CSF models as functions of spatial and temporal frequency, luminance, area and eccentricity

Fitting error

Dataset Fitting error Sensitivity adjustment
castleCSF original Barten's CSF (1999) VDP CSF stelaCSF castleCSF original Barten's CSF (1999) VDP CSF stelaCSF
Average training 3.14 ± 0.08 [dB] 4.42 ± 0.07 [dB] 10.81 ± 0.12 [dB] 4.19 ± 0.09 [dB] N/A N/A N/A N/A
Average testing 3.33 ± 0.23 [dB] 4.50 ± 0.31 [dB] 10.86 ± 0.49 [dB] 4.37 ± 0.41 [dB] N/A N/A N/A N/A
modelfest 2.67 [dB] 3.09 [dB] 2.79 [dB] 2.77 [dB] 1.019 1.251 1.346 1.240
hdrvdp_csf 2.36 [dB] 4.89 [dB] 5.39 [dB] 2.23 [dB] 1.351 1.456 2.914 1.645
rovamo1993 2.80 [dB] 3.93 [dB] 7.43 [dB] 2.98 [dB] 1.664 2.294 5.258 1.990
laird2006 5.70 [dB] 6.73 [dB] 7.40 [dB] 5.97 [dB] 0.873 0.579 0.455 0.809
snowden1995 3.99 [dB] 4.69 [dB] 14.93 [dB] 5.74 [dB] 1.172 0.981 1.815 1.150
robson1966 2.33 [dB] 2.71 [dB] 10.77 [dB] 2.87 [dB] 1.097 0.846 1.717 1.107
virsu1979 4.06 [dB] 5.53 [dB] 10.97 [dB] 5.80 [dB] 1.601 2.414 1.950 1.360
virsu1982 2.42 [dB] 5.77 [dB] 9.57 [dB] 4.14 [dB] 0.965 1.699 0.624 1.118
wright1983 2.27 [dB] 3.63 [dB] 10.27 [dB] 2.42 [dB] 0.823 1.157 1.198 0.867
colorfest 3.29 [dB] 3.48 [dB] 5.39 [dB] 3.42 [dB] 0.889 0.794 1.351 0.857
hdr_csf 3.45 [dB] 4.13 [dB] 5.16 [dB] 3.59 [dB] 1.000 1.000 1.000 1.000
hdr_csf_disc 2.07 [dB] 3.15 [dB] 1.14 [dB] 1.65 [dB] 1.189 1.071 1.087 0.921
hansen2009 2.86 [dB] 8.43 [dB] 3.01 [dB] 10.17 [dB] 1.291 2.596 0.753 1.943

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 19.625 726 N/A N/A -2761.64
original Barten's CSF (1999) 13 38.461 766 17.4194 0.0000 ✓ -2317.52
VDP CSF 6 228.460 773 164.37 0.0000 ✓ -943.561
stelaCSF 21 35.001 758 17.775 0.0000 ✓ -2374.94

