Abstract

A demand for division of focal plane (DoFP) polarization imaging technology grows rapidly as nanofabrication technologies become mature. For real-time polarization imaging, a DoFP polarimeter often trades off its spatial resolution, which may cause instantaneous field of view (IFoV) errors. To deal with such problems, interpolation methods are often used to fill the missing polarization information. This paper presents an interpolation technique using Newton’s polynomial for DoFP polarimeter demosaicking. The interpolation is performed in the polarization difference domain with the interpolation error taken into consideration. The proposed method uses an edge classifier based on polarization difference and a fusion scheme to recover more accurate boundary features. Experiments using both synthetic and real DoFP images in visible and long-wave infrared spectrum demonstrate that the proposed interpolation method outperforms the state-of-the-art techniques quantitatively as well as visually to reduce nonconformities caused by high-frequency energy.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2018 (7)

N. Li, Y. Zhao, Q. Pan, and S. G. Kong, “Removal of reflections in LWIR image with polarization characteristics,” Opt. Express 26, 16488–16504 (2018).
[Crossref] [PubMed]

M. Garcia, C. Edmiston, T. York, R. Marinov, S. Mondal, N. Zhu, G. P. Sudlow, W. J. Akers, J. Margenthaler, S. Achilefu, R. Liang, M. A. Zayed, M. Y. Pepino, and V. Gruev, “Bio-inspired imager improves sensitivity in near-infrared fluorescence image-guided surgery,” Optica 5, 413–422 (2018).
[Crossref] [PubMed]

J. Zhang, H. Luo, R. Liang, A. Ahmed, X. Zhang, B. Hui, and Z. Chang, “Sparse representation-based demosaicing method for microgrid polarimeter imagery,” Opt. Lett. 43, 3265–3268 (2018).
[Crossref] [PubMed]

J. Zhang, J. Shao, H. Luo, X. Zhang, B. Hui, Z. Chang, and R. Liang, “Learning a convolutional demosaicing network for microgrid polarimeter imagery,” Opt. Lett. 43, 4534–4537 (2018).
[Crossref] [PubMed]

S. Mihoubi, P. J. Lapray, and L. Bigué, “Survey of demosaicking methods for polarization filter array images,” Sensors 18, 3688 (2018).
[Crossref]

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

2017 (6)

M. Garcia, C. Edmiston, R. Marinov, A. Vail, and V. Gruev, “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4, 1263–1271 (2017).
[Crossref]

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
[Crossref] [PubMed]

J. Zhang, H. Luo, R. Liang, W. Zhou, B. Hui, and Z. Chang, “PCA-based denoising method for division of focal plane polarimeters,” Opt. Express 25, 2391–2400 (2017).
[Crossref]

M. Reda, Y. Zhao, and J. C.-W. Chan, “Polarization guided autoregressive model for depth recovery,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

2016 (4)

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

S. Li, W. Jin, R. Xia, L. Li, and X. Wang, “Radiation correction method for infrared polarization imaging system with front-mounted polarizer,” Opt. Express 24, 26414–26430 (2016).
[Crossref] [PubMed]

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24, 20799–20807 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

E. Gilboa, J. P. Cunningham, A. Nehorai, and V. Gruev, “Image interpolation and denoising for division of focal plane sensors using Gaussian processes,” Opt. Express 22, 15277–15291 (2014).
[Crossref] [PubMed]

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

2013 (4)

2011 (3)

2009 (1)

2008 (1)

2006 (2)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[Crossref] [PubMed]

K. H. Chung and Y. H. Chan, “Color demosaicing using variance of color differences,” IEEE Trans. Image Process. 15, 2944–2955 (2006).
[Crossref] [PubMed]

2002 (1)

1998 (1)

Abubakar, A.

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

Achilefu, S.

Ahmed, A.

Akers, W. J.

Ando, K.

Anisetti, M.

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

Arakawa, S.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Asada, N.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Au, O. C.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Baba, M.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Bastaki, E.

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

Bayer, B. E.

B. E. Bayer, “Color imaging array,” (1976). US Patent 3,971,065.

Bello, D. S. S.

M. Sarkar, D. S. S. Bello, C. van Hoof, and A. J. Theuwissen, “Biologically inspired CMOS image sensor for fast motion and polarization detection,” IEEE Sens. J. 13, 1065–1073 (2013).
[Crossref]

Bermak, A.

