Abstract

Polarization is an important characteristic of electromagnetic waves, which can not be detected by either the human visual system or traditional image sensors. Motivated by various animal species with polarization vision as well as by the prospect of improving the image quality of the imaging systems, we are exploring the potential of polarization for microscope imaging. The most powerful techniques for molecule monitoring requires complex preprocessing for labeling the sample with different dyes. In this paper, we propose a cell detection method using polarization imaging without any need for staining target cell samples with any chemical dye. The motivation for this work is to develop an optical imaging technique that is simple and that can be used on live cells. The polarization sensitivity of cell samples is studied in this paper. A definition for the quantity called “polarization deviation” is proposed in order to identify clearer the difference between target cells and the background. Based on the polarization deviation detection method, a three-parameter polarization imaging method is employed to further simplify the image capture procedure for the proposed label-free cell detection. A color imaging methodology based on the well-known color space is utilized in order to represent the captured polarization information using computer graphics.

© 2012 OSA

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  28. Y. Y. Schechner, J. Shamir, and N. Kiryati, “Vision through semireflecting media: polarization analysis,” Opt. Lett.24, 1088–1090 (1999).
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  35. R. C. Gonzalez and P. Wintz, “Image Enhancement,” in Digital Image Processing (Addison Weslwy, 1987), pp. 146–152.
  36. G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res.17, 1019–1028 (1977).
    [CrossRef] [PubMed]
  37. M. P. Rowe, E. N. Pugh, and N. Engheta, “Polarizationdifference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett.20, 608–610 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  39. J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representation for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A15, 367–374 (1998).
    [CrossRef]
  40. S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express14, 7099–7108 (2006).
    [CrossRef] [PubMed]
  41. S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization-based and specular-reflection-based noncontact latent fingerprint imaging and lifting,” J. Opt. Soc. Am. A23, 2137–2153 (2006).
    [CrossRef]
  42. K. M. Yemelyanov, S. S. Lin, W. Q. Luis, E. N. Pugh, and N. Engheta, “Bio-inspired display of polarization information using selected visual cues,” Proc. of SPIE5158, 71–84 (2003).
    [CrossRef]

2009 (1)

T. Treibitz and Y. Y. Schechner, “Active polarization descattering,” IEEE Trans. Pattern Anal. Mach. Intell.31, 385–399 (2009).
[CrossRef] [PubMed]

2006 (5)

2005 (2)

Y. Y. Schechner and N. Karpel, “Recovery of Underwater Visibility and Structure by Polarization Analysis,” IEEE J Oceanic Eng.30, 570–587 (2005).
[CrossRef]

R. Nothdurft and G. Yao, “Expression of target optical properties in subsurface polarization-gated imaging,” Opt. Express13, 4185–4195 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (4)

2001 (4)

F. Goudail and P. Rfrgier, “Statistical algorithms for target detection in coherent active polarimetric images,” J. Opt. Soc. Am. A18, 3049–3060 (2001).
[CrossRef]

F. Goudail and P. Rfrgier, “Statistical techniques for target detection in polarization diversity images,” Opt. Lett.26, 644–646 (2001).
[CrossRef]

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull.200, 177–183 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1998 (2)

1996 (2)

1995 (1)

1991 (1)

1989 (1)

R. Wehner, “Neurobiology of polarization vision,” TRENDS NEUROSCI12, 353–359, (1989).
[CrossRef] [PubMed]

1988 (1)

T. Labhart, “Polarization opponent interneurons in the insect visual system,” Nature331, 435–437 (1988).
[CrossRef]

1977 (1)

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res.17, 1019–1028 (1977).
[CrossRef] [PubMed]

1976 (1)

R. Wehner, “Polarized-light navigation by insects,” SCI AM235, 106–114, (1976).
[CrossRef] [PubMed]

1942 (2)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica9, 686–698 (1942).
[CrossRef]

F. Zernike, “Phasecontrast, a new method for the microscopicobservation of transparentobjects part II,” Physica9, 974–980 (1942).
[CrossRef]

Alfano, R. R.

Bernard, G. D.

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res.17, 1019–1028 (1977).
[CrossRef] [PubMed]

Bigu, L.

Bron, F. M.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Chang, P. C.

Chang, P. C. Y.

Chenault, D. B.

Clark, G.

G. Clark, Staining Procedures (Williams and Wilkins, 1981).

G. Clark, F. H. Kasten, and H. J. Conn, History of Staining (Williams and Wilkins, 1981).

Conn, H. J.

H. J. Conn, Biological Stains: A Handbook on the Nature and Uses of the Dyes Employed in the Biological Laboratory (Biotech Publications, 1953).

G. Clark, F. H. Kasten, and H. J. Conn, History of Staining (Williams and Wilkins, 1981).

Cronin, T. W.

