A priori information and optimisation in polarimetry
Optics Express, Vol. 16, Issue 19, pp. 15212-15227 (2008)
http://dx.doi.org/10.1364/OE.16.015212
Acrobat PDF (264 KB)
Abstract
Polarimetric measurements are designed to obtain information pertaining to the system under study, however noise in the system limits the precision and hence information obtainable. Exploitation of a priori knowledge of the system allows for an improvement in the precision of experimental data. In this vein we present a framework for system design and optimisation based upon the Fisher information matrix, which allows easy incorporation of such a priori information. As such the proposed figure of merit is more complete than the commonly used condition number. Conditions of equivalence are considered, however a number of examples highlight the failings of the condition number under more general scenarios. Bounds on the achievable informational gains via multiple polarimeter arms are also given. Finally we present analytic results concerning error distribution in a Mueller matrix polar decomposition, allowing for a more accurate noise analysis in polarimetric experiments.
© 2008 Optical Society of America
1. Introduction
5. J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006). [CrossRef]
10. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films , 455–456112–119 (2004). [CrossRef]
13. I. J. Cox and C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 31152–1158 (1986). [CrossRef]
14. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]
2. System description
2.1. Stokes polarimeter
2.2. Mueller polarimeter
2.3. Noise model
19. M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 1513597–13606 (2007). [CrossRef] [PubMed]
21. B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 382317–2326 (1988). [CrossRef]
23. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Europ. Opt. Soc. Rap. Public. 308002 (2008). [CrossRef]
3. Information in polarimetry
3.1. Deterministic systems
26. R. Fisher, “On the mathematical foundations of theoretical statistics,” Phil. Trans. R. Soc. Lond. 222309–368 (1922) [CrossRef]
27. R. Fisher, “Theory of statistical estimation,” Proc. Cam. Phil. Soc. 22700–725 (1925) [CrossRef]
3.2. Stochastic systems
29. H. H. Barrett, J.L. Denny, R.F. Wagner, and K.J. Myers “Objective assessment of image quality. II Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995) [CrossRef]
31. E. W. Barankin, “Locally best unbiased estimates,” Ann. Math. Stat. 20477–501 (1949). [CrossRef]
33. S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 503697–3715 (2005). [CrossRef] [PubMed]
3.3. Channel capacity
34. B. R. FriedenPhysics from Fisher Information: A Unification, (Cambridge University Press1998). [CrossRef]
5. J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006). [CrossRef]
35. J. S. Tyo, “Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 251198–1200 (2000). [CrossRef]
36. A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007). [CrossRef]
34. B. R. FriedenPhysics from Fisher Information: A Unification, (Cambridge University Press1998). [CrossRef]
4. Optimisation of polarimeters
4.1. Ellipsoids of concentration
7. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef] [PubMed]
10. A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films , 455–456112–119 (2004). [CrossRef]
4.2. Nuisance parameters
5. Examples
5.1. Maximal ignorance
39. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41965–972 (2002). [CrossRef]
7. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef] [PubMed]
35. J. S. Tyo, “Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 251198–1200 (2000). [CrossRef]
39. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41965–972 (2002). [CrossRef]
41. R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44190–196 (2005). [CrossRef] [PubMed]
39. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41965–972 (2002). [CrossRef]
5.2. Matched filter
5.3. Linear polarimeter
6. J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system,” J. Opt. Soc. Am. A 15359–366 (1998). [CrossRef]
6. Extension of optimisation results
14. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]
45. B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. 43, 2824–2832 (2004) [CrossRef] [PubMed]
47. M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998). [CrossRef]
7. Noise propagation in Mueller matrix polar decomposition
7.1. Single element systems
14. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]
14. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef]
48. S. M. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 201651–1657 (2003). [CrossRef]
49. S. N. Savenkov and K. E. Yushtin, “Mueller matrix elements error distribution for polarimetric measurements,” Proc. SPIE 5158251–259 (2003). [CrossRef]
7.2. Composite systems
50. J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 292234–2236 (2004). [CrossRef] [PubMed]
