CSF models as functions of spatial, luminance, and colour directions

Fitting error

Dataset Fitting error Sensitivity adjustment
castleCSF Postreceptoral contrast Combined Chromatic Pyramid of Visibility CSF castleCSF Postreceptoral contrast Combined Chromatic Pyramid of Visibility CSF
Average training 3.12 ± 0.04 [dB] 2.93 ± 0.03 [dB] 5.22 ± 0.05 [dB] N/A N/A N/A
Average testing 3.21 ± 0.14 [dB] 3.03 ± 0.09 [dB] 5.28 ± 0.26 [dB] N/A N/A N/A
modelfest 2.01 [dB] 1.35 [dB] 3.33 [dB] 0.963 0.922 1.685
hdrvdp_csf 1.67 [dB] 2.26 [dB] 7.83 [dB] 1.243 1.219 1.997
rovamo1993 3.12 [dB] 3.26 [dB] 7.11 [dB] 1.667 1.688 3.956
virsu1979 5.33 [dB] 5.81 [dB] 9.47 [dB] 1.389 1.451 3.129
wright1983 1.43 [dB] 1.03 [dB] 5.90 [dB] 1.045 1.062 1.689
colorfest 2.55 [dB] 2.17 [dB] 5.61 [dB] 0.953 0.924 2.672
hdr_csf 3.03 [dB] 2.64 [dB] 8.09 [dB] 1.000 1.000 1.000
kim2013 3.58 [dB] 3.09 [dB] 4.57 [dB] 1.073 1.155 2.640
five_centres 3.32 [dB] 3.04 [dB] 3.70 [dB] 1.121 1.420 7.182
lucassen2018 2.79 [dB] 2.88 [dB] 2.93 [dB] 0.601 0.730 2.471
hdr_csf_disc 2.84 [dB] 2.47 [dB] 5.46 [dB] 1.516 1.389 1.481
vanderHorst1969_b 3.08 [dB] 3.12 [dB] 4.76 [dB] 0.635 0.711 2.166

Model comparison statistics

Model No. of free parameters Sum of Square Errors (SS) Degrees of freedom (df) F-test AIC
F-statistic p-value
castleCSF (Reference Model) 53 26.135 1002 N/A N/A -3795.43
Postreceptoral contrast 31 22.925 1024 -5.59307 1.0000 -3977.66
Combined Chromatic Pyramid of Visibility CSF 15 71.938 1040 46.2136 0.0000 ✓ -2803.19

We use AIC and F-test to test whether the difference in fitting error is statistically significant at alpha=0.05 level. Both statistical metrics take the number of optimized parameters into account.

F-test: For F-test, we compare the fitting results from castleCSF with those of other models. The F-static is calculated using the residual sum of squares and degrees of freedom (number of data points - number of optimized parameters) from both models. The corresponding p-value indicates whether or not the null hypothesis is rejected, where H0: the castleCSF does not provide significant better fit than the other model. The p-values less than 0.05 indicates that castleCSF provides a better fit to the data at the significance level of 0.05 (marked with ✓). We performed the F-test for all individual datasets as well as for all datasets combined. For smaller datasets, where the number of data points are comparable to the number of model parameters, F-test can not provide any results since it indicates there is more variance within the models' fits than between.

AIC: Akaike information criterion is a statistical estimator of prediction error and relative quality of the models, which accounts for the number of parameters of each model. The model with the lower AIC score is considered to be better and with a good balance of error value and the number of parameters.

The sensitivity adjustment column contains a multiplier that is used to adjust the sensitivity of each datasets. It corresponds to sd in the paper.

