**4. Information optical method for modeling aero-optical imaging**

#### **4.1 Angular spectrum propagation model**

**Figure 18** depicts the propagation of the angular frequency spectrum of plane wave. As to the optical wave, the complex amplitude of the plane wave is expressed by

**Figure 18.** *The angular spectrum propagation of plane wave in uniform optical media.*

$$\begin{split} U(\mathbf{r},t) &= a \exp\left(ik \cdot \mathbf{r}\right) \\ &= a \exp\left[ik(\mathbf{x}\cos a + \mathbf{y}\cos \beta + \mathbf{z}\cos \gamma)\right] \end{split} \tag{21}$$

where *a* is the constant amplitude; cos ð Þ *α*, cos *β*, cos *γ* denotes the direction cosine; *k* ¼ 2*π=λ* is the wave vector [30].

The angular frequency spectrum is just the 2-D Fourier transformation of the complex amplitude of optical wave. Given that a monochromatic light injects the X-Y plane along with the direction of Z axis, its angular spectrum can be expressed as

$$A\left(f\_{\mathbf{x}}, f\_{\mathbf{y}}, z\right) = \iint U(\mathbf{x}, \mathbf{y}, z) \exp\left[-i2\pi \left(\mathbf{x}f\_{\mathbf{x}} + \mathbf{y}f\_{\mathbf{y}}\right)\right] d\mathbf{x}d\mathbf{y} \tag{22}$$

where *U x*ð Þ , *y*, *z* is the complex amplitude distribution. On the other hand, the inverse Fourier transform of the angular spectrum is a complex amplitude distribution. Here, the transmission of aerial optics through the flow fields can be seen as the transmission of each spectrum. Each plane wave spectrum in ð Þ *x*, *y*, 0 is denoted by *<sup>A</sup>*<sup>0</sup> *<sup>f</sup> <sup>x</sup>*, *<sup>f</sup> <sup>y</sup>*, 0 � �, and each spectrum in ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* is denoted by *<sup>A</sup>*<sup>0</sup> *<sup>f</sup> <sup>x</sup>*, *<sup>f</sup> <sup>y</sup>*, *<sup>z</sup>* � �, where *<sup>f</sup> <sup>x</sup>* <sup>¼</sup> cos *α <sup>λ</sup>* , *<sup>f</sup> <sup>y</sup>* <sup>¼</sup> cos *<sup>β</sup> <sup>λ</sup>* . According to the Helmholtz scalar equation, angular spectrum propagation function can be obtained as

$$A\_1(\frac{\cos a}{\lambda}, \frac{\cos \beta}{\lambda}, z) = A\_0\left(\frac{\cos a}{\lambda}, \frac{\cos \beta}{\lambda}, 0\right) \exp\left(j kz\sqrt{1 - \cos^2 a - \cos^2 \beta}\right) \tag{23}$$

which describes the propagation between the two parallel planes. As a matter of fact, a transfer function of frequency filtering is obtained by

$$\begin{aligned} H\_c \left( f\_x, f\_y \right) &= \frac{A \left( f\_x, f\_y \right)}{A\_0 \left( f\_x, f\_y \right)} \\ &= \left\{ \begin{array}{l} ikz \left[ ikz \sqrt{1 - \left( \left. \left. f\_x \right)^2 - \left( \left. \left. f\_y \right)^2 \right| \right. \right. \right]} & f\_x \right.^2 + f\_y \right.^2 < \frac{1}{\lambda^2} \\ & 0 & \text{others} \end{array} \tag{24}$$

The model of aero-optical imaging proposed in this chapter is inspired by the above analysis. Hence, the aero-optical transmission is translated into spatial filtering with limited bandwidth of the lights.

