# Research on the Auto-Exposure Method of an Aerial TDI Camera Based on Scene Prediction
## 3 Auto-Exposure Effect Evaluation
### Image Power Spectrum Variance  
The power spectrum reflects the characteristics of the image in the frequency domain, and it is closely related to the information content of the image. Fourier transform is used to transform the signal from the spatial domain to the frequency domain, which is defined as follows:  
\[F(u,v)=\sum_{x=0}^{M-1}\sum_{y=0}^{N-1}f(x,y)exp[-2\pi j(\frac{ux}{M}+\frac{ yy}{N})], \tag{18}\]  
where \(F(u,v)\) is the two-dimensional discrete Fourier transform of the image, \(f(x,y)\) is an image of size \(M\) by \(N\), \(u\) and \(v\) are the frequency components, and Equation (18) must be evaluated in the range \(u=0,1,2,\cdots,M-1\) and \(v=0,1,2,\cdots,N-1\). Two-dimensional discrete Fourier transforms are usually a complex function, which can be expressed in polar coordinates as:  
\[F(u,v)=\mid F(u,v)\mid e^{\phi\phi(u,v)}, \tag{19}\]  
where the amplitude \(\mid F(u,v)\mid\) is called the Fourier spectrum or spectrum.  
The power spectrum is the square of the spectrum, reflecting the energy size of the image in each spatial frequency component, and it is defined as \(P(u,v)=\mid F(u,v)\mid^{2}\).  
As shown in Figure 3, two clear and fuzzy images are used, respectively, for comparison, Figure 3(a1,b1) are the clear and fuzzy images, Figure 3(a2,b2) are the corresponding power spectra, and Figure 3(a3,b3) are the corresponding three-dimensional power spectra. As can be seen from the figure, Figure 3(a1) is a clear image and there are more bright spots in the Fourier power spectrum, while Figure 3(b1) is a fuzzy image and there are fewer bright spots in the Fourier power spectrum. Therefore, the clarity of the image determines the complexity of its power spectrum; in other words, the higher degree of dispersion of the image power spectrum, the clearer the image is.  
In the field of digital image processing, the dispersion of the image is usually represented by variance, and the variance in the image power spectrum is represented by \(\delta\),  
\[\delta=\frac{1}{M\times N}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}\left[P(u,v)-\overline {P(u,v)}\right]^{2}, \tag{20}\]  
where \(\overline{P(u,v)}\) refers to the average of the image power spectrum, and \(M\) and \(N\) are the number of rows and columns of the image, respectively.  
In order to verify the relationship between the variance in the image power spectrum and the clarity of the image, the lens blur processing of different degrees is carried out on the clear image in Figure 3(a1). The blur radius is r and the value range of r is [0, 1, 2..... 10]. The larger the value of r is, the more serious degree of the image blur will be.  
As can be seen from Figure 4, the variance in the image power spectrum gradually decreases with an increase in the blur radius. Therefore, it can be concluded that, the clearer the image is, the greater the image power spectrum variance is and the more frequency information the image power spectrum contains.  
Figure 4: Relation between image power spectrum variance and blur radius.  
Figure 3: (a1,b1) are the clear and fuzzy images, (a2,b2) are the corresponding power spectra, and (a3,b3) are the corresponding three-dimensional power spectra.