Model parameters

castleCSF
M_lms2acc = 1.0000 1.0000 0 1.0000 -2.3112 0 -1.0000 -1.0000 50.9875 p.rg.sigma_sust = 16.541; p.rg.beta_sust = 1.15549; p.rg.ch_sust.S_max = [ 560.666 18.7251 0.547119 ]; p.rg.ch_sust.f_max = 0.0383326; p.rg.ch_sust.bw = 2.09558; p.rg.A_0 = 1105.11; p.rg.f_0 = 0.100771; p.rg.ecc_drop = 0.0591431; p.rg.ecc_drop_nasal = 2.89648e-05; p.rg.ecc_drop_f = 2.04986e-69; p.rg.ecc_drop_f_nasal = 0.180118; p.yv.sigma_sust = 7.9187; p.yv.beta_sust = 0.999363; p.yv.ch_sust.S_max = [ 94.9895 23.7932 0.49588 ]; p.yv.ch_sust.f_max = 0.0278077; p.yv.ch_sust.bw = 1.97672; p.yv.A_0 = 2.65159e+07; p.yv.f_0 = 0.000558562; p.yv.ecc_drop = 0.00357397; p.yv.ecc_drop_nasal = 5.85804e-141; p.yv.ecc_drop_f = 0.0080878; p.yv.ecc_drop_f_nasal = 0.0147658; p.ach.ach_sust.S_max = [ 53.2679 0.714156 0.233877 4.0378e-07 1.22586e+10 ]; p.ach.ach_sust.f_max = [ 2.21057 444.996 0.224904 ]; p.ach.ach_sust.bw = 0.000211859; p.ach.ach_sust.a = 0.0353055; p.ach.ach_sust.A_0 = 157.103; p.ach.ach_sust.f_0 = 0.702338; p.ach.ach_trans.S_max = [ 0.194173 2772.81 ]; p.ach.ach_trans.f_max = 0.000336157; p.ach.ach_trans.bw = 3.5291; p.ach.ach_trans.a = 0.000241177; p.ach.ach_trans.A_0 = 2.96528; p.ach.ach_trans.f_0 = 2.39983; p.ach.sigma_trans = 0.07374; p.ach.sigma_sust = 10.467; p.ach.omega_trans_sl = 2.58795; p.ach.omega_trans_c = 4.45472; p.ach.ecc_drop = 0.0259781; p.ach.ecc_drop_nasal = 0.0452708; p.ach.ecc_drop_f = 0.0217926; p.ach.ecc_drop_f_nasal = 0.0068348; Parameters for Ach component: p.ach_sust.S_max = [ 53.2679 0.714156 0.233877 4.0378e-07 1.22586e+10 ]; p.ach_sust.f_max = [ 2.21057 444.996 0.224904 ]; p.ach_sust.bw = 0.000211859; p.ach_sust.a = 0.0353055; p.ach_trans.S_max = [ 0.194173 2772.81 ]; p.ach_trans.f_max = 0.000336157; p.ach_trans.bw = 3.5291; p.ach_trans.a = 0.000241177; p.ach_trans.A_0 = 2.96528; p.ach_trans.f_0 = 2.39983; p.sigma_trans = 0.07374; p.sigma_sust = 10.467; p.omega_trans_sl = 2.58795; p.omega_trans_c = 4.45472; p.ecc_drop = 0.0259781; p.ecc_drop_nasal = 0.0452708; p.ecc_drop_f = 0.0217926; p.ecc_drop_f_nasal = 0.0068348; Parameters for RG component: p.ch_sust.S_max = [ 560.666 18.7251 0.547119 ]; p.ch_sust.f_max = 0.0383326; p.ch_sust.bw = 2.09558; p.A_0 = 1105.11; p.f_0 = 0.100771; p.sigma_sust = 16.541; p.beta_sust = 1.15549; p.ecc_drop = 0.0591431; p.ecc_drop_nasal = 2.89648e-05; p.ecc_drop_f = 2.04986e-69; p.ecc_drop_f_nasal = 0.180118; Parameters for YV component: p.ch_sust.S_max = [ 94.9895 23.7932 0.49588 ]; p.ch_sust.f_max = 0.0278077; p.ch_sust.bw = 1.97672; p.A_0 = 2.65159e+07; p.f_0 = 0.000558562; p.sigma_sust = 7.9187; p.beta_sust = 0.999363; p.ecc_drop = 0.00357397; p.ecc_drop_nasal = 5.85804e-141; p.ecc_drop_f = 0.0080878; p.ecc_drop_f_nasal = 0.0147658;
Postreceptoral contrast
p.ach.S_max = [ 20499.6 21.4108 0.243117 276405 0.00303308 ]; p.ach.f_max = [ 1.62832 268.674 0.215424 ]; p.ach.bw = 0.185226; p.ach.gamma = 0.904648; p.ach.Ac_prime = 46.6295; p.rg.S_max = [ 3900.41 13.767 0.645983 ]; p.rg.f_max = 0.0116232; p.rg.bw = 2.53994; p.rg.gamma = 1.44612; p.rg.Ac_prime = 2.24541; p.yv.S_max = [ 83666.7 22.4433 0.511101 ]; p.yv.f_max = 0.0012374; p.yv.bw = 2.82036; p.yv.gamma = 1.36762; p.yv.Ac_prime = 1.92925; p.colmat = [ 0.0138517 4.22238 2.19247 0.103 3.89314e-06 0.000153138 ];
Combined Chromatic Pyramid of Visibility CSF
p.P_ach = 1.28333; p.c_w_neg_ach = 0.031704; p.c_f_neg_ach = 0.0420185; p.c_i_ach = 0.23583; p.c_0_ach = 0.510208; p.P_rg = 1.12999; p.c_w_neg_rg = 0.0481838; p.c_f_neg_rg = 0.0670723; p.c_i_rg = 0.278649; p.c_0_rg = 0.610102; p.P_yv = 0.030551; p.c_w_neg_yv = 0.039198; p.c_f_neg_yv = 0.192522; p.c_i_yv = 0.296315; p.c_0_yv = 0.0662066; Parameters for Ach subclass Parameters for RG subclass Parameters for YV subclass

