Russell C. Hardie, Michael A. Rucci, Santasri Bose-Pillai, and Richard Van Hook, "Application of tilt correlation statistics to anisoplanatic optical turbulence modeling and mitigation," Appl. Opt. 60, G181-G198 (2021)
Atmospheric optical turbulence can be a significant source of image
degradation, particularly in long range imaging applications. Many
turbulence mitigation algorithms rely on an optical transfer function
(OTF) model that includes the Fried parameter. We present
anisoplanatic tilt statistics for spherical wave propagation. We
transform these into 2D autocorrelation functions that can inform
turbulence modeling and mitigation algorithms. Using these, we
construct an OTF model that accounts for image registration. We also
propose a spectral ratio Fried parameter estimation algorithm that is
robust to camera motion and requires no specialized scene content or
sources. We employ the Fried parameter estimation and OTF model for
turbulence mitigation. A numerical wave-propagation turbulence
simulator is used to generate data to quantitatively validate the
proposed methods. Results with real camera data are also
presented.
The truth images used here may be obtained in Refs. [34,35]. The data with turbulence were simulated using the method in Ref. [22]. The real camera data have not been approved for public release by AFRL at this time.
34. E. Agustsson and R. Timofte, “NTIRE 2017 challenge on single image super-resolution: dataset and study,” in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops (2017).
22. R. C. Hardie, J. D. Power, D. A. LeMaster, D. R. Droege, S. Gladysz, and S. Bose-Pillai, “Simulation of anisoplanatic imaging through optical turbulence using numerical wave propagation with new validation analysis,” Opt. Eng. 56, 1–16 (2017). [CrossRef]
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Optical Parameters Used for the Example and Simulation Results
Parameter
Value
Aperture
Focal length
-number
Wavelength
Object distance
Nyquist pixel spacing (focal plane)
Nyquist pixel angle
Table 2.
Turbulence Parameters for Six Levels of Turbulence Generated with the Anisoplanatic Turbulence Simulator [22] to Test the Spectral Ratio Estimatora
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
(
0.1000
0.2500
0.5000
1.0000
1.5000
2.0000
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Theoretical (unitless)
1.0697
1.8536
2.8096
4.2585
5.4314
6.4547
Isoplanatic angle (pixels)
6.6174
3.8188
2.5194
1.6622
1.3033
1.0966
RMS tilt (pixels)
0.9026
1.4272
2.0183
2.8543
3.4958
4.0367
Tilt variance (pixels)
0.8147
2.0368
4.0736
8.1473
12.2209
16.2946
Patch tilt variance (pixels)
0.5333
1.3333
2.6666
5.3333
7.9999
10.6666
Residual tilt variance (pixels)
0.2154
0.5385
1.0770
2.1541
3.2311
4.3082
The patch and residual tilt variance values are for $M = 100$ ($201 \times 201\; {\rm pixel}$ patch).
Table 3.
Spectral Ratio Fried Parameter Estimation Error Analysis for the Six Levels of Turbulence Detailed in Table 2 and the Truth Image Boats
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Stationary (m)
0.1830
0.1039
0.0697
0.0475
0.0371
0.0302
Percent error
−3.75%
−5.34%
−3.73%
−0.46%
−1.01%
−4.17%
Global (m) ()
0.2913
0.1738
0.1168
0.0734
0.0575
0.0495
Percent error
53.20%
58.37%
61.40%
53.66%
53.66%
57.23%
Global (m) ()
0.1863
0.1112
0.0747
0.0469
0.0368
0.0317
Percent error
−2.01%
1.30%
3.24%
−1.72%
−1.71%
0.57%
BMA (m) ()
0.3189
0.2128
0.1402
0.0905
0.0689
0.0597
Percent error
67.74%
93.96%
93.69%
89.55%
84.04%
89.56%
BMA (m) ()
0.1436
0.0958
0.0631
0.0408
0.0310
0.0269
Percent error
−24.50%
−12.69%
−12.81%
−14.68%
−17.16%
−14.67%
BMA (m) ()
0.1692
0.1129
0.0744
0.0480
0.0366
0.0317
Percent error
−11.01%
2.91%
2.77%
0.57%
−2.35%
0.57%
Table 4.
