T},$$
$$\textrm{ }J \equiv \left|{\frac{{\partial [{\xi ,\eta ,\zeta } ]}}{{\partial [{x,y,z} ]}}} \right|,$$
$$R ({\textbf Q} )\equiv{-} \left[ {\frac{{\partial {\textbf F}}}{{\partial \xi }} + \frac{{\partial {\textbf G}}}{{\partial \eta }} + \frac{{\partial {\textbf H}}}{{\partial \zeta }}} \right] + {\textbf Q}{R_{GCL}},$$
$${R_{GCL}} \equiv \frac{\partial }{{\partial t}}\left( {\frac{1}{J}} \right) + \frac{\partial }{{\partial \xi }}\left( {\frac{{\dot{\xi }}}{J}} \right) + \frac{\partial }{{\partial \eta }}\left( {\frac{{\dot{\eta }}}{J}} \right) + \frac{\partial }{{\partial \zeta }}\left( {\frac{{\dot{\zeta }}}{J}} \right),$$
where F, G, and H are the discrete fluxes in the ξ, η, and ζ directions, respectively. Additional items are used for satisfying the geometric conservation law. Using the second-order backward Euler equation to discretize the fluid domain in terms of time, we get [20–22]
$${\dot{{\textbf Q}}^{k + 1}} \approx \frac{{3{{\textbf Q}^{k + 1}} - 4{{\textbf Q}^k} + {{\textbf Q}^{k - 1}}}}{{2\Delta t}} + o({\Delta {t^2}} ),$$
where $o$ indicates the order, and k is the structural time step counter. The spatial discretization of the structural domain can be expressed as [23]
$$[{\textbf M} ]{\ddot{\textbf{u}} + }[{{\textbf C}({{\textbf u,t}} )} ]{\dot{\textbf{u}} + }[{{\textbf K}({{\textbf u,T}} )} ]{\textbf u = }{{\textbf F}^{\textbf T}}({\textbf T} ){\textbf + }{{\textbf F}^{\textbf a}},$$
where [M] denotes the mass matrix, [C] denotes the damping matrix, [K] denotes the stiffness matrix, Fa denotes the mechanical load vector, and FT represents the thermal load vector, ${\textbf u}$ denotes the discretized structural displacement.

Using the Newmark-β method to discretize the structure domain in terms of time, we get [23,24]

$${{\textbf u}^{m + 1}} \approx {{\textbf u}^m} + \Delta t{\dot{{\textbf u}}^m} + \frac{1}{2}\Delta {t^2}\frac{{{{\ddot{{\textbf u}}}^{m + 1}} + {{\ddot{{\textbf u}}}^m}}}{2} + o({\Delta {t^3}} ),$$
$${\dot{{\textbf u}}^{m + 1}} \approx {\dot{{\textbf u}}^m} + \Delta t\frac{{{{\ddot{{\textbf u}}}^{m + 1}} + {{\ddot{{\textbf u}}}^m}}}{2} + o({\Delta {t^3}} ),$$
where γ and β are numerical parameters, and m is the structural time step counter.

The spatial discretization of the thermal domain can be expressed as [24,25]

$$[{{{\textbf C}_{\textbf T}}({\textbf T} )} ]{\dot{\textbf T} + }[{{{\textbf K}_{\textbf T}}({\textbf T} )} ]{\textbf T = }{{\textbf Q}_{\textbf T}},$$
where T, [CT], [KT], and QT denote the temperature vector, thermal capacitance matrix, thermal conductivity matrix, and heat load vector, respectively.

Using the Crank-Nicolson method to discretize the thermal domain in terms of time, we get [25]

$${\dot{{\textbf T}}^{n + {1 / 2}}} \approx \frac{{{{\textbf T}^{n + 1}} - {{\textbf T}^n}}}{{\Delta t}} + o({\Delta {t^2}} ),$$
where n is the thermal time step counter.

