# Non-Rotationally Symmetric Field Mapping for Back-Scanned Step/Stare Imaging System
## 1 Introduction  
Modern intelligence, surveillance, and reconnaissance systems strongly require to have wide area coverage and high-resolution imaging abilities. High-resolution imaging has often been achieved at the cost of a small field of view (FOV) for current electro-optical and infrared sensors [1]. In certain imaging applications, such as infrared search and track (IRST) systems, it is desirable for the imaging sensor to scan large fields of regard at a high rate and with diffraction limited performance. The second-generation IRST systems utilize long line detectors with time delay and integration capability to achieve wide area coverage. However, due to systems limitations of line detectors,the scanning mode of IRST cannot be stopped at any desired position to scan a small sector of interest. They also suffer from more false alarms [2]. The third-generation IRST systems utilize the FPAs with step/stare imaging to achieve wide area coverage. The new IRST achieves greater performances than the previous generation thanks to their staring nature, i.e., to longer integration times (milliseconds rather than microseconds) [3; 4; 5; 6; 7; 8]. In step/stare imaging applications, a high-resolution narrow FOV sensor is rapidly stepped or indexed across an area of interest, stopping at intervals to stare and collect imagery. Then the images at each location are stitched together to form a large but high-resolution image. This process requires a series of small angular steps and a high angular acceleration [9].However, the image motion occurs during the scanning within the exposure time. In order to achieve the line of sight stability, a special optical mechanism has been required to compensate the scanning image motion to achieve the \"stare\" result when searching and tracking [10; 11]. Among different technical methods to achieve step/stare imaging, the back-scanning optical compensation technology is the most commonly utilized, in which a plane mirror within the optical system moves reversely to achieve the line of sight stability when searching and tracking [12; 13; 14; 15; 16]. For example, the camera on the Global Hawk unmanned aerial vehicle utilizes the fast steering mirror (FSM) behind the afocal optics to compensate the scanning image motion. FSMs are utilized to point for line of sight stability in optical systems. With small size, high accuracy, high bandwidth and fast response speed, FSMs are widely applied in electro-optical and infrared sensors [17; 18; 19]. However, it is frequently assumed that the virtual of a target viewed through a rotating mirror moves with respect to the observer at twice the angular rate of mirror rotation, which is inaccurate and leads to imprecise treatment of open-loop tracking systems. The reason is that the sample of rays reflected from a target to an arbitrary tracker position changes as a function of mirror orientation, and the angle of incidence is not a linear function of mirror orientation, but depends on the relative positions of the target, mirror, and tracker, as well as the orientation of the mirror [20; 21].Although the equations describing the line-of-sight kinematics derive entirely from the simple plane mirror law of reflection, they are non-linearly- and axis-coupled and these effects increase as the FSM angular displacement increases, which would contribute to pointing errors in certain modes of operation [9]. As discussed above, the back-scanned step/stare imaging system can scan a large field of regard at a high rate and with diffraction limited performance. However, the traditional optical design forms can hold only one field point relatively stable on the FPA, typically the central or on-axis field point; all other off-axis field points may wander during the exposure time due to image distortion characteristics of the imaging systems, which reduces the SNR of the target.