A Smooth Non-Iterative Local Polynomial (SNILP) Model of Image Vignetting
Abstract
:1. Introduction
1.1. Problem of Vignetting
1.2. Vignetting Correction
1.3. Objective of the Work
2. Vignetting Models Based on Decomposition
2.1. Local Parabolic Model
2.2. Local Polynomial (LP) Model
2.3. Smooth Local Polynomial (SLP) Model
2.4. Essential Properties of the LP and SLP Models
3. SNILP—A New Vignetting Model
3.1. Model Description
3.2. Selected Properties of the SNILP Model
3.3. Comparison of the SLP and SNILP Models
4. Experimental Results
4.1. Assumptions and Conditions of the Experiment
4.2. Results of the Experiment
4.3. Discussion of the Results
5. Comparison of Others Features of the Tested Vignetting Models
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
IQR | InterQuartile Range |
LP | Local Polynomial (vignetting model) |
LSCD | Local Slope Corrected Deviation |
LSD | Local Standard Deviation |
MAE | Mean Absolute Error |
MLSD | Mean Local Standard Deviation |
MLSCD | Mean Local Slope Corrected Deviation |
P2D | Polynomial 2D (vignetting model) |
Polynomial Approximation of degree s among horizontal (x) or vertical (y) line of the input image | |
RMSE | Root Mean Square Error |
RP | Radial Polynomial (vignetting model) |
SLP | Smooth Local Polynomial (vignetting model) |
SNILP | Smooth Non-Iterative Local Polynomial (vignetting model) |
STD | STandard Deviation |
VM | Vignetting Model |
Appendix A. Data Preparation for Numerical Tests
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Percentile | Focal Length [mm] | ||
---|---|---|---|
24 | 50 | 300 | |
0 | 0.0 | 0.0 | 0.0 |
0.05 | 1.11e-16 | 0.0 | 0.0 |
0.25 | 4.44e-16 | 1.11e-16 | 1.11e-16 |
0.5 | 1.67e-15 | 1.11e-16 | 1.11e-16 |
0.75 | 7.44e-15 | 2.22e-16 | 3.33e-16 |
0.95 | 3.26e-14 | 4.44e-16 | 4.44e-16 |
1 | 7.04e-13 | 1.89e-15 | 1.55e-15 |
mean | 7.13e-15 | 1.80e-16 | 1.83e-16 |
Parameters | Webcam | ||
---|---|---|---|
Cam-A | Cam-B | Cam-C | |
Logitech C920 | Xiaomi IMILAB CMSXJ22A | A4Tech PK-910H | |
Diagonal field of view | 78° | 85° | 70° |
Maximal resolution | |||
Maximal frame rate @ 1080p | 30 fps | 30 fps | 30 fps |
Focus type | auto focus | fixed focus | fixed focus |
Focus range | — | — | >60 cm |
Camera | Model | Order | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
Cam-A | P2D | 0.4831 | 0.4802 | 0.3932 | 0.3371 | 0.2930 | 0.2837 | 0.2746 | 0.2677 | 0.2577 |
RP | 0.6181 | 0.6181 | 0.5858 | 0.5858 | 0.5837 | 0.5837 | 0.5828 | 0.5828 | 0.5805 | |
SLP | 0.4485 | 0.4338 | 0.3254 | 0.3148 | 0.2796 | 0.2732 | 0.2520 | 0.2493 | 0.2458 | |
SNILP | 0.4485 | 0.4338 | 0.3254 | 0.3148 | 0.2796 | 0.2732 | 0.2520 | 0.2493 | 0.2458 | |
Cam-B | P2D | 2.7785 | 2.7503 | 0.9788 | 0.9395 | 0.9299 | 0.9224 | 0.6145 | 0.6017 | 0.5447 |
RP | 2.8143 | 2.8143 | 1.2296 | 1.2296 | 1.2322 | 1.2322 | 1.0423 | 1.0423 | 1.0143 | |
SLP | 2.2618 | 2.2276 | 0.9267 | 0.9222 | 0.6818 | 0.6665 | 0.5295 | 0.5178 | 0.4965 | |
SNILP | 2.2618 | 2.2276 | 0.9267 | 0.9222 | 0.6818 | 0.6665 | 0.5295 | 0.5178 | 0.4965 | |
Cam-C | P2D | 2.0798 | 1.8989 | 1.7279 | 1.6951 | 0.8810 | 0.8221 | 0.7058 | 0.6857 | 0.5785 |
RP | 2.8762 | 2.8762 | 2.6539 | 2.6539 | 2.2816 | 2.2816 | 2.2510 | 2.2510 | 2.2501 | |
SLP | 2.0720 | 1.8825 | 1.5142 | 1.4789 | 0.6578 | 0.6446 | 0.5635 | 0.5408 | 0.4378 | |
SNILP | 2.0720 | 1.8825 | 1.5142 | 1.4789 | 0.6578 | 0.6446 | 0.5635 | 0.5408 | 0.4378 |
Camera | Model | Order | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
Cam-A | P2D | 0.6565 | 0.6629 | 0.5102 | 0.4101 | 0.3563 | 0.3512 | 0.3363 | 0.3302 | 0.3088 |