The equations for the interaction between the fluid and the structural domain at the fluid-structure interface can be expressed as

$${\sigma _F} \cdot \vec{n} = {\sigma _S} \cdot \vec{n},$$
$${\vec{q}_F} \cdot \vec{n} ={-} ({\kappa \vec{\nabla }T} )s \cdot \vec{n},$$
$${\vec{x}_F} = ({\vec{u} + \vec{x}} )s,$$
$${\dot{v}_F} = {\vec{\dot{u}}_S},$$
$${T_F} = {T_T},$$
where $\vec{n}$ represents the unit direction along the flow field direction on the boundary, T is the temperature, $\vec{u}$ is the deformation of the material from the initial configuration, and $\kappa$ is the thermal conductivity of the material. The subscript F represents the fluid domain, the subscript S represents the structural domain, and the subscript T represents the thermal field. These equations characterize that the stress, heat flux, displacement, velocity, and temperature are equal at the boundary of the fluid and structure domains.

Following Fermat’s principle, the calculation of the imaging domain is completed, and the optical path of the target signal to the infrared detection system is obtained [11,14,15].

Various time integrators can be applied in the solver in every physical field. The boundary conditions need to be exchanged no more than once per time step. The data exchange between the solvers is completed by the grid association and data interpolation. The grid association diagram is shown in Fig. 4. Using the projection points corresponding to the unknown data points in each calculation domain, the grid associated with the unknown data points can be determined. Using the physical quantities of the associated grid nodes, the physical quantities of the unknown points are calculated using the inverse distance interpolation algorithm. The interpolation algorithm can be expressed as:

$$q = \sum\nolimits_{i = 1}^n {} q(i) \cdot \omega (i),$$
$${\omega _i} = \frac{{h_j^{ - p}}}{{\sum\nolimits_{j = 1}^n {h_j^{ - p}} }},$$
where $\omega$ is the weighting function, h is the distance from the point to be interpolated to the associated grid node, p is any positive real number, and n is the number of grid nodes.

 figure: Fig. 4.

Fig. 4. The grid association diagram.

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3. Image quality evaluation index

In this study, point spread function (PSF) and image shift have been combined with the peak signal-to-noise ratio (PSNR) for evaluating the imaging quality of the infrared detection system.

3.1 PSF of the infrared detection system

The refractive index in the external flow field depends on the relevant density. Using the Gladstone-Dale formula, the refractive index in the flow field is written as

$$N = 1 + {K_{\textrm{GD}}}\rho ,$$
$${K_{\textrm{GD}}}(\lambda )= 2.23 \times {10^{ - 4}}\left( {1 + \frac{{7.52 \times {{10}^{ - 3}}}}{{{\lambda^2}}}} \right),$$
where N is the refractive index and $\lambda$ is the wavelength.

Refractive index changes in the optical window may arise from thermo-optical and elasto-optical effects. The thermo-optical effect exerts a stronger impact on the refractive index changes as compared to the elasto-optical effect. Here, we only consider the influence of the thermo-optical effect. Thus, refractive index changes in the optical material with temperature can be defined as the thermo-optical effect [3].

$$n[{\lambda ,t(x,y,z)} ]= n(\lambda ,{t_0}) + \frac{{\textrm{d}(\lambda ,t)}}{{\textrm{d}t}}\Delta t(x,y,z), $$
where $n(\lambda ,{t_0})$ represents the refractive index at any grid node of the reference temperature ${t_0}$, ${{\textrm{d}n(\lambda ,t)} / {\textrm{d}t}}$ represents the thermo-optical coefficient, and $\Delta t(x,y,z)$ represents the temperature change value in $t(x,y,z)$ relative to ${t_0}$.

