The visual appearance of natural scenes is governed by a surprisingly simple hidden structure. The distributions of contrast values in natural images generally follow a Weibull distribution, with beta and gamma as free parameters. Beta and gamma seem to structure the space of natural images in an ecologically meaningful way, in particular with respect to the fragmentation and texture similarity within an image. Since it is often assumed that the brain exploits structural regularities in natural image statistics to efficiently encode and analyze visual input, we here ask ourselves whether the brain approximates the beta and gamma values underlying the contrast distributions of natural images. We present a model that shows that beta and gamma can be easily estimated from the outputs of X-cells and Y-cells. In addition, we covaried the EEG responses of subjects viewing natural images with the beta and gamma values of those images. We show that beta and gamma explain up to 71% of the variance of the early ERP signal, substantially outperforming other tested contrast measurements. This suggests that the brain is strongly tuned to the image's beta and gamma values, potentially providing the visual system with an efficient way to rapidly classify incoming images on the basis of omnipresent low-level natural image statistics.

*distribution*of contrast may provide additional information. It has been shown that the more the geometry and the surface structure of a scene are coherent in space, the higher the correlation will be among the corresponding image contrast values (Rousselet, Pernet, Bennett, & Sekuler, 2008; Thomson, 2001). Contrast values in coherent scenes will therefore generally be highly correlated, and any device that records contrasts over a patch of the visual field (such as the receptive fields of neurons in the visual system) thus produces a sum over correlated values. As sums over correlated values follow a Weibull distribution (Meeker & Escobar, 1998), it can be predicted that the distribution of contrasts in natural images of coherent surface structures should follow a Weibull distribution when recorded with receptive fields of a given extent. Indeed, it has been observed (Geusebroek & Smeulders, 2002, 2005; Simoncelli, 1999) that the distributions of contrast values in natural images follow a Weibull function of the following form:

*c*is a normalization constant that transforms the frequency distribution into a probability distribution. The parameter

*μ,*denoting the origin of the contrast distribution, is generally close to zero for natural images. We normalize out this parameter by subtracting the smallest contrast value from the contrast data, leaving two free parameters per image,

*β*(beta) and

*γ*(gamma).

*I*(

*x, y*) and subsequently convolved (denoted with ⊗) with first-order directional Gaussian derivative filters. These filters were designed to respond maximally to edges running vertically and horizontally relative to the image grid, one filter for each of the two perpendicular directions

*x*and

*y*. The filters were applied separately to the input images, producing separate measurements of the gradient component in each direction. The directional image derivatives were combined in the following way to obtain a gradient magnitude per image:

*G*(

*x, y*) is the two-dimensional Gaussian distribution function

*μ, β*(beta), and

*γ*(gamma) represent the origin, scale, and shape of the distribution, respectively, and

*c*is a normalization constant. We estimated

*μ*and normalized it out to achieve illumination invariance, leaving only parameters beta and gamma.

*I*(

*x, y*), with this DoG filter resulted in the filtered images

*C*/ (1.038 +

*C*)) corresponding to that of parvocellular X-cells (Croner & Kaplan, 1995). Summation of the resulting intensity values yielded the “X-output.”

*μ*V) or gradient (a larger voltage step than 50

*μ*V/sample). The average responses (ERPs) were converted to a Current Source Density (CSD) response. The CSD conversion gives a signal that is more localized in space than a regular ERP and therefore has the advantage of reflecting more reliably the activity of neural tissue directly underlying the recording electrode (Nunez & Srinivasan, 2006).

*r*= 0.95;

*p*(1598) = 3.69e − 237; see Figure 2b).

*r*= 0.70;

*p*(1598) = 3.69e − 237; see Figure 2d). An even better approximation of gamma could be obtained by combining the outputs of both the X- and Y-systems. Figure 2e shows how gamma is defined by a combination of X- and Y-outputs. Stepwise regression of gamma versus X- and Y-outputs yielded a combined correlation of

*r*(1597) = 0.82.

