The ModelFest data set was created to provide a public source of data to test and calibrate models of foveal spatial contrast detection. It consists of contrast thresholds for 43 foveal achromatic contrast stimuli collected from each of 16 observers. We have fit these data with a variety of simple models that include one of several contrast sensitivity functions, an oblique effect, a spatial sensitivity aperture, spatial frequency channels, and nonlinear Minkowski summation. While we are able to identify one model, with particular parameters, as providing the lowest overall residual error, we also note that the differences among several good-fitting models are small. We find a strong reciprocity between the size of the spatial aperture and the value of the summation exponent: both are effective means of limiting the extent of spatial summation. The results demonstrate the power of simple models to account for the visibility of a wide variety of spatial stimuli and suggest that special mechanisms to deal with special classes of stimuli are not needed. But the results also illustrate the limited power of even this large data set to distinguish among similar competing models. We identify one model as a possible standard, suitable for simple theoretical and applied predictions.

*g*in the image was converted to luminance

*L*on the display according to the formula

*c*is the contrast of the stimulus and

*L*

_{0}is the mean luminance. In each lab,

*L*

_{0}was fixed to a value in the range 30 ± 5 cd m

^{−2}. The mathematical notation used in this paper is summarized in 3.

*L*

_{0}). Fixation guides were presented continuously in the form of “L”-shaped marks at the four corners of the stimulus image.

Index | Type | Parameters |
---|---|---|

1 | Gabor, fixed size | 1.12 cycles/degree |

2 | Gabor, fixed size | 2 cycles/degree |

3 | Gabor, fixed size | 2.83 cycles/degree |

4 | Gabor, fixed size | 4 cycles/degree |

5 | Gabor, fixed size | 5.66 cycles/degree |

6 | Gabor, fixed size | 8 cycles/degree |

7 | Gabor, fixed size | 11.3 cycles/degree |

8 | Gabor, fixed size | 16 cycles/degree |

9 | Gabor, fixed size | 22.6 cycles/degree |

10 | Gabor, fixed size | 30 cycles/degree |

11 | Gabor, fixed cycles | 2 cycles/degree, b_{x} = b_{y} = 1 octave |

12 | Gabor, fixed cycles | 4 cycles/degree, b_{x} = b_{y} = 1 octave |

13 | Gabor, fixed cycles | 8 cycles/degree, b_{x} = b_{y} = 1 octave |

14 | Gabor, fixed cycles | 16 cycles/degree, b_{x} = b_{y} = 1 octave |

15 | Gabor, elongated | 4 cycles/degree, σ_{x} = 0.05°, b_{y} = 0.5 octave |

16 | Gabor, elongated | 8 cycles/degree, σ_{x} = 0.05°, b_{y} = 0.5 octave |

17 | Gabor, elongated | 16 cycles/degree, σ_{x} = 0.05°, b_{y} = 0.5 octave |

18 | Gabor, elongated | 4 cycles/degree, b_{x} = 2 octave, b_{y} = 1 octave |

19 | Gabor, elongated | 4 cycles/degree, σ_{x} = 0.05°, b_{y} = 1 octave |

20 | Gabor, elongated | 4 cycles/degree, b_{x} = 1 octave, b_{y} = 2 octave |

21 | Gabor, elongated | 4 cycles/degree, b_{x} = 1 octave, σ_{y} = 0.5° |

22 | Compound Gabor | 2 and 2√2 cycles/degree |

23 | Compound Gabor | 2 and 4 cycles/degree |

24 | Compound Gabor | 4 and 4√2 cycles/degree |

25 | Compound Gabor | 4 and 8 cycles/degree |

26 | Gaussian | σ_{x} = σ_{y} = 30 min |

27 | Gaussian | σ_{x} = σ_{y} = 8.43 min |

28 | Gaussian | σ_{x} = σ_{y} = 2.106 min |

29 | Gaussian | σ_{x} = σ_{y} = 1.05 min |

30 | Edge × Gaussian | |

31 | Line × Gaussian | 0.5 min (1 pixel) wide horizontal line |

32 | Dipole × Gaussian | 3 pixels wide |

33 | 5 collinear Gabors | 8 cycles/degree, in phase, b_{x} = b_{y} = 1 octave, separation = 5 σ_{x} |