Spectral Ratio Fried Parameter Estimation Error Analysis for the Six Levels of Turbulence Detailed in Table 2 and the Truth Image Stream and Bridge
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Stationary (m)
0.1823
0.1058
0.0692
0.0473
0.0371
0.0298
Percent error
−4.15%
−3.56%
−4.41%
−0.99%
−1.00%
−5.33%
Global (m) ()
0.2803
0.1684
0.1121
0.0716
0.0556
0.0466
Percent error
47.41%
53.50%
54.80%
49.94%
48.42%
47.80%
Global (m) ()
0.1793
0.1077
0.0717
0.0458
0.0356
0.0298
Percent error
−5.71%
−1.81%
−0.98%
−4.10%
−5.07%
−5.46%
BMA (m) ()
0.3096
0.2071
0.1415
0.0899
0.0663
0.0565
Percent error
62.81%
88.71%
95.41%
88.24%
77.13%
79.42%
BMA (m) ()
0.1394
0.0932
0.0637
0.0405
0.0299
0.0254
Percent error
−26.71%
−15.05%
−12.04%
−15.27%
−20.27%
−19.24%
BMA (m) ()
0.1643
0.1099
0.0751
0.0477
0.0352
0.0300
Percent error
−13.62%
0.12%
3.68%
−0.13%
−6.02%
−4.81%
Table 5.
Turbulence Mitigation PSNR (dB) Results for the Boats Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
22.8144
21.0166
19.5873
19.4271
19.3396
17.1386
Avg
24.0733
22.3642
21.0926
19.9563
19.3304
18.9094
Global Avg
24.8010
23.2296
21.8923
20.5321
19.8463
19.3972
BMA
25.5550
24.3532
22.9716
21.3339
20.4592
19.8855
Avg Wiener
33.9471
28.0064
24.9611
23.1497
22.1659
21.5099
Global Avg Wiener
34.8527
31.3373
27.5469
24.3288
23.2257
22.4591
BMA
36.2669
35.0499
30.9621
26.1062
24.9845
23.9241
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 6.
Turbulence Mitigation PSNR (dB) Results for the Stream and Bridge Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
23.1948
22.0832
19.5952
20.0127
20.2346
17.5237
Avg
24.1645
22.7428
21.6037
20.5968
20.0119
19.5737
Global Avg
24.7622
23.4439
22.2945
21.1216
20.4964
20.0397
BMA
25.4163
24.4391
23.2534
21.8289
21.0319
20.4842
Avg Wiener
31.2960
27.0460
24.7814
23.1652
22.3452
21.8384
Global Avg Wiener
32.3660
29.2104
26.4898
24.0227
23.0100
22.3299
BMA
33.5384
32.3853
28.8911
25.5475
24.2221
23.1938
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 7.
Turbulence Mitigation SSIM Results for the Boats Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
0.6965
0.6269
0.5259
0.5001
0.5003
0.4328
Avg
0.7379
0.6347
0.5550
0.4909
0.4602
0.4415
Global Avg
0.7749
0.6885
0.6058
0.5225
0.4850
0.4626
BMA
0.8134
0.7614
0.6859
0.5800
0.5238
0.4901
Avg Wiener
0.9557
0.8695
0.7649
0.6693
0.6094
0.5704
Global Avg Wiener
0.9659
0.9261
0.8588
0.7343
0.6675
0.6191
BMA
0.9776
0.9728
0.9424
0.8368
0.7714
0.7093
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 8.
