Abstract

A theoretical and experimental study of the imaging of latent fingerprints by a phase-resolved fluorescence technique along with associated signal-processing analysis is described. The system configuration is optimized by incorporation of a novel approach of homodyne-assisted even-step phase shifting in a signal-processing concept. The excitation laser source and gain of the detection device, which are modulated at megahertz frequency followed by sensitive signal-processing concepts, are employed to separate the fingerprint fluorescence from background fluorescence. Experiments are carried out with fingerprints deposited upon different types of substrate surfaces. Later, a quantitative image-quality assessment is carried out, which confirms the improvement in the quality of the phase-resolved fingerprint image. Imaging of older fingerprints with better contrast is also carried out with the proposed novel technique.

© 2005 Optical Society of America

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References

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  1. P. Margot, C. Lennard, Fingerprint Detection Techniques (Institut de Police Scientifique et de Criminologie, Université de Lausanne, Lausanne, Switzerland, 1994).
  2. R. Saferstein, Criminalistics: An Introduction to Forensic Science (Prentice-Hall, Englewood Cliffs, N.J., 1981).
  3. J. F. Cowger, Friction Ridge Skin: Comparison and Identification of Fingerprints (Elsevier, New York, 1983).
  4. B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).
  5. E. R. Menzel, “Recent advances in photoluminescence detection of fingerprints,” Sci. World J. 1, 498–509 (2001).
    [CrossRef]
  6. R. H. Murdock, E. R. Menzel, “A computer interfaced time-resolved luminescence imaging system,” J. Forensic Sci. 38, 521–529 (1993).
  7. L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
    [CrossRef]
  8. L. B. McGown, D. S. Kreiss, “Spectral fingerprinting using phase-resolved fluorescence spectroscopy,” in Fluorescence Detection II, E. R. Menzel, ed., Proc. SPIE910, 73–79 (1988).
    [CrossRef]
  9. U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
    [CrossRef]
  10. J. A. Steinkamp, H. A. Crissman, “Resolution of heterogenous fluorescence emission signals and decay lifetime measurement on fluorochrome-labelled cells by phase sensitive FCM,” in Advances in Fluorescence Sensing Technology, J. R. Lakowicz, R. B. Thompson, eds., Proc. SPIE1885, 278–289 (1993).
    [CrossRef]
  11. J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic, New York, 1999).
    [CrossRef]
  12. Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).
  13. D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
    [CrossRef]
  14. M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
    [CrossRef]
  15. L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

2004 (1)

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

2003 (1)

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

2001 (2)

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

E. R. Menzel, “Recent advances in photoluminescence detection of fingerprints,” Sci. World J. 1, 498–509 (2001).
[CrossRef]

2000 (1)

M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
[CrossRef]

1993 (1)

R. H. Murdock, E. R. Menzel, “A computer interfaced time-resolved luminescence imaging system,” J. Forensic Sci. 38, 521–529 (1993).

1977 (1)

B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).

Baumgartner, R.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Chao, Z. X.

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Cowger, J. F.

J. F. Cowger, Friction Ridge Skin: Comparison and Identification of Fingerprints (Elsevier, New York, 1983).

Crissman, H. A.

J. A. Steinkamp, H. A. Crissman, “Resolution of heterogenous fluorescence emission signals and decay lifetime measurement on fluorochrome-labelled cells by phase sensitive FCM,” in Advances in Fluorescence Sensing Technology, J. R. Lakowicz, R. B. Thompson, eds., Proc. SPIE1885, 278–289 (1993).
[CrossRef]

Dalrymple, B. E.

B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).

Dinish, U. S.

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Duff, J. M.

B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).

Erimberger, D.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Fajdiga, I.

M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
[CrossRef]

Hofstetter, A.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Knuchel, R.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Kreiss, D. S.

L. B. McGown, D. S. Kreiss, “Spectral fingerprinting using phase-resolved fluorescence spectroscopy,” in Fluorescence Detection II, E. R. Menzel, ed., Proc. SPIE910, 73–79 (1988).
[CrossRef]

Lakowicz, J. R.

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic, New York, 1999).
[CrossRef]

Lennard, C.

P. Margot, C. Lennard, Fingerprint Detection Techniques (Institut de Police Scientifique et de Criminologie, Université de Lausanne, Lausanne, Switzerland, 1994).

Margot, P.

P. Margot, C. Lennard, Fingerprint Detection Techniques (Institut de Police Scientifique et de Criminologie, Université de Lausanne, Lausanne, Switzerland, 1994).

McGown, L. B.

L. B. McGown, D. S. Kreiss, “Spectral fingerprinting using phase-resolved fluorescence spectroscopy,” in Fluorescence Detection II, E. R. Menzel, ed., Proc. SPIE910, 73–79 (1988).
[CrossRef]

Menzel, E. R.

