Optimal selection of commercial sensors for linear model representation of daylight spectra

Neng-Chung Hu; Chin-Chuan Wu

doi:10.1364/AO.47.003114

Applied Optics
Vol. 47,
Issue 17,
pp. 3114-3123
(2008)
•https://doi.org/10.1364/AO.47.003114

Optimal selection of commercial sensors for linear model representation of daylight spectra

Neng-Chung Hu and Chin-Chuan Wu

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Neng-Chung Hu and Chin-Chuan Wu, "Optimal selection of commercial sensors for linear model representation of daylight spectra," Appl. Opt. 47, 3114-3123 (2008)

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Abstract

The average spectral power distribution of a set of measured daylight spectra, $E_{av} (λ)$ , is used for preliminary screening to select optimal sensor sets for daylight recovery. Spectra quite different from $E_{av} (λ)$ are applied to the screened sets to obtain minimum total spectral error, which is closely related to recovery metrics but not to the coefficient of error. All basis functions should be utilized to make these two errors equal, to predict precisely the best sensor set, and to extend a set of few sensors to a set of many sensors. These are not acquirable by an exhaustive full search method.

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SPDs	Dawn	Noon	Dusk
In the training set
Color temperature (K)	3508	6958	3418
GFC	0.99909	0.99915	0.99966
Spectral error	0.001999	0.001708	0.000679
$Δ E$	3.9208	1.7636	0.75363
Not in the training set
Color temperature (K)	3607	6601	3218
GFC	0.99876	0.99882	0.99891
Spectral error	0.002879	0.002783	0.0002541
$Δ E$	5.101	4.7934	4.4525

Sensor Set No.	Error
Sensor Set No.	$‖ {\vec{δ}}_{av} ‖$	ε	$\hat{ε}$
25	0.030737	0.036513	0.034713
40	0.030315	0.036823	0.034858
2	0.033206	0.036973	0.036304
4	0.033715	0.037296	0.037005
45	0.032552	0.037459	0.037041
80	0.033961	0.037591	0.037339
11	0.033567	0.037811	0.037347
5	0.034122	0.037851	0.037354
35	0.033697	0.037904	0.037493
13	0.033575	0.038031	0.037527

Sensor Set No.	Error^b Original Versus Reconstructed	Averaged GFC (S.D)	Averaged Error (S.D)	Averaged $Δ E$ (S.D)	Average ε of Spectra Not in Training Set (S.D)	Averaged GFC, of Spectra Not in Training Set (S.D)
25	0.036513	0.99927 (0.00048586)	0.036513 (0.011065357)	0.430733 (0.288045)	0.041322 (0.01254234)	0.99901 (0.0006123)
40	0.036823	0.99928 (0.00041771)	0.036823 (0.009660086)	0.440445 (0.228941)	0.041632 (0.012103535)	0.999 (0.0006215)
2	0.036973	0.99928 (0.00037038)	0.036973 (0.008564635)	0.447471 (0.30467)	0.041781 (0.011245231)	0.99899 (0.0005924)
4	0.037296	0.99926 (0.00041765)	0.037296 (0.009427938)	0.452398 (0.185165)	0.042103 (0.0118234455)	0.99897 (0.0006125)
45	0.037459	0.99924 (0.00051636)	0.037459 (0.010860239)	0.453989 (0.310414)	0.042265 (0.0125694211)	0.99896 (0.0006429)
80	0.03759	0.99924 (0.00044203)	0.03759 (0.009840196)	0.493536 (0.346251)	0.042419 (0.012386542)	0.99894 (0.0006543)
11	0.037811	0.99924 (0.00039581)	0.037811 (0.008956544)	0.499538 (0.220432)	0.04263 (0.0113445901)	0.99893 (0.0006312)
5	0.037851	0.99924 (0.00042266)	0.037851 (0.009441143)	0.500057 (0.519831)	0.042657 (0.0124021897)	0.99891 (0.0007164)
35	0.037904	0.99924 (0.00041559)	0.037904 (0.009359716)	0.968214 (0.857549)	0.042613 (0.0111422145)	0.99891 (0.0005928)
13	0.038031	0.99923 (0.00043012)	0.038031 (0.009644981)	0.971252 (0.573222)	0.042834 (0.0124428891)	0.9989 (0.0007321)

