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Aerospace Systems

, Volume 2, Issue 1, pp 61–70 | Cite as

OPD analysis and prediction in aero-optics based on dictionary learning

  • Zijia Li
  • Yuanxiang LiEmail author
  • Boyang Xing
  • Bin Zhang
  • Hongya Tuo
  • Hong Liu
Original Paper
  • 168 Downloads

Abstract

When aircraft flying at a high speed, the density and reflective index of atmosphere around it become uneven. Thus images or videos observed from the observation window on the aircraft are usually blur or quivering, which is called the aero-optical effect. To recover the images from poor quality, it is necessary to learn about the wavefront distortion of the light, described as optical path difference (OPD). Among the existing methods, the method of computational fluid dynamics (CFD) simulation followed by ray tracing is very time consuming, and the method of real-time OPD measurement with OPD sensor has a certain lag for OPD with high frequency. In this paper, a reconstructible dimension reduction method based on dictionary learning is employed to map the high-dimensional OPD data into a low-dimensional space, and the OPD data are calculated when rays travel across the supersonic shear layer. All the parameters of training and test datasets remain the same except the convective Mach numbers (Mc number). According to the dimension reduction results of training sets, we find that OPD is obviously periodic and its distribution characteristics have a strong correlation with Mc number. By fitting the OPD data in the low-dimensional space, every point on the fitting curve can be reconstructed to the original high-dimensional space, which works as prediction. Compared with the truthful data, the average similarity coefficient of the prediction for the test datasets is up to 83%, which means that the prediction result is credible.

Keywords

Aero-optics Optical path difference Prediction Couple dictionary learning 

1 Introduction

When vehicles flying at high speed, an uneven refractive index field is generated outside the observation window because of the inhomogeneous density field in the surrounding atmosphere. Thus, images acquired by the camera inside the vehicle from the observation window are usually blurred, offset or jitter, which is called aero-optical effect (AOE) [1]. AOE makes it difficult to achieve the follow-up tasks, such as target observation and identification. It is necessary to recover the clear images from the ones with poor quality. The mechanism of image degradation with AOE is an important key.

Up to now, AOE image correction methods can be roughly divided into two categories. One category is the adaptive optics systems [2, 3, 4, 5, 6], which apply feedback mechanism to measure the wavefront aberration signal in real time and correct it. The wavefront aberration is usually represented with optical path difference (OPD). Adaptive optics systems with complex mechanical structure are usually employed on ground-based telescopes, in which wavefront aberration occurs on the order of 1–10 Hz. However, the frequency of OPD at supersonic speed is on the order of kilohertz approximately. In consideration of the response speed, the result of image recovery with adaptive optics systems for supersonic aero-optical is not very satisfactory. The other category is image processing. In 2005, Iterative Blind Deconvolution (IBD) algorithm was applied by Philips laboratories for AOE image correction [7]. Hong Hanyu et al. firstly decomposed the degenerated image into multiple scales with wavelet decomposition algorithm; then IBD algorithm was used to achieve the optimal estimation of the clear image and the point spread function (PSF) [8].

For above two kinds of methods, image correction or restoration is achieved by measuring or estimating the degradation parameters, including OPD and PSF. The mechanism of image degradation has not been considered thoroughly. According to the basic principles of camera imaging [9] and wave optics [10], PSF is mainly determined by OPD, which describes the wavefront aberration and reflects how images degenerate. OPD can be obtained with computational fluid dynamics (CFD) simulation followed by ray tracing [11]. Then, PSF can be further obtained from OPD, which is much useful for image recovery. Wu et al. [12] designed a Neural Network model and realized the prediction of OPD by measuring the temperature of several positions in the flow flied. The prediction results are close to those obtained by ray tracing and the prediction error is small. However, CFD is also needed because it is not practical to arrange lots of temperature sensors on the surface of the observation window, which means it is just a theoretical development and cannot be applied to the actual situation.

In this paper, we will explore another way for OPD prediction when lights passing through the shear layer, by fitting a curve of OPD with respect to several flow parameter in a much lower dimension. Convective Mach number (Mc number) is selected as the only flow parameter for the sake of simplicity. This method is fast and accurate enough for real-time application than the above two categories of methods and it will make a huge boost for AOE image restoration.

In most situations, OPD has an irregular shape and high dimension in spatial domain and it is nonlinear over time, which raises the difficulty for feature extraction. In recent years, data analysis methods based on machine learning are developing fast. New methods for data dimension reduction and reconstruction have played a significant role for data analysis such as classification, identification, prediction, visualization and other tasks [13]. There are several kinds of algorithms commonly used for dimension reduction, including linear algorithms [13] such as PCA (Principle Component Analysis), LDA (Linear Discriminant Analysis), LPP (Locality Preserving Projection), and nonlinearities algorithms such as ISOMAP (Isometric Mapping) and LLE (Locally Linear Embedding). All of them focus on mining the intrinsic properties of the original data in low-dimensional space, but the ability of data recovery is relatively poor. Couple dictionary learning (CDL) method [14] based on compressed sensing can achieve data reconstruction effectively from low-dimensional data after nonlinear dimension reduction [15]. Thus, CDL method is more suitable for OPD analysis and prediction due to its nonlinearity and reversibility.