After obtaining the refractive index field of the flow field and the optical window, the infrared signal transmitted in the non-uniform gradient refractive index field needs to be traced. For the same, the optical path of the infrared signal to the exit pupil of the infrared detection system needs to be calculated. In accordance with Fermat’s principle, the light equation is given by [26]

$$\frac{\textrm{d}}{{\textrm{d}s}}\left[ {n(r )\frac{{\textrm{d}r}}{{\textrm{d}s}}} \right] = \nabla n(r ), $$
where r represents the position vector related to ray propagation, $\textrm{d}s$ represents the step size of the propagation path, and $n(r )$ represents the refractive index distribution. The fourth-order Runge-Kutta method is employed for solving the Fermat formula. The coordinates at the signal propagation point are given by
$$\left\{ \begin{array}{l} {{\boldsymbol r}_{i + 1}} = {{\boldsymbol r}_i} + \frac{h}{6}({{{\boldsymbol K}_1} + 2{{\boldsymbol K}_2} + 2{{\boldsymbol K}_3} + {{\boldsymbol K}_4}} )\\ {{\boldsymbol T}_{i + 1}} = {{\boldsymbol T}_i} + \frac{h}{6}({{{\boldsymbol L}_1} + 2{{\boldsymbol L}_2} + 2{{\boldsymbol L}_3} + {{\boldsymbol L}_4}} )\end{array} \right.,$$
where
$$\left\{ \begin{array}{l} \frac{{\textrm{d}{\boldsymbol r}}}{{\textrm{d}p}} = {\boldsymbol T}\\ \frac{{\textrm{d}{\boldsymbol T}}}{{\textrm{d}p}} = n\nabla n \end{array} \right.,$$
where $p = \int {\frac{{\textrm{d}s}}{n}}$

The coefficients can be expressed as

$$\left\{ {\begin{array}{{l}} {{K_1} = {T_i}}\\ {{K_2} = {T_i} + {{h{L_1}} / 2}}\\ {{K_3} = {T_i} + {{h{L_2}} / 2}}\\ {{K_4} = {T_i} + h{L_3}} \end{array}} \right., $$
where
$$\left\{ {\begin{array}{{l}} {{L_1} = n(r)\nabla n(r){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (at{\kern 1pt} {\kern 1pt} {r_i})}\\ {{L_2} = n(r)\nabla n(r){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (at{\kern 1pt} {\kern 1pt} {r_i} + {{h{K_1}} / 2}, \textrm{in the direction of}\;{K_1})}\\ {{L_3} = n(r)\nabla n(r){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (at{\kern 1pt} {\kern 1pt} {r_i} + {{h{K_2}} / 2}, \textrm{in the direction of}\textrm{ }{K_2})}\\ {{L_4} = n(r)\nabla n(r){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (at{\kern 1pt} {\kern 1pt} {r_i} + h{K_3}, \textrm{in the direction of}\textrm{ }{K_3})} \end{array}} \right..$$
When the signal is at the ith step of ray tracing, the optical path length (OPL) is given by
$$\textrm{OP}{\textrm{L}_i} = \int_{{t_0}}^{{t_0} + \Delta t} {\left( {{N_i}\frac{{\partial {l_i}}}{{\partial t}} + {l_i}\frac{{\partial {N_i}}}{{\partial t}}} \right)} \textrm{d}t + \int_{{\varepsilon _0}}^{{\varepsilon _0} + \Delta \varepsilon } {\left( {{N_i}\frac{{\partial {l_i}}}{{\partial \varepsilon }} + {l_i}\frac{{\partial {N_i}}}{{\partial \varepsilon }}} \right)\textrm{d}\varepsilon } ,$$
where l is the signal transmission distance.

Using the OPL obtained by ray tracing, the wave aberration, Wj, of the exit pupil is calculated as follows:

$$\textrm{OP}{\textrm{L}_0} = \frac{1}{{{N_r}}}\sum\limits_j {\textrm{OP}{\textrm{L}_j}} ,$$
$${W_j}({x,y} )= \frac{{2\mathrm{\pi }}}{\lambda }({\textrm{OP}{\textrm{L}_j} - \textrm{OP}{\textrm{L}_0}} ),$$
where ${N_r}$ is the number of rays.

$\textrm{PSF}({x^{\prime},y^{\prime}} )$ is calculated from the pupil function. The pupil function can be expressed as

$$A(x^{\prime},y^{\prime}) = \left\{ {\begin{array}{{cc}} {a(x,y)\textrm{exp} [{jW(x,y)} ]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {x^2} + {y^2} \le {{({D / 2})}^2}}\\ {{\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {x^2} + {y^2} \ge {{({D / 2})}^2}} \end{array}} \right., $$
where $a(x,y)$ represents the amplitude distribution followed by the pupil function, and D is the pupil diameter.