Sensor Set No.	Sensor No.	Sensor No.	Sensor No.	Sensor No.	Sensor No.	Sensor No.	GFC	Spectral Error
Full search method
103217	1	10	24	25	26	27	0.99927	0.036513
473415	7	10	11	24	25	27	0.99928	0.036823
554960	10	23	24	25	26	27	0.99928	0.036973
543319	10	11	23	24	25	27	0.99926	0.037296
477221	7	10	22	24	25	27	0.99924	0.037459
543264	10	11	22	23	24	27	0.99924	0.03759
554729	10	20	24	25	26	27	0.99924	0.037811
554891	10	22	23	24	25	27	0.99924	0.037851
543284	10	11	22	24	25	27	0.99924	0.037904
543144	10	11	20	24	25	27	0.99923	0.038031
Proposed method
25	10	24	27	25	26	1	0.99927	0.036513
40	10	24	27	25	11	7	0.99928	0.036823
2	10	24	27	25	23	26	0.99928	0.036973
4	10	24	27	25	23	11	0.99926	0.037296
45	10	24	27	25	22	7	0.99924	0.037459
80	10	24	27	23	11	22	0.99924	0.03759
11	10	24	25	25	20	26	0.99924	0.037811
5	10	24	27	25	23	22	0.99924	0.037851
35	10	24	27	25	11	22	0.99924	0.037904
13	10	24	27	25	20	11	0.99923	0.038031

SNR	Error
SNR	$‖ Δ {\vec{Y}}_{av}^{'} ‖$ , Eq. (7a), All Basis Functions Used	$‖ Δ {\vec{Y}}_{av}^{"} ‖ = ‖ [T^{w}] {\vec{c}}_{i}^{*} ‖$ , Eq. (7b), Six Basis Functions Used	$‖ [P^{w}] {\vec{f}}_{av}^{*} ‖$	Increasing Deviation Rate, $(‖ Δ {\vec{Y}}_{av}^{'} ‖ - ‖ Δ {\vec{Y}}_{av}^{"} ‖) / ‖ Δ {\vec{Y}}_{av}^{'} ‖$
30	0.000357	0.000356	0.0000171	0.28%
20	0.000523	0.000522	0.0000182	0.19%
10	0.000851	0.000829	0.0000393	2.59%
5	0.001412	0.001358	0.0000496	3.82%

Set No.	ε without noise	Set No.	ε with 30 SNR	Set No.	ε with 20 SNR	Set No.	ε with 10 SNR	Set No.	ε with 5 SNR
25	0.036513	25	0.037292	40	0.037511	40	0.038966	40	0.043575
40	0.036823	40	0.037455	25	0.037621	25	0.039885	25	0.043609
2	0.036973	2	0.038894	2	0.038833	2	0.040073	80	0.044347
4	0.037296	45	0.03920	4	0.039384	80	0.040455	5	0.044383
45	0.037459	11	0.039237	11	0.039615	13	0.04063	13	0.044542
80	0.03759	4	0.039385	5	0.039675	4	0.040826	45	0.045829
11	0.037811	35	0.039399	45	0.039679	5	0.041057	2	0.046469
5	0.037851	5	0.039522	80	0.03975	45	0.04116	11	0.047173
35	0.037904	80	0.03968	35	0.03975	11	0.0418	4	0.047219
13	0.038031	13	0.039813	13	0.039907	35	0.042139	35	0.047716

Sensor Nos.	Averaged GFC	Averaged error
27, 25, 16	0.9989	0.046101
10, 25, 26, 1	0.99906	0.04286
27, 11, 16, 1, 12	0.9991	0.041971
10, 25, 23, 8, 22, 1	0.99923	0.038662

Sensor Nos.	Averaged GFC	Averaged Spectral Error	Averaged $Δ E$
10, 24, 27	0.99495	0.10008	7.2192
10, 24, 27, 25	0.99888	0.045237	1.1168
10, 24, 27, 25, 26	0.99914	0.039892	0.89881
10, 24, 27, 25, 26, 1	0.99927	0.036513	0.430733

Abstract

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Tables (8)

Equations (31)

Applied Optics