Bigué, L.

S. Mihoubi, P. J. Lapray, and L. Bigué, “Survey of demosaicking methods for polarization filter array images,” Sensors 18, 3688 (2018).
[Crossref]

Boussaid, F.

Bulirsch, R.

J. Stoer and R. Bulirsch, Introduction to numerical analysis, vol. 12 (Springer Science & Business Media, 2013).

Bull, D. R.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

Cain, S. C.

M. W. Hyde, S. C. Cain, J. D. Schmidt, and M. J. Havrilla, “Material classification of an unknown object using turbulence-degraded polarimetric imagery,” IEEE Trans. Geosci. Remote. Sens. 49, 264–276 (2011).
[Crossref]

Chan, J. C.-W.

M. Reda, Y. Zhao, and J. C.-W. Chan, “Polarization guided autoregressive model for depth recovery,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
[Crossref]

Chan, Y. H.

K. H. Chung and Y. H. Chan, “Color demosaicing using variance of color differences,” IEEE Trans. Image Process. 15, 2944–2955 (2006).
[Crossref] [PubMed]

Chang, Z.

Charbois, J. M.

Chen, Z.

Chenault, D. B.

Cheng, Y.

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

Chigrinov, V. G.

Chung, K. H.

K. H. Chung and Y. H. Chan, “Color demosaicing using variance of color differences,” IEEE Trans. Image Process. 15, 2944–2955 (2006).
[Crossref] [PubMed]

Cui, N.

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

Cunningham, J. P.

Daly, I. M.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

Damiani, E.

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

Devlaminck, V.

Edmiston, C.

Engheta, N.

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

Fang, L.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Feng, B.

H. Liu, Z. Shi, B. Feng, B. Hui, and Y. Zhao, “Non-uniformity calibration for MWIR polarization imagery obtained with integrated microgrid polarimeters,” in Selected Papers of the Chinese Society for Optical Engineering Conferences held October and November 2016, vol. 10255 (International Society for Optics and Photonics, 2017), p. 102554D.

Fest, E.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Furukawa, R.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Gao, S.

S. Gao and V. Gruev, “Gradient-based interpolation method for division-of-focal-plane polarimeters,” Opt. Express 21, 1137–1151 (2013).
[Crossref] [PubMed]

S. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express 19, 26161–26173 (2011).
[Crossref]

S. Gao and V. Gruev, “Image interpolation methods evaluation for division of focal plane polarimeters,” in Infrared Technology and Applications XXXVII, vol. 8012 (International Society for Optics and Photonics, 2011), p. 80120N.
[Crossref]

Garcia, M.

Gartley, M.

D. Vorobiev, Z. Ninkov, and M. Gartley, “Polarization in a snap: imaging polarimetry with micropolarizer arrays,” in Polarization: Measurement, Analysis, and Remote Sensing XI, vol. 9099 (International Society for Optics and Photonics, 2014), p. 909904.

Gilboa, E.

Goldstein, D. H.

D. H. Goldstein, Polarized Light, Revised and Expanded (Marcel Dekker, Inc, 2003).
[Crossref]

Goldstein, D. L.

Graham, R.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Gruev, V.

M. Garcia, C. Edmiston, T. York, R. Marinov, S. Mondal, N. Zhu, G. P. Sudlow, W. J. Akers, J. Margenthaler, S. Achilefu, R. Liang, M. A. Zayed, M. Y. Pepino, and V. Gruev, “Bio-inspired imager improves sensitivity in near-infrared fluorescence image-guided surgery,” Optica 5, 413–422 (2018).
[Crossref] [PubMed]

M. Garcia, C. Edmiston, R. Marinov, A. Vail, and V. Gruev, “Bio-inspired color-polarization imager for real-time in situ imaging,” Optica 4, 1263–1271 (2017).
[Crossref]

A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
[Crossref] [PubMed]

E. Gilboa, J. P. Cunningham, A. Nehorai, and V. Gruev, “Image interpolation and denoising for division of focal plane sensors using Gaussian processes,” Opt. Express 22, 15277–15291 (2014).
[Crossref] [PubMed]

S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
[Crossref] [PubMed]

S. Gao and V. Gruev, “Gradient-based interpolation method for division-of-focal-plane polarimeters,” Opt. Express 21, 1137–1151 (2013).
[Crossref] [PubMed]

S. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express 19, 26161–26173 (2011).
[Crossref]

S. Gao and V. Gruev, “Image interpolation methods evaluation for division of focal plane polarimeters,” in Infrared Technology and Applications XXXVII, vol. 8012 (International Society for Optics and Photonics, 2011), p. 80120N.
[Crossref]

Gunturk, B.