E. R. Menzel, “Recent advances in photoluminescence detection of fingerprints,” Sci. World J. 1, 498–509 (2001).
[CrossRef]

R. H. Murdock, E. R. Menzel, “A computer interfaced time-resolved luminescence imaging system,” J. Forensic Sci. 38, 521–529 (1993).

B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).

Murdock, R. H.

R. H. Murdock, E. R. Menzel, “A computer interfaced time-resolved luminescence imaging system,” J. Forensic Sci. 38, 521–529 (1993).

Murukeshan, V. M.

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Ong, L. S.

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Ong, S. K.

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Saferstein, R.

R. Saferstein, Criminalistics: An Introduction to Forensic Science (Prentice-Hall, Englewood Cliffs, N.J., 1981).

Schmeller, N.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Schneede, P.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Seah, L. K.

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

Smid, L.

M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
[CrossRef]

Steinkamp, J. A.

J. A. Steinkamp, H. A. Crissman, “Resolution of heterogenous fluorescence emission signals and decay lifetime measurement on fluorochrome-labelled cells by phase sensitive FCM,” in Advances in Fluorescence Sensing Technology, J. R. Lakowicz, R. B. Thompson, eds., Proc. SPIE1885, 278–289 (1993).
[CrossRef]

Stepp, H.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Zaak, D.

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Zargi, M.

M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
[CrossRef]

Eur. Arch. Otolaryng. Rhino-L (1)

M. Zargi, I. Fajdiga, L. Smid, “Autofluorescence imaging in the diagnosis of laryngeal cancer,” Eur. Arch. Otolaryng. Rhino-L 257, 17–23 (2000).
[CrossRef]

J. Forensic Sci. (2)

R. H. Murdock, E. R. Menzel, “A computer interfaced time-resolved luminescence imaging system,” J. Forensic Sci. 38, 521–529 (1993).

B. E. Dalrymple, J. M. Duff, E. R. Menzel, “Inherent fingerprint luminescence—detection by laser,” J. Forensic Sci. 22, 106–115 (1977).

Opt. Commun. (1)

U. S. Dinish, L. K. Seah, V. M. Murukeshan, L. S. Ong, “Theoretical analysis of phase-resolved fluorescence emission from fingerprint samples,” Opt. Commun. 223, 55–60 (2003).
[CrossRef]

Opt. Laser Technol. (1)

L. K. Seah, U. S. Dinish, S. K. Ong, Z. X. Chao, V. M. Murukeshan, “Time-resolved imaging of latent fingerprints with nanosecond resolution,” Opt. Laser Technol. 36, 371–376 (2004).
[CrossRef]

Sci. World J. (1)

E. R. Menzel, “Recent advances in photoluminescence detection of fingerprints,” Sci. World J. 1, 498–509 (2001).
[CrossRef]

Urology (1)

D. Erimberger, D. Zaak, H. Stepp, R. Knuchel, R. Baumgartner, P. Schneede, N. Schmeller, A. Hofstetter, “Autofluorescence imaging to optimise 5-ALA- induced fluorescence endoscopy of bladder carcinoma,” Urology 58, 372–375 (2001).
[CrossRef]

Other (8)

L. B. McGown, D. S. Kreiss, “Spectral fingerprinting using phase-resolved fluorescence spectroscopy,” in Fluorescence Detection II, E. R. Menzel, ed., Proc. SPIE910, 73–79 (1988).
[CrossRef]

L. K. Seah, V. M. Murukeshan, L. S. Ong, Z. X. Chao, U. S. Dinish, S. K. Ong, “Method and apparatus for imaging latent fingerprints,” Singapore patent file number 200307736-9 (29December2003).

P. Margot, C. Lennard, Fingerprint Detection Techniques (Institut de Police Scientifique et de Criminologie, Université de Lausanne, Lausanne, Switzerland, 1994).

R. Saferstein, Criminalistics: An Introduction to Forensic Science (Prentice-Hall, Englewood Cliffs, N.J., 1981).

J. F. Cowger, Friction Ridge Skin: Comparison and Identification of Fingerprints (Elsevier, New York, 1983).

J. A. Steinkamp, H. A. Crissman, “Resolution of heterogenous fluorescence emission signals and decay lifetime measurement on fluorochrome-labelled cells by phase sensitive FCM,” in Advances in Fluorescence Sensing Technology, J. R. Lakowicz, R. B. Thompson, eds., Proc. SPIE1885, 278–289 (1993).
[CrossRef]

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Kluwer Academic, New York, 1999).
[CrossRef]

Z. X. Chao, U. S. Dinish, L. K. Seah, V. M. Murukeshan, “Homodyne and heterodyne signal processing assisted phase resolved optical technique for latent fingerprint imaging: a theoretical study,” J. Mod. Opt. (to be published).