34 | 5 collinear Gabors | 8 cycles/degree, out of phase, b_{x} = b_{y} = 1 octave, separation = 5 σ_{x} |

35 | Binary noise | 1 × 1 min samples |

36 | Oriented Gabor | 4 cycles/degree, 45°, b_{x} = b_{y} = 1 octave |

37 | Oriented Gabor | 4 cycles/degree, 0°, b_{x} = b_{y} = 1 octave |

38 | Compound Gabor | 4 cycles/degree, 0° and 90°, b_{x} = b_{y} = 1 octave |

39 | Compound Gabor | 4 cycles/degree, 45° and 90°, b_{x} = b_{y} = 1 octave |

40 | Disk | 1/4° diameter |

41 | Bessel × Gaussian | 4 cycles/degree |

42 | Checkerboard | 4 cycles/degree fundamental |

43 | Natural image | Image of San Francisco |

_{10}(

*c*), where

*c*is contrast as defined in Equation 1. Each value has been rounded to three decimal places. This data set is provided as a supplement to this paper, as the text file modelfestbaselinedata.csv , described more completely in 2.

_{10}

*c*). In those units, each threshold is

*t*

_{s,o,r}, where the indices refer to stimulus (

*s*= 1,…,

*S*), observer (

*o*= 1,…,

*O*), and replication (

*r*= 1,…,

*R*). The mean for each observer over replications can be written

*t*

_{s,o}, and these are shown for all 16 observers in Figure 2, plotted as a function of the arbitrary index number. Each observer is represented by a different color. We write

*t*

_{o}for the mean of

*t*

_{s,o}over stimuli for each observer, and

*t*

_{s}for the mean over observers for each stimulus. The variability among observers can be represented by

_{0}for these data is 3.46 dB, indicating considerable variation among observers. Some of this variance is accounted for by the different mean sensitivities of the observers. We can construct a second measure of error,

*t*

_{o}from each threshold

*t*

_{s,o}, and the grand mean

*t*

_{0}from each stimulus mean

*t*

_{s}. This error has a value of 2.29 dB. The RMS error associated with the observers,

*t*

_{s}−

*t*

_{0}− (

*τ*

_{s}−

*τ*

_{s}), where

*τ*is the corresponding true value. If the models were correct, the models' predictions would be that the

*τ*and the RMS error of the model would be another estimate of this same standard deviation. Our best possible model RMS error is thus 0.56 dB.

^{−6}deg

^{−2}s

^{−1}. Zero dBB is defined so as to approximate the minimum visible contrast energy for a sensitive human observer (Watson, 2000; Watson, Barlow, & Robson, 1983; Watson, Borthwick, & Taylor, 1997). A virtue of the dBB unit is that it takes into account the contrast energy of the stimulus. One quick observation we may make from Figure 3B is that for the average observer, the best thresholds are about 7 dBB above (less sensitive than) the canonical “sensitive human observer.” Curiously, the ModelFest stimulus that the eye sees best is not a Gabor but a small Gaussian (stimulus 28). This differs from the classical result (Watson, Barlow, & Robson, 1983), but that result was obtained with moving rather than stationary targets.

*N*

_{y}= 256 rows and

*N*

_{x}= 256 columns. The output of the model was a contrast threshold.

_{10}ratio of thresholds for 0° and 45° oriented gratings at various spatial frequencies.

_{10}units at 25 cycles/degree. We fit a linear function (red line) to these data but truncate it when the function goes above 0 log attenuation (green line). These two lines form the frequency-dependent part of our oblique effect model. We assume in addition that at any given spatial frequency, sensitivity varies as a sinusoidal function of orientation. The resulting model for the oblique effect is then given by

*γ*= 3.48 cycles/degree,

*λ*= 13.57 cycles/degree. This function has two parameters corresponding to the frequency at which sensitivity begins to decline (

*γ*) and the slope of the linear-log decline (

*λ*). From this function we can create a discrete FIR digital oblique effect filter (OEF), as shown in Figure 7.

