Abstract

In active scalar polarimetric imaging systems, the illumination and analysis polarization states are degrees of freedom that can be used to maximize the performance. These optimal states depend on the statistics of the noise that perturbs image acquisition. We investigate the problem of optimization of discrimination ability (contrast) of such imagers in the presence of three different types of noise statistics frequently encountered in optical images (Gaussian, Poisson, and Gamma). To compare these different situations within a common theoretical framework, we use the Bhattacharyya distance and the Fisher ratio as measures of contrast. We show that the optimal states depend on a trade-off between the target/background intensity difference and the average intensity in the acquired image, and that this trade-off depends on the noise statistics. On a few examples, we show that the gain in contrast obtained by implementing the states adapted to the noise statistics actually present in the image can be significant.

© 2012 Optical Society of America

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References

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2011 (2)

2010 (2)

2009 (2)

2004 (1)

2002 (2)

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

P. Réfrégier and F. Goudail, “Invariant polarimetric contrast parameters for coherent light,” J. Opt. Soc. Am. A 19, 1223–1233 (2002).
[CrossRef]

2000 (1)

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

1999 (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806  nm,” Proc. SPIE 3707, 449–460 (1999).
[CrossRef]

1998 (1)

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

1996 (2)

1988 (1)

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

1987 (2)

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991 (1987).
[CrossRef]

K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
[CrossRef]

1981 (2)

J. E. Solomon, “Polarization imaging,” Appl. Opt. 20, 1537–1544 (1981).
[CrossRef]

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).

Anna, G.

Antonelli, M.

Benali, A.

Boerner, W. M.

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991 (1987).
[CrossRef]

Boerner, W.-M.

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

Breugnot, S.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806  nm,” Proc. SPIE 3707, 449–460 (1999).
[CrossRef]

Cariou, J.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Chipman, R. A.

Clémenceau, P.

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806  nm,” Proc. SPIE 3707, 449–460 (1999).
[CrossRef]

Cover, T. M.

T. M. Cover and A. Thomas, Elements of Information Theory (Wiley, 1991).

De Martino, A.

Delyon, G.

DeMartino, A.

Dolfi, D.

Engheta, N.

Floc’h, M.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Gayet, B.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Goudail, F.

Jain, A.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

Kieleck, C.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Kim, K.

Kong, J. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Kostinski, A. B.

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991 (1987).
[CrossRef]

Le Brun, G.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Lin, S.

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

Lotrian, J.

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Lu, S. Y.

Mandel, L.

Miller, M. I.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

Moulin, P.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

Novak, L. M.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Novikova, T.

Orlik, X.

Pierangelo, A.

Pugh, E. N.

Ramchandran, K.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

Réfrégier, P.

Richert, M.

Rowe, M. P

Shin, R. T.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Shribak, M.

Solomon, J. E.

Swartz, A. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Thomas, A.

T. M. Cover and A. Thomas, Elements of Information Theory (Wiley, 1991).

Tyo, J. S.

Validire, P.

Walraven, R.

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).

Wolf, E.

Yamaguchi, Y.

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

Yang, J.

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

Yueh, H. A.

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

A. B. Kostinski and W. M. Boerner, “On the polarimetric contrast optimization,” IEEE Trans. Antennas Propag. 35, 988–991 (1987).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

J. Yang, Y. Yamaguchi, W.-M. Boerner, and S. Lin, “Numerical methods for solving the optimal problem of contrast enhancement,” IEEE Trans. Geosci. Remote Sens. 38, 965–971 (2000).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded image data,” IEEE Trans. Pattern Anal. Machine Intell. 24, 1153–1166 (2002).
[CrossRef]

J. Geophys. Res. (1)

A. A. Swartz, H. A. Yueh, J. A. Kong, L. M. Novak, and R. T. Shin, “Optimal polarizations for achieving maximal contrast in radar images,” J. Geophys. Res. 93, 15252–15260 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

R. Walraven, “Polarization imagery,” Opt. Eng. 20, 14–18 (1981).

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (1)

S. Breugnot and P. Clémenceau, “Modeling and performances of a polarization active imager at λ=806  nm,” Proc. SPIE 3707, 449–460 (1999).
[CrossRef]

Pure Appl. Opt. (1)

M. Floc’h, G. Le Brun, C. Kieleck, J. Cariou, and J. Lotrian, “Polarimetric considerations to optimize lidar detection of immersed targets,” Pure Appl. Opt. 7, 1327–1340 (1998).
[CrossRef]

Other (2)

T. M. Cover and A. Thomas, Elements of Information Theory (Wiley, 1991).

J. W. Goodman, Statistical Optics (Wiley, 1985).

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Figures (8)

Fig. 1.
Fig. 1.