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Figures (11)

Fig. 1
Fig. 1

Graphic representation of modulated excitation and homodyne detector modulation: a, modulated excitation; c, homodyne detector modulation.

Fig. 2
Fig. 2

Graphic representations of modulated fluorescence emissions from the sample: b, fluorescence emission from the background; f, fluorescence emission from the fingerprint.

Fig. 3
Fig. 3

Graphic representation of even-step phase shifting to cut off the dc term from the fluorescence emission signal: b, fluorescence emission from the background; f, fluorescence emission from the fingerprint.

Fig. 4
Fig. 4

Graphic representation of fluorescence emission from the sample after removal of the dc term: b, fluorescence emission from the background; f, fluorescence emission from the fingerprint.

Fig. 5
Fig. 5

Fingerprint image with the representative diagonal sampling line.

Fig. 6
Fig. 6

Representation of the pixel arrays that undergo diagonal sampling.

Fig. 7
Fig. 7

Schematic diagram of the experimental setup.

Fig. 8
Fig. 8

Sweat fingerprint deposited upon an aluminum substrate: (a) image under room light, (b) conventional digital image, and (c) phase-resolved image.

Fig. 9
Fig. 9

Oily fingerprint deposited upon an aluminum substrate: (a) image under room light, (b) conventional digital image, and (c) phase-resolved image.

Fig. 10
Fig. 10

Fingerprint deposited upon a fluorescing fruit juice carton: (a) image under room light, (b) conventional digital image, and (c) phase-resolved image.

Fig. 11
Fig. 11

Blitz-Green-treated fingerprint deposited upon fluorescing smooth calendar paper: (a) image under room light, (b) conventional digital image, and (c) phase-resolved image.

Tables (1)

Tables Icon

Table 1 Quantitative Comparison of Fingerprint Image Quality Obtained with PR and Conventional Optical Filtering Methods

Equations (16)

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

I ( t ) = A [ 1 + m e x sin ( ω t ) ] ,
ω τ f = tan ϕ f , m f = [ 1 + ( ω τ f ) 2 ] - 1 / 2 ,
ω τ b = tan ϕ b , m b = [ 1 + ( ω τ b ) 2 ] - 1 / 2 ,
F ( t ) = { A f [ 1 + m ex m f sin ( ω t - ϕ f ) ] from the fingerprint , A b [ 1 + m ex m b sin ( ω τ - ϕ b ) ] from the background ,
G ( t ) = B [ 1 + m g sin ( ω t - ϕ g ) ] .
H ( ϕ , ϕ g ) = { A f B [ 1 + ½ m g m ex m f cos ( ϕ f - ϕ g ) ] A b B [ 1 + ½ m g m ex m b cos ( ϕ b - ϕ g ) ] .
H ( ϕ , ϕ g ) = { A f B [ 1 + ½ m g m ex m f cos ( ϕ f ) ] , A b B [ 1 + ½ m g m ex m b cos ( ϕ b ) ] , ϕ g = 0 ;
H ( ϕ , ϕ g ) = { A f B [ 1 + ½ m g m ex m f cos ( ϕ f - π / 2 ) ] , A b B [ 1 + ½ m g m ex m b cos ( ϕ b - π / 2 ) ] , ϕ g = π / 2 ;
H ( ϕ , ϕ g ) = { A f B [ 1 + ½ m g m ex m f cos ( ϕ f - π ) ] , A b B [ 1 + ½ m g m ex m b cos ( ϕ b - π ) ] , ϕ g = π ;
H ( ϕ , ϕ g ) = { A f B [ 1 + ½ m g m ex m f cos ( ϕ f - 3 π / 2 ) ] , A b B [ 1 + ½ m g m ex m b cos ( ϕ b - 3 π / 2 ) ] , ϕ g = 3 π / 2.
H ( ϕ , ϕ g ) avg = { A f B from the fingerprint , A b B from the background .
H 0 ( ϕ , ϕ g ) = { ½ B m g m ex A f m f cos ( ϕ f - ϕ g ) ½ B m g m ex A b m b cos ( ϕ b - ϕ g ) .
H 0 ( ϕ , ϕ g ) = ½ B m g m ex A f m f sin ( ϕ b - ϕ f ) .
H 0 ( ϕ , ϕ g ) = ½ B m g m ex A f m f sin ( ϕ f - ϕ b ) .
[ x 11 , x 22 , x 33 , x 44 , x 55 , x 66 , x 77 , x 88 ] .
Score = [ x 22 - x 11 + x 33 - x 22 + x 44 - x 33 + x 55 - x 44 + x 66 - x 55 + x 77 - x 66 + x 88 - x 77 ] max [ x 11 , x 22 , x 33 , x 44 , x 55 , x 66 , x 77 , x 88 ] .

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