The target imaging process of one supersonic aircraft generally meets the far-field approximation requirement. In accordance with the Huygens’ principle, the amplitude distribution of the light field on the image plane is measured via a Fourier transform as expressed below :

$$U(x^{\prime},y^{\prime}) = \int\!\!\!\int {A(x,y)\textrm{exp} \left[ {\textrm{ - }j\frac{{2\pi }}{{\lambda f}}(xx^{\prime} + yy^{\prime})} \right]} \textrm{d}x\textrm{d}y.$$
Since the light intensity is proportional to the square of the amplitude, the PFS as:
$$\textrm{PSF}({x^{\prime},y^{\prime}} )= {|{U({x^{\prime},y^{\prime}} )} |^2} = U({x^{\prime},y^{\prime}} ){U^ \ast }({x^{\prime},y^{\prime}} ).$$

3.2 Image offset of the target image

The optical system is generally considered a linear system. When the rays emitted from the object transmit via the linear system to the image space, their frequency will not be changed, but the contrast decreases, and phase shifts occur simultaneously.

The difference between the centroids of the distorted target image and the original target image is called the image shift. Based on the calculation method in Section 3.1, the PSF of the infrared detection system affected by the aero-optical effect has been obtained. The original image is convolved with the PSF of the infrared detection system, and the distorted image affected by the aero-optical effect is obtained. The centroid positions of the original image and the distorted image can be obtained by image processing calculations. The image shift as

$${\delta _x} = \frac{{X - {X_{\textrm{origin}}}}}{{f^{\prime}}},\textrm{ }{\delta _y} = \frac{{Y - {Y_{\textrm{origin}}}}}{{f^{\prime}}},$$
where $f^{\prime}$ is the focal length of the optical system. X and Y is the centroid position of the distorted image. ${X_{\textrm{origin}}}$ and ${Y_{\textrm{origin}}}$ is the centroid position of the original image.

3.3 PSNR of the target image

In the working state of an infrared detection system, the optical window heated to a high temperature will emit a large amount of infrared radiation, resulting in the aero-thermal radiation effect. In the process of thermal radiation illuminance calculation, the optical window is a medium for signal transmission and a radiation source. An aerodynamically heated optical window produces a background noise signal on the target image and reduces the signal-to-noise ratio of the optical seeker. In severe cases, “thermal barrier” phenomenon is formed, and the target signal is overwhelmed by noise signal [14]. A schematic showing the radiation transmission is shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Schematic showing the radiation transmission through the optical window.

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According to the radiation law, the panel radiance, ${L_\lambda }$, can be expressed as

$${L_\lambda } = \frac{{2\varepsilon h{v_c}^2{\lambda ^{ - 5}}}}{{\textrm{exp} ({{{h{v_c}} / {k{T_i}\lambda }}} )- 1}}, $$
where $\varepsilon $ denotes the surface element emission rate, h is the Planck constant, ${v_c}$ is the speed of light in the medium, $k$ is the Boltzmann constant, and $\lambda $ is the wavelength.

Any radiation surface element is a Lambert radiator that has identical radiant luminance along each direction. The radiation rays are traced by ray tracing. The amount of light received by the detector pixels and the energy are measured to calculate the irradiance on the detector surface. Suppose that the infrared radiated ray direction was at an angle with respect to the normal of the surface element and the area of the surface element was ds, then the radiation power of the infrared rays from the surface element at the spatial angle and in the waveband is given by [27]

$$\textrm{d}W = {L_\lambda }\textrm{cos}\theta \textrm{d}\lambda \textrm{d}s\textrm{d}\varOmega .$$

The irradiance on the detector surface is thus calculated using the above equation. Based on the optical dome radiation power, the output voltage of every pixel of the detector is calculated using the following equation:

$${V_{ij}} = G \cdot {R_{ij}} \cdot \textrm{d}W + {V_{Nij}}, $$
where Vij represents the output voltage related to the radiation power received at the pixel (i, j), G denotes the gain of the preamplifier of the detector, ${R_{ij}}$ denotes the response rate at the pixel (i, j), $\textrm{d}W$ denotes the radiative power related to the optical dome, and ${V_{Nij}}$ denotes the root mean square noise at the pixel (i, j). The output voltage can be transferred to the gray value via the preprocessing circuit, and thus a distorted image is obtained owing to the thermal radiation interference from the optical dome.