X. Li, B. Gunturk, and L. Zhang, “Image demosaicing: A systematic survey,” in Visual Communications and Image Processing 2008, vol. 6822 (International Society for Optics and Photonics, 2008), p. 68221J.
[Crossref]

Havrilla, M. J.

M. W. Hyde, S. C. Cain, J. D. Schmidt, and M. J. Havrilla, “Material classification of an unknown object using turbulence-degraded polarimetric imagery,” IEEE Trans. Geosci. Remote. Sens. 49, 264–276 (2011).
[Crossref]

Hirasawa, Y.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Hiura, S.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Hui, B.

Hyde, M. W.

M. W. Hyde, S. C. Cain, J. D. Schmidt, and M. J. Havrilla, “Material classification of an unknown object using turbulence-degraded polarimetric imagery,” IEEE Trans. Geosci. Remote. Sens. 49, 264–276 (2011).
[Crossref]

Jaiswal, S. P.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Jakhetiya, V.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Jeon, G.

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

Jin, W.

Kakiuchi, K.

Kennedy, A.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Komori, K.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Kong, S. G.

N. Li, Y. Zhao, Q. Pan, and S. G. Kong, “Removal of reflections in LWIR image with polarization characteristics,” Opt. Express 26, 16488–16504 (2018).
[Crossref] [PubMed]

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
[Crossref]

LaCasse, C. F.

Lapray, P. J.

S. Mihoubi, P. J. Lapray, and L. Bigué, “Survey of demosaicking methods for polarization filter array images,” Sensors 18, 3688 (2018).
[Crossref]

Li, L.

Li, N.

Li, S.

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

S. Li, W. Jin, R. Xia, L. Li, and X. Wang, “Radiation correction method for infrared polarization imaging system with front-mounted polarizer,” Opt. Express 24, 26414–26430 (2016).
[Crossref] [PubMed]

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Li, X.

X. Li, B. Gunturk, and L. Zhang, “Image demosaicing: A systematic survey,” in Visual Communications and Image Processing 2008, vol. 6822 (International Society for Optics and Photonics, 2008), p. 68221J.
[Crossref]

Liang, H.

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Liang, R.

Liu, H.

H. Liu, Z. Shi, B. Feng, B. Hui, and Y. Zhao, “Non-uniformity calibration for MWIR polarization imagery obtained with integrated microgrid polarimeters,” in Selected Papers of the Chinese Society for Optical Engineering Conferences held October and November 2016, vol. 10255 (International Society for Optics and Photonics, 2017), p. 102554D.

Lou, X.

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Luo, H.

Malone, N. R.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Margenthaler, J.

Marinov, R.

Maruyama, Y.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Matoba, Y.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Matsuoka, H.

Mihoubi, S.

S. Mihoubi, P. J. Lapray, and L. Bigué, “Survey of demosaicking methods for polarization filter array images,” Sensors 18, 3688 (2018).
[Crossref]

Miyazaki, D.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Mondal, S.

Mueller, K.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Nakamura, M.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Nehorai, A.

Ninkov, Z.

D. Vorobiev, Z. Ninkov, and M. Gartley, “Polarization in a snap: imaging polarimetry with micropolarizer arrays,” in Polarization: Measurement, Analysis, and Remote Sensing XI, vol. 9099 (International Society for Optics and Photonics, 2014), p. 909904.

Noda, T.

Ohba, Y.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Ohta, J.

Pan, Q.

N. Li, Y. Zhao, Q. Pan, and S. G. Kong, “Removal of reflections in LWIR image with polarization characteristics,” Opt. Express 26, 16488–16504 (2018).
[Crossref] [PubMed]

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

Pan, X.

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Pang, J.

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

Peng, Q.

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
[Crossref]

Pepino, M. Y.

Powell, S. B.

Ratliff, B. M.

Reda, M.

