## 3 Analysis of the Analytical MTF Expression for Complex Image Motion
### Convergence of the Approximate Analytical MTF Expression  
To achieve Equation (14), in practice, the infinitely many terms involving the Bessel function of the first kind in Equation (14) needs to be approximated to the sum of the finite terms, which brings the truncation error. The MTF can be calculated by truncating up to the 20 th order Bessel functions of the first kind. The truncation error can be described as follows:  
\[\begin{array}{l}\text{E(N)}=\sqrt{\left[\frac{\text{J}_{0}(2\pi\text{fD}) \sin(2\pi\text{fv}_{\text{e}})}{2\pi\text{fv}_{\text{e}}}\!+\!\text{A}+\text{B }\right]^{2}+\left[\frac{\text{J}_{0}(2\pi\text{fD})\sin^{2}(\pi\text{fv}_{ \text{e}})}{\pi\text{fv}_  
Substituting in Equation (17), Equation (18) can be written as follows:  
\[\begin{array}{ll}\mathrm{E(N)}&<2\sum\limits_{n=2\mathrm{N}+1}^{\infty}\left| \mathrm{J}_{n}(2\pi\mathrm{fD})\right|\left[\frac{\sin(n\omega+2\pi\mathrm{rfv })\mathrm{t}_{e}}{n\omega+2\pi\mathrm{rfv}}+\frac{\sin(n\omega-2\pi\mathrm{rfv })\mathrm{t}_{e}}{n\omega-2\pi\mathrm{rfv}}\right]\\ &+2\sum\limits_{n=2\mathrm{N}+1}^{\infty}\left|\mathrm{J}_{n}(2\pi\mathrm{fD}) \right|\left[\frac{1-\cos(n\omega+2\pi\mathrm{rfv})\mathrm{t}_{e}}{n\omega+2 \pi\mathrm{rfv}}-\frac{1-\cos(n\omega-2\pi\mathrm{rfv})\mathrm{t}_{e}}{n\omega -2\pi\mathrm{rfv}}\right]\right|\end{array}. \tag{19}\]  
Based on the properties of the trigonometric functions and the range of parameters of the image motion, the approximation error is given as follows:  
\[\mathrm{E(N)}<8\sum\limits_{n=2\mathrm{N}+1}^{\infty}\left|\mathrm{J}_{n}(2 \pi\mathrm{fD})\right|. \tag{20}\]  
The integral order Bessel function of the first kind can be expressed in the form of a power series, as follows:  
\[\mathrm{J}_{n}(\mathrm{x})=\sum\limits_{m=0}^{\infty}\left(-1\right)^{m}\frac {1}{\Gamma(m+1)\Gamma(m+n+1)}\left(\frac{\mathrm{x}}{2}\right)^{2m+n}. \tag{21}\]  
Substituting Equation (21) into Equation (20), the truncation error can be expressed as follows:  
\[\mathrm{E(N)}<8\sum\limits_{n=2\mathrm{N}+1}^{\infty}\left|\sum\limits_{m=0}^{ \infty}\left(-1\right)^{m}\frac{1}{m!(m+n)!}(\pi\mathrm{fD})^{2m+n}\right|<8 \sum\limits_{n=2\mathrm{N}+1}^{\infty}\left|\sum\limits_{m=0}^{\infty}\left(- 1\right)^{m}\frac{1}{m!n!}(\pi\mathrm{fD})^{2m+n}\right|. \tag{22}\]  
According to the [PERSON]'s series of \(\mathrm{e}^{\mathrm{x}}\), which is presented as the following:  
\[\mathrm{e}^{\mathrm{x}}=\sum\limits_{k=0}^{2\mathrm{N}}\frac{\mathrm{x}^{k}}{ k!}+\frac{\mathrm{e}^{\mathrm{e}_{k}}}{(2\mathrm{N}+1)!}\mathrm{x}^{2\mathrm{N}+1}, \mathrm{\xi}\in(0,\mathrm{x}), \tag{23}\]  
the upper bound to the absolute error is expressed as follows:  
\[\mathrm{E(N)}<8\left|\sum\limits_{n=2\mathrm{N}+1}^{\infty}\frac{1}{n!}(\pi \mathrm{fD})^{n}\sum\limits_{m=0}^{\infty}\frac{1}{m!}(\pi\mathrm{fD})^{2m} \right|<\left|\frac{8\exp\left[\left(\pi\mathrm{fD}\right)^{2}+(\pi\mathrm{fD })\right]}{(2\mathrm{N}+1)!}(\pi\mathrm{fD})^{2\mathrm{N}+1}\right|. \tag{24}\]  
With Equation (24), the upper bound error can be calculated. However, it is necessary to emphasize that fewer order Bessel functions are required generally. A plot of the upper limit error for 2 Nth order approximation at the spatial frequency \(\mathrm{f}=1/\mathrm{D}\) is shown in Figure 1. As can be seen, up to the 20 th order Bessel functions are needed for an approximation error less than \(10^{-20}\). In this paper, up to 20 th order Bessel functions are used to calculate the MTF for the image motion.