In this paper, we propose a new method for OPD prediction based on CDL algorithm. Compared with the direct interpolation result of OPD, the prediction result with the proposed method has shown a greater accuracy in the experience. Considering obvious randomness of the flow and OPD, correlation coefficient R, mean square error MSE and peak signal-to-noise ratio PSNR are selected as evaluation indicators of the prediction results. According to the experiments, the average correlation coefficient R is up to 83% for almost all Mc numbers, which means that the prediction results are reliable and the training data are sufficient.

The paper is organized as follows: In Sect. 2, the simplified aero-optical model is presented and the data structure of OPD is described. In Sect. 3, dimensionality reduction and reconstruction methods are introduced and fitting algorithm and evaluation methods for prediction accuracy are presented as well. Section 4 shows the experimental results. Finally, the conclusion is given in Sect. 5.

2 Aero-optical model and OPD data

To facilitate the research, some simplifications are made on the real physical situation. OPD analysis and prediction are all performed based on the simplified model. As shown in Fig. 1, for the real situation, cooling jet is required to reduce the temperature of the observation window caused by aerodynamic thermal effect, which is upstream of the observation window and can protect it from deformation [16].
Fig. 1

Schematic of the real situation

Fig. 2

Simplified shear layer aero-optical model

Fig. 3

Contour map of OPD for 3-dimensional flow

Since the cooling jet has a low velocity relative to the incoming atmosphere, shear layer is formed from the junction of the two streams above the observation window. Then, the density becomes not uniform within the shear layer and affect the refractive index of air to be uneven according to Gladstone–Dale relationship [17]:where n and \(\rho \) are the refraction index and density of medium, respectively, \( \lambda \) is the wavelength of light, and \(K_{\mathrm{GD}}\) is a constant related to \(\lambda \). Light beam from the target in the distance is deflected as passing through the shear layer and the wavefront becomes distorted, making the quality of target image poor.
Wavefront distortion is generally expressed with OPD. If we define the initial direction of light as direction y and observation window is on the plane \(y=0\), for any position Q(x, 0, z) on the observation window, the OPD when the light beam reaches point Q at time t, can be expressed as \(\mathrm{{OPD}}(x,z,t)=\mathrm{{OPD}}(x,0,z,t)\):
$$\begin{aligned} \mathrm{{OPD}}(x,z,t)=\mathrm{{OPL}}(x,z,t)-\overline{\mathrm{{OPL}}(x,z,t)} \end{aligned}$$
(2)
where \(\overline{\mathrm{{OPL}}(x,z,t)}\) is the average of \(\mathrm{{OPL}}(x,z,t)\) and \(\mathrm{{OPL}}(x,z,t)= \mathrm{{OPL}}(x,0,z,t)\) means the optical path length from light source to position Q(x, 0, z) at time t:
$$\begin{aligned} \mathrm{{OPL}}(x,z,t)=\int _{{\varvec{l}}} n(x,y,z,t)\mathrm{{d}}l \end{aligned}$$
(3)
where \({\varvec{{l}}}\) is the light path from light source to position Q(x, 0, z). To facilitate the analysis, the shear layer aero-optical model is simplified as shown in Fig. 2. The incoming flow is parallel to the observation window. The angle between the cooling jet and the incoming flow is set to be sinusoidal and periodical to achieve flow control. The flow control method will make the vortex structure rolled up by the shear layer more regular, so that the periodicity and characteristics of the OPD are more obvious, which is convenient for image correction and restoration. Define the junction position of two streams as the coordinate origin position \(x=0\), and the observation window is in the range of \(x \in [0.2,0.3]\) m. All parameters between the two streams are set to be same except the speed, including sonic speed, then the convection Mach number becomes: \(Mc=(Ma_1-Ma_2)/2\).
Since the two streams have different Mach numbers (Ma number), there exists shear layer at the interface. Supposing target is infinitely far from the observation window and the light coming from the target is perpendicular to the window before affected by the shear layer. That is, the wavefront of the beam is parallel to the observation window and the undistorted OPD is zero. After light passing through the shear layer region, the wavefront of the light will becomes aberrated and OPD will change. The research target is to use several sets of known OPD data to achieve OPD prediction under other Mc number conditions.
Table 1

Experimental parameters of datasets

Datasets

Index

Mc

\( Ma_1 \)

The other parameters

Training sets

1

0.300

1.8

(1) \( Ma_2=1.2 \);

2

0.350

1.9

(2) Sample on spatial domain \( x_i \):

3

0.400

2.0

      (a) Range (m) : \( x_i \in [0.2,0.3] \)