*r*is the distance from fixation in degrees, and

*σ*is the standard deviation of the Gaussian (which we also refer to as its size), also in degrees. This function was chosen primarily for mathematical convenience: its rate of decline is easily controlled and it never goes to zero. The peak value of the Gaussian is 1, so that the aperture defines the attenuation of sensitivity relative to that at the point of fixation. The Gaussian aperture multiplies the image produced by the CSF and OEF elements of the model. The aperture was centered on the image, which corresponds to an assumption that the observer fixated the center of each target.

*β*(see below), the channel gains were adjusted to yield approximately flat contrast sensitivity over frequency. This means that variations in contrast sensitivity over frequency are controlled primarily by the CSF. In this report, we did not vary any of the parameters of the Gabor channel component.

Number of frequencies | 11 |
---|---|

Number of orientations | 4 |

Number of phases | 2 (odd and even) |

Bandwidth | 1.4 octaves |

Highest center frequency | 30 cycles/degree |

Lowest center frequency | 0.9375 cycles/degree |

Frequency spacing | 1/2 octave |

Orientation spacing | 45° |

Pyramid sampling | Yes |

*c*

_{T}is the contrast threshold,

*r*

_{x,y}are the processed pixel values, prior to pooling, to a stimulus of unit peak contrast, and

*p*

_{x}and

*p*

_{y}are the width and height of each pixel in degrees. These latter terms are introduced to make the result independent of the specific resolution at which the calculation is performed.

*β*= 2) probability summation (

*β*∼ 3), and peak detection (

*β*= ∞) (Watson, 1979).

*q*, and the results are combined over the

*Q*channels,

*f*

_{0}that scales frequency in the high-frequency lobe, a parameter

*f*

_{1}that scales frequency in the low-frequency lobe, and a parameter

*a*that determines the weight of the low-frequency lobe. Note that

*f*

_{0}and

*f*

_{1}may also be thought of as specifying widths of subtractive center and surround components of the space domain convolution kernel corresponding to the filter (which is in turn sometimes thought of as the receptive field corresponding to the filter). Because many of the “lobe” functions (exp, Gaussian, sech) have a value of 1 at

*f*= 0, the DC gain is in these cases equal to 1 −

*a*.

CSF | Parameters | RMS error (dB) |
---|---|---|

LSI | 11 | 1.0243 |

HPmH | 5 | 1.0329 |

HPmG | 5 | 1.0468 |

YQM | 4 | 1.0694 |

EmG | 4 | 1.0755 |

LP | 4 | 1.0916 |

HmG | 4 | 1.0959 |

HmH | 4 | 1.1104 |

MS | 4 | 1.2009 |

DoG | 4 | 1.7830 |

Constant | 1 | 5.8607 |

*a*:

*β*= 2,

*β*= free,

*β*= ∞). For each tested configuration, we estimated all free parameters and recorded the residual error. We have not evaluated the fit of every possible configuration but have rather tried to understand the contribution of each component, the best version of each option, and the best obtainable overall fit.

*β*= ∞). Alternatively, we may say that it has no CSF, no pooling, no oblique effect, no aperture, and no channels. It has a single parameter (sensitivity) and assumes a target is detected whenever its peak contrast equals a certain value. While not a reasonable model, it provides a useful error benchmark, of about 8 dB, against which other fits may be compared.

*β*= ∞) is replaced with energy detection (

*β*= 2), the error is again reduced by about a factor of two to a value of about 2 dB, as shown by the third point in the series. Allowing the pooling exponent to vary (

*β*= free), which we call generalized energy, results in yet another drop in error by about a factor of two, as shown by the fourth point. The RMS error at this point is in the neighborhood of 1, which we can characterize as a “good” fit (see below).

*β*= 2, while a circular symbol indicates that

*β*was free to vary. Finally, when channels are included, a dashed line is use to connect the points.

*β*

*β*ranged between 2 and 3. Among high quality fits (error < 1.2 dB), the mean

*β*was 2.58 (

*SD*= 0.18,

*n*= 21). As we will see in greater detail below, estimates of

*β*interact with the presence and size of the spatial aperture. Without an aperture, high quality fits of

*β*average 2.7 (

*SD*= 0.02,

*n*= 7) while with an aperture the average was 2.52 (

*SD*= 0.19,

*n*= 14).