8. Conclusions
A. Fisher information for random parameters
References and links
1. | Planets Stars and Nebulae Studied With PhotopolarimetryT. Gehrels ed. (University of Arizona Press1974). |
2. | J. Tinbergen, “Interstellar polarization in the immediate solar neighbourhood,” Astron. Astrophys. 105, 53–64 (1982). |
3. | R. A. Chipman, Polarimetry, Handbook of OpticsVol 2, (McGraw-Hill, New York, 1995). |
4. | R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987). |
5. | J. Zallat, S. Aïnouz, and M. Ph. Stoll “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A: Pure Appl. Opt. 8807–814 (2006). [CrossRef] |
6. | J. S. Tyo, “Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system,” J. Opt. Soc. Am. A 15359–366 (1998). [CrossRef] |
7. | J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41619–630 (2002). [CrossRef] [PubMed] |
8. | A. Ambirajan and D.C. Look, “Optimum angles for a polarimeter,” Opt. Eng. 341651–1658 (1995). [CrossRef] |
9. | M. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 412488–2493 (2002). [CrossRef] [PubMed] |
10. | A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, “General methods for optimized design and calibration of Mueller polarimeters,” Thin Solid Films , 455–456112–119 (2004). [CrossRef] |
11. | D. Mendlovic and A. W. Lohmann “Spacebandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14 (1997). |
12. | T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973). |
13. | I. J. Cox and C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 31152–1158 (1986). [CrossRef] |
14. | S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996). [CrossRef] |
15. | R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light,” J. Mod. Opt. 29685–689 (1982). |
16. | E. Collett, “Automatic determination of the polarization state of nanosecond laser pulses,” U.S. Patent No. 4158506 (1979). |
17. | H. Mueller, “The foundations of optics” J. Opt. Soc. Am. 38, 661–661 (1948). |
18. | V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” J. Mod. Opt. 29685–689 (1982) |
19. | M. R. Foreman, S. S. Sherif, and P. Török, “Photon statistics in single molecule orientational imaging,” Opt. Express 1513597–13606 (2007). [CrossRef] [PubMed] |
20. | A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-Resolution Full-Field Optical Coherence Tomography,” Appl. Opt. 43, 2874–2883 (2004). [CrossRef] [PubMed] |
21. | B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 382317–2326 (1988). [CrossRef] |
22. | J. W. Goodman, Statistical Optics (Wiley, New York, 2000). |
23. | A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises,” J. Europ. Opt. Soc. Rap. Public. 308002 (2008). [CrossRef] |
24. | H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6. |
25. | C. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 3781–89 (1945) |
26. | R. Fisher, “On the mathematical foundations of theoretical statistics,” Phil. Trans. R. Soc. Lond. 222309–368 (1922) [CrossRef] |
27. | R. Fisher, “Theory of statistical estimation,” Proc. Cam. Phil. Soc. 22700–725 (1925) [CrossRef] |
28. | |
29. | H. H. Barrett, J.L. Denny, R.F. Wagner, and K.J. Myers “Objective assessment of image quality. II Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995) [CrossRef] |
30. | L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991) |
31. | E. W. Barankin, “Locally best unbiased estimates,” Ann. Math. Stat. 20477–501 (1949). [CrossRef] |
32. | M. F. Kijewski, S. P. Müller, and S. C. Moore, “The Barankin bound: a model of detection with location uncertainty,” Proc. SPIE 1768153 (1992). [CrossRef] |
33. | S. P. Müller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 503697–3715 (2005). [CrossRef] [PubMed] |
34. | B. R. FriedenPhysics from Fisher Information: A Unification, (Cambridge University Press1998). [CrossRef] |
35. | J. S. Tyo, “Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. 251198–1200 (2000). [CrossRef] |
36. | A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, “Precision of degree of polarization estimation in the presence of additive Gaussian detector noise,” Opt. Commun. 278264–269 (2007). [CrossRef] |
37. | R. Mehra, “Optimal input signals for parameter estimation in dynamic systems—Survey and new results,” IEEE T. Automat. Contr. 19753–768 (1974). [CrossRef] |
38. | E. Walter and L. Pronzatom “Qualitative and quantitative experiment design for phenomenological models - a survey,” Automatica 26195–213 (1990). [CrossRef] |
39. | S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41965–972 (2002). [CrossRef] |
40. | V. Bhapkar and C. Srinivasan, “On Fisher information inequalities in the presence of nuisance parameters,” Ann. Inst. Statist. Math 46593–604 (1994). |
41. | R. M. A. Azzam and F. F. Sudradjat, “Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter,” Appl. Opt. 44190–196 (2005). [CrossRef] [PubMed] |