4

0.450

2.1

      (b) Interval (m): \( \mathrm{{d}}x=1\times 10^{-3} \,\,{\text {m}} \)

5

0.500

2.2

      (c) Number : \( m_{x0}=100 \);

6

0.550

2.3

(3) Sample on time domain \( t_i \):

7

0.600

2.4

      (a) Range (ms) : \( t_i \in [0.5,2] \)

Testing sets

1

0.325

1.85

      (b) Interval (s): \( \mathrm{{d}}t=2\times 10^{-6} \)

2

0.375

1.95

      (c) Number : \( m_t=750 \);

3

0.425

2.05

(4) Sample dimension: \( m_0=m_t \cdot m_{x0}=100\times 750 \);

4

0.525

2.25

(5) Angle between cooling jet and incoming air \( A(t)=\alpha _0 \cdot \sin (2\pi t /T),\)

   

      in which \(\alpha _0=0.3^{\circ } \) and \( T=30 \,\mathrm{{d}}t \)

Figure 3 has shown the contour map of OPD for 3-dimensional flow simulated by CFD and ray tracing algorithm. The Mc numbers for (a), (b), (c) and (d) are 0.3, 0.5, 0.7 and 0.9, respectively, and the moment \(t_i\) is selected randomly after the flow becomes stable. According to the result of simulation, when \(Mc<0.6\), the 3-dimensional shear layer can be simplified as 2-dimensional for the above simplified model, which is shown in Fig. 3. The distribution of OPD on the observation window is independent of z and OPD is just a variable about x, which means the direction of incoming flow. That is:
$$\begin{aligned} \mathrm{{OPD}}(x,z,t)=\mathrm{{OPD}}(x,t) \end{aligned}$$
(4)
Therefore, this paper conducts OPD prediction for two-dimensional shear layer. The main parameters of OPD data are shown in Table 1. Considering the actual situation and the purpose of hoping to have a large variation range for Mc number, 11 sets of OPD data with different Mc numbers were calculated with CFD simulation followed by ray tracing. Among them, 7 sets are selected as training sets \( ({{\varvec{Mc}}}_{\mathrm{{train}}}=\{0.3,0.35,\ldots ,0.55,0.6\} \), which are uniform and have a large variation relatively). The other 4 sets are selected as test sets \( ({{\varvec{Mc}}}_{\mathrm{{test}}}=\{0.325,0.375,0.425,0.525\})\) to verify the accuracy of the prediction. While regarding Mc number as the label, the number of samples in training sets is \(n_{\mathrm{{train}}}=7\) and the one in test sets is \(n_{\mathrm{{test}}}=4\).
Each set of OPD data can be seen as a distribution both on time domain and spatial domain. The spatial domain means different position on the observation window, expressed as \({{\varvec{x}}}_i, i=1, 2, \ldots , m_{x0}\). \(m_{x0}=100\) is the dimension of OPD on original spatial domain. The sampling interval is \(\mathrm{{d}}x=1 \times 10^{-3}\) m, and the total length of the observation window is \(L=m_{x0} \times \mathrm{{d}}x=100 \times 10^{-3} {\text { m}}=0.1{\text { m}} \). The time domain means different moments \(t_i, i=1,2,\ldots ,m_t\), and \(m_t=750\) means the dimension of OPD on time domain. The sampling interval is \(\mathrm{{d}}t=2 \times 10^{-6}\) s and the sampling duration is \(T=m_t \times \mathrm{{d}}t=1.5 \times 10^{-3}\) s. Considering that the flow field and OPD are still in the development stage before 0.5 ms according to the simulation, the flow flied becomes stable and the regularity of OPD is more obvious after 0.5 ms. Hence, the sample range in time domain is selected from 0.5 to 2 ms. Therefore, the OPD datasets can be expressed as matrixes with \(m_{x0}\) rows and \(m_t\) columns:
$$\begin{aligned} \mathrm{{Training \ sets}}&:\mathrm{{OPD}}(Mc_1) \in {{\varvec{R}}}^{m_{x0} \times m_t} \end{aligned}$$
(5)
$$\begin{aligned} \mathrm{{Test sets}}&: \mathrm{{OPD}}(Mc_2) \in {{\varvec{R}}}^{m_{x0} \times m_t} \end{aligned}$$
(6)
where \(Mc_1\in {{\varvec{Mc}}}_{\mathrm{{train}}}\) and \(Mc_2\in {{\varvec{Mc}}}_{\mathrm{{test}}}\).
Fig. 4