*β*do vary somewhat with the CSF. In Figure 16 we show the summation exponents

*β*estimated for the no-channel model for each CSF, plotted versus the estimated size,

*σ*. For all except the poorly fitting DoG, the variations in

*β*are modest. But the reciprocity between these two parameters

*β*and

*σ*is striking. We return to this reciprocity in the following section.

*β*from 2.39 to 2.70 (averaged over all CSF functions except DoG;

*SD*= 0.06 and 0.02, respectively). This suggests that the aperture and a higher

*β*both serve to reduce the efficiency of spatial summation. This notion is confirmed when

*β*is fixed at 2. The absence of an aperture now causes a marked increase in error (red filled squares), while the presence of an aperture yields a fit which is only slightly poorer than when

*β*is free to vary (black open squares).

*σ*, when

*β*is free or when it is fixed at 2. In the latter case, inefficient summation must rely on the aperture, so a relatively small size is estimated (

*σ*= 0.364 degrees,

*SD*= 0.003), while in the former case a

*β*greater than 2 can do the same job, so a larger aperture is found (

*σ*= 0.615 degrees,

*SD*= 0.06) (in both cases, averaged over all CSFs except DoG).

*β*and

*σ*is provided in Figure 18. Here we have fit the standard model, but fixed

*β*at a particular value between 2 and 3, and re-estimated all remaining parameters. We plot the parameter

*σ*and the error. This shows that as

*β*increases, the estimated value of size

*σ*also increases, so that at a

*β*of 3 the aperture is effectively absent. This is further evidence that

*β*and

*σ*both act to limit the efficiency of spatial summation.

*β*is free to vary, the average size of the aperture is 0.615 degrees. This corresponds to a decline by a factor of 2 in 0.724 degrees. This rate of decline is consistent with Robson and Graham's rule at a spatial frequency of 16.6 cycles/degree. This is well within the range of ModelFest frequencies, which suggests a compromise between a larger aperture (suitable for lower frequencies) and a smaller one (suitable for higher frequencies).

*β*and size

*σ*, a reasonable conclusion is that parameter estimates of either

*β*or

*σ*should be adopted with caution. Further research will be required to constrain better these two mechanisms for restricting foveal summation.

*β*. Before discussing this figure further, we note that if the channels consist of an orthonormal transform, whose individual kernels were orthogonal and whose joint effect has no influence on contrast energy, then when

*β*= 2, introduction of channels can have no effect. The Gabor channels that we use do not quite meet these conditions, but approximate them, so we should expect little effect of channels when

*β*= 2. And indeed, the square symbols in Figure 19 confirm this expectation.

*β*= 2 also reaffirm the observation made above regarding the trade-off between

*β*and the aperture: either an aperture or

*β*> 2 is required to produce a good fit. If both are absent (solid red squares), the error doubles from about 1 to 2 dB.

*β*is free to vary (circular symbols), the addition of channels produces a modest but significant improvement in the fit. The error declines by 0.26 dB when the aperture is present (open red circles), and about 0.18 dB when it is not (solid red circles). The estimated values of

*β*are higher when channels are present (2.87 with aperture, 3.40 without) than when they are absent (2.40 with aperture, 2.71 without) and also show some of the trade-off between

*β*and aperture.

*β*free or fixed at 2, and the no-channel model with

*β*free or fixed at 2, in all cases with an aperture. As noted above, when

*β*= 2, we expect little difference between channel and no-channel models (red and black squares) and this is borne out here.

*β*is free to vary, addition of channels results in a reduction of error for the best CSFs of about 0.25 dB (black versus red circles).

*N*is the number of parameters of a model. In using this measure, we do not treat the addition of channels or the oblique effect as adding a parameter because no parameters were estimated in those cases. When comparisons are based on this measure, the best-fitting models are generally those with channels, a fixed oblique effect, a Gaussian aperture, and

*β*> 2. To allow additional comparisons, the fifty conditions yielding the lowest NRMS values are shown in Table 4.

