Abstract

We discuss denoising in Hadamard transform spectrometry (HTS) in terms of sensor noise, photon noise, and the sparsity of the source. An analysis based on spectra classification is proposed to estimate the signal-to-noise ratio (SNR) of both HTS and slit-based spectrometry. In contrast with previous theory, it is shown that HTS can improve the sensitivity of the sensor and that HTS outperforms slit-based spectrometry when the signal is dominated by photon noise and the source is sparse. Numerical simulations show that HTS is a good method for improving the poor SNR associated with weak or sparse signals.

© 2014 Optical Society of America

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2013 (2)

A. Lefkimmiatis and M. Unser, IEEE Trans. Image Process. 22, 4314 (2013).
[CrossRef]

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

2012 (3)

2011 (1)

2010 (2)

2009 (1)

2008 (1)

D. Needell and J. A. Tropp, Appl. Comput. Harmon. Anal. 26, 301 (2008).

2007 (1)

2006 (2)

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

N. Shimano, IEEE Trans. Image Process. 15, 1848 (2006).
[CrossRef]

2005 (1)

1995 (1)

1993 (1)

R. Damaschini, Pure Appl. Opt. 2, 173 (1993).
[CrossRef]

1987 (1)

1973 (1)

B. R. Kowalski and C. F. Bender, Anal. Chem. 45, 2234 (1973).
[CrossRef]

Barducci, A.

Bender, C. F.

B. R. Kowalski and C. F. Bender, Anal. Chem. 45, 2234 (1973).
[CrossRef]

Bian, X.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Bowles, J. H.

Brady, D. J.

Burling-Claridge, G. R.

Butcher, S. D.

Chen, D. T.

Clowers, B. H.

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

Corson, M.

Corson, M. R.

Cree, M. J.

Damaschini, R.

R. Damaschini, Pure Appl. Opt. 2, 173 (1993).
[CrossRef]

Davis, C. O.

Faisal, M.

Fang, H.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Fateley, W. G.

Gehm, M. E.

Guzzi, D.

Hammaker, R. M.

Harmany, Z. T.

Z. T. Harmany, R. F. Marcia, and R. M. Willett, IEEE Trans. Image Process. 21, 1084 (2012).
[CrossRef]

Harwit, M.

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).

Helstrom, C. W.

Hill, H. H.

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

Korwan, D. R.

Kowalski, B. R.

B. R. Kowalski and C. F. Bender, Anal. Chem. 45, 2234 (1973).
[CrossRef]

Künnemeyer, R.

Lanterman, A. D.

Lastri, C.

Lefkimmiatis, A.

A. Lefkimmiatis and M. Unser, IEEE Trans. Image Process. 22, 4314 (2013).
[CrossRef]

Li, R. R.

Liu, H.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Lucke, R. L.

Marcia, R. F.

Z. T. Harmany, R. F. Marcia, and R. M. Willett, IEEE Trans. Image Process. 21, 1084 (2012).
[CrossRef]

Marcoionni, P.

Marks, D. L.

Massick, S. M.

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

McGlothlin, N. R.

Moses, W. J.

Mrozack, A.

Nardino, V.

Needell, D.

D. Needell and J. A. Tropp, Appl. Comput. Harmon. Anal. 26, 301 (2008).

Pippi, I.

Shimano, N.

N. Shimano, IEEE Trans. Image Process. 15, 1848 (2006).
[CrossRef]

Siems, W. F.

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

Sloane, N. J. A.

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).

Snyder, D. L.

Snyder, W. A.

Streeter, L.

Tilotta, D. C.

Tropp, J. A.

D. Needell and J. A. Tropp, Appl. Comput. Harmon. Anal. 26, 301 (2008).

Unser, M.

A. Lefkimmiatis and M. Unser, IEEE Trans. Image Process. 22, 4314 (2013).
[CrossRef]

Wagadarikar, A. A.

White, R. L.

Willett, R. M.

Z. T. Harmany, R. F. Marcia, and R. M. Willett, IEEE Trans. Image Process. 21, 1084 (2012).
[CrossRef]

Wood, D. L.

Wuttig, A.

Yan, L.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Zhang, T.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Zhang, X.