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull.200, 177–183 (2001).
[CrossRef] [PubMed]

Demos, S. G.

Deuz, J. L.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Devaux, C.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

DeVlaminck, V.

Egan, W. G.

Engheta, N.

S. S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization-based and specular-reflection-based noncontact latent fingerprint imaging and lifting,” J. Opt. Soc. Am. A23, 2137–2153 (2006).
[CrossRef]

S. S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express14, 7099–7108 (2006).
[CrossRef] [PubMed]

S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express14, 7099–7108 (2006).
[CrossRef] [PubMed]

S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization-based and specular-reflection-based noncontact latent fingerprint imaging and lifting,” J. Opt. Soc. Am. A23, 2137–2153 (2006).
[CrossRef]

K. M. Yemelyanov, S. S. Lin, W. Q. Luis, E. N. Pugh, and N. Engheta, “Bio-inspired display of polarization information using selected visual cues,” Proc. of SPIE5158, 71–84 (2003).
[CrossRef]

J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representation for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A15, 367–374 (1998).
[CrossRef]

J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt.35, 1855–1870 (1996).
[CrossRef] [PubMed]

J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scatter media by polarization-difference imaging”, Appl. Opt.35, 1855–1870 (1996).
[CrossRef] [PubMed]

M. P. Rowe, E. N. Pugh, and N. Engheta, “Polarizationdifference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett.20, 608–610 (1995).
[CrossRef] [PubMed]

Fasolka, M.

T. Germer and M. Fasolka, “Characterizing surface roughness of thin films by polarized light,” Proc. SPIE518, 264–275 (2003).
[CrossRef]

Flitton, J. C.

Galland, F.

Germer, T.

T. Germer and M. Fasolka, “Characterizing surface roughness of thin films by polarized light,” Proc. SPIE518, 264–275 (2003).
[CrossRef]

Goldstein, D.

D. Goldstein, “Polarized Light” (Marcel Dekker: New York, 2003).
[CrossRef]

Goldstein, D. L.

Goloub, P.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and P. Wintz, “Image Enhancement,” in Digital Image Processing (Addison Weslwy, 1987), pp. 146–152.

Goudail, F.

Hara, K.

D. Miyazaki, R. Tan, K. Hara, and K. Ikeuchi, “Polarization-based Inverse Rendering from a Single View,” in Proceedings of Ninth IEEE International Conference on Computer Vision (IEEE2003), pp. 982–987.
[CrossRef]

Herman, M.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Hopcraft, K. I.

Ikeuchi, K.

D. Miyazaki, R. Tan, K. Hara, and K. Ikeuchi, “Polarization-based Inverse Rendering from a Single View,” in Proceedings of Ninth IEEE International Conference on Computer Vision (IEEE2003), pp. 982–987.
[CrossRef]

Israel, T.

Y. Y. Schechner, N. Karpel, T. Israel, and I. Technology, “Clear Underwater Vision,” Pattern Recogn1, 536–543 (2004).

Jakeman, E.

Johnson, W. R.

Jordan, D. L.

Karpel, N.

Y. Y. Schechner and N. Karpel, “Recovery of Underwater Visibility and Structure by Polarization Analysis,” IEEE J Oceanic Eng.30, 570–587 (2005).
[CrossRef]

Y. Y. Schechner, N. Karpel, T. Israel, and I. Technology, “Clear Underwater Vision,” Pattern Recogn1, 536–543 (2004).

Kasten, F. H.

G. Clark, F. H. Kasten, and H. J. Conn, History of Staining (Williams and Wilkins, 1981).

Kiang, Y. W.

Kiryati, N.

Labhart, T.

T. Labhart, “Polarization opponent interneurons in the insect visual system,” Nature331, 435–437 (1988).
[CrossRef]

Lafrance, B.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Lin, S. S.

Lin, S.-S.

Luis, W. Q.

K. M. Yemelyanov, S. S. Lin, W. Q. Luis, E. N. Pugh, and N. Engheta, “Bio-inspired display of polarization information using selected visual cues,” Proc. of SPIE5158, 71–84 (2003).
[CrossRef]

Lyot, B.

B. Lyot, Research on the Polarization of Light from Planets and Some Substances (Ann. Observatoire de Paris, Section de Meudon1929).

Maignan, F.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Marchand, A.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Marshall, J.

T. W. Cronin and J. Marshall, “Parallel processing and image analysis in the eyes of mantis shrimps,” Biol. Bull.200, 177–183 (2001).
[CrossRef] [PubMed]

Miyazaki, D.

D. Miyazaki, R. Tan, K. Hara, and K. Ikeuchi, “Polarization-based Inverse Rendering from a Single View,” in Proceedings of Ninth IEEE International Conference on Computer Vision (IEEE2003), pp. 982–987.
[CrossRef]

Nadal, F.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Namer, E.

