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摘要: 后向投影(BP)算法是合成孔径雷达成像算法发展的重要方向之一。然而，由于BP算法具有较大的计算量，阻碍了其在工程应用上的发展。因此，近年来如何有效地提高BP算法的运算效率受到了广泛的重视。该文讨论了基于多种成像面坐标系的快速BP算法，包括距离-方位平面坐标系、地平面坐标系和非欧氏坐标系。该文首先简要介绍了原始BP算法的原理和不同坐标系对加速BP算法的影响，并对BP算法的发展历程进行梳理。然后讨论了基于不同成像面坐标系的快速BP算法的研究进展，并重点介绍了作者所在研究团队近年来在快速BP成像方面完成的研究工作。最后介绍了快速BP算法在工程上的应用，并展望了未来快速BP成像算法的研究发展趋势。Abstract: The Back Projection (BP) algorithm is an important direction in the development of synthetic aperture radar imaging algorithms. However, the large computational load of the BP algorithm has hindered its development in engineering applications. Therefore, techniques to enhance the computational efficiency of the BP algorithm have recently received widespread attention. This paper discusses the fast BP algorithm based on various imaging plane coordinate systems, including the distance-azimuth plane coordinate system, the ground plane coordinate system, and the non-Euclidean coordinate system. First, the principle of the original BP algorithm and the impact of different coordinate systems on accelerating the BP algorithm are introduced, and the development history of the BP algorithm is sorted out. Then, the research progress of the fast BP algorithm based on different imaging plane coordinate systems is examined, focusing on the recent research work completed by the author’s research team. Finally, the application of fast BP algorithm in engineering is introduced, and the research development trend of the fast BP imaging algorithm is discussed.
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图 10 距离-方位平面坐标系成像模型
Figure 10. Imaging model in the distance-azimuth plane coordinate system
图 15 FFBP算法与CFBP算法计算量对比
Figure 15. Comparison of computational load between FFBP algorithm and CFBP algorithm
图 25 GCBP算法对地表曲面成像仿真结果
Figure 25. Simulation results of GCBP algorithm for surface imaging
表 1 各类坐标系下成像优缺点对比
Table 1. Comparison of advantages and disadvantages of imaging in various coordinate systems
坐标系 优点 缺点 局部极坐标系 可以以较低的采样率进行子孔径低分辨成像而
不会造成频谱混叠各级图像相干融合时需要进行插值操作，会使得误差积累，
分级次数越多，误差积累越大，成像质量越差全局虚拟极坐标系 不再需要进行插值处理，成像质量高 成像条件限制大，要求子图像波数谱无模糊，
并且在最后的图像融合阶段，计算量较大混合坐标系 只需对角度维进行插值，不需要对距离维插值，
减少了插值次数，提高了成像质量需要插值处理，对成像质量依旧有影响 全局直角坐标系 成像时无需插值与逐点运算，成像质量更好，
更利于工程化实现直角坐标系对子孔径图像采样率要求过高 
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表 2 FFBP算法与CFBP算法计算量对比
Table 2. Comparison of computational burden between FFBP algorithm and CFBP algorithm
算法 计算量 FFBP $ \dfrac{{8{N^2}{L_{\text{a}}}}}{n} + 64{N^2}{\log _2}n $ CFBP $ \dfrac{{8{N^2}{L_{\text{a}}}}}{n} + \displaystyle\sum\limits_{i = 1}^{{{\log }_2}n} {\left( {{N^2}\left( \begin{gathered} 3{\log _2}\sqrt N + {\log _2}\sqrt {\frac{{N \times {2^{i - 1}}}}{n}} + \\ 2{\log _2}\sqrt {\frac{{N \times {2^i}}}{n}} + 6 \\ \end{gathered} \right)} \right)} $
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表 3 文献[26]、文献[36]与文献[37]所提算法计算量对比
Table 3. Comparison of computational burden between algorithm Proposed in Ref. [26], Ref. [36] and Ref. [37]
算法 计算量 文献[26]所提算法 $ \dfrac{{8{N^2}{L_{\text{a}}}}}{n} + 8{N^2}{\log _2}n $ 文献[36]所提算法 $ 50{N^2}{\log _2}N + 30{N^2} + \dfrac{{8{N^2}{L_{\text{a}}}}}{n} $ 文献[37]所提算法 $ \begin{gathered} \dfrac{{8{N^2}{L_{\text{a}}}}}{n} + \left( {2n + 2} \right){N^2} \\ +\left( {n + 1} \right){N^2}{\log _2}N - {N^2}{\log _2}\sqrt n \\ \end{gathered} $
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表 4 文献[42]、文献[52]与文献[58]所提算法计算量对比
Table 4. Comparison of computational burden between algorithm proposed in Ref. [42], Ref. [52] and Ref. [58]
算法 计算量 文献[42]所提算法 $ \begin{aligned} & \frac{{16mD{N^2}{L_{\text{a}}}}}{{m + 1}}{{ + m{{D}}}}{N^2} + \rho mDN_{{\mathrm{DCT}}}^2 \\& + \rho \left( {2{D^2} + D + 27} \right){L_{\mathrm{a}}} \end{aligned} $ 文献[52]所提算法 $ \begin{aligned} & \frac{{16mD{N^2}{L_{\text{a}}}}}{{n\left( {m + 1} \right)}} + 64mD{N^2}{\log _2}n \\ & + 2mD{N^2}{\log _2}N + 4mD{N^2} \\ & + \left( {6{D^2} + 21D} \right){L_{\mathrm{a}}} \end{aligned} $ 文献[58]所提算法 $ \begin{aligned} & \frac{{16mD{N^2}{L_{\text{a}}}}}{{n\left( {m + 1} \right)}} + mD\left( {n + 4} \right){N^2}{\log _2}N \\ & - mD{N^2}{\log _2}\sqrt n + mD\left( {2n + 8} \right){N^2} \\& + m8N + \left( {6{D^2} + 21D} \right){L_{\mathrm{a}}} \end{aligned} $
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