Diagram of prediction method of OPD

3 OPD analysis and prediction

As shown in Eqs. 5 and 6, the dimension of OPD in original space is high \( (m_0=m_{x0} \times m_t=100\times 750) \). Considering the significant nonlinearity of OPD, it will result in a large prediction error while fitting a curve of OPD with respect to Mc number directly in the original space. Therefore, CDL algorithm is used firstly to reduce the dimension of the training sets only on spatial domain from the original spatial dimension \( (m_{x0}=100) \) to a much lower spatial dimension \( (m_{x1}=8) \), as shown in Fig.  4. In Fig. 4, DR means dimensionality reduction, REC means reconstruction, \( m_{x0}=100, m_{x1}=8\), and \(d=6 \). The results are expressed as \( {{\varvec{OPD}}}_1(Mc_1) \in {{\varvec{R}}}^{m_{x1} \times m_t} \). After that, PCA algorithm is applied to extract the first d principal components of \( {{\varvec{OPD}}}_1 (Mc_1 ) \). The coefficients are expressed as vector \( {{\varvec{OPD}}}_2 (Mc_1) \in {{\varvec{R}}}^d \), which is convenient to be fitted as a curve with respect to different Mc number using spline interpolation algorithm for its low dimension. Any point on the fitting curve corresponds to different Mc number and can be reconstructed back to the original dimension (name the prediction of OPD as \( {{\varvec{OPD}}}_{\mathrm{{pre}}} \in {{\varvec{R}}}^{m_{x0} \times m_t} \)), including \( Mc_2 \in {{\varvec{Mc}}}_{test } \), which can be considered as prediction result of the test sets. Finally, the prediction results are compared with test sets and several evaluation standards are selected to verify the predicted effect.

3.1 CDL algorithm

Among all the dimension reduction algorithms, compressed sensing has the advantages of nonlinearity and reconfigurability [14]. In this paper, CDL algorithm [15] is applied to achieve dimension reduction for OPD data on spatial domain by training the high-dimensional dictionary D and low-dimensional dictionary P with the training sets. The data information in high-dimensional space is gathered in its low-dimensional subspace and the distance, angle, inner product and other properties of the original signal in the high-dimensional space are also preserved in the low-dimensional space. CDL algorithm is described as:
$$\begin{aligned} {{\varvec{x}}}&= {{\varvec{D}}} \alpha \end{aligned}$$
(7)
$$\begin{aligned} {{\varvec{y}}}&= {{\varvec{P}}}\alpha \end{aligned}$$
(8)
where \( {{\varvec{x}}}\in {{\varvec{R}}}^m \) is the original signal, \( {{\varvec{y}}}\in {{\varvec{R}}}^d \) is its dimensionality reduction result, \( {{\varvec{D}}} \in {{\varvec{R}}}^{m \times k} \) is a high-dimensional dictionary, and \( {{\varvec{P}}}\in {{\varvec{R}}}^{d \times k} \) is a low-dimensional dictionary. \( \alpha \in {{\varvec{R}}}^k \) is the sparse representation of the signal \( {{\varvec{x}}} \) with high dimensional dictionary \( {{\varvec{D}}} \) and it is also the sparse representation of the signal \( {{\varvec{y}}} \) with the low-dimensional dictionary \( {{\varvec{P}}} \). When \( {{\varvec{D}}} \) is full rank and the energy of its SVD eigenvalue \(\lambda _i\) is concentrated in its front d dimension, it can be approximated that \( {{\varvec{P}}} \) is the dimensionality reduction result of \( {{\varvec{D}}} \) with PCA algorithm [18].
In this paper, the high-dimensional dictionary \( {{\varvec{D}}} \) and the low-dimensional dictionary \( {{\varvec{P}}} \) are first trained with the improved K-SVD algorithm proposed in [18]. To increase the sample numbers for training dictionary, the spatial distribution of OPD at different times in training sets is regarded as different training samples \( {{\varvec{x}}} \), which means that both Mc number and time are seen as labels. Then, the total number of samples for the CDL training rises to \( n_{{\text {CDL}}}=m_t \times n_{\mathrm{{train}}}=750 \times 7 \) from \( n_{\mathrm{{train}}}=7 \), and the dimension of sample becomes \( m_{{\text {CDL}}}=m_{x0}=100 \) from \( m_0=m_{x0} \times m_t=100 \times 750 \). The training datasets \({{\varvec{X}}} \) for CDL become:
$$\begin{aligned} \begin{aligned} {{\varvec{X}}} =&\{ {{\varvec{x}}}\in {{\varvec{R}}}^{m_{x0}} | {{\varvec{x}}}={{\varvec{OPD}}}(Mc_1,t_i ), \\&Mc_1\in {{\varvec{Mc}}}_{\mathrm{{train}}}, i=1,2,\ldots ,m_t \} \end{aligned} \end{aligned}$$
(9)
After dictionary \( {{\varvec{D}}} \) and \( {{\varvec{P}}} \) are acquired with K-SVD algorithm, the main steps of CDL dimension reduction algorithm \( {{\varvec{y}}}={\text {CDL}}[{{\varvec{x}}}] \) can be described as:
  1. (1)
    Sparse coding:
    $$\begin{aligned} \alpha \leftarrow \min _{\alpha } (\Vert {{\varvec{x}}}-{{\varvec{D}}}\alpha \Vert _2^2 + \lambda \Vert \alpha \Vert _1) \end{aligned}$$
    (10)
     
