Abstract

The basic measurement equation r = B + αd + n is solved for α (the weight or abundance of the spectral target vector d) by two methods: (a) by subtracting the stochastic spectral background vector B from the spectral measurement’s vector r (subtraction solution) and (b) by orthogonal subspace projection (OSP) of the measurements to a subspace orthogonal to B (the OSP solution). The different geometry of the two solutions and in particular the geometry of the noise vector n is explored. The angular distribution of the noise angle between B and n is the key factor for determining and predicting which solution is better. When the noise-angle distribution is uniform, the subtraction solution is always superior regardless of the orientation of the spectral target vector d. When the noise is more concentrated in the direction parallel to B, the OSP solution becomes better (as expected). Simulations and one-dimensional hyperspectral measurements of vapor concentration in the presence of background radiation and noise are given to illustrate these two solutions.

© 2005 Optical Society of America

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References

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  1. A. Ben-David, H. Ren, “Detection, identification, and estimation of biological aerosols and vapors with a Fourier-transform infrared spectrometer,” Appl. Opt. 42, 4887–4900 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. D. R. Morgan, “Spectral absorption pattern detection and estimation,” Appl. Spectrosc. 31, 404–424 (1977).
    [CrossRef]
  4. G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 2.
  5. T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Upper Saddle River, N. J., 2000), Chap. 2.
  6. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N. J., 1993), Chap. 8.
  7. A. Hayden, E. Niple, B. Boyce, “Determination of trace-gas amounts in plumes by the use of orthogonal digital filtering of thermal-emission spectra,” Appl. Opt. 35, 2802–2809 (1996).
    [CrossRef] [PubMed]
  8. J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sensing 32, 779–785 (1994).
    [CrossRef]
  9. C.-I. Chang, H. Ren, “An experiment-based quantitative and comparative analysis of target detection and image classification algorithms for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sensing 38, 1044–1063 (2000).
    [CrossRef]
  10. J. J. Settle, “On the relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 42, 1045–1046 (1996).
    [CrossRef]
  11. C.-I. Chang, “Further results on relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 36, 1030–1032 (1998).
    [CrossRef]
  12. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Pa., 1991).

2003 (1)

2000 (1)

C.-I. Chang, H. Ren, “An experiment-based quantitative and comparative analysis of target detection and image classification algorithms for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sensing 38, 1044–1063 (2000).
[CrossRef]

1998 (1)

C.-I. Chang, “Further results on relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 36, 1030–1032 (1998).
[CrossRef]

1996 (2)

J. J. Settle, “On the relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 42, 1045–1046 (1996).
[CrossRef]

A. Hayden, E. Niple, B. Boyce, “Determination of trace-gas amounts in plumes by the use of orthogonal digital filtering of thermal-emission spectra,” Appl. Opt. 35, 2802–2809 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sensing 32, 779–785 (1994).
[CrossRef]

1977 (1)

Ben-David, A.

Boyce, B.

Chang, C.-I.

C.-I. Chang, H. Ren, “An experiment-based quantitative and comparative analysis of target detection and image classification algorithms for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sensing 38, 1044–1063 (2000).
[CrossRef]

C.-I. Chang, “Further results on relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 36, 1030–1032 (1998).
[CrossRef]

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sensing 32, 779–785 (1994).
[CrossRef]

Golub, G.

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 2.

Hall, J. L.

Harsanyi, J.

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sensing 32, 779–785 (1994).
[CrossRef]

Hassibi, B.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Upper Saddle River, N. J., 2000), Chap. 2.

Hayden, A.

Herr, K. C.

Kailath, T.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Upper Saddle River, N. J., 2000), Chap. 2.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N. J., 1993), Chap. 8.

Morgan, D. R.

Niple, E.

Polak, M. L.

Ren, H.

