Abstract

Multivariate optical computing (MOC) is a method of performing chemical analysis using a multilayer thin-film structure known as a multivariate optical element (MOE). Recently we have been advancing MOC for imaging problems by using an imaging MOE (IMOE) in a normal-incidence geometry and employing normalization by the 1-norm. There are several important differences between the previously described 45° and the normal-incidence imaging, one of which is the measurement precision due to photon counting. We compare this precision to 45° MOC. We also discuss how MOE models with similar values of standard errors of calibration and prediction and similar gain values may vary in precision because of the sign or offset of the regression vector encoded in the IMOE spectrum. Experimental verification of a key result is provided by near-infrared imaging of slides coated with a dye-doped polymer film.

© 2007 Optical Society of America

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  1. M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
    [CrossRef] [PubMed]
  2. O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
    [CrossRef]
  3. K. R. Beebe, R. J. Pell, and M. B. Seasholtz, Chemometrics: A Practical Guide (Wiley, 1998).
  4. H. Martens and T. Naes, Multivariate Calibration (Wiley, 1989).
  5. O. O. Soyemi, F. G. Haibach, P. J. Gemperline, and M. L. Myrick, "A nonlinear optimization algorithm for multivariate optical element design," Appl. Spectrosc. 56, 477-487 (2002).
    [CrossRef]
  6. F. G. Haibach and M. L. Myrick, "Precision in multivariate optical computing," Appl. Opt. 43, 2130-2140 (2004).
    [CrossRef] [PubMed]
  7. O. O. Soyemi, P. J. Gemperline, and M. L. Myrick, "Design of angle-tolerant multivariate optical elements for chemical imaging," Appl. Opt. 41, 1936-1941 (2002).
    [CrossRef] [PubMed]
  8. V. Saptari and K. Youcef-Tourni, "Design of a mechanical-tunable filter spectrometer for noninvasive glucose measurement," Appl. Opt. 43, 2680-2688 (2004).
    [CrossRef] [PubMed]
  9. A. F. Michels, T. Menegotto, and F. Horowitz, "Interferometric monitoring of dip coating," Appl. Opt. 43, 820-823 (2004).
    [CrossRef] [PubMed]

2004 (3)

2002 (2)

2001 (1)

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

1998 (1)

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Aust, J. F.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Beebe, K. R.

K. R. Beebe, R. J. Pell, and M. B. Seasholtz, Chemometrics: A Practical Guide (Wiley, 1998).

Dobrowolski, J. A.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Eastwood, D.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Gemperline, P.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Gemperline, P. J.

Haibach, F. G.

Horowitz, F.

Karunamuni, J.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Li, H.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Martens, H.

H. Martens and T. Naes, Multivariate Calibration (Wiley, 1989).

Menegotto, T.

Michels, A. F.

Myrick, M. L.

F. G. Haibach and M. L. Myrick, "Precision in multivariate optical computing," Appl. Opt. 43, 2130-2140 (2004).
[CrossRef] [PubMed]

O. O. Soyemi, F. G. Haibach, P. J. Gemperline, and M. L. Myrick, "A nonlinear optimization algorithm for multivariate optical element design," Appl. Spectrosc. 56, 477-487 (2002).
[CrossRef]

O. O. Soyemi, P. J. Gemperline, and M. L. Myrick, "Design of angle-tolerant multivariate optical elements for chemical imaging," Appl. Opt. 41, 1936-1941 (2002).
[CrossRef] [PubMed]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Naes, T.

H. Martens and T. Naes, Multivariate Calibration (Wiley, 1989).

Nelson, M. P.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Pell, R. J.

K. R. Beebe, R. J. Pell, and M. B. Seasholtz, Chemometrics: A Practical Guide (Wiley, 1998).

Saptari, V.

Seasholtz, M. B.

K. R. Beebe, R. J. Pell, and M. B. Seasholtz, Chemometrics: A Practical Guide (Wiley, 1998).

Soyemi, O.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Soyemi, O. O.