Principle of active scalar polarimetric imaging. PSG, polarization state generator; PSA, polarization state analyzer.

Fig. 2.
Fig. 2.

Evolution of Bgam(x) in function of |x|=|TTDS|/TTMS

Fig. 3.
Fig. 3.

Variation of θopt as a function of ε for different values of diattenuation coefficient d in the presence of Gaussian additive noise (Subsection 3.A).

Fig. 4.
Fig. 4.

(a) Variation of θopt as a function of ε in the scenario of Subsection 3.A, for diattenuation d=1, in the presence of Gaussian, Poisson, and Gamma noise sources. (b) Azimuth of the optimal states for the three types of noises, ε=20°, and d=1.

Fig. 5.
Fig. 5.

Evolution of the contrast with parameter ε in the presence of (a) Gaussian noise, (b) Poisson noise, and (c) Gamma noise, in the scenario of Subsection 3.A, for d=1. Figure (c) has been obtained with S and T partially polarized (s=t=0.99) in order to have contrasts that are not infinite.

Fig. 6.
Fig. 6.

Evolution of the Bhattacharyya distance with the parameter ρ in the presence of (a) Gaussian noise, (b) Poisson noise, and (c) Gamma noise, for the scenario of Subsection 3.B.

Fig. 7.
Fig. 7.

Evolution of the ellipticities of the optimal states S and T with the parameter ρ, for the scenario of Subsection 3.B.

Fig. 8.
Fig. 8.

Images in the presence of (a)–(c) Gaussian, (d)–(f) Poisson, and (h)–(j) Gamma noise of a target characterized by a Mueller matrix Mb and a background characterized by a Mueller matrix Ma (ρ=0.14), for the scenario of Subsection 3.B. In each case, the image is obtained with optimal states computed with the hypothesis of the presence of Gaussian, Poisson, and Gamma noise (respectively from left to right). The number of photons by pixel is I0=100 and the standard deviation of the Gaussian noise is equal to 10. The Gamma noise is of order L=1.

Equations (25)

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Su=MuSwithu={a,b}.
iu=ηI02TTMuS,
B=ln[D[Pa(x)Pb(x)]1/2dx],
F=(mamb)2vara+varb,
Bgau=(iaib)28σ2.
M=Ma+Mb2,D=MaMb2.
Bgau(S,T)=η2I02(TTDS)28σ2.
(Sgau,Tgau)=argmaxS,T[|TTDS|].
Bpoi=12(iaib)2.
Bpoi(S,T)=ηI04(TTMS+TTDSTTMSTTDS)2.
(Spoi,Tpoi)=argmaxS,T[Bpoi(S,T)].
Bgam=Lln[12(r+1r)],
Bgam(S,T)=Lln[12(TTMS+TTDSTTMSTTDS+TTMSTTDSTTMS+TTDS)].
Bgam(x)=Lln[12g(x)],
g(x)=1+x1x+1x1+x,x=TTDSTTMS.
(Sgam,Tgam)=argmaxS,T[|TTDS|TTMS].
Bpoi(S,T)ηI04(TTDS)2TTMS=12(ηI0/2×TTDS)2ηI0/2×TTMS,
Bgam(S,T)L2(TTDSTTMS)2=12(ηI0/2×TTDS)2(ηI0/2×TTMS)2/L.
P(γ,d)=11+d(1dvγTdvγP˜γ).
M=11+d(1dcosε00Dcosεα+(1α)cos2ε0000α+(1α)sin2ε0000α),D=11+d(00dsinε000(1α)cosεsinε0dsinε(1α)cosεsinε000000).
Bgau=12[ηI0σ(1+d)]2{sinεsin2θ(d+(1α)cosεcos2θ)}2.
ia=ηI02(1+d)[(1+α)+2dcos(ε2θ)+(1α)cos2(ε2θ)],ib=ηI02(1+d)[(1+α)+2dcos(ε+2θ)+(1α)cos2(ε+2θ)].
Ma=(0.700000.35+ρ00000.35+ρ00000.2),Mb=(0.800000.700000.700000.45),
M=(0.7500000.525+ρ/200000.525+ρ/200000.325),D=(0.0500000.175ρ/200000.175ρ/200000.125).
M=(M0000M˜),D=(D0000D˜),

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