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Anal. Chem. (2)

B. H. Clowers, W. F. Siems, H. H. Hill, and S. M. Massick, Anal. Chem. 78, 44 (2006).
[CrossRef]

B. R. Kowalski and C. F. Bender, Anal. Chem. 45, 2234 (1973).
[CrossRef]

Appl. Comput. Harmon. Anal. (1)

D. Needell and J. A. Tropp, Appl. Comput. Harmon. Anal. 26, 301 (2008).

Appl. Opt. (6)

IEEE Trans. Image Process. (3)

Z. T. Harmany, R. F. Marcia, and R. M. Willett, IEEE Trans. Image Process. 21, 1084 (2012).
[CrossRef]

A. Lefkimmiatis and M. Unser, IEEE Trans. Image Process. 22, 4314 (2013).
[CrossRef]

N. Shimano, IEEE Trans. Image Process. 15, 1848 (2006).
[CrossRef]

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

Opt. Express (3)

Opt. Lett (1)

X. Bian, T. Zhang, L. Yan, X. Zhang, H. Fang, and H. Liu, Opt. Lett 38, 815 (2013).
[CrossRef]

Pure Appl. Opt. (1)

R. Damaschini, Pure Appl. Opt. 2, 173 (1993).
[CrossRef]

Other (2)

M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (Academic, 1979).

MultiSpec, “AVIRIS image Indian Pine, Indiana,” https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html .

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Figures (4)

Fig. 1.
Fig. 1.

Comparison of the SNRs for different sparsity signals. (a) SNR of a reconstructed signal using different methods. (b) Comparison of the SNRs as a function of sparsity.

Fig. 2.
Fig. 2.

SNR performance for HTS and slit-based spectrometry associated with light intensity. (a) Comparison of the SNRs for slit-based spectrometry and HTS noise with that of photon noise only. (b) Ratios of the photon-noise-only and HTS SNRs to the slit-based-spectrometry SNR.

Fig. 3.
Fig. 3.

(a) and (f) Original images from the two datasets. (b) and (g) Reimaging using HTS for SNR=20. (c) and (h) Reimaging using slit-based spectrometry for SNR=20. (d) and (i) Reimaging using HTS for SNR=10. (e) and (j) Reimaging using slit-based spectrometry for SNR=10.

Fig. 4.
Fig. 4.

Accuracy statistics of the regenerated synthetic dataset: Sf, S1Sf in the HTS and f measured using slit-based spectrometry via classification method (a) K-NN or (b) SVM.

Tables (1)

Tables Icon

Table 1. Average OA Confidence Interval of the Three Types of Dataset, Sf, S1Sf, and f, as Determined by SVMa

Equations (16)

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g=Sf+nHTS,
(SNR)RMS=(σ2/ε)1/2=(n+1)/2N,
SNR=signal(noisephoton2)noisephoton2+noisedetector270,
DE=|(fafb)T(fafb)|1/2=|ΔfTΔf|1/2.
SNRslitDE=|ΔfTΔfΔnslitTΔnslit|1/2,
SNRHTSDE=|ΔfTΔf(S1ΔnHTS)TS1ΔnHTS|1/2,
(S1ΔnHTS)TS1ΔnHTS=4(N+1)ΔnHTSTIΔnHTS4(N+1)2ΔnHTSTJNΔnHTS,
(ΔnHTS)TJNΔnHTS=(i=1NΔnHTS(i))2.
(S1ΔnHTS)TS1ΔnHTS=4(N+1)2{ΔnHTSTIΔnHTS+i=1Nj=1N[ΔnHTS(i)ΔnHTS(j)]2},
|ΔfTΔf(ΔnHTSk)TΔnHTSk|1/2=SNRHTSDE
nHTSphoton(i)(N+1)2nslitphoton(i).
nHTSrestore(k)=2N+1(iAΔnHTS(i)jBΔnHTS(j)),A={x|S1(k,x)>0},B={x|S1(k,x)<0},
eSNRHTSDE=|(SΔf)TSΔfΔnHTSTΔnHTS|1/2=(|ΔfTΔf+ΔfTJNΔf(VΔnHTS)TVΔnHTS+(VΔnHTS)TJNVΔnHTS|1/2),
(VΔnHTS)TJNVΔnHTS=(ΔnHTS(N+1)/2)TJNΔnHTS(N+1)/2.
ni,photonfi,ni,slit=ni,photon2+ni,detector2.
ni,HTSrestore2(N+1)ni,HTSmeasured2ni,photon22+ni,detector2(N+1),

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