#### **4.2 Linear filter model of aero-optical imaging**

The light source described in this chapter may be too far away from the built-in detector, causing the output wave to appear as a plane wave. Therefore, the transmission of light at hypersonic speed can be considered as the transmission of plane waves. The term "diffraction" is conveniently described by Sommerfeld as " any deviation of light rays from rectilinear paths that cannot be interpreted as reflection or

*Perspective Chapter: Computational Modeling for Predicting the Optical Distortions… DOI: http://dx.doi.org/10.5772/intechopen.106591*

refraction.". Furthermore, the results obtained from the scalar diffraction theory approximate the real effect if the wavelength is smaller than the diffraction aperture and the observation point is far from the diffraction aperture [31]. The supersonic flow field transmission process considering the above discussion, aero-optics transmission can be seen as a scalar diffraction problem, so angular spectrum propagation can be used for aero-optics research.

From the point of view of information optics, each cubic grid can be seen as an optical system that composes any optical filtering system that characterizes the flow fields. After these considerations, the supersonic flow fields can be divided into 63 � 63 � 79 optical filtration subsystems, which are serially connected in the negative direction along the Z axis. **Figure 19** maps a sketch of the plane optical wave through CFD cubic grids.

**Figure 20** shows the structure of one cubic CFD grid. Each cubic grid has eight points with the determined index of refraction. The following equation gives out the characteristic parameter *ni oc* of the cubic optical filtering system.

$$n^i\_{\
oc} = \frac{(n\_1 + n\_2 + \dots + n\_8)}{8} \tag{25}$$

where *i* is the order number of the optical filtering system (*i* ¼ 63 � 63 � 79). One of these serial systems is described in **Figure 21**. *Hi f <sup>x</sup>*, *f <sup>y</sup>*, *zi* denotes the transfer

**Figure 19.** *The sketch map of CFD cubic grids.*

**Figure 20.** *The distribution of CFD grid points in one cubic optic system.*

**Figure 21.** *The diagram of serially connected systems representing the transmission of angle spectrum.*

function of the *i*th optical system, which can be gained through Eq. (23) based on angular spectrum propagation model.

In the frequency domain, the output of such serially linear filtering system can be obtained through

$$A\_i(f\_{\,\,x}, f\_{\,\,y}, z\_i) = A\_0\left(f\_{\,\,x}, f\_{\,\,y}, \mathbf{0}\right) H\_1 H\_2 \cdots H\_i \tag{26}$$

However, Eq. (23) is actually called as coherence transfer function (CTF). As to aero-optics, it should be viewed as a noncoherent imaging system, which is a linear system concerning on the distribution of light intensity. To the noncoherent imaging system, optical transfer function (OTF) is utilized for the study of light propagation. The OTF can be derived from CTF through the following equation

$$H(u,v) = \frac{H\_{\epsilon}(u,v) \otimes H\_{\epsilon}(u,v)}{\int \int\_{-\infty}^{\infty} |H\_{\epsilon}(a,\ \beta)| da d\beta} \tag{27}$$

where CTF is ½ �¼ *hc*ð Þ *x*, *y Hc*ð Þ *u*, *v* , *u* ¼ *f <sup>x</sup>*,*v* ¼ *f <sup>y</sup>* and OTF is ½ �¼ *h x*ð Þ , *y H u*ð Þ , *v* . Thus, OTF of the flow fields is equal to the 2D Fourier transformation of the pointspread function (PSF, *PSF* ¼ *h x*ð Þ , *y* ) of light intensity distributions. Relationship between the input light intensity distribution *Io*ð Þ *x*, *y* and output light intensity distribution *I x*ð Þ , *y* is obtained by

$$\mathbb{E}[I\_o(\mathbf{x},\ \mathcal{y})] \cdot \mathbb{E}[h(\mathbf{x},\ \mathcal{y})] = \mathbb{E}[I(\mathbf{x},\ \mathcal{y})] \tag{28}$$

where *I x*ð Þ¼ , *y* j j *U x*ð Þ , *y* <sup>2</sup> is the light intensity distribution.

