Abstract

Linear and nonlinear components analysis of data from a monostatic laser polarimeter is developed and applied to the task of remote, nonimaging discrimination among different textures on paint and polymer coupons independent of their spatial orientations. Both principal-components analysis and nonlinear components analysis are applied to multidimensional laser data in measured Mueller matrices, with discrimination via cluster segmentation in derived linear and nonlinear constant channels. Textures on the discriminated coupons are generated by heating and illustrated in optical micrographs.

© 2007 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  2. R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.
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    [CrossRef]
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    [CrossRef]
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  14. D. C. Dayton, B. G. Hoover, and J. D. Gonglewski, "Full-order Mueller matrix polarimeter using liquid-crystal phase retarders and active illumination," in Optics in Atmospheric Propagation and Adaptive Systems V, Proc. SPIE 4884, 40-48 (2002).
  15. D. G. Jones, D. H. Goldstein, and J. C. Spaulding, "Reflective and polarimetric characteristics of urban materials," in Polarization: Measurement, Analysis, and Remote Sensing VII, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 6240, 62400A (2006).
    [CrossRef]
  16. K. M. Yemelyanov, S.-S. Lin, E. N. Pugh, Jr., and N. Engheta, "Adaptive algorithms for two-channel polarization sensing under various polarization statistics with nonuniform distributions," Appl. Opt. 45, 5504-5520 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. B. G. Hoover, R. A. Peredo, L. F. DeSandre, and L. J. Ulibarri, "Active polarimetric assessment of surface weathering," in Laser Radar Techniques for Atmospheric Sensing, U. N. Singh, ed., Proc. SPIE 5575, 38-43 (2004).
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    [CrossRef]
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    [CrossRef]
  21. G. D. Lewis and D. L. Jordan, "Remote sensing of polarimetric speckle," J. Phys. D 34, 1399-1407 (2001).
    [CrossRef]
  22. H. L. van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, 1968).
  23. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).
  24. A. Hyvärinen and P. Pajunen, "Nonlinear independent component analysis: Existence and uniqueness results," Neural Networks 12, 429-439 (1999).
    [CrossRef]
  25. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
    [CrossRef]
  26. N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods (Cambridge U. Press, 2000).
  27. S. T. Roweis and L. K. Saul, "Nonlinear dimensionality reduction by locally linear embedding," Science 290, 2323-2326 (2000).
    [CrossRef] [PubMed]
  28. J. S. Tyo and B. G. Hoover, "Laser polarimeter as an invariant monitor," in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 6682, 66820S (2007).
    [CrossRef]

2007 (2)

F. A. Sadjadi, "Invariants of polarization transformations," Appl. Opt. 46, 2914-2921 (2007).
[CrossRef] [PubMed]

J. S. Tyo and B. G. Hoover, "Laser polarimeter as an invariant monitor," in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 6682, 66820S (2007).
[CrossRef]

2006 (2)

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, "Reflective and polarimetric characteristics of urban materials," in Polarization: Measurement, Analysis, and Remote Sensing VII, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 6240, 62400A (2006).
[CrossRef]

K. M. Yemelyanov, S.-S. Lin, E. N. Pugh, Jr., and N. Engheta, "Adaptive algorithms for two-channel polarization sensing under various polarization statistics with nonuniform distributions," Appl. Opt. 45, 5504-5520 (2006).
[CrossRef] [PubMed]

2005 (2)

2001 (1)

G. D. Lewis and D. L. Jordan, "Remote sensing of polarimetric speckle," J. Phys. D 34, 1399-1407 (2001).
[CrossRef]

2000 (2)

S. T. Roweis and L. K. Saul, "Nonlinear dimensionality reduction by locally linear embedding," Science 290, 2323-2326 (2000).
[CrossRef] [PubMed]

S. Breugnot and P. Clemenceau, "Modeling and performances of a polarization active imager at λ = 806 nm," Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

1999 (2)

A. Hyvärinen and P. Pajunen, "Nonlinear independent component analysis: Existence and uniqueness results," Neural Networks 12, 429-439 (1999).
[CrossRef]

G. D. Lewis, D. L. Jordan, and P. J. Roberts, "Backscattering target detection in a turbid medium by polarization discrimination," Appl. Opt. 38, 3937-3944 (1999).
[CrossRef]

1998 (2)

1997 (1)

M. P. Silverman and W. Strange, "Object delineation within turbid media by backscattering of phase modulated light," Opt. Commun. 144, 7-11 (1997).
[CrossRef]