  2. (2)
    Dimension reduction:
    $$\begin{aligned} {{\varvec{y}}}={{\varvec{P}}}\alpha \end{aligned}$$
    (11)
    Similarly, the main step of CDL reconstruction algorithm \( {{\varvec{x}}}={\text {CDL}}^{-1} [{{\varvec{y}}}] \) is:
     
  3. (1)
    Sparse coding:
    $$\begin{aligned} \alpha \leftarrow \min _{\alpha }(\Vert {{\varvec{y}}}-{{\varvec{P}}}\alpha \Vert _2^2+ \lambda \Vert \alpha \Vert _1) \end{aligned}$$
    (12)
     
  4. (2)
    Dimension reduction:
    $$\begin{aligned} {{\varvec{x}}}={{\varvec{D}}}\alpha \end{aligned}$$
    (13)
    With the help of CDL algorithm, \( {{\varvec{OPD}}}(Mc_1)\in {{\varvec{R}}}^{m_{x0} \times m_t} \) will be first dimension reduced reversibly only in spatial domain. The result is expressed as \( {{\varvec{OPD}}}_1(Mc_1 ) \in {{\varvec{R}}}^{m_{x1} \times m_t} \) and the process of dimension reduction and reconstruction can be expressed as:
    $$\begin{aligned} {{\varvec{OPD}}}_1 (Mc_1)&={\text {CDL}}[{{\varvec{OPD}}}(Mc_1)] \end{aligned}$$
    (14)
    $$\begin{aligned} {{\varvec{OPD}}}_{{\text {rec}}}(Mc_1)&={\text {CDL}}^{-1} [{{\varvec{OPD}}}_1(Mc_1)] \end{aligned}$$
    (15)
    where \( Mc_1\in {{\varvec{Mc}}}_{\mathrm{{train}}} \).
     

3.2 Principal component extraction

The dimension reduction result of the OPD in the spatial domain \( {{\varvec{OPD}}}_1(Mc_1)\in {{\varvec{R}}}^{m_{x1}\times m_t} \) still has a high dimension in the time domain \( (m_t=750) \). While Mc number is regarded as the only label, the quantity of training samples is small \( (n=n_{\mathrm{{train}}}=7) \). It is still difficult to fit the curve of \( {{\varvec{OPD}}}_1 \) with respect to \( Mc_1 \).

Therefore, in the next step, PCA algorithm [13] is used to extract the first d principal components of \( {{\varvec{OPD}}}_1 (Mc_1 ) \), whose coefficients are described as a d-dimensional vector named \( {{\varvec{OPD}}}_2 (Mc_1)\in {{\varvec{R}}}^d \). Then, the curve of \( {{\varvec{OPD}}}_2 (Mc_1) \) with respect of Mc number is fitted in the d-dimensional space. Every point on the fitting curve (expressed as \( {{\varvec{OPD}}}_{2{\mathrm{{pre}}}}(Mc)\in {{\varvec{R}}}^d) \) can be reconstructed into the original domain with PCA and CDL reconstruction algorithm in turn (whose reconstruction results are referred to as \( {{\varvec{OPD}}}_{1{\mathrm{{pre}}}} \in {{\varvec{R}}}^{m_{x1} \times m_t}) \)and \( {{\varvec{OPD}}}_{{\mathrm{{pre}}}}\in {{\varvec{R}}}^{m_{x0}\times m_t} \).

PCA algorithm for dimension reduction \( {{\varvec{y}}}=PCA[{{\varvec{x}}}] \) can be expressed as:
$$\begin{aligned} {{\varvec{y}}}={{\varvec{U}}}_d^\mathrm{{T}}\cdot \vec {{{\varvec{x}}}} \end{aligned}$$
(16)
where \( {{\varvec{X}}} \) is the training sets, \( {{\varvec{x}}} \in {{\varvec{X}}} \) is the training sample, \( \vec {{x}} \) is the sample vector of the sample \( {{\varvec{x}}} \) expanded by column. \( {{\varvec{y}}}\in {{\varvec{R}}}^d \) is the low-dimensional representation of sample \( {{\varvec{x}}} \). \( {{\varvec{U}}}_d\in {{\varvec{R}}}^{m\times d} \) is the principal component projection matrix of \( {{\varvec{X}}} \), which is the first d column vector of \( {{\varvec{U}}} \) obtained with PCA algorithm:
$$\begin{aligned} {{\varvec{E}}}({{\varvec{x}}}{{\varvec{x}}}^\mathrm{{T}})={{\varvec{U}}}\varLambda {{\varvec{U}}}^\mathrm{{T}}={{\varvec{U}}} \left\{ \begin{matrix} \lambda _1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \lambda _2 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ddots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \lambda _n \\ \end{matrix} \right\} {{\varvec{U}}}^\mathrm{{T}} \end{aligned}$$
(17)
In which \( \lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _n \). PCA algorithm for data reconstruction \( {{\varvec{x}}}=PCA^{-1}[{{\varvec{y}}}] \) can be expressed as:
$$\begin{aligned} {\vec {x}}={{\varvec{U}}}_d \cdot {{\varvec{y}}}, \quad {\vec {x}} \rightarrow {{\varvec{x}}} \end{aligned}$$
(18)
In which \( {{\varvec{x}}} \) is the final reconstruction result by resizing the vector \( {\vec {x}} \) to the correct rows and columns numbers.
The first d principal components of \( {{\varvec{OPD}}}_1(Mc_1) \in {{\varvec{R}}}^{m_{x1} \times m_t}\) are extracted with PCA algorithm, which can be seen as the further dimension reduction result, expressed as \( {{\varvec{OPD}}}_2(Mc_1)\in {{\varvec{R}}}^d \):
$$\begin{aligned} {{\varvec{OPD}}}_2(Mc_1)=PCA[{{\varvec{OPD}}}_1(Mc_1)] \end{aligned}$$
(19)
where \( Mc_1 \in {{\varvec{Mc}}}_{\mathrm{{train}}} \).