S. Shwartz, E. Namer, and Y. Schechner, “Blind Haze Seperation,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE2006), pp. 1984–1991.

Nothdurft, R.

Perry, G.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Pugh, E. N.

S. S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization-based and specular-reflection-based noncontact latent fingerprint imaging and lifting,” J. Opt. Soc. Am. A23, 2137–2153 (2006).
[CrossRef]

S. S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express14, 7099–7108 (2006).
[CrossRef] [PubMed]

S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Separation and contrast enhancement of overlapping cast shadow components using polarization,” Opt. Express14, 7099–7108 (2006).
[CrossRef] [PubMed]

S.-S. Lin, K. M. Yemelyanov, E. N. Pugh, and N. Engheta, “Polarization-based and specular-reflection-based noncontact latent fingerprint imaging and lifting,” J. Opt. Soc. Am. A23, 2137–2153 (2006).
[CrossRef]

K. M. Yemelyanov, S. S. Lin, W. Q. Luis, E. N. Pugh, and N. Engheta, “Bio-inspired display of polarization information using selected visual cues,” Proc. of SPIE5158, 71–84 (2003).
[CrossRef]

J. S. Tyo, E. N. Pugh, and N. Engheta, “Colorimetric representation for use with polarization-difference imaging of objects in scattering media,” J. Opt. Soc. Am. A15, 367–374 (1998).
[CrossRef]

J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scattering media by polarization-difference imaging,” Appl. Opt.35, 1855–1870 (1996).
[CrossRef] [PubMed]

J. S. Tyo, M. P. Rowe, E. N. Pugh, and N. Engheta, “Target detection in optically scatter media by polarization-difference imaging”, Appl. Opt.35, 1855–1870 (1996).
[CrossRef] [PubMed]

M. P. Rowe, E. N. Pugh, and N. Engheta, “Polarizationdifference imaging: a biologically inspired technique for observation through scattering media,” Opt. Lett.20, 608–610 (1995).
[CrossRef] [PubMed]

Rfrgier, P.

Rowe, M. P.

Schechner, Y.

S. Shwartz, E. Namer, and Y. Schechner, “Blind Haze Seperation,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE2006), pp. 1984–1991.

Schechner, Y. Y.

T. Treibitz and Y. Y. Schechner, “Active polarization descattering,” IEEE Trans. Pattern Anal. Mach. Intell.31, 385–399 (2009).
[CrossRef] [PubMed]

Y. Y. Schechner and N. Karpel, “Recovery of Underwater Visibility and Structure by Polarization Analysis,” IEEE J Oceanic Eng.30, 570–587 (2005).
[CrossRef]

Y. Y. Schechner, N. Karpel, T. Israel, and I. Technology, “Clear Underwater Vision,” Pattern Recogn1, 536–543 (2004).

Y. Y. Schechner, J. Shamir, and N. Kiryati, “Vision through semireflecting media: polarization analysis,” Opt. Lett.24, 1088–1090 (1999).
[CrossRef]

Shamir, J.

Shaw, J. A.

Shwartz, S.

S. Shwartz, E. Namer, and Y. Schechner, “Blind Haze Seperation,” in Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE2006), pp. 1984–1991.

Stephens, D.

D. Stephens, Cell Imaging (Scion, 2006).

Sun, C. W.

Takakura, Y.

Tan, R.

D. Miyazaki, R. Tan, K. Hara, and K. Ikeuchi, “Polarization-based Inverse Rendering from a Single View,” in Proceedings of Ninth IEEE International Conference on Computer Vision (IEEE2003), pp. 982–987.
[CrossRef]

Tanr, D.

J. L. Deuz, F. M. Bron, C. Devaux, P. Goloub, M. Herman, B. Lafrance, F. Maignan, A. Marchand, F. Nadal, G. Perry, and D. Tanr, “Remote sensing of aerosols over land surfaces from POLDER-ADEOS-1 polarized measurements”, J. Geophys. Res.106, 4913–4926 (2001).
[CrossRef]

Technology, I.

Y. Y. Schechner, N. Karpel, T. Israel, and I. Technology, “Clear Underwater Vision,” Pattern Recogn1, 536–543 (2004).

Terrier, P.

Treibitz, T.

T. Treibitz and Y. Y. Schechner, “Active polarization descattering,” IEEE Trans. Pattern Anal. Mach. Intell.31, 385–399 (2009).
[CrossRef] [PubMed]

Tyo, J. S.

Walker, J. G.

Wang, H. H.

Wang, W. B.

Wang, Y. M.

Waterman, T. H.