1991 (1)

C. Jutten and J. Hérault, "Blind separation of sources, part I: an adaptive algorithm based on a neuromimetic architecture," Signal Process. 24, 1-10 (1991).
[CrossRef]

1985 (1)

R. Barakat, "The statistical properties of partially polarized light," Opt. Acta 32, 295-312 (1985).
[CrossRef]

1942 (1)

F. Perrin, "Polarization of light scattered by isotropic opalescent media," J. Chem. Phys. 10, 415-427 (1942).
[CrossRef]

Appl. Opt. (5)

J. Chem. Phys. (1)

F. Perrin, "Polarization of light scattered by isotropic opalescent media," J. Chem. Phys. 10, 415-427 (1942).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. D (1)

G. D. Lewis and D. L. Jordan, "Remote sensing of polarimetric speckle," J. Phys. D 34, 1399-1407 (2001).
[CrossRef]

Neural Networks (1)

A. Hyvärinen and P. Pajunen, "Nonlinear independent component analysis: Existence and uniqueness results," Neural Networks 12, 429-439 (1999).
[CrossRef]

Opt. Acta (1)

R. Barakat, "The statistical properties of partially polarized light," Opt. Acta 32, 295-312 (1985).
[CrossRef]

Opt. Commun. (1)

M. P. Silverman and W. Strange, "Object delineation within turbid media by backscattering of phase modulated light," Opt. Commun. 144, 7-11 (1997).
[CrossRef]

Opt. Eng. (1)

S. Breugnot and P. Clemenceau, "Modeling and performances of a polarization active imager at λ = 806 nm," Opt. Eng. 39, 2681-2688 (2000).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

D. G. Jones, D. H. Goldstein, and J. C. Spaulding, "Reflective and polarimetric characteristics of urban materials," in Polarization: Measurement, Analysis, and Remote Sensing VII, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 6240, 62400A (2006).
[CrossRef]

J. S. Tyo and B. G. Hoover, "Laser polarimeter as an invariant monitor," in Polarization Science and Remote Sensing III, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 6682, 66820S (2007).
[CrossRef]

Science (1)

S. T. Roweis and L. K. Saul, "Nonlinear dimensionality reduction by locally linear embedding," Science 290, 2323-2326 (2000).
[CrossRef] [PubMed]

Signal Process. (1)

C. Jutten and J. Hérault, "Blind separation of sources, part I: an adaptive algorithm based on a neuromimetic architecture," Signal Process. 24, 1-10 (1991).
[CrossRef]

Other (11)

B. G. Hoover, D. C. Dayton, J. E. Havey, J. D. Gonglewski, V. L. Gamiz, and L. J. Ulibarri, "Active detection of off-diagonal Mueller elements of rough targets," in Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, 226-238 (2003).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

R. A. Chipman, "Polarimetry," in Handbook of Optics (McGraw-Hill, 1994), Chap. 22.

B. G. Hoover, R. A. Peredo, L. F. DeSandre, and L. J. Ulibarri, "Active polarimetric assessment of surface weathering," in Laser Radar Techniques for Atmospheric Sensing, U. N. Singh, ed., Proc. SPIE 5575, 38-43 (2004).

B. Jähne, Digital Image Processing (Springer, 1997).

J. A. Richards and X. Jia, Remote Sensing Digital Image Analysis (Springer, 1999).

D. C. Dayton, B. G. Hoover, and J. D. Gonglewski, "Full-order Mueller matrix polarimeter using liquid-crystal phase retarders and active illumination," in Optics in Atmospheric Propagation and Adaptive Systems V, Proc. SPIE 4884, 40-48 (2002).

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods (Cambridge U. Press, 2000).

H. L. van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, 1968).

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1983).

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Figures (9)

Fig. 1
Fig. 1

Schematic training procedure for derivation of constant channels from N-dimensional data. Transformation inputs are data projections (solid arrows) and transformation outputs are channels (split arrows).

Fig. 2
Fig. 2

(Color online) Derivation of an angle-invariant channel from monostatic laser-polarimeter data. The horizontal axes are the probe angle in degrees. (a) Normalized Mueller matrices due to polymer coupons sample 1 ( ) and sample 2 (×). (b) Data projections in the constant channel c 1 = m 33 ( m 22 m 11 ) / 2 .

