基于Boltzmann函数刃边拟合的光栅成像系统点扩展函数模型 
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Point Spread Function Model of Grating Imaging System Based on Boltzmann Function Edge Fitting
1 广西大学电气工程学院, 广西 南宁 530004
2 广西大学物理科学与工程技术学院, 广西 南宁 530004
图 & 表
图 1. 实验光路示意图
Fig. 1. Diagram of experimental light path
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图 2. 部分入射角下光栅所成图像
Fig. 2. Images formed by grating at partial incident angles
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图 3. 刃边图像和不同函数拟合的效果。(a)刃边图像;(b)不同函数的拟合结果;(c)局部放大
Fig. 3. Edge image and fitting effects of different functions. (a) Edge image; (b) fitting results of different functions; (c) local enlargement
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图 4. 部分入射角下光栅成像PSF
Fig. 4. PSF of grating imaging at partial incident angles
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图 5. 序列PSF示意图
Fig. 5. Diagram of sequential PSF
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图 6. 轴向水平剖面能量分布示意图
Fig. 6. Schematic of axial horizontal profile energy distribution
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图 7. 不同能量下PSF半径的变化
Fig. 7. Change of PSF radius under different light intensities
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图 8. 光栅系统所成图像
Fig. 8. Images formed by grating system
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图 9. 复原图像
Fig. 9. Restored images
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图 10. 不同拟合函数的复原对比。(a) Gauss函数;(b) Hyperbolic函数;(c) Boltzmann函数
Fig. 10. Restoration comparison of different fitting functions. (a) Gauss function; (b) Hyperbolic function; (c) Boltzmann function
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表 1不同函数对ESF的拟合优度
Table1. Fitting goodness of different functions on ESF
| θ /(°) | Gauss | Hyperbolic | Boltzmann |
|---|
| SSE | R-square | RMSE | SSE | R-square | RMSE | SSE | R-square | RMSE |
|---|
| 4 | | 0.0076 | 0.9988 | 0.0127 | 0.0047 | 0.9991 | 0.0091 | | 2 | | 0.0180 | 0.9978 | 0.0195 | 0.0130 | 0.9984 | 0.0155 | | 0 | | 0.0028 | 0.9997 | 0.0085 | 0.0069 | 0.9994 | 0.0110 | | -34 | 0.1164 | 0.9845 | 0.0492 | 0.0264 | 0.9972 | 0.0215 | 0.0120 | 0.9984 | 0.0147 | | -36 | 0.0161 | 0.9960 | 0.0206 | 0.0206 | 0.9960 | 0.0164 | 0.0070 | 0.9988 | 0.0111 |
|
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表 2不同能量下曲线的拟合参数
Table2. Fitting parameters of curves under different light intensities
| E | ω0 /pixel | θR /(°) | α /(°) |
|---|
| 6.5×10-3 | 20.08 | 10.45 | 62.51 | | 1.3×10-2 | 19.13 | 10.94 | 60.24 | | 2.6×10-2 | 17.85 | 12.65 | 54.68 | | 5.2×10-2 | 14.28 | 12.36 | 49.12 | | 1.0×10-1 | 12.25 | 15.16 | 38.94 |
|
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表 3不同能量下曲线的拟合优度
Table3. Fitting goodness of curves under different light intensities
| E | 6.5×10-3 | 1.3×10-2 | 2.6×10-2 | 5.2×10-2 | 1.0×10-1 |
|---|
| R-square | 0.9609 | 0.9683 | 0.9723 | 0.9699 | 0.9600 |
|
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表 4图像复原前后的质量评价
Table4. Quality evaluation before and after image restoration
| θ /(°) | GMG | SSIM |
|---|
| Before restoration | After restoration | Increase rate | Before restoration | After restoration | Increase rate |
|---|
| 4 | 0.0086 | 0.0150 | 74.42% | 44.94% | 89.31% | 98.73% | | 2 | 0.0082 | 0.0145 | 76.83% | 51.36% | 88.63% | 72.57% | | 0 | 0.0091 | 0.0157 | 72.53% | 53.31% | 91.94% | 72.47% | | -34 | 0.0088 | 0.0141 | 60.23% | 53.92% | 89.76% | 66.47% | | -36 | 0.0085 | 0.0143 | 68.23% | 48.34% | 87.49% | 80.99% |
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表 5不同拟合函数的质量评价
Table5. Quality evaluation of different fitting functions
| Function | GMG | SSIM |
|---|
| Beforerestoration | Afterrestoration | Increaserate | Beforerestoration | Afterrestoration | Increaserate |
|---|
| Gauss | | 0.0091 | 3.41% | | 58.29% | 8.10% | | Hyperbolic | 0.0088 | 0.0095 | 7.95% | 53.92% | 69.38% | 28.67% | | Boltzmann | | 0.0141 | 60.23% | | 89.76% | 66.47% |
|
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卢泉, 张泽昊, 张卫平, 刘诣荣. 基于Boltzmann函数刃边拟合的光栅成像系统点扩展函数模型[J]. 光学学报, 2020, 40(14): 1405003. Quan Lu, Zehao Zhang, Weiping Zhang, Yirong Liu. Point Spread Function Model of Grating Imaging System Based on Boltzmann Function Edge Fitting[J]. Acta Optica Sinica, 2020, 40(14): 1405003.
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