Javier Hernández-Andrés, Juan L. Nieves, Eva M. Valero, and Javier Romero, "Spectral-daylight recovery by use of only a few sensors," J. Opt. Soc. Am. A 21, 13-23 (2004)
Linear models have already been proved accurate enough to recover spectral functions. We have resorted to such linear models to recover spectral daylight with the response of no more than a few real sensors. We performed an exhaustive search to obtain the best set of Gaussian sensors with a combination of optimum spectral position and bandwidth. We also examined to what extent the accuracy of daylight estimation depends on the number of sensors and their spectral properties. A set of 2600 daylight spectra [J. Opt. Soc. Am. A 18, 1325 (2001)] were used to determine the basis functions in the linear model and also to evaluate the accuracy of the search. The estimated spectra are compared with the original ones for different spectral daylight and skylight sets of data within the visible spectrum. Spectral similarity, colorimetric differences, and integrated spectral irradiance errors were all taken into account. We compare our best results with those obtained by using a commercial CCD, revealing the CCD’s potential as a daylight-estimation device.
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Optimum Spectral Sensitivities of the Gaussian Sensors for Daylight Recovery
Number of Sensors
Gaussian Sensor Characteristics
Spectral Sensitivity (nm)
First Sensor
Second Sensor
Third Sensor
Fourth Sensor
Fifth Sensor
3
Peak sensitivity
425
440
740
FWHM
250
120
300
4
Peak sensitivity
430
445
520
750
FWHM
290
100
40
330
5
Peak sensitivity
450
485
505
590
750
FWHM
230
110
110
15
370
Table 2
Mean, Standard Deviations, and 99th Percentile Results Obtained with the Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against a Set of 2600 Daylight Spectraa
Numberof Sensors
GFC
Integrated Irradiance Error
Fractional Deviation
RMS Error
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
3
0.9997 (0.0004)
0.9983
0.3360 (0.2983)
1.1855
0.0133% (0.0116%)
0.0572%
0.0173 (0.0092)
0.0497
0.3715 (0.4078)
1.6072
CCDcamera
0.9991 (0.0015)
0.9921
0.1967 (0.1740)
0.6825
1.1874% (1.0116%)
5.1053%
0.0298 (0.0198)
0.1083
0.6295 (0.7686)
2.6421
4
0.9998 (0.0003)
0.9990
0.1186 (0.1208)
0.4354
0.0056% (0.0051%)
0.0209%
0.0112 (0.0061)
0.0328
0.2620 (0.3001)
1.086
5
0.9999 (0.0002)
0.9995
0.1072 (0.1180)
0.3972
0.0058% (0.0052%)
0.0222%
0.0076 (0.0044)
0.0226
0.1771 (0.2234)
0.7225
Also included are the results obtained with a JVC TK-1270E CCD color camera. Standard deviations (SD) are given in parentheses.
Table 3
Means and Standard Deviations Obtained with the Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against a Set of 2600 Daylight Spectra, Compared (in Italics) with the Results Obtained with a Theoretical Eigenvector Expansion [(Eq. 4)]a
Mean GFC (SD)
Mean RMS Error (SD)
3 sensors
0.99971 (0.00039)
0.3715 (0.4078)
3 eigenvectors
0.99973
(0.00038)
0.3442
(0.3775)
4 sensors
0.99983 (0.00030)
0.2620 (0.3001)
4 eigenvectors
0.99986
(0.00025)
0.2503
(0.2868)
5 sensors
0.99992 (0.00017)
0.1771 (0.2234)
5 eigenvectors
0.99995
(0.00008)
0.1516
(0.1881)
Standard deviations are given in parentheses.
Table 4
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against Nine Typical CIE Daylight Spectra of Different CCTsa
Number of Sensors
Mean GFC (SD)
Mean
(SD)
Mean Integrated Irradiance Error (SD)
Mean Fractional Deviation (SD)
3
0.9946 (0.0033)
24.8328 (16.1775)
0.0416% (0.0074%)
0.0420 (0.0121)
CCDcamera
0.9831 (0.0143)
14.8362 (9.0366)
3.8200% (2.1130%)
0.1420 (0.0820)
4
0.9979 (0.0011)
8.3701 (4.1103)
0.0078% (0.0048%)
0.0298 (0.0062)
5
0.9980 (0.0011)
4.9978 (1.9412)
0.0328% (0.0177%)
0.0322 (0.0022)
4000, 6000, 8000, 10,000, 12,000, 14,000, 16,000, 18,000, and 20,000 K.
Table 5
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) versus 12 Daylight Spectra Measured in the US by Lee37a
Number of Sensors
Mean GFC (SD)
Mean
(SD)
Mean Integrated Irradiance Error (SD)
Mean Fractional Deviation (SD)
3
0.9992 (0.0006)
0.2847 (0.2141)
0.0323% (0.0319%)
0.0296 (0.0145)
CCDcamera
0.9967 (0.0054)
0.1776 (0.1512)
2.2306% (2.9605%)
0.0501 (0.0472)
4
0.9995 (0.0003)
0.1136 (0.0818)
0.0110% (0.0060%)
0.0208 (0.0066)
5
0.9996 (0.0002)
0.1531 (0.1364)
0.0060% (0.0061%)
0.0181 (0.0061)
These 12 spectral daylight measurements have different CCTs from 5750 to 16,780 K and a sampling rate of 5 nm. Standard deviations are given in parentheses.