3.3 OPD fitting and prediction

After fitting \( {{\varvec{OPD}}}_2 (Mc_1 ) \) with respect of Mc number with cubic spline algorithm, a prediction for \( {{\varvec{OPD}}}_2 (Mc_2 ) \) can be obtained, named \( {{{\varvec{OPD}}}}_{2{\mathrm {{pre}}}} (Mc_2 )\in {{\varvec{R}}}^d \):
$$\begin{aligned} {{\varvec{OPD}}}_{2{\mathrm{{pre}}}} (Mc_2 )={\text {Spline}}[{{\varvec{OPD}}}_2 ({{\varvec{Mc}}}_{{\text {train}}})] \end{aligned}$$
(20)
where function \( f_{{\mathrm{{pre}}}} (y)={\text {Spline}}[f({{\varvec{X}}})] \) means the prediction for f(y) with cubic spline algorithm, while input \( {{\varvec{X}}}=\{x_1,x_2,\ldots ,x_n\} \) and the outputs \( f(x_1), f(x_2), \ldots , f(x_n) \) are all known.
By performing PCA and CDL reconstruction in sequence, the prediction result of the test sets in the original space \( {{\varvec{OPD}}}_{{\mathrm{{pre}}}} (Mc_2 ) \) is finally obtained:
$$\begin{aligned} {{\varvec{OPD}}}_{1{\mathrm{{pre}}} }(Mc_2 )&=PCA^{-1} [{{\varvec{OPD}}}_{2{\mathrm{{pre}}}} (Mc_2 )] \end{aligned}$$
(21)
$$\begin{aligned} {{\varvec{OPD}}}_{{\mathrm{{pre}}}} (Mc_2 )&={\text {CDL}}^{-1} [{{\varvec{OPD}}}_{1{\mathrm{{pre}}}} (Mc_2 )] \end{aligned}$$
(22)
where \( {{\varvec{OPD}}}_{1{\mathrm{{pre}}}} \in {{\varvec{R}}}^{m_{x1} \times m_t} \) and \( {{\varvec{OPD}}}_{{\mathrm{{pre}}}} \in {{\varvec{R}}}^{m_{x0} \times m_t} \).

3.4 Prediction evaluation

To make a compare between the prediction and the test sets, it is necessary to select several evaluation indicators to evaluate the prediction error and similarity. Considering that the signal is clearly periodic, correlation coefficient R, mean square error MSE and peak signal-to-noise ratio PSNR are selected as the evaluation indicators finally. When R is close to 1, MSE is close to 0 and PSNR is a big number, it means that the prediction result has high similarity, the prediction error is small, and the prediction result is good. These three evaluation indexes are calculated as follows in which \( {{\varvec{x}}} \) means the original signal and \( \hat{{{\varvec{x}}}} \) means the prediction:
$$\begin{aligned} R({{\varvec{x}}},\hat{{{\varvec{x}}}})&= \frac{cov({{\varvec{x}}},\hat{{{\varvec{x}}}})}{\sqrt{{\text {var}}({{\varvec{x}}}) \cdot {\text {var}}(\hat{{{\varvec{x}}}})}} \end{aligned}$$
(23)
$$\begin{aligned} {\text {MSE}}({{\varvec{x}}},\hat{{{\varvec{x}}}})&= {\text {mean}}[({{\varvec{x}}}-\hat{{{\varvec{x}}}})^2] \end{aligned}$$
(24)
$$\begin{aligned} {\text {PSNR}}({{\varvec{x}}},\hat{{{\varvec{x}}}})&= 10log_{10} \frac{\max ({{\varvec{x}}})}{{\text {MSE}}({{\varvec{x}}},\hat{{{\varvec{x}}}})} \end{aligned}$$
(25)
Fig. 5

The distribution of \( {{\varvec{OPD}}}_1 \) vs different Mc numbers at different moments

