## 2 Theory of Video SAR
### Overview  
ViSAR is a SAR imaging technique in which the radar is operated in spotlight mode on a circular flight path, as shown in Figure 1. Radar data is collected on a region of interest, and images are generated at high frame rates. The ViSAR system can generate electromagnetic SAR images at frame rates similar to conventional video formats, which are typically \(2\sim 5\) Hz or more.  
Figure 1 shows the ViSAR geometry when it is operated on a circular path. There are two kinds of image formation methods used in ViSAR with circular antenna motion: full synthetic aperture methods [13; 15] and sub-aperture methods [14; 16]. As the name suggests, in the full synthetic aperture method, SAR images are generated from the full aperture, which corresponds to the complete circular path. In this case, an ultimate resolution of \(\frac{\lambda}{4}\) can be obtained. Another advantage of this method is that images are acquired over 360 degrees. If a full aperture image is synthesized every time data is obtained, the frame rate becomes very small. Also, as the BPA is typically used in forming a full aperture image, the computation power required is very large. Therefore, a fast-factorized BPA is used in order to increase the frame rate and reduce the computation load, by factorizing the full synthetic aperture into sub-apertures, and updating the SAR image every time sub-aperture data is obtained. A higher frame rate can be obtained because the SAR image is updated in a sliding window fashion;  
Figure 1: The collection geometry of video SAR on a circular path. SAR: synthetic aperture radar.  
the update is performed every time sub-aperture data is modified instead of waiting for a full aperture data update. Conventional ViSAR systems often use this method [13]. Since fast-factorized BPA updates the image using the sliding window method, it has the disadvantage that an afterimage is retained in the SAR image. This is also seen with circular SAR when there is a large amount of overlap between input data. The fast-factorized BPA can achieve a high frame rate, even at relatively low frequencies [15; 16].  
The sub-aperture method for circular antenna motion is a method for synthesizing SAR images that uses only data obtained from the sub-aperture, which is a fraction of the full aperture. This is the same as operation in spotlight SAR mode, with partial measurement of a circular path. In this case, it is possible to update the SAR image at a high frame rate without an afterimage. However, since the SAR image must be synthesized using only sub-aperture data, the synthetic aperture time must be short to achieve a high frame rate. Therefore, at a given resolution, the transmitted frequency or the speed of the radar should be high. In addition, as only sub-aperture data is used, this technique is less restricted by the flight path than the full aperture method. Therefore, it is suitable for applications such as weapon assignment [15; 16]. In this paper, we will focus mainly on the sub-aperture method.  
When ViSAR is operated in spotlight SAR mode on a circular path, the direction of the image changes according to the aspect angle of the SAR image. Because of this, stationary targets appear to rotate in the SAR video. In this case, it would be convenient to interpret the SAR images as if they were taken from an optical camera at a fixed point. To do this, a reference angle is defined. SAR images obtained at different aspect angles are subsequently rotated and aligned to this reference. This reference angle is called the cardinal direction up (CDU), as shown in Figure 2. \(\theta_{as}\) denotes the aspect angle with respect to the cardinal direction.