Oblique | Aperture | Channels | β | CSF | RMS | N | NRMS |
---|---|---|---|---|---|---|---|

Fixed | Gaussian | Gabor | Free | HPmH | 0.772 | 7 | 0.844 |

Fixed | Gaussian | Gabor | Free | LP | 0.787 | 6 | 0.848 |

Fixed | Gaussian | Gabor | Free | LSI | 0.763 | 13 | 0.914 |

Fixed | Gaussian | Gabor | Free | HPmG | 0.867 | 7 | 0.948 |

Fixed | Gaussian | Gabor | Free | EmG | 0.881 | 6 | 0.950 |

Fixed | Gaussian | Gabor | Free | YQM | 0.966 | 6 | 1.041 |

Fixed | Gabor | Free | LSI | 0.903 | 12 | 1.064 | |

Fixed | Gaussian | Gabor | Free | HSmG | 1.024 | 6 | 1.104 |

Fixed | Gaussian | Gabor | Free | MS | 1.036 | 6 | 1.117 |

Fixed | Gaussian | Free | HPmH | 1.033 | 7 | 1.129 | |

Fixed | Gaussian | Gabor | Free | HmH | 1.050 | 6 | 1.132 |

Fixed | Gaussian | Free | HPmG | 1.047 | 7 | 1.144 | |

Fixed | Gaussian | Gabor | 2 | LP | 1.080 | 5 | 1.148 |

Fixed | Gaussian | Free | YQM | 1.069 | 6 | 1.153 | |

Fixed | Gaussian | Free | EmG | 1.075 | 6 | 1.159 | |

Fixed | Gaussian | Gabor | 2 | HPmH | 1.083 | 6 | 1.168 |

Fixed | Gaussian | Free | LP | 1.092 | 6 | 1.177 | |

Fixed | Gaussian | Free | HSmG | 1.096 | 6 | 1.181 | |

Fixed | Gaussian | Gabor | 2 | EmG | 1.111 | 5 | 1.181 |

Fixed | Free | HPmH | 1.098 | 6 | 1.183 | ||

Fixed | Free | HPmG | 1.098 | 6 | 1.184 | ||

Fixed | Free | EmG | 1.122 | 5 | 1.193 | ||

Fixed | Gaussian | Free | HmH | 1.110 | 6 | 1.197 | |

Fixed | Free | LP | 1.133 | 5 | 1.206 | ||

Fixed | Gaussian | 2 | HPmH | 1.122 | 6 | 1.209 | |

Fixed | Gaussian | 2 | YQM | 1.139 | 5 | 1.211 | |

Fixed | Gaussian | Gabor | 2 | YQM | 1.149 | 5 | 1.222 |

Fixed | Gaussian | Free | LSI | 1.024 | 13 | 1.226 | |

Fixed | Free | YQM | 1.154 | 5 | 1.227 | ||

Fixed | Gaussian | 2 | HPmG | 1.142 | 6 | 1.231 | |

Fixed | Gaussian | 2 | HSmG | 1.157 | 5 | 1.231 | |

Fixed | Gaussian | Gabor | 2 | LSI | 1.046 | 12 | 1.232 |

Fixed | Gaussian | 2 | HmH | 1.162 | 5 | 1.236 | |

Gaussian | Free | HPmH | 1.137 | 7 | 1.242 | ||

Fixed | Gaussian | Gabor | 2 | HSmG | 1.171 | 5 | 1.246 |

Fixed | Free | HSmG | 1.189 | 5 | 1.264 | ||

Fixed | Gaussian | 2 | EmG | 1.191 | 5 | 1.266 | |

Gaussian | Free | EmG | 1.175 | 6 | 1.267 | ||

Gaussian | Free | YQM | 1.176 | 6 | 1.267 | ||

Fixed | Gaussian | Gabor | 2 | HmH | 1.198 | 5 | 1.274 |

Fixed | Free | LSI | 1.083 | 12 | 1.275 | ||

Gaussian | Free | LP | 1.184 | 6 | 1.276 | ||

Fixed | Gaussian | 2 | LP | 1.207 | 5 | 1.283 | |

Fixed | Free | HmH | 1.215 | 5 | 1.293 | ||

Fixed | Gaussian | Free | MS | 1.201 | 6 | 1.295 | |

Fixed | Gaussian | 2 | LSI | 1.100 | 12 | 1.296 | |

Gaussian | Free | HSmG | 1.203 | 6 | 1.297 | ||

Fixed | Gaussian | 2 | MS | 1.230 | 5 | 1.309 | |

Gaussian | Free | HmH | 1.217 | 6 | 1.312 |

*β*= 2 is generically described as an energy model. Energy models predict that at threshold all targets have the same filtered contrast energy. Such models can arise from several different mechanisms. In the energy-only model (Manahilov & Simpson, 2001), targets are filtered, their energy collected, and noise added to account for the variability of detection. Manahilov & Simpson (2001) have shown that the energy-only model is consistent with their data on summation between Gabor patches with frequencies a factor of three apart. As we have shown, the energy metric (without an aperture) is not consistent with the ModelFest data. The energy metric with an aperture, while not as good a fit as the channel model, is still consistent with the data.

