## 1 Introduction  
Nowadays, optical remote-sensing imaging is of interest for a wide variety of applications [1]. In many types of vehicular or airborne imaging and in robotic systems, despite the use of high-quality sensors, the resolution is limited by the image motion [2; 3; 4] and, as a result, the high-resolution capability of the sensor may be wasted. Therefore, it is considered that the influence of the image displacement on image quality is necessary in order to design the optical mechanical structure of the instrument, and is significant for further image restoration. The degradation of image quality caused by image motion is generally described by the modulation transfer function (MTF) [5; 6]. The image motion can take several forms, such as a linear image motion, because of the attitude change of a craft, and sinusoidal image motion, because of the turbines and motors that give rise to mechanical vibrations. It is very important to calculate the MTF for complex image motion accurately in order to provide significant theoretical guidance for instrument design and to increase the effect of image restoration. Specifically, in the design stage of the remote-sensing imaging instrument, an exact MTF calculation model for complex image motion can make the allocation of instrument indicators more suitable [7; 8]. Moreover, in the process of the fuzzy restoration of remote-sensing images in the later stage, Wiener filtering and other restoration methods also need to obtain accurate MTF values [9; 10; 11]. However, the complicated image motion expression makesit difficult for the existing methods to calculate MTF accurately, which leads to an inappropriate instrument design and poor image-restoration results. Therefore, the calculation of the MTF for complex image motion is essential for the reasonable distribution of errors and image restoration. Earlier MTF calculation methods were focused on a single form of motion. [PERSON] established an analytical MTF expression based on spatial domain spread function analysis [12]. [PERSON] mainly analyzed the impact of arbitrary frequency sinusoidal vibration on MTF [13]. Some studies created a mathematical model of a vibrating modulate transfer function for a remote-sensing camera using statistical averages and statistical moments [14; 15; 16]. The aforementioned literature proposed MTF calculation methods that only led to the MTF expression for a simple form of motion. There were a few models that were aimed at more complicated motion functions. [PERSON] proposed that the shorter the time exposure, the smaller the blur radius caused by the image motion [17]. Therefore, the MTF calculation model for the complex image motion was accomplished using the assumption that for short time exposures, the image velocity was essentially uniform. Based on this method, the MTF for linear motion is currently being used in the design of remote-sensing cameras to calculate the MTF for complex image motion. However, the existing MTF analysis methods are incapable of calculating the MTF for mixed image motion accurately, which is not equal to the product of the MTFs for a single form of image motion. We noted that the MTF calculation method mentioned in the literature [12; 13; 14; 15; 16] is available for any arbitrary image motion in principle, which includes complex motion. But, in the case of the mixing of multiple forms of motion, the expression of MTF is much more complicated, which means that the analytical MTF expression is almost impossible to deduce using the calculation method presented above. The MTF calculation model in the literature [17] for complex motion is inaccurate because of the linear motion approximation. Therefore, the MTF calculation method for the mixed image motion of the linear motion and sinusoidal vibration needs further research. Furthermore, few studies have been carried out on the simulation method aimed at the MTF for image motion, which makes the calculation methods hard to verify.