Turbulence Mitigation SSIM Results for the Stream and Bridge Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
0.6142
0.5320
0.4143
0.3962
0.3940
0.3323
Avg
0.6470
0.5318
0.4452
0.3831
0.3547
0.3379
Global Avg
0.6896
0.5896
0.4983
0.4130
0.3778
0.3562
BMA
0.7373
0.6774
0.5916
0.4746
0.4151
0.3811
Avg Wiener
0.9330
0.8214
0.6909
0.5811
0.5107
0.4734
Global Avg Wiener
0.9473
0.8918
0.8023
0.6460
0.5637
0.5120
BMA
0.9597
0.9536
0.9065
0.7768
0.6794
0.5889
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 9.
Optical Parameters for the Real Camera Data
Parameter
Value
Aperture
Focal length
-number
Wavelength
Object distance
Actual pixel spacing (focal plane)
Tables (9)
Table 1.
Optical Parameters Used for the Example and Simulation Results
Parameter
Value
Aperture
Focal length
-number
Wavelength
Object distance
Nyquist pixel spacing (focal plane)
Nyquist pixel angle
Table 2.
Turbulence Parameters for Six Levels of Turbulence Generated with the Anisoplanatic Turbulence Simulator [22] to Test the Spectral Ratio Estimatora
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
(
0.1000
0.2500
0.5000
1.0000
1.5000
2.0000
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Theoretical (unitless)
1.0697
1.8536
2.8096
4.2585
5.4314
6.4547
Isoplanatic angle (pixels)
6.6174
3.8188
2.5194
1.6622
1.3033
1.0966
RMS tilt (pixels)
0.9026
1.4272
2.0183
2.8543
3.4958
4.0367
Tilt variance (pixels)
0.8147
2.0368
4.0736
8.1473
12.2209
16.2946
Patch tilt variance (pixels)
0.5333
1.3333
2.6666
5.3333
7.9999
10.6666
Residual tilt variance (pixels)
0.2154
0.5385
1.0770
2.1541
3.2311
4.3082
The patch and residual tilt variance values are for $M = 100$ ($201 \times 201\; {\rm pixel}$ patch).
Table 3.
Spectral Ratio Fried Parameter Estimation Error Analysis for the Six Levels of Turbulence Detailed in Table 2 and the Truth Image Boats
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Stationary (m)
0.1830
0.1039
0.0697
0.0475
0.0371
0.0302
Percent error
−3.75%
−5.34%
−3.73%
−0.46%
−1.01%
−4.17%
Global (m) ()
0.2913
0.1738
0.1168
0.0734
0.0575
0.0495
Percent error
53.20%
58.37%
61.40%
53.66%
53.66%
57.23%
Global (m) ()
0.1863
0.1112
0.0747
0.0469
0.0368
0.0317
Percent error
−2.01%
1.30%
3.24%
−1.72%
−1.71%
0.57%
BMA (m) ()
0.3189
0.2128
0.1402
0.0905
0.0689
0.0597
Percent error
67.74%
93.96%
93.69%
89.55%
84.04%
89.56%
BMA (m) ()
0.1436
0.0958
0.0631
0.0408
0.0310
0.0269
Percent error
−24.50%
−12.69%
−12.81%
−14.68%
−17.16%
−14.67%
BMA (m) ()
0.1692
0.1129
0.0744
0.0480
0.0366
0.0317
Percent error
−11.01%
2.91%
2.77%
0.57%
−2.35%
0.57%
Table 4.
