Javier Hernández-Andrés, Javier Romero, Antonio García-Beltrán, and Juan L. Nieves, "Testing linear models on spectral daylight measurements," Appl. Opt. 37, 971-977 (1998)
We have analyzed the results of the reconstruction quality of 252
daylight spectral curves measured at Granada, Spain, using four bases
obtained from measurements in different areas of the world. For
these reconstructions we used two different methods (orthogonality of
characteristic vectors and chromaticity coordinates) to study the
influence of the wavelength range and spectral resolution. The
reconstruction method from chromaticity coordinates presents
difficulties for the spectral recovery of daylight spectral power
distributions regardless of the basis used. The orthogonality
method makes clear that the best bases were those proposed by the CIE,
but more than two characteristic CIE vectors were needed for good
reconstruction.
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Average GFC for the 252 Curves from x,
y Data with Different Bases, Spectral Ranges, and Spectral
Resolutionsa
Studies
Spectral Range and Spectral Resolution
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
CIE
0.99615
0.99615
0.99538
0.99505
0.99632
0.99638
0.99500
0.99476
Sastri and Das
0.98701
0.98851
n.a.
n.a.
0.98755
0.98984
n.a.
n.a.
Bendigo
0.99561
0.99586
0.99434
0.99462
0.99658
0.99696
n.a.
n.a.
Coburg
0.99294
0.99338
0.95342
0.95891
0.99354
0.99416
n.a.
n.a.
n.a., not available.
Table 3
Percentage of Reconstructions (from x,
y Data) that Surpass a Given GFC
Valuea
Studies
Spectral Range
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
≥0.99
≥0.999
≥0.99
≥0.999
≥0.99
≥0.999
≥0.99
≥0.999
CIE
96.4
0
95.2
0
96.8
0
94.4
0
Sastri and Das
45.6
0
n.a.
n.a.
66.7
0
n.a.
n.a.
Bendigo
91.7
34.5
88.5
28.6
94.8
42.5
n.a.
n.a.
Coburg
82.9
37.7
23.0
1.6
83.7
40.5
n.a.
n.a.
Spectral resolution: 5 nm; n.a., not
available.
Table 4
Average GFC for the 252 Curves by use of
Orthogonalitya
Studies
Spectral Range and Spectral Resolution
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
CIE [2]
0.99744
0.99795
0.99729
0.99734
0.99855
0.99848
0.99751
0.99715
CIE [3]
0.99963
0.99951
CIE [4]
0.99977
0.99964
Sastri and Das [2]
0.99088
0.99407
n.a.
n.a.
0.99115
0.99441
n.a.
n.a.
Sastri and Das [3]
0.99124
0.99435
0.99150
0.99467
Sastri and Das [4]
0.99129
0.99440
0.99155
0.99469
Bendigo [2]
0.73422
0.72321
0.76604
0.75941
0.74275
0.73626
n.a.
n.a.
Bendigo [3]
0.62875
0.60401
0.75072
0.74807
0.64284
0.62713
Coburg [2]
0.90833
0.91006
0.93758
0.94555
0.90928
0.91106
n.a.
n.a.
Coburg [3]
0.49860
0.49729
0.67643
0.69886
0.51619
0.52204
Coburg [4]
0.48197
0.48069
0.65940
0.67416
0.50235
0.50993
Granada [2]
0.99932
0.99933
0.99910
0.99910
0.99939
0.99942
0.99904
0.99904
Granada [3]
0.99978
0.99979
0.99959
0.99960
0.99980
0.99981
0.99954
0.99954
Granada [4]
0.99994
0.99994
0.99984
0.99984
0.99995
0.99995
0.99982
0.99982
Granada [5]
0.99996
0.99996
0.99991
0.99991
0.99998
0.99998
0.99990
0.99990
Granada [6]
0.99998
0.99998
0.99996
0.99996
0.99998
0.99999
0.99996
0.99996
For bases formed by characteristic
vectors, the number of characteristic vectors used is indicated in
brackets (without including the mean vector); in the Granada basis,
the number in brackets indicates the number of eigenvectors. When
we calculated the GFC in a spectral range smaller than that used for
the inner product, the results are in italic; n.a., not available.
Table 5
Percentage of Reconstructions from Orthogonality
Properties that Surpass a Given GFC Valuea
Studies
Spectral Range
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
CIE [2]
95.2
71.0
0
95.2
33.3
0
96.8
81.0
0
95.2
11.1
0
CIE [3]
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
100
92.9
0
n.a.
n.a.
n.a.
CIE [4]
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
100
99.2
0
n.a.
n.a.
n.a.
Sastri and Das [2]
97.6
0
0
n.a.
n.a.
n.a.
98.8
0
0
n.a.
n.a.
n.a.
Sastri and Das [3]
99.6
0
0
n.a.
n.a.
n.a.
99.6
0
0
n.a.
n.a.
n.a.
Sastri and Das [4]
99.6
0
0
n.a.
n.a.
n.a.
99.6
0
0
n.a.
n.a.
n.a.
Granada [2]
98.8
83.7
27.0
98.8
79.0
7.1
99.2
87.7
35.3
99.2
78.2
5.2
Granada [3]
99.6
97.6
61.5
99.2
93.7
23.4
100
97.6
65.1
99.6
90.5
19.4
Granada [4]
100
99.2
90.1
99.6
99.2
64.3
100
99.6
94.8
99.6
98.8
59.1
Granada [5]
100
99.2
95.6
100
99.6
84.1
100
100
98.0
100
99.6
80.2
Granada [6]
100
100
98.8
100
100
94.8
100
100
98.8
100
100
94.0
Spectral resolution: 5 nm; n.a., not
available. For bases formed by characteristic vectors, the number
of characteristic vectors used is indicated in brackets (without
including the mean vector); in the Granada basis, the number in
brackets indicates the number of eigenvectors. When we calculated
the GFC in a spectral range smaller than that used for the inner
product, the results are in italic.