The PSNR was calculated for evaluating the quality of the distorted images. This is the most common and objective imaging evaluation parameter. It is given by [28]

$$\textrm{PSNR}(f,G) = 10\log 10\left[ {\frac{{{{(M - 1)}^2}}}{{\textrm{MSE}(f,G)}}} \right], $$
where M represents the pixel number in the images and MSE represents the mean squared error in the images.

4. Experimental verification

For verifying the reliability of the imaging quality prediction method for the infrared detection system, the results obtained from the numerical calculations using the prediction method were compared with the experimental results obtained from the arc wind tunnel experiment conducted in our laboratory. During the experiment, the optical window involved in imaging was used as the experimental object. This optical window was in the shape of a flat plate. To make a comparision, only the optical window was used as the prediction object in the numerical calculation, and the calculation boundary conditions were unified with the experimental conditions. The parameters used in the experiment and the numerical calculations are listed in Table 1.

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Table 1. Parameters used in the experiment and numerical calculations

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The wind tunnel selected for the experiment was the hypersonic arc wind tunnel of the China Aerospace Aerodynamics Technology Research Institute. A schematic of the arc wind tunnel experimental test system is shown in Fig. 6. The blackbody heat source radiates infrared signals outwardly and emits parallel light through the collimator to simulate a target at infinity. The infrared rays pass through the hypersonic fluid and the optical window heated pneumatically and are finally received by the detector.

 figure: Fig. 6.

Fig. 6. Schematic of the experimental setup of the arc wind tunnel test system.

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From the measurement from the wind tunnel experiment, the heat flux density at the optical window external surface varies in the range of 6.0 × 105∼ 1.7 × 106W/m2, and the result of the numerical calculation using predictive simulation is in the range of 5.82 × 105∼ 1.64 × 106W/m2. The error between the numerical simulation result and the experimentally measured heat flux density result is less than 3%. This shows that the optical windows evaluated in the numerical simulation and wind tunnel tests are in the same thermal environment. The comparison result of the imaging quality evaluation for the infrared detection system is valuable. The numerical calculation results are presented in Fig. 7.

 figure: Fig. 7.

Fig. 7. The heat flux results obtained from the numerical calculation.

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4.1 Comparison of the PSF

In the experiment, the detection results obtained by the infrared detector are shown in Fig. 8. When a black body is used as a signal source for experiments, additional background noise will be introduced, interfering with the detector's infrared signal. Hence, in the experiment, the Wiener filter method is used to obtain PSF of the flow field and the optical window under the influence of aero-optical effects. The PSF results obtained by numerical simulation and experiment are compared, and the result is shown in Fig. 9. No imaging system was built in the experiment, and the rays were directly received by the detector. In order to match the experimental state, the PSF result obtained in the simulation process is obtained by tracing the rays directly to the infrared detector.

 figure: Fig. 8.

Fig. 8. Images obtained by the optical window at different times in the experiment.

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 figure: Fig. 9.

Fig. 9. The PSF comparison result of experiment and simulation under the influence of aero-optical effect.

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From the comparison result, the change law of the PSF obtained by the experiment and the simulation method is consistent with time. In particular, at the same time point, the distribution of the PSF is similar. It can be seen from the distribution of the PSF that with the passage of working time, the signal received by the infrared imaging system will appear ghost, blur, and jitter. Moreover, when t>8s, the quality of the signal result is severely degraded, and when t>12s, the target signal is difficult to be recognized.