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 = 16.5458; p.rg.beta_sust = 1.15414; p.rg.ch_sust.S_max = [ 463.398 38.8259 0.477632 ]; p.rg.ch_sust.f_max = 0.0119347; p.rg.ch_sust.bw = 1.03471; p.rg.A_0 = 2827.45; p.rg.f_0 = 0.0679329; p.rg.ecc_drop = 0.0591361; p.rg.ecc_drop_nasal = 2.89648e-05; p.rg.ecc_drop_f = 2.04986e-69; p.rg.ecc_drop_f_nasal = 0.180122; p.yv.sigma_sust = 337.471; p.yv.beta_sust = 0.00899757; p.yv.ch_sust.S_max = [ 87.0195 3.34009 0.42683 ]; p.yv.ch_sust.f_max = 0.00960582; p.yv.ch_sust.bw = 2.59622; p.yv.A_0 = 4.27496e+07; p.yv.f_0 = 0.00145138; p.yv.ecc_drop = 0.00365422; p.yv.ecc_drop_nasal = 5.85804e-141; p.yv.ecc_drop_f = 0.00903915; p.yv.ecc_drop_f_nasal = 0.0125592; p.ach.ach_sust.S_max = [ 54.6746 2.2499 0.183109 4.51332e-07 1.34951e+10 ]; p.ach.ach_sust.f_max = [ 1.91731 116.884 0.259557 ]; p.ach.ach_sust.bw = 0.000217476; p.ach.ach_sust.a = 0.12163; p.ach.ach_sust.A_0 = 157.103; p.ach.ach_sust.f_0 = 0.702338; p.ach.ach_trans.S_max = [ 0.213643 2468.52 ]; p.ach.ach_trans.f_max = 0.000228223; p.ach.ach_trans.bw = 2.71853; p.ach.ach_trans.a = 0.000241177; p.ach.ach_trans.A_0 = 7.91945; p.ach.ach_trans.f_0 = 2.24404; p.ach.sigma_trans = 0.0768356; p.ach.sigma_sust = 10.6147; p.ach.omega_trans_sl = 2.38362; p.ach.omega_trans_c = 4.94128; p.ach.ecc_drop = 0.026449; p.ach.ecc_drop_nasal = 0.0610948; p.ach.ecc_drop_f = 0.0181267; p.ach.ecc_drop_f_nasal = 0.00589758; Parameters for Ach component: p.ach_sust.S_max = [ 54.6746 2.2499 0.183109 4.51332e-07 1.34951e+10 ]; p.ach_sust.f_max = [ 1.91731 116.884 0.259557 ]; p.ach_sust.bw = 0.000217476; p.ach_sust.a = 0.12163; p.ach_trans.S_max = [ 0.213643 2468.52 ]; p.ach_trans.f_max = 0.000228223; p.ach_trans.bw = 2.71853; p.ach_trans.a = 0.000241177; p.ach_trans.A_0 = 7.91945; p.ach_trans.f_0 = 2.24404; p.sigma_trans = 0.0768356; p.sigma_sust = 10.6147; p.omega_trans_sl = 2.38362; p.omega_trans_c = 4.94128; p.ecc_drop = 0.026449; p.ecc_drop_nasal = 0.0610948; p.ecc_drop_f = 0.0181267; p.ecc_drop_f_nasal = 0.00589758; Parameters for RG component: p.ch_sust.S_max = [ 463.398 38.8259 0.477632 ]; p.ch_sust.f_max = 0.0119347; p.ch_sust.bw = 1.03471; p.A_0 = 2827.45; p.f_0 = 0.0679329; p.sigma_sust = 16.5458; p.beta_sust = 1.15414; p.ecc_drop = 0.0591361; p.ecc_drop_nasal = 2.89648e-05; p.ecc_drop_f = 2.04986e-69; p.ecc_drop_f_nasal = 0.180122; Parameters for YV component: p.ch_sust.S_max = [ 87.0195 3.34009 0.42683 ]; p.ch_sust.f_max = 0.00960582; p.ch_sust.bw = 2.59622; p.A_0 = 4.27496e+07; p.f_0 = 0.00145138; p.sigma_sust = 337.471; p.beta_sust = 0.00899757; p.ecc_drop = 0.00365422; p.ecc_drop_nasal = 5.85804e-141; p.ecc_drop_f = 0.00903915; p.ecc_drop_f_nasal = 0.0125592;
original Barten's CSF (1999)
p.k = 6.85118; p.eta0 = 0.0354405; p.sigma0 = 0.52239; p.eg = 97.8251; p.u00 = 3.21885; p.Phi00 = 3e-08; p.T = 0.0661693; p.Xmax0 = 7.37859; p.Nmax = 15.7171; p.tau10 = 0.0441063; p.tau20 = 0.03434; p.n1 = 4.46403; p.n2 = 2.53013;
VDP CSF
p.P = 133.9; p.ob = 1.44757; p.k = 0.206707; p.epsilon = 1.39533; p.a_l_m = 1.967; p.b_l_m = 13.5589;
stelaCSF
p.ach_sust.S_max = [ 44.4763 0.713964 0.277857 7.54866e-07 8.50217e+09 ]; p.ach_sust.f_max = [ 1.55909 103.604 0.251815 ]; p.ach_sust.bw = 0.177674; p.ach_sust.a = 1.74265e-102; p.ach_trans.S_max = [ 0.411266 211.504 ]; p.ach_trans.f_max = 0.00115423; p.ach_trans.bw = 2.52628; p.ach_trans.a = 0.000273289; p.sigma_trans = 0.156257; p.sigma_sust = 15.8409; p.ecc_drop = 0.0277526; p.ecc_drop_nasal = 0.0275591; p.ecc_drop_f = 0.0175072; p.ecc_drop_f_nasal = 0.00987064;

↸CSF model: castleCSF

L+M

↸CSF model: original Barten's CSF (1999)

L+M

↸CSF model: VDP CSF

L+M

↸CSF model: stelaCSF

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: [laird2006] Laird et al. 2006

Achromatic CSF as a function of temporal frequency for different spatial frequencies

↸Dataset: [snowden1995] Snowden et al. 1995

Temporal contrast sensitivity at different spatial frequencies and luminance levels

↸Dataset: [robson1966] Robson 1966

Spatial CSF for different temporal frequencies

Temporal CSF for different spatial frequencies

↸Dataset: [virsu1979] Virsu & Rovamo 1979

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

↸Dataset: [virsu1982] Virsu et al. 1982

Contrast sensitivity as the function of frequency

↸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: [hdr_csf_disc] High Dynamic Range Disc CSF

CSF as the function of size at different luminance levels

↸Dataset: [hansen2009] Hansen et al. 2009

CSF as the function of eccentricity