Abstract

A ground-based full-sky imaging polarimeter based on liquid crystal variable retarders (LCVRs) is proposed in this paper. Our proposed method can be used to realize the rapid detection of the skylight polarization information with hemisphere field-of-view for the visual band. The characteristics of the incidence angle of light on the LCVR are investigated, based on the electrically controlled birefringence. Then, the imaging polarimeter with hemisphere field-of-view is designed. Furthermore, the polarization calibration method with the field-of-view multiplexing and piecewise linear fitting is proposed, based on the rotation symmetry of the polarimeter. The polarization calibration of the polarimeter is implemented with the hemisphere field-of-view. This imaging polarimeter is investigated by the experiment of detecting the skylight image. The consistency between the obtained experimental distribution of polarization angle with that due to Rayleigh scattering model is 90%, which confirms the effectivity of our proposed imaging polarimeter.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]

2013

2011

A. R. Dahlberg, N. J. Pust, J. A. Shaw, “Effects of surface reflectance on skylight polarization measurements at the Mauna Loa Observatory,” Opt. Express 19(17), 16008–16021 (2011).
[CrossRef] [PubMed]

N. J. Pust, A. R. Dahlberg, M. J. Thomas, J. A. Shaw, “Comparison of full-sky polarization and radiance observations to radiative transfer simulations which employ AERONET products,” Opt. Express 19(19), 18602–18613 (2011).
[CrossRef] [PubMed]

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

2006

2004

2002

2001

J. Gál, G. Horvath, A. Barta, R. Wehner, “Polarization of the moonlit clear night sky measured by full-sky imaging polarimeter at full moon: comparison of the polarization of moonlit and sunlit skies,” J. Geophys. Res. 106(D19), 22647–22653 (2001).
[CrossRef]

2000

D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. 25(11), 802–804 (2000).
[CrossRef] [PubMed]

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 25(16), 1198–1200 (2000).
[CrossRef] [PubMed]

1999

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

1997

1993

1987

1982

A. T. Young, “Rayleigh scattering,” Phys. Today 35(1), 42–48 (1982).
[CrossRef]

1980

1913

H. H. Kimball, “The effect of the atmospheric turbidity of 1912 on solar radiation intensities and skylight polarization,” Bull. Mt. Weather Obs. 5, 295–312 (1913).

Adams, C. N.

Akesson, S.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

Barta, A.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[CrossRef] [PubMed]

J. Gál, G. Horvath, A. Barta, R. Wehner, “Polarization of the moonlit clear night sky measured by full-sky imaging polarimeter at full moon: comparison of the polarization of moonlit and sunlit skies,” J. Geophys. Res. 106(D19), 22647–22653 (2001).
[CrossRef]

Breon, F. M.

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

Bréon, F. M.

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

Brogniez, G.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

Castaño, D. J.

Chepfer, H.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

Coulson, K. L.

Dahlberg, A. R.

Dereniak, E. L.

Descour, M. R.

Deuzé, J. L.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

Duggin, M. J.

Flamant, P. H.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

Gál, J.

G. Horváth, A. Barta, J. Gál, B. Suhai, O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[CrossRef] [PubMed]

J. Gál, G. Horvath, A. Barta, R. Wehner, “Polarization of the moonlit clear night sky measured by full-sky imaging polarimeter at full moon: comparison of the polarization of moonlit and sunlit skies,” J. Geophys. Res. 106(D19), 22647–22653 (2001).
[CrossRef]

Goloub, P.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

H. Chepfer, G. Brogniez, P. Goloub, F. M. Bréon, P. H. Flamant, “Observations of horizontally oriented ice crystals in cirrus clouds with POLDER-1/ADEOS-1,” J. Quant. Spectrosc. Radiat. Transf. 63(2–6), 521–543 (1999).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

Gordon, H. R.

Haiman, O.

Hegedüs, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

Herman, M.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

Horvath, G.

J. Gál, G. Horvath, A. Barta, R. Wehner, “Polarization of the moonlit clear night sky measured by full-sky imaging polarimeter at full moon: comparison of the polarization of moonlit and sunlit skies,” J. Geophys. Res. 106(D19), 22647–22653 (2001).
[CrossRef]

Horváth, G.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

B. Suhai, G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[CrossRef] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[CrossRef] [PubMed]

Kattawar, G. W.

Kemme, S. A.

Kimball, H. H.

H. H. Kimball, “The effect of the atmospheric turbidity of 1912 on solar radiation intensities and skylight polarization,” Bull. Mt. Weather Obs. 5, 295–312 (1913).

Lavorato, M.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

Li, N.

Liu, Y.

Marchand, A.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

Meyer-Rochow, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

North, J. A.

Pelon, J.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

Perry, G.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

Phipps, G. S.

Pomozi, I.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

Pust, N. J.

Sabatke, D. S.

Sauvage, L.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

Shaw, J. A.

Spinhirne, J.

H. Chepfer, P. Goloub, J. Spinhirne, P. H. Flamant, M. Lavorato, L. Sauvage, G. Brogniez, J. Pelon, “Cirrus Cloud Properties Derived from POLDER-1/ADEOS Polarized Radiances: First Validation Using a Ground-Based Lidar Network,” J. Appl. Meteorol. 39(2), 154–168 (2000).
[CrossRef]

Suhai, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

B. Suhai, G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[CrossRef] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[CrossRef] [PubMed]

Susana, S.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

Sweatt, W. C.