4.2 Comparison of the image offset

For further quantitative comparison, the centroid offset of the image received by the infrared detector was calculated by image processing. In case the experiment time exceeding 12 s, the image received by the detector is highly affected by noise, and the centroid offset of the signal received by the detector cannot be calculated. The comparison between the experimental results and the simulation results is shown in Table 2 below. The comparison results show that the maximum error between the experimental and simulation results is less than 4.5%. As time progresses, the deviation of the image obtained by the infrared detection system is getting larger and larger.

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Table 2. Centroid offset obtained from the experiment and numerical simulation

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4.3 Comparison of the PSNR

In order to verify the reliability of the method proposed in this paper for the imaging background noise prediction, the PSNR of the target image obtained from simulation and experiment were compared. The comparison results are shown in Table 3. The comparison results show that the maximum error between the experimental and simulation results is less than 4.34%.

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Table 3. The PSNR of the target image obtained from the experiment and numerical simulation

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The curve of the PSNR of the target image with time is shown in Fig. 10. As time progresses, the PSNR of the target image shows a nearly linear rapid decline. With the advancement of working hours, the background noise of the image becomes more and more serious, and the target information was submerged by the noise.

 figure: Fig. 10.

Fig. 10. The PSNR comparison result of experiment and simulation under the influence of aero-optical effect.

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5. Summary

In conclusion, a scheme accompanied by the time-accuracy technique was developed to predict the imaging quality of the infrared detection system under hypersonic conditions in this study. Three evaluation parameters, namely, PSF, image shift, and PSNR of the distorted images, were proposed for evaluating the imaging quality of the infrared detection system. With a numerical simulation calculation model and wind tunnel test, three parameters are obtained and compared. The comparison result shows that the error of offset is 4.5%, and the error of PSNR is 4.34%. The change of PSF over time follows a consistent trend, and the distribution pattern of each time point is also consistent. The reliability of the method is proved in this paper. The method is promising in evaluating the precise guidance capability for infrared detection technology under hypersonic conditions. Moreover, the result obtained by this method promises to provide a valuable basis for the improvement of the precise guidance capability of infrared detection technology.

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 but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Schematic showing the signal transmission in the infrared detection system.

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Fig. 2.
Fig. 2. Joint prediction scheme incorporating the aero-optical transmission and aero-thermal radiation effect.

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Fig. 3.
Fig. 3. Schematic showing the coupling scheme base on time-discreteness.

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Fig. 4.
Fig. 4. The grid association diagram.

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Fig. 5.
Fig. 5. Schematic showing the radiation transmission through the optical window.

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Fig. 6.
Fig. 6. Schematic of the experimental setup of the arc wind tunnel test system.

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Fig. 7.
Fig. 7. The heat flux results obtained from the numerical calculation.

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Fig. 8.
Fig. 8. Images obtained by the optical window at different times in the experiment.

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Fig. 9.
Fig. 9. The PSF comparison result of experiment and simulation under the influence of aero-optical effect.

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Fig. 10.
Fig. 10. The PSNR comparison result of experiment and simulation under the influence of aero-optical effect.

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Tables (3)

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Table 1. Parameters used in the experiment and numerical calculations

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Table 2. Centroid offset obtained from the experiment and numerical simulation

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Table 3. The PSNR of the target image obtained from the experiment and numerical simulation

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Equations (40)