Average GFC for the 252 Curves from x,
y Data with Different Bases, Spectral Ranges, and Spectral
Resolutionsa
Studies
Spectral Range and Spectral Resolution
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
CIE
0.99615
0.99615
0.99538
0.99505
0.99632
0.99638
0.99500
0.99476
Sastri and Das
0.98701
0.98851
n.a.
n.a.
0.98755
0.98984
n.a.
n.a.
Bendigo
0.99561
0.99586
0.99434
0.99462
0.99658
0.99696
n.a.
n.a.
Coburg
0.99294
0.99338
0.95342
0.95891
0.99354
0.99416
n.a.
n.a.
n.a., not available.
Table 3
Percentage of Reconstructions (from x,
y Data) that Surpass a Given GFC
Valuea
Studies
Spectral Range
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
≥0.99
≥0.999
≥0.99
≥0.999
≥0.99
≥0.999
≥0.99
≥0.999
CIE
96.4
0
95.2
0
96.8
0
94.4
0
Sastri and Das
45.6
0
n.a.
n.a.
66.7
0
n.a.
n.a.
Bendigo
91.7
34.5
88.5
28.6
94.8
42.5
n.a.
n.a.
Coburg
82.9
37.7
23.0
1.6
83.7
40.5
n.a.
n.a.
Spectral resolution: 5 nm; n.a., not
available.
Table 4
Average GFC for the 252 Curves by use of
Orthogonalitya
Studies
Spectral Range and Spectral Resolution
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
10 nm
5 nm
CIE [2]
0.99744
0.99795
0.99729
0.99734
0.99855
0.99848
0.99751
0.99715
CIE [3]
0.99963
0.99951
CIE [4]
0.99977
0.99964
Sastri and Das [2]
0.99088
0.99407
n.a.
n.a.
0.99115
0.99441
n.a.
n.a.
Sastri and Das [3]
0.99124
0.99435
0.99150
0.99467
Sastri and Das [4]
0.99129
0.99440
0.99155
0.99469
Bendigo [2]
0.73422
0.72321
0.76604
0.75941
0.74275
0.73626
n.a.
n.a.
Bendigo [3]
0.62875
0.60401
0.75072
0.74807
0.64284
0.62713
Coburg [2]
0.90833
0.91006
0.93758
0.94555
0.90928
0.91106
n.a.
n.a.
Coburg [3]
0.49860
0.49729
0.67643
0.69886
0.51619
0.52204
Coburg [4]
0.48197
0.48069
0.65940
0.67416
0.50235
0.50993
Granada [2]
0.99932
0.99933
0.99910
0.99910
0.99939
0.99942
0.99904
0.99904
Granada [3]
0.99978
0.99979
0.99959
0.99960
0.99980
0.99981
0.99954
0.99954
Granada [4]
0.99994
0.99994
0.99984
0.99984
0.99995
0.99995
0.99982
0.99982
Granada [5]
0.99996
0.99996
0.99991
0.99991
0.99998
0.99998
0.99990
0.99990
Granada [6]
0.99998
0.99998
0.99996
0.99996
0.99998
0.99999
0.99996
0.99996
For bases formed by characteristic
vectors, the number of characteristic vectors used is indicated in
brackets (without including the mean vector); in the Granada basis,
the number in brackets indicates the number of eigenvectors. When
we calculated the GFC in a spectral range smaller than that used for
the inner product, the results are in italic; n.a., not available.
Table 5
Percentage of Reconstructions from Orthogonality
Properties that Surpass a Given GFC Valuea
Studies
Spectral Range
300–700 nm
300–780 nm
330–700 nm
300–830 nm
Basis
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
≥0.99
≥0.999
≥0.9999
CIE [2]
95.2
71.0
0
95.2
33.3
0
96.8
81.0
0
95.2
11.1
0
CIE [3]
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
100
92.9
0
n.a.
n.a.
n.a.
CIE [4]
n.a.
n.a.
n.a.
n.a.
n.a.
n.a.
100
99.2
0
n.a.
n.a.
n.a.
Sastri and Das [2]
97.6
0
0
n.a.
n.a.
n.a.
98.8
0
0
n.a.
n.a.
n.a.
Sastri and Das [3]
99.6
0
0
n.a.
n.a.
n.a.
99.6
0
0
n.a.
n.a.
n.a.
Sastri and Das [4]
99.6
0
0
n.a.
n.a.
n.a.
99.6
0
0
n.a.
n.a.
n.a.
Granada [2]
98.8
83.7
27.0
98.8
79.0
7.1
99.2
87.7
35.3
99.2
78.2
5.2
Granada [3]
99.6
97.6
61.5
99.2
93.7
23.4
100
97.6
65.1
99.6
90.5
19.4
Granada [4]
100
99.2
90.1
99.6
99.2
64.3
100
99.6
94.8
99.6
98.8
59.1
Granada [5]
100
99.2
95.6
100
99.6
84.1
100
100
98.0
100
99.6
80.2
Granada [6]
100
100
98.8
100
100
94.8
100
100
98.8
100
100
94.0
Spectral resolution: 5 nm; n.a., not
available. For bases formed by characteristic vectors, the number
of characteristic vectors used is indicated in brackets (without
including the mean vector); in the Granada basis, the number in
brackets indicates the number of eigenvectors. When we calculated
the GFC in a spectral range smaller than that used for the inner
product, the results are in italic.