Table 6
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Daylight Eigenvectors (Fig. 1) versus Our Set of 1567 Skylight Spectra with a Sampling Rate of 5 nma
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Skylight Eigenvectors (see Ref. 22) versus Our Set of 1567 Skylight Spectra with a Sampling Rate of 5 nm
Number of Sensors
Mean GFC (SD)
Mean
(SD)
Mean Integrated Irradiance Error (SD)
Mean Fractional Deviation (SD)
3
0.9979 (0.0049)
0.3898 (0.8592)
0.0701% (0.0640%)
0.0403 (0.0303)
CCDcamera
0.7800 (0.2323)
2.0551 (8.5778)
14.1358% (12.4656%)
0.5711 (0.4939)
4
0.9994 (0.0008)
0.1812 (0.2821)
0.0096% (0.0078%)
0.0240 (0.0157)
5
0.9999 (0.0001)
0.0513 (0.0996)
0.0061% (0.0049%)
0.0099 (0.0052)
Tables (7)
Table 1
Optimum Spectral Sensitivities of the Gaussian Sensors for Daylight Recovery
Number of Sensors
Gaussian Sensor Characteristics
Spectral Sensitivity (nm)
First Sensor
Second Sensor
Third Sensor
Fourth Sensor
Fifth Sensor
3
Peak sensitivity
425
440
740
FWHM
250
120
300
4
Peak sensitivity
430
445
520
750
FWHM
290
100
40
330
5
Peak sensitivity
450
485
505
590
750
FWHM
230
110
110
15
370
Table 2
Mean, Standard Deviations, and 99th Percentile Results Obtained with the Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against a Set of 2600 Daylight Spectraa
Numberof Sensors
GFC
Integrated Irradiance Error
Fractional Deviation
RMS Error
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
Mean (SD)
99th Percentile
3
0.9997 (0.0004)
0.9983
0.3360 (0.2983)
1.1855
0.0133% (0.0116%)
0.0572%
0.0173 (0.0092)
0.0497
0.3715 (0.4078)
1.6072
CCDcamera
0.9991 (0.0015)
0.9921
0.1967 (0.1740)
0.6825
1.1874% (1.0116%)
5.1053%
0.0298 (0.0198)
0.1083
0.6295 (0.7686)
2.6421
4
0.9998 (0.0003)
0.9990
0.1186 (0.1208)
0.4354
0.0056% (0.0051%)
0.0209%
0.0112 (0.0061)
0.0328
0.2620 (0.3001)
1.086
5
0.9999 (0.0002)
0.9995
0.1072 (0.1180)
0.3972
0.0058% (0.0052%)
0.0222%
0.0076 (0.0044)
0.0226
0.1771 (0.2234)
0.7225
Also included are the results obtained with a JVC TK-1270E CCD color camera. Standard deviations (SD) are given in parentheses.
Table 3
Means and Standard Deviations Obtained with the Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against a Set of 2600 Daylight Spectra, Compared (in Italics) with the Results Obtained with a Theoretical Eigenvector Expansion [(Eq. 4)]a
Mean GFC (SD)
Mean RMS Error (SD)
3 sensors
0.99971 (0.00039)
0.3715 (0.4078)
3 eigenvectors
0.99973
(0.00038)
0.3442
(0.3775)
4 sensors
0.99983 (0.00030)
0.2620 (0.3001)
4 eigenvectors
0.99986
(0.00025)
0.2503
(0.2868)
5 sensors
0.99992 (0.00017)
0.1771 (0.2234)
5 eigenvectors
0.99995
(0.00008)
0.1516
(0.1881)
Standard deviations are given in parentheses.
Table 4
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) Tested against Nine Typical CIE Daylight Spectra of Different CCTsa
Number of Sensors
Mean GFC (SD)
Mean
(SD)
Mean Integrated Irradiance Error (SD)
Mean Fractional Deviation (SD)
3
0.9946 (0.0033)
24.8328 (16.1775)
0.0416% (0.0074%)
0.0420 (0.0121)
CCDcamera
0.9831 (0.0143)
14.8362 (9.0366)
3.8200% (2.1130%)
0.1420 (0.0820)
4
0.9979 (0.0011)
8.3701 (4.1103)
0.0078% (0.0048%)
0.0298 (0.0062)
5
0.9980 (0.0011)
4.9978 (1.9412)
0.0328% (0.0177%)
0.0322 (0.0022)
4000, 6000, 8000, 10,000, 12,000, 14,000, 16,000, 18,000, and 20,000 K.
Table 5
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Daylight Eigenvectors (Fig. 1) versus 12 Daylight Spectra Measured in the US by Lee37a
Number of Sensors
Mean GFC (SD)
Mean
(SD)
Mean Integrated Irradiance Error (SD)
Mean Fractional Deviation (SD)
3
0.9992 (0.0006)
0.2847 (0.2141)
0.0323% (0.0319%)
0.0296 (0.0145)
CCDcamera
0.9967 (0.0054)
0.1776 (0.1512)
2.2306% (2.9605%)
0.0501 (0.0472)
4
0.9995 (0.0003)
0.1136 (0.0818)
0.0110% (0.0060%)
0.0208 (0.0066)
5
0.9996 (0.0002)
0.1531 (0.1364)
0.0060% (0.0061%)
0.0181 (0.0061)
These 12 spectral daylight measurements have different CCTs from 5750 to 16,780 K and a sampling rate of 5 nm. Standard deviations are given in parentheses.
Table 6
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Daylight Eigenvectors (Fig. 1) versus Our Set of 1567 Skylight Spectra with a Sampling Rate of 5 nma
Means and Standard Deviations Obtained with Our Optimum Sensors (Table 1) and Our Skylight Eigenvectors (see Ref. 22) versus Our Set of 1567 Skylight Spectra with a Sampling Rate of 5 nm
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