42. | A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971). |
43. | P. Török, M. Salt, E.E. Kriezis, P.R.T. Munro, H.P. Herzig, and C. Rockstuhl, “Optical disk and reader therefor,” Worldwide Patent No. WO 2006/010882 (2006). |
44. | P. R. T. Munro and P. Török, “Properties of confocal Mueller-matrix polarimeters,” (submitted to Opt. Lett.). |
45. | B. Laude-Boulesteix, A. De Martino, B. Drévillon, and L. Schwartz, “Mueller Polarimetric Imaging System with Liquid Crystals,” Appl. Opt. 43, 2824–2832 (2004) [CrossRef] [PubMed] |
46. | J. M. Bueno, “Depolarization effects in the human eye,” Vision Research 412687–2696 (2001). [CrossRef] [PubMed] |
47. | M. Floc’h, G. Le Brun, J. Cariou, and J. Lotrian, “Experimental characterization of immersed targets by polar decomposition of the Mueller matrices,” Eur. Phys. J. AP 3349–358 (1998). [CrossRef] |
48. | S. M. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 201651–1657 (2003). [CrossRef] |
49. | S. N. Savenkov and K. E. Yushtin, “Mueller matrix elements error distribution for polarimetric measurements,” Proc. SPIE 5158251–259 (2003). [CrossRef] |
50. | J. Morio and F. Goudail, “Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices,” Opt. Lett. 292234–2236 (2004). [CrossRef] [PubMed] |
OCIS Codes
(120.5410) Instrumentation, measurement, and metrology : Polarimetry
(220.4830) Optical design and fabrication : Systems design
(110.3055) Imaging systems : Information theoretical analysis
ToC Category:
Instrumentation, Measurement, and Metrology
History
Original Manuscript: July 24, 2008
Revised Manuscript: September 4, 2008
Manuscript Accepted: September 8, 2008
Published: September 11, 2008
Citation
Matthew R. Foreman, Carlos Macias Romero, and Peter Török, "A priori information and optimisation in
polarimetry," Opt. Express 16, 15212-15227 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15212
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References
- T. Gehrels, ed., Planets Stars and Nebulae Studied With Photopolarimetry, (University of Arizona Press 1974).
- J. Tinbergen, "Interstellar polarization in the immediate solar neighbourhood," Astron. Astrophys. 105, 53-64 (1982).
- R. A. Chipman, Polarimetry, Handbook of Optics Vol 2, (McGraw-Hill, New York, 1995).
- R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarised light, (Elsevier, North Holland, 1987).
- J. Zallat, S. Aï?nouz, and M. Ph. Stoll "Optimal configurations for imaging polarimeters: impact of image noise and systematic errors," J. Opt. A: Pure Appl. Opt. 8, 807-814 (2006). [CrossRef]
- J. S. Tyo, "Optimum linear combination strategy for an N-channel polarization sensitive imaging or vision system," J. Opt. Soc. Am. A 15,359-366 (1998). [CrossRef]
- J. S. Tyo, "Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error," Appl. Opt. 41, 619-630 (2002). [CrossRef] [PubMed]
- A. Ambirajan and D. C. Look, "Optimum angles for a polarimeter," Opt. Eng. 34,1651-1658 (1995). [CrossRef]
- M. Smith, "Optimization of a dual-rotating-retarder Mueller matrix polarimeter," Appl. Opt. 41, 2488-2493 (2002). [CrossRef] [PubMed]
- A. De Martino, E. Garcia-Caurel, B. Laude, and B. Drévillon, "General methods for optimized design and calibration of Mueller polarimeters," Thin Solid Films 455-456,112-119 (2004). [CrossRef]
- D. Mendlovic and A. W. Lohmann "Spacebandwidth product adaptation and its application to superresolution: fundamentals," J. Opt. Soc. Am. A 14, 558-562 (1997).
- T. J. Rothenberg, Efficient estimation with a priori information (New Haven, Yale University Press, 1973).
- I. J. Cox and C. J. R. Sheppard, "Information capacity and resolution in an optical system," J. Opt. Soc. Am. A 3, 1152-1158 (1986). [CrossRef]
- S. Y. Lu and R. A. Chipman, "Interpretation of Mueller matrices based on polar decomposition," J. Opt. Soc. Am. A 13, 1106-1113 (1996). [CrossRef]
- R. M. A. Azzam, "Division-of-amplitude Photopolarimeter (DOAP) for the simultaneous measurement of all four Stokes parameters of light," J. Mod. Opt. 29, 685-689 (1982).
- E. Collett, "Automatic determination of the polarization state of nanosecond laser pulses," U.S. Patent No. 4158506 (1979).
- H. Mueller, "The foundations of optics" J. Opt. Soc. Am. 38, 661-661 (1948).
- V. Delaubert, N. Treps, C. Fabre, H. A. Bachor, and P. Réfrégier, "Quantum limits in image processing," J. Mod. Opt. 29, 685-689 (1982)
- M. R. Foreman, S. S. Sherif, and P. Török, "Photon statistics in single molecule orientational imaging," Opt. Express 15, 13597-13606 (2007). [CrossRef] [PubMed]
- A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, "Ultrahigh-Resolution Full-Field Optical Coherence Tomography," Appl. Opt. 43, 2874-2883 (2004). [CrossRef] [PubMed]
- B. J. Meers, "Recycling in laser-interferometric gravitational-wave detectors," Phys. Rev. D 38, 2317-2326 (1988). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley, New York, 2000).