Fig. 6

The reconstruction of training sets from \( m_1 \)-dimensional to \( m_0 \)-dimensional with CDL algorithm

4 Experimental analysis

4.1 Dimension reduction of training sets

Since OPD has a high dimension in the original space (whose dimension \( m_0=m_{x0} \times m_t=100 \times 750 \)) and it is not convenient for interpolation accurately, CDL method is firstly used to reduce the dimension of the OPD in spatial domain. The spatial distribution of OPD at any moment is mapped to a low-dimensional subspace in the spatial domain (whose dimension \( m_{x1}=8 \)) from the original spatial domain (whose dimension \( m_{x0}=100 \)). In that low-dimensional subspace, the distribution of \( {{\varvec{OPD}}}_1 (Mc_1 ) \in {{\varvec{R}}}^{m_{x1} \times m_t} \) in the first two dimension is shown in Fig. 5. Different color means different Mc numbers (\( n_{\mathrm{{train}}}=7 \) in total but only the first two dimensions are shown for better observability and simplicity) and different points with the same color mean different moments. Obviously, for any Mc number, \( {{\varvec{OPD}}}_1 \) has a good periodicity with time and moves along a circle as time goes on in the first two dimensions. The centre of the circle remains the same approximately but the radius increases while the Mc number gets bigger. Therefore, it can be preliminarily predicted that in the low-dimensional space shown in Fig. 5, the test sets of OPD are also circularly distributed, whose centre is the same as the training sets and radius can be estimated using interpolation method with the radius in the training sets. For example, the distribution radius of \( {{\varvec{OPD}}}_1 \) in 2-dimensional space when \( Mc=0.325 \) might be \( R_{Mc=0.325}=(R_{Mc=0.300}+R_{Mc=0.350})/2 \) roughly, in which \( R_{Mc=0.300} \) and \( R_{Mc=0.350} \)mean the distribution radius of \( {{\varvec{OPD}}}_1 \) .
Table 2

Evaluation for the reconstruction result of training sets with CDL algorithm

Mc number

0.3

0.4

0.5

0.6

Average R

90.0%

90.3%

95.4%

93.6%

Average MSE

\( 9.8 \times 10^{-4} \)

\( 1.2 \times 10^{-3} \)

\( 1.4 \times 10^{-3} \)

\( 3.5 \times 10^{-3} \)

Average PSNR (dB)

8.04

8.93

12.19

12.32

Fig. 7

The fitting result of \( {{\varvec{OPD}}}_2 \)

Fig. 8

Prediction of test sets at two random moment \( t_i, i \in \{1,2,\ldots ,m_t |m_t=750\} \)

4.2 Reconstruction of training sets

With the help of CDL reconstruction algorithm, the low-dimensional representation in the spatial domain of the training sets \( {{\varvec{OPD}}}_1 (Mc_1) \), whose dimension is \( m_1=m_{x1}\times m_t=8 \times 750 \), can be remapped to the original spatial domain, whose dimension is \( m_0=m_{x0} \times m_t=100 \times 750 \). The reconstruction result is shown in Fig. 6, in which \( m_1=m_{x1} \times m_t=8 \times 750 \) and \( m_0=m_{x0} \times m_t=100 \times 750 \). (a), (b), (c) and (d) mean the reconstruction result with different Mc number and the time \( t_i \) in the figure is randomly selected from \( i \in [1,750] \). The evaluation indicators of the training sets reconstruction are shown in Table 2. R between the reconstruction result and the truth keeps high (\( R \ge 90 \% \)). Therefore, the dimension reduction and reconstruction method are both reasonable. However, when the Mc number is low such as \( Mc=0.3 \) and 0.4, R is small relative to \( Mc=0.5 \) or 0.6 and PSNR is slightly lower but MSE in the case of 0.3 and 0.4 is small. This might be because the signal energy is weak due to the small fluctuation range of the OPD.
Table 3

Evaluation for the prediction result of test sets with CDL algorithm

Mc number

0.325

0.375

0.425

0.525

Average R

75.9%

87.8%

87.2%

81.6%

Average MSE

\( 3.3 \times 10^{-3} \)

\( 2.1 \times 10^{-3} \)

\( 5.1 \times 10^{-3} \)

\( 2.2 \times 10^{-2} \)

Average PSNR (dB)

6.06

9.03

9.59

8.36

Fig. 9

Direct interpolation result of test sets at two random moment \(t_i, i \in \{1,2,\ldots ,m_t |m_t=750\} \)

Table 4

Evaluation for the direct interpolation result of test sets

Mc number

0.325

0.375

0.425

0.525

Average R

20.6%

47.6%

66.5%

71.6%

Average MSE

\( 9.1 \times 10^{-3} \)

\( 1.28 \times 10^{-2} \)

\( 1.19 \times 10^{-2} \)

\( 3.2 \times 10^{-2} \)

Average PSNR (dB)