M. Reda, Y. Zhao, and J. C.-W. Chan, “Polarization guided autoregressive model for depth recovery,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

Roberts, N. W.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

Sarkar, M.

M. Sarkar, D. S. S. Bello, C. van Hoof, and A. J. Theuwissen, “Biologically inspired CMOS image sensor for fast motion and polarization detection,” IEEE Sens. J. 13, 1065–1073 (2013).
[Crossref]

Sasagawa, K.

Schmidt, J. D.

M. W. Hyde, S. C. Cain, J. D. Schmidt, and M. J. Havrilla, “Material classification of an unknown object using turbulence-degraded polarimetric imagery,” IEEE Trans. Geosci. Remote. Sens. 49, 264–276 (2011).
[Crossref]

Shao, J.

Shaw, J. A.

Shen, L.

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
[Crossref]

Shi, Z.

H. Liu, Z. Shi, B. Feng, B. Hui, and Y. Zhao, “Non-uniformity calibration for MWIR polarization imagery obtained with integrated microgrid polarimeters,” in Selected Papers of the Chinese Society for Optical Engineering Conferences held October and November 2016, vol. 10255 (International Society for Optics and Photonics, 2017), p. 102554D.

Shigetomi, T.

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Shishido, S.

Sienicki, J.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Stark, J.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Stoer, J.

J. Stoer and R. Bulirsch, Introduction to numerical analysis, vol. 12 (Springer Science & Business Media, 2013).

Sudlow, G. P.

Takruri, M.

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

Terada, T.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Terrier, P.

Thai, Y.

N. R. Malone, A. Kennedy, R. Graham, Y. Thai, J. Stark, J. Sienicki, and E. Fest, “Staring MWIR, LWIR and 2-color and scanning LWIR polarimetry technology,” in Infrared Remote Sensing and Instrumentation XIX, vol. 8154 (International Society for Optics and Photonics, 2011), p. 81540T.
[Crossref]

Theuwissen, A. J.

M. Sarkar, D. S. S. Bello, C. van Hoof, and A. J. Theuwissen, “Biologically inspired CMOS image sensor for fast motion and polarization detection,” IEEE Sens. J. 13, 1065–1073 (2013).
[Crossref]

Tibbs, A. B.

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

Tokuda, T.

Tyo, J. S.

Uesaka, Y.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Vail, A.

Van der Spiegel, J.

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

van Hoof, C.

M. Sarkar, D. S. S. Bello, C. van Hoof, and A. J. Theuwissen, “Biologically inspired CMOS image sensor for fast motion and polarization detection,” IEEE Sens. J. 13, 1065–1073 (2013).
[Crossref]

Vorobiev, D.

D. Vorobiev, Z. Ninkov, and M. Gartley, “Polarization in a snap: imaging polarimetry with micropolarizer arrays,” in Polarization: Measurement, Analysis, and Remote Sensing XI, vol. 9099 (International Society for Optics and Photonics, 2014), p. 909904.

Wang, X.

Wu, J.

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

Wu, W.

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

Wu, X.

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

Xia, R.

Yamazaki, T.

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

Ye, W.

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Yi, C.

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

York, T.

Zayed, M. A.

Zhang, J.

Zhang, L.

X. Li, B. Gunturk, and L. Zhang, “Image demosaicing: A systematic survey,” in Visual Communications and Image Processing 2008, vol. 6822 (International Society for Optics and Photonics, 2008), p. 68221J.
[Crossref]

Zhang, M.

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

Zhang, X.

Zhao, X.

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
[Crossref] [PubMed]

X. Zhao, F. Boussaid, A. Bermak, and V. G. Chigrinov, “High-resolution thin “guest-host” micropolarizer arrays for visible imaging polarimetry,” Opt. Express 19, 5565–5573 (2011).
[Crossref] [PubMed]

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

Zhao, Y.

N. Li, Y. Zhao, Q. Pan, and S. G. Kong, “Removal of reflections in LWIR image with polarization characteristics,” Opt. Express 26, 16488–16504 (2018).
[Crossref] [PubMed]

M. Reda, Y. Zhao, and J. C.-W. Chan, “Polarization guided autoregressive model for depth recovery,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
[Crossref]

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

H. Liu, Z. Shi, B. Feng, B. Hui, and Y. Zhao, “Non-uniformity calibration for MWIR polarization imagery obtained with integrated microgrid polarimeters,” in Selected Papers of the Chinese Society for Optical Engineering Conferences held October and November 2016, vol. 10255 (International Society for Optics and Photonics, 2017), p. 102554D.