Synowicki, R. A.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Verly, P. G.

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

Youcef-Tourni, K.

Zhang, L.

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Anal. Chem. (2)

M. P. Nelson, J. F. Aust, J. A. Dobrowolski, P. G. Verly, and M. L. Myrick, "Multivariate optical computation for predictive spectroscopy," Anal. Chem. 70, 73-82 (1998).
[CrossRef] [PubMed]

O. Soyemi, D. Eastwood, L. Zhang, H. Li, J. Karunamuni, P. Gemperline, R. A. Synowicki, and M. L. Myrick, "Design and testing of a multivariate optical element: the first demonstration of multivariate optical computing for predictive spectroscopy," Anal. Chem. 73, 1069-1079 (2001).
[CrossRef]

Appl. Opt. (4)

Appl. Spectrosc. (1)

Other (2)

K. R. Beebe, R. J. Pell, and M. B. Seasholtz, Chemometrics: A Practical Guide (Wiley, 1998).

H. Martens and T. Naes, Multivariate Calibration (Wiley, 1989).

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Figures (10)

Fig. 1
Fig. 1

Comparison of 45° and normal incidence configurations. (A) Left, at an average of 45° incidence, a 6° cone of light strikes an IMOE at angles ranging between 42° and 48° incidence. Right, spectral shift expected for the range of incident angles is relatively large. (B) Left, when the average angle of incidence is normal to the IMOE surface, the same cone of light gives angles ranging from 0° to 3° off normal. Right, reduced angular spread and reduced sensitivity to angular variations causes the spectral shift to be much reduced. A factor of 45 difference between (A) and (B) is illustrated.

Fig. 2
Fig. 2

Configurations for imaging MOC analyzed in this paper. M represents the MOE or IMOE position; B represents a beam splitter; D represents a detector for transmission (subscript T), reflection (subscript R) or open (subscript 0) measurements. (a) Configuration used for all 45° degree incidence calculations. (b) Generic configuration used for normal incidence calculations except for those using a filter wheel. (c) Configuration used for filter-wheel calculations.

Fig. 3
Fig. 3

Optimal beam-split fraction, f, as a function of the weighted vector mean, 〈Δ〉, of the IMOE used for imaging in the configuration shown in Fig. 2(c).

Fig. 4
Fig. 4

Schematic of apparatus for measuring prediction precision for an IMOE in reflection and transmission modes. Components are as follows. L, lamp to illuminate samples from behind; D, diffusing screen; S, samples; A, aperture; N1, N2-Inconel beam splitters; W, filter wheel with IMOE and opening for 0 measurement; M, IMOE; G, silver mirror; F, FGL 715 long-pass Schott glass filter (Thorlabs Inc.); C, camera with lens for imaging. N1, N2, W, M, G, F, and C were mounted on a translation stage. System is shown in the configuration for IMOE reflection measurement. IMOE transmission measurements were made by translating the camera unit to the right until N1 was beneath the aperture.

Fig. 5
Fig. 5

Typical image of a slide using the MOE in reflection mode. The right slide has an arrow for focus and alignment, the middle slide is the coated slide, and the left slide is a reference slide, which is the same for all images. See text for details on how information in the images was used for calibration purposes.

Fig. 6
Fig. 6

Photon-noise behavior of detector. Linear relation between the standard deviation of detector counts over several identical measurements versus the square root of detector A∕D counts follows the behavior expected for noise limited by photon counting. Each point represents the result for a single pixel measured 64 times on the same scene. The best-fit line shown in solid black yields a slope of 0.1817 ± 0.0009 , equivalent to 5.50 ± 0.03 detected photons per count. The intercept is 148 ± 22 , equivalent to a noise of 12 counts at zero signal. This plot became nonlinear above 45,000 count units.