*β*> 2, may be regarded as the prediction of a probability-summation-only model, where the probability of detection is the probability that any of the noisy outputs of the filter is greater than a constant (Quick, 1974; Robson & Graham, 1981). Note that the probability summation model uses the maximum rule (summation with

*β*= ∞) but the predictive metric for the model has

*β*< ∞.

*β*= 2, channel and no-channel curves in Figure 20).

*β*free) or 290 at 3.44 cycles/degree (

*β*= 2) (Table 5). Note that these values, especially the peak gain, are not independent of the other parameters used in the metric.

_{0}value of 3.46 dB (Equation 2) shows that there is considerable variation with the ModelFest population.

*β*> 2. The parameters of this metric are shown in one line of Table 5 and are labeled Standard A. Because it provides nearly as good a fit, and is computationally much simpler, we also provide second Standard B in which

*β*= 2 (orange highlighting). The user's particular application will determine which of these two standards is appropriate. We hope that these standards will provide useful benchmarks for both future theoretical modeling as well as practical calculations of foveal spatial pattern thresholds.

*β*and the sizes of the spatial aperture. About the same fit is obtained by including a spatial aperture, or an exponent grater than 2. We interpret this as a consequence of both serving to restrict the efficiency of spatial summation.

_{10}of the threshold contrast and are rounded to three decimal places.

Term | Definition | Units |
---|---|---|

L(g) | Luminance as a function of gray level g | cd/m^{2} |

L_{0} | Mean luminance | cd/m^{2} |

c | Contrast | |

g | Digital graylevel | |

t_{s,o,r} | Threshold for stimulus s, observer o, and replication r | dB contrast |

O(f,θ) | Oblique effect filter | |

f | Spatial frequency | cycles/degree |

θ | Orientation | radians |

γ | Oblique effect parameter | cycles/degree |

λ | Oblique effect parameter | cycles/degree |

A(r) | Aperture | |

r | Radial distance from fixation | degree |

σ | Size (standard deviation) of Gaussian aperture | degree |

β | Summation exponent | |

p_{x}, p_{y} | Width and height of each pixel in the input image | degree |

N_{x}, N_{y} | Number of rows and columns in the input image | pixels |

r_{x,y} | Processed pixel value | |

S(f) | CSF | |

f_{0} | High-frequency scale, used in several CSFs | cycles/degree |

f_{1} | Low-frequency scale, used in several CSFs | cycles/degree |

a | Parameters used in several CSFs, usually attenuation at low frequencies | |

p | Parameter used in several CSFs |

*β*was estimated and in which it was fixed at 2. Within these two categories, conditions are sorted by NRMS. The proposed standards A and B are highlighted. See the definition of each CSF for a definition of the parameters. We also show two derived parameters for each function: the peak gain (max) and the peak frequency (

*f*

_{max}).

Standard | CSF | RMS | NP | NRMS | Gain | f_{0} | f_{1} | a | β | p | σ | f_{max} | max |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | HPmH | 1.0329 | 7 | 1.1288 | 373.08 | 4.1726 | 1.3625 | 0.8493 | 2.4081 | 0.7786 | 0.6273 | 3.45 | 217.3 |

HPmG | 1.0468 | 7 | 1.1440 | 289.45 | 5.3459 | 1.9793 | 0.7983 | 2.4054 | 0.8609 | 0.6311 | 3.32 | 221.8 | |