Through the above theoretical analysis, a discrete OTF matrix 63 � 63 could be acquired. In other words, PSF can be gained for the spatial filtering of image. The relationship between PSF and the light intensity distributions is satisfied by

$$I(\mathbf{x}, \boldsymbol{y}) = I\_o(\mathbf{x}, \boldsymbol{y}) \* h(\mathbf{x}, \boldsymbol{y}) \tag{29}$$

Here, the transmission of right incident light through the supersonic flow fields is considered. And position of the image centroid can be calculated by

$$\mathbf{x}\_{\varepsilon} = \frac{\sum\_{i} i \cdot I(i, j)}{\sum\_{i} \sum\_{j} I(i, j)}, \mathbf{y}\_{\varepsilon} = \frac{\sum\_{j} j \cdot I(i, j)}{\sum\_{i} \sum\_{j} I(i, j)} \tag{30}$$

where ð Þ *i*, *j* is the coordinate of the corresponding image pixel at the image coordinate, and *I i*ð Þ , *j* is the corresponding pixel value. Thus, the centroid shift of the degraded image can be evaluated through,

*Perspective Chapter: Computational Modeling for Predicting the Optical Distortions… DOI: http://dx.doi.org/10.5772/intechopen.106591*

$$
\Delta \mathbf{x} = \mathbf{x}'\_c - \mathbf{x}''\_c,\\
\Delta \mathbf{y} = \mathbf{y}'\_c - \mathbf{y}''\_c \tag{31}
$$

where *x*<sup>0</sup> *<sup>c</sup>*, *<sup>y</sup>*<sup>0</sup> � � *c* , *xo <sup>c</sup>*, *y<sup>o</sup> c* � � are respectively the centroids of the degraded image and original image. To estimate the total aberrations induced by the flow fields, Euclidean distance (ED) of image shift is used for an implicit assessment through

$$\mathbf{r}\_{\rm ED} = \sqrt{\left(\mathbf{x'} - \mathbf{x}\right)^2 + \left(\mathbf{y'} - \mathbf{y}\right)^2} \tag{32}$$

#### **4.3 Simulation results**

Digital images of aircraft obtained from the Internet are used to simulate aerooptical images. To study the shift of image centroid without considering the effect of temporal integration on the image, only snapshots were numerically investigated. **Figure 22** shows the original image. **Figures 23**–**25** show the degradation results of a supersonic flow fields with the same Mach number (Ma = 7) and height of 30, 35 and 40 km, respectively. **Figures 23** and **26** show the results when the height is 30 km and the Mach numbers are 7 and 5, respectively. **Table 1** shows the results of calculating the shifts of image centroid. Although the evaluation method cannot calculate the total

**Figure 22.** *The original image.*

**Figure 23.** *The aero-optically degraded image (H = 30 km, Ma = 7).*

**Figure 24.** *The aero-optically degraded image (H = 35 km, Ma = 7).*

**Figure 25.** *The aero-optically degraded image (H = 40 km, Ma = 7).*

### **Figure 26.** *The aero-optically degraded image (H = 30 km, Ma = 5).*

aero-optical distortion, the validity of the proposed model of the aero-optical imaging method based on optical information can be easily verified.

It must qualitatively satisfy the aero-optical effect of light propagation in a supersonic flow fields. That is, at the same Mach number, the lower the height, the stronger *Perspective Chapter: Computational Modeling for Predicting the Optical Distortions… DOI: http://dx.doi.org/10.5772/intechopen.106591*


#### **Table 2.**

*Image centroid shifts related to the original image.*

the aero-optical effect. The higher the Mach number at the same height, the stronger the optical effect of air. The image is blurred and the centroid shift is larger in the quantitative analysis.

Furthermore, from the results in **Table 2**, it can be seen that the Mach number in the aero-optical images has a greater weight than the flight altitude. That is, the compressive effect of the flow fields by the flight speed compared with atmospheric density at different altitudes is a key factor influencing the change in the density field. Through the above analysis, the simulation results are consistent with the basic facts of the aero-optic effect.
