## 2 Algorithm for MBS
### Division Mode in FBP
The size of the original imaging area, i.e., \(\text{M}\times\text{N}\) (N is pixels along the range direction and M is the pixels along the azimuth direction), will be divided into \(\alpha_{2}\times\alpha_{3}\) small sub-areas; the size of every sub-area is \(m\times n\), where \(m=\frac{\text{M}}{\alpha_{2}}\), \(m=\frac{\text{N}}{\alpha_{3}}\). Here, the dark area in Figure 3a represents the corresponding pixels in the sub-imaging area. After the sub-aperture and sub-imaging area been divided, the subsequent task is to handle the complex data in each sub-object. As shown in Figure 3b, a group of concentric circles centered on sub-aperture's point can be obtained for lines from every sub-aperture to pixels in the sub-image area. Different circles form their respective sector areas and indicate unequal echo; other back-projection data of points on an identical circle are exactly the same. Thus, within an acceptable range error, it is not necessary to project echo data to all pixels in the imaging area, as the centerline represents all pixels distributed in the sub-image area. For two adjacent sub-apertures, the distance to pixels in the sub-image area can be almost the same. For the \(\alpha(1\leq\alpha\leq\alpha_{1})\) aperture and its adjacent apertures, it can be considered that they exhibit approximately the same concentric circles. Here, the echo data in the sector areas can be considered to be consistent with the data of the line from the sub-aperture's center to the sub-image's area center. Therefore, a centerline can be used as an approximation for the sector areas within an allowable range of error, thus reducing the computational cost required by the FBP algorithm. For the scenario of MBS, the scale change in slant range undulate insignificantly along with the gaze angle. Figure 3b shows the calculation of range error; the lines in this figure do not have the same scale as the actual scenario, in order to create a clear display. The lengths of line segments BC and BE are equal, centered on point B, and form an equal radius arc L\({}_{1}\); line segments AC and AD are equal, centered on point A, and form another equal radius arc L\({}_{2}\); point D is the true position of the pixel in the sub-image area; point C  
Figure 3: The schematic diagram of FBP in the division of aperture and imaging area (**a**), and in the range error calculation (**b**). is the corresponding projected position; and point E is the intersection between segment AD and equal radius arc L\({}_{1}\). According to the approximation principle, the points near the centerline use points on the centerline instead, although the approximate processing will thus produce some errors and affect the imaging quality. In this paper, the centerline is generally selected, without loss of generality, from the center of the sub-aperture to the center of the sub-imaging area, but only segments of the centerline are used. On one hand, if the segment is too short, all the pixels in the sub-imaging areas cannot be effectively projected on the line, undoubtedly increasing the range error before and after projection. On the other hand, if the segment is too long, much redundant data will be generated, in opposition to the requirement of less intermediate data. In this paper, the segment equal to the diagonal of the sub-image area is selected to ensure that each pixel in the sub-image area can be projected without generating too much redundant data. After the pixels in the sub-image area have been replaced by points on the centerline, e.g., points in Figure 3b, the range error between the real range AD and the replaced BC on the centerline can be expressed as follows:  
\[\Delta R=R(\theta+\Delta\theta)-R(\theta) \tag{3}\]  
where \(\theta\) is angle between the azimuth direction and point E; \(\Delta\theta\) is the angle between the centerline and point E; \(R\) is the true slant range (i.e., the length of AC), and \(\Delta R\) is the slant range error between length of AC and BC (i.e., the error between the true range and the projected range).