Zhou, W.

Zhu, N.

Appl. Opt. (2)

Bioinspir. Biomim. (1)

A. B. Tibbs, I. M. Daly, D. R. Bull, and N. W. Roberts, “Noise creates polarization artefacts,” Bioinspir. Biomim. 13, 015005 (2017).
[Crossref] [PubMed]

IEEE Photonics J. (1)

M. Reda, Y. Zhao, and J. C.-W. Chan, “Polarization guided autoregressive model for depth recovery,” IEEE Photonics J. 9, 1–16 (2017).
[Crossref]

IEEE Sens. J. (2)

A. Abubakar, X. Zhao, S. Li, M. Takruri, E. Bastaki, and A. Bermak, “A Block-Matching and 3-D filtering algorithm for Gaussian noise in DoFP polarization images,” IEEE Sens. J. 18, 7429–7435 (2018).
[Crossref]

M. Sarkar, D. S. S. Bello, C. van Hoof, and A. J. Theuwissen, “Biologically inspired CMOS image sensor for fast motion and polarization detection,” IEEE Sens. J. 13, 1065–1073 (2013).
[Crossref]

IEEE Trans. Electron Devices (1)

Y. Maruyama, T. Terada, T. Yamazaki, Y. Uesaka, M. Nakamura, Y. Matoba, K. Komori, Y. Ohba, S. Arakawa, and Y. Hirasawa, “3.2-MP back-illuminated polarization image sensor with four-directional Air-Gap wire grid and 2.5-μm pixels,” IEEE Trans. Electron Devices 65, 2544–2551 (2018).
[Crossref]

IEEE Trans. Geosci. Remote. Sens. (1)

M. W. Hyde, S. C. Cain, J. D. Schmidt, and M. J. Havrilla, “Material classification of an unknown object using turbulence-degraded polarimetric imagery,” IEEE Trans. Geosci. Remote. Sens. 49, 264–276 (2011).
[Crossref]

IEEE Trans. Image Process. (3)

S. P. Jaiswal, L. Fang, V. Jakhetiya, J. Pang, K. Mueller, and O. C. Au, “Adaptive multispectral demosaicking based on frequency-domain analysis of spectral correlation,” IEEE Trans. Image Process. 26, 953–968 (2017).
[Crossref]

K. H. Chung and Y. H. Chan, “Color demosaicing using variance of color differences,” IEEE Trans. Image Process. 15, 2944–2955 (2006).
[Crossref] [PubMed]

J. Wu, M. Anisetti, W. Wu, E. Damiani, and G. Jeon, “Bayer demosaicking with polynomial interpolation,” IEEE Trans. Image Process. 25, 5369–5382 (2016).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

D. Miyazaki, T. Shigetomi, M. Baba, R. Furukawa, S. Hiura, and N. Asada, “Surface normal estimation of black specular objects from multiview polarization images,” Opt. Eng. 56, 041303 (2016).
[Crossref]

Opt. Express (13)

N. Li, Y. Zhao, Q. Pan, and S. G. Kong, “Removal of reflections in LWIR image with polarization characteristics,” Opt. Express 26, 16488–16504 (2018).
[Crossref] [PubMed]

Z. Chen, X. Wang, and R. Liang, “Snapshot phase shift fringe projection 3D surface measurement,” Opt. Express 23, 667–673 (2015).
[Crossref] [PubMed]

B. M. Ratliff, C. F. LaCasse, and J. S. Tyo, “Interpolation strategies for reducing IFOV artifacts in microgrid polarimeter imagery,” Opt. Express 17, 9112–9125 (2009).
[Crossref] [PubMed]

S. Li, W. Jin, R. Xia, L. Li, and X. Wang, “Radiation correction method for infrared polarization imaging system with front-mounted polarizer,” Opt. Express 24, 26414–26430 (2016).
[Crossref] [PubMed]

S. Gao and V. Gruev, “Bilinear and bicubic interpolation methods for division of focal plane polarimeters,” Opt. Express 19, 26161–26173 (2011).
[Crossref]