Tanré, D.

J. L. Deuzé, P. Goloub, M. Herman, A. Marchand, G. Perry, S. Susana, D. Tanré, “Estimate of the aerosol properties over the ocean with POLDER,” J. Geophys. Res. 105(D12), 15329–15346 (2000).
[CrossRef]

J. L. Deuzé, M. Herman, P. Goloub, D. Tanré, A. Marchand, “Characterization of aerosols over ocean from POLDER/ADEOS-1,” Geophys. Res. Lett. 26(10), 1421–1424 (1999).
[CrossRef]

P. Goloub, D. Tanré, J. L. Deuzé, M. Herman, A. Marchand, F. M. Breon, “Validation of the first algorithm applied for deriving the aerosol properties over the ocean using the POLDER/ADEOS measurements,” IEEE Trans. Geosci. Rem. Sens. 37(3), 1586–1596 (1999).
[CrossRef]

Thomas, M. J.

Tyo, J. S.

Voss, K.

Voss, K. J.

Wehner, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Akesson, B. Meyer-Rochow, R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[CrossRef] [PubMed]

J. Gál, G. Horvath, A. Barta, R. Wehner, “Polarization of the moonlit clear night sky measured by full-sky imaging polarimeter at full moon: comparison of the polarization of moonlit and sunlit skies,” J. Geophys. Res. 106(D19), 22647–22653 (2001).
[CrossRef]

Young, A. T.

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Figures (10)

Fig. 1
Fig. 1

Schematic of Rayleigh scattering model.

Fig. 2
Fig. 2

The polarization of E vector of P point in the sky.

Fig. 3
Fig. 3

The geometric model for the incident angle of light and the molecule axis of the liquid crystal.

Fig. 4
Fig. 4

The relationship between phase delay of LCVR and applied voltage in the cases of different incident angles.

Fig. 5
Fig. 5

Schematic of the proposed imaging polarimeter.

Fig. 6
Fig. 6

polarization calibration in full field-of-view.

Fig. 7
Fig. 7

The STDV of DOP (A) and AOP (B) by different wavelengths.

Fig. 8
Fig. 8

Implement of experiment.

Fig. 9
Fig. 9

The polarization angle images by experimental and Rayleigh model.

Fig. 10
Fig. 10

Experimental DOP and AOP in different wavelengths with the solar altitude angle of 35.8 °.

Tables (2)

Tables Icon

Table 1 Circumference Average of the Mueller Matrix Condition Numbers in Full Polarization Detection Components

Tables Icon

Table 2 Experimental Results for STDV of the Polarization Calibration (λ = 476 nm)

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

S=( I Q U V )=( < E x 2 (t)>+< E y 2 (t)> < E x 2 (t)>< E y 2 (t)> 2< E x 2 (t)>< E y 2 (t)>cos( δ y (t) δ x (t)) 2< E x 2 (t)>< E y 2 (t)>sin( δ y (t) δ x (t)) )=( I 0 + I 90 I 0 I 90 I 45 I 135 I R I L )
DOLP= Q 2 + U 2 I
DOCP= | V | I
DOP= Q 2 + U 2 + V 2 I
S out =M×S in
M comb = M N M N1 ... M 2 M
sin ( Z P S ) sin ( 90 ° h s ) = sin ( | A s A p | ) sin θ
cosα=sinξcosβcos(s)-cosξsinβ
ξ=arcsin(sinΦ/n)
M PMC = M LP × M LCVR2 × M LCVR1
Con d ave ( Φ )= M PMC ( Φ ) 2 M PMC ( Φ ) 1 2
M INS =( a 00 a 01 a 02 a 03 a 10 a 11 a 12 a 13 a 20 a 21 a 22 a 23 a 30 a 31 a 32 a 33 )
I OUT =( I 0 I 1 I 2 I 3 )= M INS × S IN
M INS(i,j) =( ( a 00 ) (i,j) ( a 01 ) (i,j) ( a 02 ) (i,j) ( a 03 ) (i,j) ( a 10 ) (i,j) ( a 11 ) (i,j) ( a 12 ) (i,j) ( a 13 ) (i,j) ( a 20 ) (i,j) ( a 21 ) (i,j) ( a 22 ) (i,j) ( a 23 ) (i,j) ( a 30 ) (i,j) ( a 31 ) (i,j) ( a 32 ) (i,j) ( a 33 ) (i,j) )
M INS( x N , y N ) M INS(0, [n] ) n[n] = M INS(0, [n+1] ) M INS(0, [n] ) 1
M INS( x N , y N ) =( M INS(0, [n+1] ) M INS(0, [n] ) )(n[n])+ M INS(0, [n] )
M ¯ INS(0,j) = M INS(-1,j) + M INS(0,j) + M INS(1,j) 3
M ¯ INS( x N , y N ) =( M ¯ INS(0, [n+1] ) M ¯ INS(0, [n] ) )(n[n])+ M ¯ INS(0, [n] )

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