RP | 0.8365 | 0.8365 | 0.7544 | 0.7544 | 0.7459 | 0.7459 | 0.7415 | 0.7415 | 0.7421 | |
SLP | 0.6156 | 0.5762 | 0.3974 | 0.3818 | 0.3480 | 0.3408 | 0.2989 | 0.2951 | 0.2901 | |
SNILP | 0.6156 | 0.5762 | 0.3974 | 0.3818 | 0.3480 | 0.3408 | 0.2989 | 0.2951 | 0.2901 | |
Cam-B | P2D | 2.7444 | 2.6071 | 1.2456 | 1.1831 | 1.0988 | 1.0824 | 0.6802 | 0.6854 | 0.6690 |
RP | 2.8675 | 2.8675 | 1.6103 | 1.6103 | 1.6080 | 1.6080 | 1.3072 | 1.3072 | 1.3121 | |
SLP | 3.2378 | 3.2712 | 1.1637 | 1.1685 | 0.8412 | 0.8102 | 0.6674 | 0.6548 | 0.6277 | |
SNILP | 3.2378 | 3.2712 | 1.1637 | 1.1685 | 0.8412 | 0.8102 | 0.6674 | 0.6548 | 0.6277 | |
Cam-C | P2D | 2.9902 | 2.9323 | 2.2331 | 2.2332 | 1.0541 | 1.0078 | 0.9222 | 0.8944 | 0.7722 |
RP | 4.0800 | 4.0800 | 3.1657 | 3.1657 | 2.7657 | 2.7657 | 2.6463 | 2.6463 | 2.6344 | |
SLP | 2.8820 | 2.8441 | 2.1795 | 2.1818 | 0.8612 | 0.8455 | 0.7420 | 0.7182 | 0.5849 | |
SNILP | 2.8820 | 2.8441 | 2.1795 | 2.1818 | 0.8612 | 0.8455 | 0.7420 | 0.7182 | 0.5849 |
Image Resolution | Image Size [Mpx] | Image Format | Computional Time [s] for Model | |||
---|---|---|---|---|---|---|
P2D | RP | SLP | SNILP | |||
0.08 | QVGA | 0.03 | 0.02 | 0.10 | 0.01 | |
0.31 | VGA | 0.11 | 0.06 | 1.43 | 0.14 | |
0.48 | Super VGA | 0.17 | 0.10 | 1.81 | 0.18 | |
0.79 | XGA | 0.30 | 0.18 | 2.44 | 0.24 | |
1.31 | SXGA | 0.54 | 0.30 | 3.42 | 0.33 | |
1.92 | UXGA | 0.74 | 0.44 | 4.14 | 0.40 | |
2.07 | HDV | 0.80 | 0.48 | 4.43 | 0.43 | |
2.21 | 2K Digital Cinema | 0.94 | 0.50 | 4.68 | 0.45 | |
8.29 | 4K UHDTV | 3.36 | 2.10 | 11.70 | 1.14 | |
8.85 | Canon PowerShot G9 X Mark II | 3.78 | 2.18 | 12.22 | 1.19 | |
19.96 | 4K Digital Cinema | 20.85 | 4.60 | 23.40 | 2.29 | |
33.18 | 8K UHDTV | 61.03 | 8.43 | 34.21 | 3.35 | |
50.32 | Canon EOS 5DS | 158.97 | 17.48 | 52.16 | 5.10 |
Model | Size of X | Remarks |
---|---|---|
P2D | ||
RP | In the case of this model, the stage of searching for an optical center of the image can require more memory than the stage of approximation of the radial function . For the approach to searching for the optical center, which is used in the experiment above, the statement is true when | |
SLP | In the SPL model, there is a stage of calculation vignetting estimate by averaging (4) the previously calculated values and , which requires additional space of a size to keep these variables. Because of the small value of s, which is commonly used, the stage of averaging usually requires more memory than 1D approximation needed for calculation of and . | |
SNILP |
Model | Type of Parameters | Number of Parameters |
---|---|---|
P2D | parameters of 2D approximation polynomial | |
RP | parameters of 1D radial polynomial function + coordinates of image optical center | |
SLP | parameters of 1D approximation of each line in the horizontal and the vertical direction of the input image | |
SNILP | parameters of 1D approximation of each line along the larger side of the input image | , where |
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Bal, A.; Palus, H. A Smooth Non-Iterative Local Polynomial (SNILP) Model of Image Vignetting. Sensors 2021, 21, 7086. https://doi.org/10.3390/s21217086
Bal A, Palus H. A Smooth Non-Iterative Local Polynomial (SNILP) Model of Image Vignetting. Sensors. 2021; 21(21):7086. https://doi.org/10.3390/s21217086
Chicago/Turabian StyleBal, Artur, and Henryk Palus. 2021. "A Smooth Non-Iterative Local Polynomial (SNILP) Model of Image Vignetting" Sensors 21, no. 21: 7086. https://doi.org/10.3390/s21217086