S. Gao and V. Gruev, “Gradient-based interpolation method for division-of-focal-plane polarimeters,” Opt. Express 21, 1137–1151 (2013).
[Crossref] [PubMed]

J. Zhang, H. Luo, B. Hui, and Z. Chang, “Image interpolation for division of focal plane polarimeters with intensity correlation,” Opt. Express 24, 20799–20807 (2016).
[Crossref] [PubMed]

A. Ahmed, X. Zhao, V. Gruev, J. Zhang, and A. Bermak, “Residual interpolation for division of focal plane polarization image sensors,” Opt. Express 25, 10651–10662 (2017).
[Crossref] [PubMed]

X. Zhao, F. Boussaid, A. Bermak, and V. G. Chigrinov, “High-resolution thin “guest-host” micropolarizer arrays for visible imaging polarimetry,” Opt. Express 19, 5565–5573 (2011).
[Crossref] [PubMed]

E. Gilboa, J. P. Cunningham, A. Nehorai, and V. Gruev, “Image interpolation and denoising for division of focal plane sensors using Gaussian processes,” Opt. Express 22, 15277–15291 (2014).
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J. Zhang, H. Luo, R. Liang, W. Zhou, B. Hui, and Z. Chang, “PCA-based denoising method for division of focal plane polarimeters,” Opt. Express 25, 2391–2400 (2017).
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K. Sasagawa, S. Shishido, K. Ando, H. Matsuoka, T. Noda, T. Tokuda, K. Kakiuchi, and J. Ohta, “Image sensor pixel with on-chip high extinction ratio polarizer based on 65-nm standard CMOS technology,” Opt. Express 21, 11132–11140 (2013).
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S. B. Powell and V. Gruev, “Calibration methods for division-of-focal-plane polarimeters,” Opt. Express 21, 21039–21055 (2013).
[Crossref] [PubMed]

Opt. Lett. (2)

Optica (2)

Proc. IEEE (1)

M. Zhang, X. Wu, N. Cui, N. Engheta, and J. Van der Spiegel, “Bioinspired focal-plane polarization image sensor design: from application to implementation,” Proc. IEEE 102, 1435–1449 (2014).
[Crossref]

Sensors (1)

S. Mihoubi, P. J. Lapray, and L. Bigué, “Survey of demosaicking methods for polarization filter array images,” Sensors 18, 3688 (2018).
[Crossref]

Other (11)

D. H. Goldstein, Polarized Light, Revised and Expanded (Marcel Dekker, Inc, 2003).
[Crossref]

Y. Zhao, C. Yi, S. G. Kong, Q. Pan, and Y. Cheng, Multi-band Polarization Imaging and Applications (SpringerBerlin Heidelberg, 2016).
[Crossref]

J. Stoer and R. Bulirsch, Introduction to numerical analysis, vol. 12 (Springer Science & Business Media, 2013).

D. Vorobiev, Z. Ninkov, and M. Gartley, “Polarization in a snap: imaging polarimetry with micropolarizer arrays,” in Polarization: Measurement, Analysis, and Remote Sensing XI, vol. 9099 (International Society for Optics and Photonics, 2014), p. 909904.

S. Li, W. Ye, H. Liang, X. Pan, X. Lou, and X. Zhao, “K-SVD based denoising algorithm for DoFP polarization image sensors,” in Circuits and Systems (ISCAS), 2018 IEEE International Symposium on, (IEEE, 2018), pp. 1–5.

L. Shen, Y. Zhao, Q. Peng, J. C.-W. Chan, and S. G. Kong, “An iterative image dehazing method with polarization,” IEEE Trans. Multimed. (2018) (in press).
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B. E. Bayer, “Color imaging array,” (1976). US Patent 3,971,065.