A. Ben-David, H. Ren, “Detection, identification, and estimation of biological aerosols and vapors with a Fourier-transform infrared spectrometer,” Appl. Opt. 42, 4887–4900 (2003).
[CrossRef] [PubMed]

C.-I. Chang, H. Ren, “An experiment-based quantitative and comparative analysis of target detection and image classification algorithms for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sensing 38, 1044–1063 (2000).
[CrossRef]

Sayed, A. H.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Upper Saddle River, N. J., 2000), Chap. 2.

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Pa., 1991).

Settle, J. J.

J. J. Settle, “On the relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 42, 1045–1046 (1996).
[CrossRef]

Van Loan, C. F.

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 2.

Appl. Opt. (3)

Appl. Spectrosc. (1)

IEEE Trans. Geosci. Remote Sensing (4)

J. Harsanyi, C.-I. Chang, “Hyperspectral image classification and dimensionality reduction: an orthogonal subspace projection approach,” IEEE Trans. Geosci. Remote Sensing 32, 779–785 (1994).
[CrossRef]

C.-I. Chang, H. Ren, “An experiment-based quantitative and comparative analysis of target detection and image classification algorithms for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sensing 38, 1044–1063 (2000).
[CrossRef]

J. J. Settle, “On the relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 42, 1045–1046 (1996).
[CrossRef]

C.-I. Chang, “Further results on relationship between spectral unmixing and subspace projection,” IEEE Trans. Geosci. Remote Sensing 36, 1030–1032 (1998).
[CrossRef]

Other (4)

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, Pa., 1991).

G. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996), Chap. 2.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, Upper Saddle River, N. J., 2000), Chap. 2.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Upper Saddle River, N. J., 1993), Chap. 8.

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Figures (15)

Fig. 1
Fig. 1

Geometry of the errors, eosp and esub, for a two-band (2-D) case where eosp > 0 and esub > 0. The gray-shaded area is the allowable domain for the noise vector n, and the hatched gray area (a subsection of the gray-shaded area) is the noise domain where | eosp | < | esub |. The background vector B is along the positive y axis; the orthogonal subspace PB is the x axis. For convenience we place the target spectrum d in the first quadrant (x > 0, y > 0), and Pd is the orthogonal subspace to d. The errors | esub | and | eosp | are given by the ratio of the segments oa/ob and oc/of, respectively, where and thus | eosp | < | esub |.

Fig. 2
Fig. 2

Same as Fig. 1 but for the domain eosp < 0 and esub > 0.

Fig. 3
Fig. 3

Same as Fig. 1 but for the domain eosp < 0 and esub < 0.

Fig. 4
Fig. 4

Same as Fig. 1 but for the domain eosp > 0 and esub < 0.

Fig. 5
Fig. 5

Percentage of samples where |esub| < | eosp | as a function of the angle between B and d when the pdf of the angle of the noise is uniformly distributed. The superiority of the solution α ̂ sub over α ̂ osp is shown.

Fig. 6
Fig. 6

Noise domain for which α ̂ sub is superior to α ̂ osp (gray area) and for which α ̂ osp is superior to α ̂ sub (hatched gray area) for one realization of d in the fourth quadrant when the pdf for the noise angles is a uniform distribution.

Fig. 7
Fig. 7

Probability density function for the noise angle, which results [case (a)] in a superior solution, α ̂ osp to α ̂ sub.

Fig. 8
Fig. 8

Probability density function for the noise angle, which results [case (b)] in a superior solution, α ̂ sub to α ̂ osp.

Fig. 9
Fig. 9

Probability density function of the noise | n | for E(|r|)/E(|σn|) = 25 and 150, where the pdf (|n|) curves apply to any of the angular noise distribution (Figs. 7 and 8); case (a), case (b), and case (c) (uniform angular distribution).

Fig. 10
Fig. 10

Probability density function of the simulated measurements for case (a) (Fig. 7).