T. H. Waterman, “Polarization sensitivity” in The Handbook of Sensory Physiology, vol. VII/6B Vision in Invertebrates, edited by H. Autrum, (Springer-Verlag, New York, 1981).
[CrossRef]

Wehner, R.

R. Wehner, “Neurobiology of polarization vision,” TRENDS NEUROSCI12, 353–359, (1989).
[CrossRef] [PubMed]

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res.17, 1019–1028 (1977).
[CrossRef] [PubMed]

R. Wehner, “Polarized-light navigation by insects,” SCI AM235, 106–114, (1976).
[CrossRef] [PubMed]

Whitehead, V. S.

Wintz, P.

R. C. Gonzalez and P. Wintz, “Image Enhancement,” in Digital Image Processing (Addison Weslwy, 1987), pp. 146–152.

Yang, C. C.

Yao, G.

Yemelyanov, K. M.

Zernike, F.

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Figures (11)

Fig. 1
Fig. 1

(a) Raw image captured under unpolarized illumination. (b) A zoom-in view of local zone, two subregions are labeled as Region I and Region II, respectively.

Fig. 2
Fig. 2

(a) The digitized value of randomly selected pixels captured under different polarization angles of the linearly polarized illumination light, from Region I (as illustrated in red) and Region II (as illustrated in blue), respectively. (b) The distribution of the polarization Deviation σp for Region I (as illustrated in red) and Region II (as illustrated in blue), respectively.

Fig. 3
Fig. 3

(a) Raw captured intensity image under unpolarized illumination; (b) enhanced intensity image after histogram equalization, and (c) the image formed by distribution of the polarization deviation. The sensitivity index d′ is listed under each image.

Fig. 4
Fig. 4

Categorization results of “signal present” and “signal absent” employing a σth equals to (a) 30; (b) 40; and (c) 50.

Fig. 5
Fig. 5

A comparison of sensitivity index of different standard Deviation threshold values. (a) various “signal-present” regions are selected as testing samples; (b) The sensitivity index as shown on the y-axis changes as the threshold σth is increased.

Fig. 6
Fig. 6

Value of the captured intensity of pixels under different angles of linearly polarized light. The pixels, represented by different colors, are randomly selected from the “signal present” region.

Fig. 7
Fig. 7

Microphotography of the human pulmonary vascular smooth muscle cells illuminated under three different linear polarization angles of incident beam (a) χ1 = 0°, (b) χ2 = 60°, (c) χ3 = 120°.

Fig. 8
Fig. 8

(a) The degree of linear polarization p which varies from 0 to 1; (b) the orientation angle Ψ calculated varying from π 2 to π 2.

Fig. 9
Fig. 9

The post-processing results of the polarization degree calculated from three-parameter polarization imaging, using a threshold filter employing different cut-off threshold value σth equals to (a) 30; (b) 40; and (c) 50.

Fig. 10
Fig. 10

Three-parameter polarization color plotting while (a) no filter is employed, and a high-pass filter with a cut-off threshold equals to (b) 30, (c) 40 and (d) 50 is used, respectively.

Fig. 11
Fig. 11

Live cell detection under unpolarized illumination (first row), and using the three-parameter polarization methodology without (second row) or with (third row) a high pass filter. The results from the three-parameter polarization detection method are plotted in HSV pseudo color domain. The sample cells include (a) human pulmonary vascular smooth muscle cells, and (b) human oral epithelial cavity cells.

Tables (2)

Tables Icon

Table 1 A Comparison of Detection Sensitivity of the Unpolarizedly Illuminated Image (I), the Standard Deviation Image (σp), and the three-parameter polarization image (Lp)

Tables Icon

Table 2 A Comparison of Detection Sensitivity while Using Different Combination of Angles

Equations (7)

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σ p ( x , y ) = 1 n i = 0 n 1 [ I ( x , y | χ = χ i ) μ p ( x , y ) ] 2
μ p ( x , y ) = 1 n i = 0 n 1 [ I ( x , y | χ = χ i ) ]
d = | μ s μ n | 1 2 ( σ s 2 + σ n 2 )
= { 0 σ p ( x , y ) < σ t h 1 σ p ( x , y ) > σ t h
I χ = L ¯ [ 1 + p cos ( 2 ψ 2 χ ) ] = L ¯ + L p cos ( 2 ψ 2 χ )
[ I 0 I 60 I 120 ] = [ 1 1 0 1 cos ( 120 ° ) sin ( 120 ° ) 1 cos ( 240 ° ) sin ( 240 ° ) ] [ L ¯ L p cos 2 ψ L p sin 2 ψ ]
L ¯ = I 60 ° + I 120 ° I 0 ° L p = ( L ¯ I 0 ° ) 2 + 1 3 ( I 60 ° I 120 ° ) 2 ψ = tan 1 [ I 60 ° I 120 ° 3 ( L ¯ I 0 ° ) ]

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