Spectral Ratio Fried Parameter Estimation Error Analysis for the Six Levels of Turbulence Detailed in Table 2 and the Truth Image Stream and Bridge
Turbulence Degradation
Parameter
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Theoretical (m)
0.1901
0.1097
0.0724
0.0478
0.0374
0.0315
Stationary (m)
0.1823
0.1058
0.0692
0.0473
0.0371
0.0298
Percent error
−4.15%
−3.56%
−4.41%
−0.99%
−1.00%
−5.33%
Global (m) ()
0.2803
0.1684
0.1121
0.0716
0.0556
0.0466
Percent error
47.41%
53.50%
54.80%
49.94%
48.42%
47.80%
Global (m) ()
0.1793
0.1077
0.0717
0.0458
0.0356
0.0298
Percent error
−5.71%
−1.81%
−0.98%
−4.10%
−5.07%
−5.46%
BMA (m) ()
0.3096
0.2071
0.1415
0.0899
0.0663
0.0565
Percent error
62.81%
88.71%
95.41%
88.24%
77.13%
79.42%
BMA (m) ()
0.1394
0.0932
0.0637
0.0405
0.0299
0.0254
Percent error
−26.71%
−15.05%
−12.04%
−15.27%
−20.27%
−19.24%
BMA (m) ()
0.1643
0.1099
0.0751
0.0477
0.0352
0.0300
Percent error
−13.62%
0.12%
3.68%
−0.13%
−6.02%
−4.81%
Table 5.
Turbulence Mitigation PSNR (dB) Results for the Boats Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
22.8144
21.0166
19.5873
19.4271
19.3396
17.1386
Avg
24.0733
22.3642
21.0926
19.9563
19.3304
18.9094
Global Avg
24.8010
23.2296
21.8923
20.5321
19.8463
19.3972
BMA
25.5550
24.3532
22.9716
21.3339
20.4592
19.8855
Avg Wiener
33.9471
28.0064
24.9611
23.1497
22.1659
21.5099
Global Avg Wiener
34.8527
31.3373
27.5469
24.3288
23.2257
22.4591
BMA
36.2669
35.0499
30.9621
26.1062
24.9845
23.9241
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 6.
Turbulence Mitigation PSNR (dB) Results for the Stream and Bridge Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
23.1948
22.0832
19.5952
20.0127
20.2346
17.5237
Avg
24.1645
22.7428
21.6037
20.5968
20.0119
19.5737
Global Avg
24.7622
23.4439
22.2945
21.1216
20.4964
20.0397
BMA
25.4163
24.4391
23.2534
21.8289
21.0319
20.4842
Avg Wiener
31.2960
27.0460
24.7814
23.1652
22.3452
21.8384
Global Avg Wiener
32.3660
29.2104
26.4898
24.0227
23.0100
22.3299
BMA
33.5384
32.3853
28.8911
25.5475
24.2221
23.1938
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 7.
Turbulence Mitigation SSIM Results for the Boats Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
0.6965
0.6269
0.5259
0.5001
0.5003
0.4328
Avg
0.7379
0.6347
0.5550
0.4909
0.4602
0.4415
Global Avg
0.7749
0.6885
0.6058
0.5225
0.4850
0.4626
BMA
0.8134
0.7614
0.6859
0.5800
0.5238
0.4901
Avg Wiener
0.9557
0.8695
0.7649
0.6693
0.6094
0.5704
Global Avg Wiener
0.9659
0.9261
0.8588
0.7343
0.6675
0.6191
BMA
0.9776
0.9728
0.9424
0.8368
0.7714
0.7093
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.
Table 8.
Turbulence Mitigation SSIM Results for the Stream and Bridge Image Using the Optical Parameters in Table 1 and the Six Turbulence Levels in Table 2a
Turbulence Mitigation
Turbulence Degradation
Method
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Frame 1
0.6142
0.5320
0.4143
0.3962
0.3940
0.3323
Avg
0.6470
0.5318
0.4452
0.3831
0.3547
0.3379
Global Avg
0.6896
0.5896
0.4983
0.4130
0.3778
0.3562
BMA
0.7373
0.6774
0.5916
0.4746
0.4151
0.3811
Avg Wiener
0.9330
0.8214
0.6909
0.5811
0.5107
0.4734
Global Avg Wiener
0.9473
0.8918
0.8023
0.6460
0.5637
0.5120
BMA
0.9597
0.9536
0.9065
0.7768
0.6794
0.5889
The ${r_0}$ value used is from the global registration spectral ratio estimate for this image sequence.