Fig. 3
Fig. 3

Schematic of the laser polarimeter. PSG, polarization-state generator; PSA, polarization-state analyzer.

Fig. 4
Fig. 4

(Color online) Segmentation of data clusters plotted on the axes c 1 = m 33 ( m 22 m 11 ) / 2   and   c 2 = m 11 m 22 m 33 . (a) Discrimination between polymer coupons on the c 1 line. (b) Discrimination among polymer and paint coupons in the c 1 c 2 plane. The boxes enclose data over all probe angles in the range [ 50 ° 50 ° ] . The marker □ represents a hypothetical data point a distance D from the center of the red cluster.

Fig. 5
Fig. 5

(Color online) Cluster diagram of data due to a family of textures on a white-gloss paint projected onto three principal-component channels. The dashed line indicates where data around the specular peak of the control sample is expected to fall.

Fig. 6
Fig. 6

(Color online) Nonlinear fitting results. (a) Three-dimensional polynomial estimates overlaid on the data projections of Fig. 5. (b) Control-sample data and estimate projections onto each of the principal-component channels as functions of the probe angle.

Fig. 7
Fig. 7

(Color online) Cluster diagram of data due to the family of paint textures illustrated in Fig. 5 projected onto three nonlinear constant channels.

Fig. 8
Fig. 8

Correlation plot of the probe angle estimated from simulated test data ( θ ¯ ) due to texturized paints versus the actual probe angle ( θ l ) . At each probe angle are estimates corresponding to each of the four samples shown in Fig. 7.

Fig. 9
Fig. 9

Correlation plot of the continuous sample variable estimated from simulated test data ( s ¯ ) due to texturized paints versus the actual sample number ( s ) . At each sample number are estimates corresponding to each of the 13 probe angles.

Equations (15)

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

S out ( θ , ϕ ) = M ( θ , ϕ ) S in = [ m 00 ( θ , ϕ ) m 01 ( θ , ϕ ) m 02 ( θ , ϕ ) m 03 ( θ , ϕ ) m 10 ( θ , ϕ ) m 11 ( θ , ϕ ) m 12 ( θ , ϕ ) m 13 ( θ , ϕ ) m 20 ( θ , ϕ ) m 21 ( θ , ϕ ) m 22 ( θ , ϕ ) m 23 ( θ , ϕ ) m 30 ( θ , ϕ ) m 31 ( θ , ϕ ) m 32 ( θ , ϕ ) m 33 ( θ , ϕ ) ] × S in ,
M cal = [ 1 0.002 0.001 0.001 0.001 1.004 0.006 0.002 0.004 0.007 1.001 0.001 0.001 0.007 0.001 1.000 ] ± [ 0 0.002 0.001 0.001 0.002 0.003 0.002 0.001 0.001 0.002 0.003 0.001 0.001 0.001 0.001 0.001 ] ,
m s ( θ l ) = [ m 1 s ( θ l ) m 2 s ( θ l ) m 15 s ( θ l ) m 16 s ( θ l ) ] T [ m 00 s ( θ l ) m 01 s ( θ l ) m 32 s ( θ l ) m 33 s ( θ l ) ] T .
M = [ m 1 ( θ 1 ) m 1 ( θ L ) m 2 ( θ 1 ) m 2 ( θ L ) m S ( θ 1 ) m S ( θ L ) ] T ,
C M = M T M S L M ¯ T M ¯ ( S L ) 2 ,
p s ( θ l ) = U p T m s ( θ l ) = U p T [ m 00 s ( θ l ) m 01 s ( θ l ) m 32 s ( θ l ) m 33 s ( θ l ) ] T ,
p j = [ m 00 m 01 m 32 m 33 ] p ^ j ,
q ¯ s ( x ) = [ q ¯ 1 s ( x ) q ¯ 2 s ( x ) q ¯ n s ( x ) ] = [ f 1 s ( x ) f 2 s ( x ) f n s ( x ) ] .
D s min x j = 1 n | q j q ¯ j s ( x ) | .
D 2 = ( c 1 b 1 ) 2 + ( c 2 b 2 ) 2 ,
f j s ( x ) = a 1 j s x 4 + a 2 j s x 2 + a 3 j s x 2 + a 4 j s .
C M = U p Λ U p T ,
U p = [ p ^ 1 p ^ 16 ] .
= M U p
C P = U p T M T M U p S L ( M U p ) ¯ T ( M U p ) ¯ ( S L ) 2 = Λ ,

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