1.46

1.17

5.1

6.65

4.3 OPD interpolation

For \( {{\varvec{OPD}}}_1 (Mc_1 ) \in {{\varvec{R}}}^{m_{x1} \times m_t } \), which means that the dimension reduction results in spatial domain of the training sets with CDL method, its principal component projection matrix \( {{\varvec{U}}}_d \) is calculated with PCA algorithm, and the coefficients of the first d principal components are obtained and expressed as \( {{\varvec{OPD}}}_2 (Mc_1 ) \in {{\varvec{R}}}^d \) in which \( d=6 \). The principal component coefficients are fitted as a curve with respect to Mc numbers with cubic spline method, which means a prediction for the unknown data. The interpolation results of the first 3 principal components coeffects are shown in Fig. 7. The number of dimension of \( {{\varvec{OPD}}}_2 \) is \( d=6 \) in total, but only the first 3 dimension is shown for simplicity.

4.4 Prediction of test sets

With the help of PCA and CDL reconstruction algorithm, the fitting result, expressed as \( {{\varvec{OPD}}}_{2{\mathrm{{pre}}}} (Mc_2 ) \in {{\varvec{R}}}^d \), can be reconstructed back to the original space, expressed as \( {{\varvec{OPD}}}_2 (Mc_2 ) \in {{\varvec{R}}}^{m_{x0} \times m_t } \). The prediction of test sets is finally achieved. Figure 8a–d show the predictions of the test sets at random 2 moments \( t_i,i \in \{1,2,\ldots ,m_t |m_t=750\} \). The class of Mc number is \( n_{\mathrm{{test}}}=4 \) in total but only two are shown as examples. The prediction results are similar to the truth. The fluctuations of prediction are slightly smaller than the real datasets, but the phases maintain a high consistency.

The evaluation indexes of the prediction results are shown in Table 3. R is averaged 83% , which means that prediction accuracy is good overall. When Mc number is small, R is small relatively, while PSNR is low and MSE is relatively at a medium level. It seems the same as the reconstruction result of training sets in Sect. 4.2, which might be due to the small fluctuation range of the OPD and the weak signal energy. The direct interpolation result of test datasets in the original high-dimensional space and the evaluation indexes are shown in Fig. 9 and Table 4. Comparing Tables 3 and 4, direct interpolation results in a large prediction error, while the prediction based on dimensional reduction algorithm is more accurate.

For practical applications, the OPD distortion is less obvious when Mc number is lower, which means that the image quality is better than the one at higher Mc number. The demand for OPD prediction accuracy is lower than that of higher Mc number. Therefore, the requirements for OPD prediction accuracy under different Mc numbers conditions need to be considered in the next stage of image processing comprehensively and it needs further research. That is one of the next research priorities.

5 Conclusion

For the aero-optical effect caused by hypersonic shear layer, this paper analyses the relationship between Mc number and OPD based on a simplified aero-optical model and achieves the prediction of OPD with different Mc numbers in a certain range (\( 0.3 \le Mc \le 0.6 \)). It is of great significance for the research of the mechanism of aero-optical image degradation, and it has greatly promoted the image recovery caused by aero-optical effect in the next stage.

In this paper, the CDL dimensionality reduction algorithm and PCA algorithm are used in turn to map 7 training sets to a low-dimensional space. In the low-dimensional space, cubic spline interpolation algorithm is applied to predict the relationship between Mc number and OPD in that low-dimensional space. Any point on the fitting curve corresponds to different Mc number and can be mapped back to the original space, thereby the prediction of OPD with other Mc is realized.

Additional 4 sets of OPD data under different Mc numbers are selected as the test sets to verify the prediction. Considering correlation coefficient R, MSE and PSNR, the prediction results have a high similarity with the truth, indicating that the prediction algorithm is credible. The satisfied prediction result also illustrates that the training data are sufficient. Increasing the sample quantity of training datasets might be an effective way to increase the prediction accuracy, which needs to be lucubrated. In the meantime, according to the prediction results, when Mc number is lower, the similarity between the OPD prediction and the truth is lower than that of higher Mc number. However, considering that the image degradation with lower Mc number is less obvious than higher Mc number relatively, there are lower requirements for OPD prediction accuracy on the next stage of image recovery. Therefore, how to balance the relationship between OPD prediction accuracy and requirements for different Mc numbers needs to take image processing into consideration comprehensively and needs further research.

Notes

Acknowledgements

This work is supported in part by The Technical Innovation Project of Aerospace Advanced Technology Joint Research Center (Project No: USCAST2016-6).

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Copyright information

© Shanghai Jiao Tong University 2018

Authors and Affiliations

  • Zijia Li
    • 1
  • Yuanxiang Li
    • 1
    Email author
  • Boyang Xing
    • 1
  • Bin Zhang
    • 1
  • Hongya Tuo
    • 1
  • Hong Liu
    • 1
  1. 1.School of Aeronautics and AstronauticsShanghai Jiao Tong UniversityShanghaiChina

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