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[Crossref]

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[Crossref]

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Figures (13)

Fig. 1
Fig. 1 A micro-polarizer array of a DoFP polarimeter with four polarization orientations.
Fig. 2
Fig. 2 Test images of 12 scenes, (1)–(6) are in visible spectrum and (7)–(12) are in LWIR spectrum. And corresponding DoLP images are shown in HSV false color model for each scene.
Fig. 3
Fig. 3 Statistical results of means of high-frequency energies of intensity images and polarization difference images for 12 scenes in Fig. 2. (a) The first 6 scenes in visible spectrum. (b) The last 6 scenes in LWIR spectrum.
Fig. 4
Fig. 4 Spectrograms of intensity image and polarization difference image. (a) Test scenes. (b) Corresponding polarization difference image of I0I45. (c) Spectrograms of (a). (d) Spectrograms of (b). The first row and the second row are scenes (4) and (11) in Fig. 2 respectively.
Fig. 5
Fig. 5 A 7 × 7 MPA pattern. Parameters i and j are row and column numbers, respectively.
Fig. 6
Fig. 6 Three cases of missing polarization information interpolation at 90° sampled position. (a) Interpolation of 0° polarization information. (b) Interpolation of 45° polarization information. (c) Interpolation of 135° polarization information.
Fig. 7
Fig. 7 The procedures of the proposed algorithm.
Fig. 8
Fig. 8 The MTF of S0 for interpolation methods (a) Bilinear, (b) Bicubic spline, (c) Gradient-based, (d) Residual interpolation, (e) Our method, (f) The MTF results of S0 along fx = fy.
Fig. 9
Fig. 9 Statistical analysis of discontinuities and values over one in MTF result of the proposed method. (a) Statistics of the amount with one of d in two directions is zero at each frequency, (b) Statistics of the amount with one of φ in two directions is zero at each frequency, (c) Statistics of the amount with Φ ≤ τ at each frequency.
Fig. 10
Fig. 10 Comparison of different interpolation methods in synthetic DoFP images of four selected scenes in Fig. 2. (a) Ground-truth, (b) Bilinear, (c) Bicubic, (d) Bicubic spline, (e) Gradient-based, (f) Residual interpolation, (g) Our method. The first two rows are intensity images and error images of regions in green rectangles, and the last two rows are DoLP images of regions in black rectangles.
Fig. 11
Fig. 11 PSNR results for different standard variances of noise. (a) Intensity, (b) DoLP.
Fig. 12
Fig. 12 Two real DoFP images. (a) Intensity images, (b) DoLP images, (c) AoP images. Two scenes are captured with our developed LWIR DoFP polarization imager and Sony IMX250-MZR polarized image sensor respectively.
Fig. 13
Fig. 13 Comparison of different interpolation methods in real DoFP images. (a) Bilinear, (b) Bicubic, (c) Bicubic spline, (d) Gradient-based, (e) Residual interpolation, (f) Our method. Two rows are regions in green rectangles from two scenes in Fig. 12 respectively.

Tables (1)

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Table 1 The average RMSE performance comparisons of 12 scenes in Fig. 2.

Equations (24)