Fig. 7
Fig. 7

Transmission spectrum of slides from 650 to 1100   nm . Dip-coat rates (in units of centimeters per second) are 0.05, 0.1, 0.2, 0.5, 1, and 2 in order from the shallowest absorbance to the deepest. From top to bottom, these curves represent the spectra of calibration samples 1–6, respectively.

Fig. 8
Fig. 8

Absorbance ( 860   nm ) and inferred thickness of coating as a function of the rate of dip coating for slides coated with acrylate polymer from a methylene chloride solution. Standard deviation of absorbance is indicated by error bars. Thickness is the one-sided thickness estimated from bulk density of the polymer, mass fraction of dye, and specific absorptivity of the dye (for details see text).

Fig. 9
Fig. 9

Calibration data normalized to the area integral of the average calibration spectrum. Curves are the estimated convolved system spectral responses based on calibration transmission spectra and spectroradiometric performance and efficiency of the CCD system and illumination components and optics, excluding the IMOE. Curves, from top to bottom, are calculated for calibration samples 1–6.

Fig. 10
Fig. 10

Fabricated IMOE transmission, reflection, vectors, and averages. Solid curves correspond to transmission properties, while dashed curves correspond to reflection properties. Transmission and reflection are read from the left axis, while the corresponding vectors are read from the right axis. The right axis is obtained as 2 T - 1 or 2 R - 1 for transmission or reflection modes, respectively. The right-hand label “arbitrary units” indicates that these values are unscaled by the gain of the system. Solid and dashed straight lines refer to the transmission and reflection weighted averages, respectively. The transmission vector is assigned to the (+) terms in Eq. 33 because its weighted vector mean is + 0.0660 . The reflection vector is of equal magnitude but negative, and it is assigned to the (−) terms.

Tables (1)

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Table 1 Secondary Calibration Parameters

Equations (41)