Fig. 11
Fig. 11

Probability density function of, thin curves, solutions α ̂ sub and, thick curves, α ̂ osp for (a), (d) case (a); (b), (e), case (b); (c), (f) case (c). The true (correct) solution is α = 0.2. The SNR in the simulations is (a)–(c) 25 and (d)–(f) 150. For case (a) α ̂ osp is shown superior to α ̂ sub (more samples result in a solution closer to the true solution, α = 0.2), and for cases (b) and (c) the solution α ̂ sub is shown superior to α ̂ osp.

Fig. 12
Fig. 12

Top, solutions α ̂ sub and, bottom, α ̂ osp for 4000 TEP vapor measurements. A moving average (five sequential measurements) process was preformed on the estimated α to reduce noise. Four injections of 0.25 ml each occurred at measurements 1, 1000, 2000, and 3000.

Fig. 13
Fig. 13

Average signature (spectrum) of the background B and the target signature (TEP vapor) d.

Fig. 14
Fig. 14

Probability density function for the noise angle in the TEP vapor measurements.

Fig. 15
Fig. 15

Geometry for constructing noise n with a given angle θn and pdf (|n|) = pdf[|N(0, PMσ2)|].

Equations (27)

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r = B + α d + n
α ̂ sub = ( d T d ) 1 d T ( r B ) = α + ( d T d ) 1 d T n = α + e sub ,
α ̂ osp = ( d T P B d ) 1 d T P B r = α + ( d T P B d ) 1 d T P B n = α + e osp ,
P B = I B ( B T B ) 1 B T ,
n = n + n = P B n + ρ B ,
e sub = ( d T d ) 1 d T ( n + n ) = ( d T d ) 1 d T ( P B n + ρ B ) = d T P B n + ρ d T B d T d ,
e osp = ( d T P B d ) 1 d T P B ( n + n ) = ( d T P B d ) 1 d T P B n = d T P B n d T P B d ,
| e sub | = | d T P B n + ρ d T B d T d | = o a o b ,
| e osp | = | d T P B n d T P B d | = o c o f = o s o b ,
ρ d T P B n ( d T d d T P B d 1 ) d T B ,
ρ d T P B n ( d T d d T P B d + 1 ) d T B ,
ρ d T P B n ( d T d d T P B d 1 ) d T B ,
ρ d T P B n ( d T d d T P B d + 1 ) d T B .
θ n = cos 1 ( B T P M n | B | | P M n | ) ,
cos ( ϕ ) = ( P B d ) T P M n | P B d | | P M n | = sin ( θ n ) .
θ n = cos 1 ( B T P M n | B | | P M n | ) , cos ( ϕ ) > 0 , θ n = 360 | cos 1 ( B T P M n | B | | P M n | ) | , cos ( ϕ ) < 0 .
SNR = | r | | n | .
pdf ( α ̂ ) α ̂ d α ̂ α
θ B , d = cos 1 ( B T d | B | | d | ) = 39.31 ° ,
N = { θ n : θ 2 < θ n < θ 1 , θ 4 < θ n < θ 3 } , N osp = hist ( N ) ,
N osp = hist [ | α ̂ osp ( i ) α | < | α ̂ sub ( i ) α | ] = hist [ | e osp ( i ) | < | e sub ( i ) | ] ,
n ( θ ) = P M y + k ( θ ) B | P M y + k ( θ ) B | | P M y |
pdf ( | z | ) = 2 | z | σ 2 χ 2 ( | z | 2 σ 2 , L ) = 2 σ L 2 L / 2 Γ ( L / 2 ) | z | L 1 exp ( | z | 2 2 σ 2 )
B T oa = B T ( P M y + k B ) = B T P M y + k | B | 2 = | P M y + k B | | B | cos ( θ ) ,
o c = o a sin ( θ ) = | P M y + k B | sin ( θ ) = | P B P M y | ,
k ( θ ) = P B P M y | B | cot ( θ ) B T P M y | B | 2 ,
{ 0 < θ < 180 , cos [ ( P B d ) T P M y | P B d | | P M y | ] > 0 180 < θ < 360 , cos [ ( P B d ) T P M y | P B d | | P M y | ] < 0 } ,

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