Cone-excitation ratios for pairs of surfaces are almost invariant under changes in illumination and offer a possible basis for color constancy [Proc. R. Soc. London Ser. B 257, 115 (1994)]. We extend this idea to the perception of transparency on the basis of the close analogy between the changes in color signals that occur for surfaces when the illumination changes and the changes in color signals when the surfaces are covered by a filter. This study presents measurements and simulations to investigate the conditions under which cone-excitation ratios are statistically invariant for physically transparent systems. The invariance breaks down when the spectral transmission of the filters is low at some or all wavelengths. We suggest that cone-excitation ratios might be useful to define the stimulus conditions necessary for the perception of transparency.
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Spectral Reflectance Factors for Surfaces Used to Generate Data Shown in Table 1
nm
White
Gray
Red
Green
Blue
Yellow
400
0.3647
0.2253
0.1496
0.0907
0.4535
0.1106
410
0.6243
0.2505
0.1404
0.0900
0.5075
0.1082
420
0.8121
0.2666
0.1315
0.0909
0.5593
0.1054
430
0.9221
0.2715
0.1231
0.0921
0.6103
0.1004
440
0.9720
0.2714
0.1150
0.0977
0.6561
0.0985
450
0.9667
0.2727
0.1071
0.1101
0.6926
0.1045
460
0.9392
0.2735
0.0999
0.1327
0.7199
0.1169
470
0.9295
0.2741
0.0940
0.1704
0.7377
0.1358
480
0.9215
0.2742
0.0892
0.2209
0.7432
0.1608
490
0.9118
0.2740
0.0848
0.2921
0.7323
0.1899
500
0.9020
0.2732
0.0829
0.3580
0.7084
0.2277
510
0.8920
0.2713
0.0872
0.3944
0.6737
0.2794
520
0.8838
0.2697
0.0918
0.4031
0.6277
0.3421
530
0.8794
0.2702
0.0888
0.3697
0.5690
0.4184
540
0.8773
0.2713
0.0889
0.3178
0.5039
0.4981
550
0.8758
0.2718
0.0974
0.2660
0.4382
0.5710
560
0.8765
0.2726
0.1222
0.2153
0.3732
0.6402
570
0.8810
0.2741
0.1741
0.1724
0.3095
0.7084
580
0.8876
0.2756
0.2502
0.1385
0.2551
0.7666
590
0.8959
0.2766
0.3593
0.1182
0.2210
0.8037
600
0.9032
0.2769
0.4785
0.1068
0.1994
0.8292
610
0.9056
0.2761
0.5907
0.1010
0.1862
0.8482
620
0.9065
0.2748
0.6879
0.1006
0.1811
0.8616
630
0.9080
0.2738
0.7532
0.1048
0.1842
0.8721
640
0.9103
0.2727
0.7971
0.1109
0.1903
0.8797
650
0.9158
0.2718
0.8219
0.1156
0.1936
0.8852
660
0.9211
0.2708
0.8354
0.1196
0.1959
0.8890
670
0.9227
0.2696
0.8466
0.1205
0.1944
0.8921
680
0.9226
0.2681
0.8540
0.1250
0.1984
0.8940
690
0.9215
0.2661
0.8593
0.1398
0.2174
0.8946
700
0.9192
0.2637
0.8620
0.1625
0.2482
0.8939
Table 1
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Filter versus the Same Surfaces Viewed Directly for Three N.D. Filters and Four Color Filters
Coefficient of determination is given in parentheses.
Table 2
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Filter (with and versus the Same Surfaces Viewed Directly for Achromatic and Chromatic Filtersa
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 3
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter and versus the Same Surfaces Viewed Directly for Different Values of ra
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 4
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter versus the Same Surfaces Viewed Directly for Different Values of a
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 5
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter versus the Same Surfaces Viewed Directly with Increased Number n of Absorption Peaks in the Filtera
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Tables (6)
Table 6
Spectral Reflectance Factors for Surfaces Used to Generate Data Shown in Table 1
nm
White
Gray
Red
Green
Blue
Yellow
400
0.3647
0.2253
0.1496
0.0907
0.4535
0.1106
410
0.6243
0.2505
0.1404
0.0900
0.5075
0.1082
420
0.8121
0.2666
0.1315
0.0909
0.5593
0.1054
430
0.9221
0.2715
0.1231
0.0921
0.6103
0.1004
440
0.9720
0.2714
0.1150
0.0977
0.6561
0.0985
450
0.9667
0.2727
0.1071
0.1101
0.6926
0.1045
460
0.9392
0.2735
0.0999
0.1327
0.7199
0.1169
470
0.9295
0.2741
0.0940
0.1704
0.7377
0.1358
480
0.9215
0.2742
0.0892
0.2209
0.7432
0.1608
490
0.9118
0.2740
0.0848
0.2921
0.7323
0.1899
500
0.9020
0.2732
0.0829
0.3580
0.7084
0.2277
510
0.8920
0.2713
0.0872
0.3944
0.6737
0.2794
520
0.8838
0.2697
0.0918
0.4031
0.6277
0.3421
530
0.8794
0.2702
0.0888
0.3697
0.5690
0.4184
540
0.8773
0.2713
0.0889
0.3178
0.5039
0.4981
550
0.8758
0.2718
0.0974
0.2660
0.4382
0.5710
560
0.8765
0.2726
0.1222
0.2153
0.3732
0.6402
570
0.8810
0.2741
0.1741
0.1724
0.3095
0.7084
580
0.8876
0.2756
0.2502
0.1385
0.2551
0.7666
590
0.8959
0.2766
0.3593
0.1182
0.2210
0.8037
600
0.9032
0.2769
0.4785
0.1068
0.1994
0.8292
610
0.9056
0.2761
0.5907
0.1010
0.1862
0.8482
620
0.9065
0.2748
0.6879
0.1006
0.1811
0.8616
630
0.9080
0.2738
0.7532
0.1048
0.1842
0.8721
640
0.9103
0.2727
0.7971
0.1109
0.1903
0.8797
650
0.9158
0.2718
0.8219
0.1156
0.1936
0.8852
660
0.9211
0.2708
0.8354
0.1196
0.1959
0.8890
670
0.9227
0.2696
0.8466
0.1205
0.1944
0.8921
680
0.9226
0.2681
0.8540
0.1250
0.1984
0.8940
690
0.9215
0.2661
0.8593
0.1398
0.2174
0.8946
700
0.9192
0.2637
0.8620
0.1625
0.2482
0.8939
Table 1
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Filter versus the Same Surfaces Viewed Directly for Three N.D. Filters and Four Color Filters
Coefficient of determination is given in parentheses.
Table 2
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Filter (with and versus the Same Surfaces Viewed Directly for Achromatic and Chromatic Filtersa
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 3
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter and versus the Same Surfaces Viewed Directly for Different Values of ra
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 4
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter versus the Same Surfaces Viewed Directly for Different Values of a
Each entry is based on 1000 random samples of pairs of surfaces.
Coefficient of determination is given in parentheses.
Table 5
Gradient of Least-Squares Linear Fit for Plots of Ratios for Surfaces Viewed through a Chromatic Filter versus the Same Surfaces Viewed Directly with Increased Number n of Absorption Peaks in the Filtera