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y ^ i = b · x i .
y ^ i = G ( T - R ) · x i + O = G ( T · x i - R · x i ) + O = G ( D T , i - D R , i ) + O .
λ ( θ ) λ 0 [ 1 ( n 0 n eff ) 2 sin 2 θ ] 1 / 2 ,
y ^ i = 2 G T · x i 1 · x i + O where   O = ( O G ) .
b ˜ = b b ξ ¯ ,
y ^ = g ¯ j = 1 J b ˜ j ξ j , where   g ¯ = T X ¯ b ξ ¯ ,
b ξ ¯ J b j 2 ξ ¯ j 2 .
σ MOE = | g ¯ | ( T X ¯ ) 1 / 2 Δ ξ ¯ .
y ^ = j = 1 J b j ( x j i = 1 J x i ) = j = 1 J b j ( ξ j i = 1 J ξ i ) .
y ^ = g ¯ 1 j = 1 J b ˜ j ( ξ j i = 1 J ξ i ) , such   that   g ¯ 1 = b ξ ¯ .
y ¯ = j = 1 J b j ( ξ ¯ j i = 1 J ξ ¯ i ) = j = 1 J b j ξ ¯ j = b ,
y ^ y ¯ = J δ b j ( ξ j j ξ i ) .
g ¯ 1,min = δ b ξ ¯ ,
g ¯ 1 g ¯ 1,min = b ξ ¯ δ b ξ ¯ = b ˜ ξ ¯ δ b ˜ ξ ¯ , where b ˜ = b + δ b b ξ ¯ b ˜ + δ b ˜ .
Δ j t j r j ,
Δ ˜ j Δ j Δ ξ ¯ = b ˜ j ,
g ¯ 1 = Δ ξ ¯ δ Δ ξ ¯ g ¯ 1 ,min = Δ ξ ¯ ( Δ Δ ) ξ ¯ g ¯ 1 ,min ,
D T = T X ¯ J t j ξ j , D R = T X ¯ J r j ξ j ,
y ^ = g ¯ 1, min ( Δ Δ ) ξ ¯ ( D T D R D T + D R ) .
G 1 = g ¯ 1, min ( Δ Δ ) ξ ¯ .
σ N , 45 = 2 ( | g ¯ 1, min | ( T X ¯ ) 1 / 2 ( Δ Δ ) ξ ¯ ) 1 ( J Δ j ξ ¯ j ) 2 , = 2 ( | g ¯ 1, min | ( T X ¯ ) 1 / 2 ( Δ Δ ) ξ ¯ ) 1 Δ 2 .
D T = T X ¯ 2 J t j ξ j , D 0 = T X ¯ 2 J ξ j .
y ^ = g ¯ 1 ,min ( Δ Δ ) ξ ¯ ( 2 D T D 0 1 ) .
σ N , 0 = 6 ( | g ¯ 1 ,min | ( T X ¯ ) 1 / 2 ( Δ Δ ) ξ ¯ ) 1 + 4 3 Δ + 1 3 Δ 2 .
D T = T X ¯ ( 1 f ) J t j ξ j , D 0 = T X ¯ f J ξ j ,
y ^ = g ¯ 1 ,min ( Δ Δ ) ξ ¯ ( f 1 f ) ( 2 D T D 0 1 ) ,
σ N , 0 = ( 1 + f f ( 1 f ) ) 1 / 2 ( | g ¯ 1 ,min | ( T X ¯ ) 1 / 2 ( Δ Δ ) ξ ¯ ) × 1 + 2 1 + f Δ + 1 f 1 + f Δ 2 .
f = 1 + Δ 2 ( 1 + Δ ) Δ 1 .
D T = T X ¯ 2 J t j ξ j , D 0 = T X ¯ 2 J ξ j .
D T = T X ¯ T 2 J t j ξ j ,
D T R = T X ¯ T R 2 J t j ξ j , R ,
D 0 = T X ¯ 0 2 J ξ j ,
D 0 R = T X ¯ 0 R 2 J ξ j , R , where   X ¯ T X ¯ T R = X ¯ 0 X ¯ 0 R .
D T / D T R D 0 / D 0 R = X ¯ T / X ¯ T R X ¯ 0 / X ¯ 0 R J t j ξ j / J t j ξ j R J ξ j / J ξ j R = ( J ξ j R J t j ξ j R ) J t j ξ j J ξ j = 2 ρ J t j ξ j J ξ j = 1 ρ [ J Δ j ξ j J ξ j + 1 ]
J Δ j ξ j J ξ j = ρ D T D 0 R D 0 D T R 1 , where   ρ = 1 + Δ R ,
y ^ N , R = g ¯ 1 ,min ( Δ Δ ) ξ ¯ j = 1 J Δ j ξ j i = 1 J ξ i = g ¯ 1 ,min ( Δ Δ ) ξ ¯ ( ρ D T D 0 R D 0 D T R 1 ) .
y ^ N , R = γ D T D T R D 0 R D 0 + β .
γ = ρ D T R D 0 R G 1 ,
σ N , R = γ 2 ( 1 + Δ ) [ ( 1 + Δ ) ( 1 + Δ R ) X ¯ T R ( X ¯ T R + X ¯ T ) + 2 X ¯ 0 R ( X ¯ T + Δ X ¯ T + X ¯ T R + Δ R X ¯ T R ) ] T X ¯ T X ¯ T R X ¯ 0 R ( 1 + Δ R ) 3 .
σ N , R = 12 γ ( T X ¯ ) 1 / 2 1 + ( 1 / 3 ) Δ 1 + Δ = 12 | g ¯ 1 ,min | ( Δ Δ ) ξ ¯ ( T X ¯ ) 1 / 2 1 + 4 3 Δ + 1 3 Δ 2 .
σ N , R σ N , R + = 1 ( 4 / 3 ) | Δ | + ( 1 / 3 ) | Δ | 2 1 + ( 4 / 3 ) | Δ | + ( 1 / 3 ) | Δ | 2 ,

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