- A. Bénière, F. Goudail, M. Alouini, and D. Dolfi, "Estimation precision of degree of polarization in the presence of signal-dependent and additive Poisson noises," J. Eur. Opt. Soc. Rap. Pub. 3, 08002 (2008). [CrossRef]
- H. Cramér, Mathematical Methods of Statistics, (Princeton Univ. Press. 1946), ISBN 0-691-08004-6.
- C. Rao, "Information and the accuracy attainable in the estimation of statistical parameters," Bull. Calcutta Math. Soc. 37, 81-89 (1945)
- R. Fisher, "On the mathematical foundations of theoretical statistics," Phil. Trans. R. Soc. Lond. 222, 309-368 (1922) [CrossRef]
- R. Fisher, "Theory of statistical estimation," Proc. Cam. Phil. Soc. 22, 700-725 (1925) [CrossRef]
- http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html
- H. H. Barrett, J. L. Denny, R. F. Wagner and K. J. Myers "Objective assessment of image quality. II Fisher information, Fourier crosstalk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995) [CrossRef]
- L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991)
- E. W. Barankin, "Locally best unbiased estimates," Ann. Math. Stat. 20, 477-501 (1949). [CrossRef]
- M. F. Kijewski, S. P. Muller, and S. C. Moore, "The Barankin bound: a model of detection with location uncertainty," Proc. SPIE 1768, 153 (1992). [CrossRef]
- S. P. Muller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, "Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio," Phys. Med. Biol. 50, 3697-3715 (2005). [CrossRef] [PubMed]
- Physics from Fisher Information: A Unification, (Cambridge University Press 1998). [CrossRef]
- J. S. Tyo, "Noise equalisation in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-1200 (2000). [CrossRef]
- A . Beniere, F . Goudail, M . Alouini, and D . Dolfi, "Precision of degree of polarization estimation in the presence of additive Gaussian detector noise," Opt. Commun. 278, 264-269 (2007). [CrossRef]
- R. Mehra, "Optimal input signals for parameter estimation in dynamic systems-Survey and new results," IEEE T. Automat. Contr. 19, 753-768 (1974). [CrossRef]
- E. Walter and L. Pronzatom "Qualitative and quantitative experiment design for phenomenological models - a survey," Automatica 26, 195-213 (1990). [CrossRef]
- S. N. Savenkov, "Optimization and structuring of the instrument matrix for polarimetric measurements," Opt. Eng. 41, 965-972 (2002). [CrossRef]
- V. Bhapkar and C. Srinivasan, "On Fisher information inequalities in the presence of nuisance parameters," Ann. Inst. Statist. Math 46, 593-604 (1994).
- R. M. A. Azzam and F. F. Sudradjat, "Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter," Appl. Opt. 44, 190-196 (2005). [CrossRef] [PubMed]
- A. D. Whalen, Detection of signals in noise, (Academic Press Inc., New York, 1971).
- P. Torok, M. Salt, E. E. Kriezis, P. R. T. Munro, H. P. Herzig, and C. Rockstuhl, "Optical disk and reader therefor," Worldwide Patent No. WO 2006/010882 (2006).
- P. R. T. Munro and P. Torok, "Properties of confocal Mueller-matrix polarimeters," (submitted toOpt. Lett.).
- B. Laude-Boulesteix, A. De Martino, B. Drevillon, and L. Schwartz, "Mueller Polarimetric Imaging System with Liquid Crystals," Appl. Opt. 43, 2824-2832 (2004) [CrossRef] [PubMed]
- J. M. Bueno, "Depolarization effects in the human eye," Vision Research 41, 2687-2696 (2001). [CrossRef] [PubMed]
- M. Floc???h, G . Le Brun, J . Cariou, andJ . Lotrian, "Experimental characterization of immersed targets by polar decomposition of the Mueller matrices," Eur. Phys. J. AP 3, 349-358 (1998). [CrossRef]
- S. M. Nee, "Error analysis for Mueller matrix measurement," J. Opt. Soc. Am. A 20, 1651-1657 (2003). [CrossRef]
- S. N. Savenkov and K. E. Yushtin, "Mueller matrix elements error distribution for polarimetric measurements," Proc. SPIE 5158, 251-259 (2003). [CrossRef]
- J. Morio and F. Goudail, "Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices," Opt. Lett. 29, 2234-2236 (2004). [CrossRef] [PubMed]
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