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S 0 = 0.5 ( I ( 0 ° ) + I ( 45 ° ) + I ( 90 ° ) + I ( 135 ° ) ) S 1 = I ( 0 ° ) I ( 90 ° ) S 2 = I ( 45 ° ) I ( 135 ° )
DoLP = S 1 2 + S 2 2 S 0 AoP = 1 2 tan 1 ( S 2 S 1 )
N n ( x ) = f ( x 0 ) + f [ x 0 , x 1 ] ( x x 0 ) + + f [ x 0 , x 1 , , x n ] ( x x 0 ) ( x x 1 ) ( x x n 1 )
R n ( x ) = f ( x ) N n ( x )
R n ( x ) = 1 ( n + 1 ) ! f ( n + 1 ) ( c ) i = 0 n ( x x i )
f ( x ) f ( x 0 ) + f [ x 0 , x 1 ] ( x x 0 ) + f ( ξ ) 2 ( x x 0 ) ( x x 1 )
f [ x 0 , x 1 ] = f ( x 1 ) f ( x 0 ) x 1 x 0 = f ( x 0 + h ) f ( x 0 ) h
f ( x 0 + h 2 ) f ( x 0 ) + f ( x 0 + h ) f ( x 0 ) h h 2 + f ( ξ ) 2 h 2 ( h 2 ) = f ( x 0 + h ) + f ( x 0 ) 2 f ( ξ ) 8 h 2
I ˜ 90 ( i , j 1 ) I 90 ( i , j 2 ) + I 90 ( i , j ) 2 f ( ξ ) 8 h 2
f ( ξ ) 8 h 2 h 2 8 I 90 ( i , j + 1 ) 2 I 90 ( i , j 1 ) + I 90 ( i , j 3 ) h 2 = I 90 ( i , j + 1 ) 2 I 90 ( i , j 1 ) + I 90 ( i , j 3 ) 8
Λ = I 135 I ^ 90
f ( ξ ) 8 h 2 I 135 ( i , j + 1 ) Λ ( i , j + 1 ) 8 I 135 ( i , j 1 ) Λ ( i , j 1 ) 4 + I 135 ( i , j 3 ) Λ ( i , j 3 ) 8
f ( ξ ) 8 h 2 + I 135 ( i , j + 1 ) 2 I 135 ( i , j 1 ) + I 135 ( i , j 3 ) 8
I ˜ 90 ( i , j 1 ) I 90 ( i , j 2 ) + I 90 ( i , j ) 2 I 135 ( i , j + 1 ) 2 I 135 ( i , j 1 ) + I 135 ( i , j 3 ) 8
I ^ 0 45 ° ( i , j ) = I 90 ( i , j ) + Λ ˜ 45 ° ( i , j ) I 90 ( i , j ) + Λ ˜ ( i + 1 , j 1 ) + Λ ˜ ( i 1 , j + 1 ) 2
I ^ 0 45 ° ( i , j ) I 90 ( i , j ) + I 0 ( i + 1 , j 1 ) I ˜ 90 ( i + 1 , j 1 ) + I 0 ( i 1 , j + 1 ) I ˜ 90 ( i 1 , j + 1 ) 2 = I 90 ( i , j ) + I 0 ( i + 1 , j 1 ) + I 0 ( i 1 , j + 1 ) 2 I ˜ 90 ( i + 1 , j 1 ) + I ˜ 90 ( i 1 , j + 1 ) 2
I ^ 0 45 ° ( i , j ) I 90 ( i , j ) + I 0 ( i 1 , j 1 ) I ˜ 90 ( i 1 , j 1 ) + I 0 ( i + 1 , j + 1 ) I ˜ 90 ( i + 1 , j + 1 ) 2 = I 90 ( i , j ) + I 0 ( i 1 , j 1 ) + I 0 ( i + 1 , j + 1 ) 2 I ˜ 90 ( i 1 , j 1 ) + I ˜ 90 ( i + 1 , j + 1 ) 2
φ 45 ° = m = { 2 , 0 , 2 } n = { 2 , 0 , 2 } | I ^ 0 45 ° ( i + m , j + n ) I 90 ( i + m , j + n ) | φ 45 ° = m = { 2 , 0 , 2 } n = { 2 , 0 , 2 } | I ^ 0 45 ° ( i + m , j + n ) I 90 ( i + m , j + n ) |
Φ = max ( φ 45 ° φ 45 ° , φ 45 ° φ 45 ° )
I ^ 0 ( i , j ) = I ^ 0 45 ° ( i , j ) if ( Φ > τ ) & ( Φ = φ 45 ° φ 45 ° ) I ^ 0 ( i , j ) = I ^ 0 45 ° ( i , j ) if ( Φ > τ ) & ( Φ = φ 45 ° φ 45 ° ) I ^ 0 ( i , j ) = ω 45 ° I ^ 0 45 ° ( i , j ) + ω 45 ° I ^ 0 45 ° ( i , j ) ω 45 ° + ω 45 ° if ( Φ τ )
ω 45 ° = 1 d 45 ° + ε ω 45 ° = 1 d 45 ° + ε
d 45 ° = | I 0 ( i + 1 , j 1 ) I 0 ( i 1 , j + 1 ) | + | 2 I 90 ( i , j ) I 90 ( i 2 , j + 2 ) I 90 ( i + 2 , j 2 ) | d 45 ° = | I 0 ( i 1 , j 1 ) I 0 ( i + 1 , j + 1 ) | + | 2 I 90 ( i , j ) I 90 ( i 2 , j 2 ) I 90 ( i + 2 , j + 2 ) |
I 0 ( x , y ) = cos ( 2 π f x x + 2 π f y y ) + 1 I 45 ( x , y ) = 2 cos ( 2 π f x x + 2 π f y y ) + 2 I 90 ( x , y ) = cos ( 2 π f x x + 2 π f y y ) + 1 I 135 ( x , y ) = 0
RMSE = 1 M N i = 1 M j = 1 N ( I true ( i , j ) I interpolated ( i , j ) ) 2

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