YQM | 1.0694 | 6 | 1.1529 | 466.38 | 7.0629 | 0.6951 | 7.7712 | 2.3557 | 0.5790 | 3.32 | 219.8 | ||

EmG | 1.0755 | 6 | 1.1594 | 360.24 | 7.5237 | 1.8972 | 0.8155 | 2.4725 | 0.7071 | 3.18 | 218.4 | ||

LP | 1.0916 | 6 | 1.1767 | 214.46 | 3.2316 | 0.7127 | 2.4902 | 0.8081 | 0.7118 | 3.50 | 213.6 | ||

HmG | 1.0959 | 6 | 1.1814 | 258.17 | 6.8432 | 1.7483 | 0.7778 | 2.3277 | 0.5579 | 3.20 | 225.3 | ||

HmH | 1.1104 | 6 | 1.1970 | 271.71 | 6.7770 | 1.0461 | 0.8082 | 2.2950 | 0.5311 | 3.39 | 223.8 | ||

MS | 1.2009 | 6 | 1.2946 | 551.29 | 1.7377 | 1.0465 | 2.3643 | 0.6937 | 0.5702 | 3.06 | 215.0 | ||

DoG | 1.7830 | 6 | 1.9222 | 272.74 | 15.3870 | 1.3456 | 0.7622 | 1.9960 | 0.3548 | 2.90 | 261.2 | ||

B | HPmH | 1.1216 | 6 | 1.2091 | 501.20 | 4.3469 | 1.4476 | 0.8514 | 2 | 0.7929 | 0.3652 | 3.62 | 289.0 |

YQM | 1.1387 | 5 | 1.2113 | 621.38 | 7.0856 | 0.7285 | 8.0721 | 2 | 0.3656 | 3.46 | 284.0 | ||

HPmG | 1.1416 | 6 | 1.2307 | 359.87 | 6.0728 | 1.9505 | 0.7931 | 2 | 0.9186 | 0.3655 | 3.37 | 292.0 | |

HmG | 1.1572 | 5 | 1.2310 | 329.93 | 6.9248 | 1.8045 | 0.7827 | 2 | 0.3662 | 3.29 | 286.6 | ||

HmH | 1.1620 | 5 | 1.2361 | 345.78 | 6.7581 | 1.1210 | 0.8128 | 2 | 0.3688 | 3.52 | 279.4 | ||

EmG | 1.1905 | 5 | 1.2664 | 504.43 | 7.6399 | 1.9788 | 0.8163 | 2 | 0.3635 | 3.30 | 302.0 | ||

LP | 1.2065 | 5 | 1.2835 | 299.21 | 3.3578 | 0.7193 | 2 | 0.8009 | 0.3612 | 3.50 | 298.9 | ||

MS | 1.2301 | 5 | 1.3085 | 707.51 | 2.4887 | 0.9846 | 2 | 0.7748 | 0.3596 | 3.41 | 273.6 | ||

DoG | 1.7830 | 5 | 1.8967 | 271.70 | 15.3852 | 1.3412 | 0.7615 | 2 | 0.3563 | 2.89 | 260.3 |

*M*

_{1}), a parameter is free to vary while in the other (

*M*

_{0}) it is fixed, we say that the two models are nested, in that

*M*

_{1}is a more general version of (and includes)

*M*

_{0}. In such cases, it is possible to construct simple statistical tests.

*F*ratio distribution with 1 and

*df*

_{1}degrees of freedom. In the table, we provide 1 −

*p*values for various nested comparisons. All except the final test are significant at the .05 level.

M_{0} | M_{1} | 1 − p | Channels | Aperture | β | CSF | Figure |
---|---|---|---|---|---|---|---|

β = 2 | β free | .039 | No | Yes | LSI | 17 | |

β = 2 | β free | .015 | No | Yes | HPmH | 17 | |

β = 2 | β free | 0 | Yes | Yes | LSI | 19 | |

β = 2 | β free | .039 | No | Yes | LSI | 19 | |

β = 2 | β free | 0 | No | No | LSI | 19 | |

β = 2 | β free | 0 | Yes | No | LSI | 19 | |

σ = inf | σ free | 0 | No | 2 | LSI | 17 | |

σ = inf | σ free | .002 | Yes | Free | LSI | 19 | |

σ = inf | σ free | .07 | No | Free | LSI | 19 |

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