↸CSF model: castleCSF

L+M
L-M
S-(L+M)

↸CSF model: Postreceptoral contrast

L+M
L-M
S-(L+M)

↸CSF model: Combined Chromatic Pyramid of Visibility CSF

L+M
L-M
S-(L+M)

Legend

To keep the plots legible, only up to 3 models are plotted.

↸Dataset: [modelfest] ModelFest

Achromatic CSF as a function of frequency

↸Dataset: [hdrvdp_csf] HDR-VDP CSF

Achromatic CSF as a function of frequency

Achromatic CSF as a function of size

↸Dataset: [rovamo1993] Rovamo et al. 1993

CSF as the funcation of stimulus area

CSF as the function of spatial frequency

↸Dataset: [virsu1979] Virsu & Rovamo 1979

Contrast sensitivity of central and peripheral vision as a function of spatial frequency and eccentricity

↸Dataset: [wright1983] Wright and Johnson 1983

CSF as function of eccentricity

↸Dataset: [colorfest] ColorFest

Chromatic CSF as a function of frequency

↸Dataset: [hdr_csf] High Dynamic Range CSF

CSF as the function of frequency at different luminance levels (fixed number of cycles)

↸Dataset: [kim2013] Chromatic CSF [Kim et al. 2013]

Chromatic CSF as a function of frequency

↸Dataset: [five_centres] Five centres [Xu et al. 2020]

Chromatic CSF as ellipses

Chromatic CSF as a function of frequency

↸Dataset: [lucassen2018] Lucassen et al. 2018

Lucassen2018 CSF sensitivity vs spatial frequency

↸Dataset: [hdr_csf_disc] High Dynamic Range Disc CSF

CSF as the function of size at different luminance levels

↸Dataset: [vanderHorst1969_b] Van der Horst & Bouman 1969

Static chromatic contrast thresholds as a function of spatial frequency

Static chromatic contrast thresholds as function of luminance

Chromatic contrast thresholds for travelling sine waves as function of spatial frequency

Chromatic contrast thresholds for travelling sine waves as function of temporal frequency