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∂ ρ ∂ t + ∇ → ⋅ ( ρ v → ) = 0 ,
∂ ( ρ v → ) ∂ t + ∇ → ⋅ ( ρ v → ⊗ v → − σ ) = 0 ,
∂ e ∂ t + ∇ → ⋅ ( e v → − σ ⋅ v → + q → ) = 0 ,
1 J ∂ Q ∂ t = R ( Q ) ,
Q ≡ [ ρ , ρ v → , e ] T ,
  J ≡ | ∂ [ ξ , η , ζ ] ∂ [ x , y , z ] | ,
R ( Q ) ≡ − [ ∂ F ∂ ξ + ∂ G ∂ η + ∂ H ∂ ζ ] + Q R G C L ,
R G C L ≡ ∂ ∂ t ( 1 J ) + ∂ ∂ ξ ( ξ ˙ J ) + ∂ ∂ η ( η ˙ J ) + ∂ ∂ ζ ( ζ ˙ J ) ,
Q ˙ k + 1 ≈ 3 Q k + 1 − 4 Q k + Q k − 1 2 Δ t + o ( Δ t 2 ) ,
[ M ] u ¨ + [ C ( u , t ) ] u ˙ + [ K ( u , T ) ] u = F T ( T ) + F a ,
u m + 1 ≈ u m + Δ t u ˙ m + 1 2 Δ t 2 u ¨ m + 1 + u ¨ m 2 + o ( Δ t 3 ) ,
u ˙ m + 1 ≈ u ˙ m + Δ t u ¨ m + 1 + u ¨ m 2 + o ( Δ t 3 ) ,
[ C T ( T ) ] T ˙ + [ K T ( T ) ] T = Q T ,
T ˙ n + 1 / 2 ≈ T n + 1 − T n Δ t + o ( Δ t 2 ) ,
σ F ⋅ n → = σ S ⋅ n → ,
q → F ⋅ n → = − ( κ ∇ → T ) s ⋅ n → ,
x → F = ( u → + x → ) s ,
v ˙ F = u ˙ → S ,
T F = T T ,
q = ∑ i = 1 n q ( i ) ⋅ ω ( i ) ,
ω i = h j − p ∑ j = 1 n h j − p ,
N = 1 + K GD ρ ,
K GD ( λ ) = 2.23 × 10 − 4 ( 1 + 7.52 × 10 − 3 λ 2 ) ,
n [ λ , t ( x , y , z ) ] = n ( λ , t 0 ) + d ( λ , t ) d t Δ t ( x , y , z ) ,
d d s [ n ( r ) d r d s ] = ∇ n ( r ) ,
{ r i + 1 = r i + h 6 ( K 1 + 2 K 2 + 2 K 3 + K 4 ) T i + 1 = T i + h 6 ( L 1 + 2 L 2 + 2 L 3 + L 4 ) ,
{ d r d p = T d T d p = n ∇ n ,
{ K 1 = T i K 2 = T i + h L 1 / 2 K 3 = T i + h L 2 / 2 K 4 = T i + h L 3 ,
{ L 1 = n ( r ) ∇ n ( r ) ( a t r i ) L 2 = n ( r ) ∇ n ( r ) ( a t r i + h K 1 / 2 , in the direction of K 1 ) L 3 = n ( r ) ∇ n ( r ) ( a t r i + h K 2 / 2 , in the direction of   K 2 ) L 4 = n ( r ) ∇ n ( r ) ( a t r i + h K 3 , in the direction of   K 3 ) .
OP L i = ∫ t 0 t 0 + Δ t ( N i ∂ l i ∂ t + l i ∂ N i ∂ t ) d t + ∫ ε 0 ε 0 + Δ ε ( N i ∂ l i ∂ ε + l i ∂ N i ∂ ε ) d ε ,
OP L 0 = 1 N r ∑ j OP L j ,
W j ( x , y ) = 2 π λ ( OP L j − OP L 0 ) ,
A ( x ′ , y ′ ) = { a ( x , y ) exp [ j W ( x , y ) ] x 2 + y 2 ≤ ( D / 2 ) 2 0 x 2 + y 2 ≥ ( D / 2 ) 2 ,
U ( x ′ , y ′ ) = ∫ ∫ A ( x , y ) exp [  -  j 2 π λ f ( x x ′ + y y ′ ) ] d x d y .
PSF ( x ′ , y ′ ) = | U ( x ′ , y ′ ) | 2 = U ( x ′ , y ′ ) U ∗ ( x ′ , y ′ ) .
δ x = X − X origin f ′ ,   δ y = Y − Y origin f ′ ,
L λ = 2 ε h v c 2 λ − 5 exp ( h v c / k T i λ ) − 1 ,
d W = L λ cos θ d λ d s d Ω .
V i j = G ⋅ R i j ⋅ d W + V N i j ,
PSNR ( f , G ) = 10 log ⁡ 10 [ ( M − 1 ) 2 MSE ( f , G ) ] ,
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