*r*= 0.94 contour lines in Figure 2f). We examined the same for the size of the Gaussian smoothing, suggested here to be executed by the Y-cells (see Figure 2g). Results indicate that the physiological sizes of the Y-cells are close to the optimum needed to estimate gamma.

*n*= 16) while EEG was recorded and calculated a grand-average per image. Next, the Weibull parameters, estimated from the individual stimuli, were correlated with the ERP responses.

*r*= −0.71 with) the measured ERP signal recorded between 80 and 200 ms at electrode Iz, which is overlying the early visual cortex. Beta explained the most variance at 113 ms after stimulus onset (

*r*= −0.71;

*p*(1598) = 3.0e − 245;

*r*

^{2}= 0.504) while gamma explained most variance at 133 ms (

*r*= −0.481;

*p*(1598) = 3.6e − 234;

*r*

^{2}= 0.231). Figure 3a shows the correlations of beta and gamma with the ERP signal over time. Figure 3b shows scatter plots of these correlations.

*F*(1, 1596) = 910,289,

*p*= 1.3e − 158, for gamma

*F*(1, 1596) = 10.9,

*p*= 0.001,

*r*

^{2}= 0.51) and 133 ms (for beta

*F*(1, 1596) = 419.8,

*p*= 7.5e − 083 and gamma

*F*(1, 1596) = 7.6,

*p*= 0.006,

*r*

^{2}= 0.41).

*r*= 0.928), the second factor explained 16% of the variance and correlated mainly with gamma (

*r*= 0.928). We subsequently correlated these factors with the EEG signal. This revealed that the “beta”-like factor peaks at 121 ms, in channel Iz (beta like

*F*(1, 1596) = 1504.891,

*p*= 1.9e − 232, gamma like

*F*(1, 1596) = 2.7e − 31,

*p*= 1.1e − 33,

*r*

^{2}= 0.507) and that the “gamma”-like factor peaks at 145 ms in channel Oz (beta like

*F*(1, 1596) = 1002.726,

*p*= 3.5e − 171, gamma like

*F*(1, 1596) = 214.611,

*p*= 1.1e − 45,

*r*

^{2}= 0.41). This confirms that beta and gamma independently explain a substantial amount of variance (see Figure 3c).

r ^{2} | ms | Channel | |
---|---|---|---|

Beta | 0.50 | 113 | Iz |

Gamma | 0.23 | 129 | Iz |

Exp | 0.04 | 133 | Iz |

Gaussian | 0.15 | 102 | Oz |

Max | 0.16 | 133 | Iz |

Median | 0.31 | 109 | Oz |

Mean | 0.17 | 102 | POz |

Michelson | 0.07 | 305 | Oz |

RMS | 0.06 | 223 | Oz |

Slope | 0.24 | 98 | Oz |

Intercept | 0.14 | 102 | POz |

*r*= 0.71, for all 1599 images). This strong correlation is not a statistical necessity, but a property of natural images, borne out from the originating mechanism of the contrast distribution discussed above. Scrambled versions of the same images yield a substantial lower correlation (

*r*= 0.33). So, the correlation between beta and gamma is the result of spatial coherence.

*r*= 0.828; Beta only vs. JPEG size yields

*r*= 0.819).

*R*-squared value is required. The algorithm we used is a variation on the subspace methods used in, for example, Stoica and Nehorai (1989). Such algorithms assume a model for the data in which the signal and noise are uncorrelated and the noise is uncorrelated in time. They then use an eigenvalue decomposition of the second-order moment matrix to estimate the signal and noise space. We can use this decomposition to project the data into the noise space, thus creating residuals based on a minimal model. These residuals are then transformed to

*R*-squared values to be compared with the

*R*-squared values obtained from the data.

*R*-squared, then we must never underestimate the amount of signal in the data, so that our maximum would end up lower than it should. We have used a hypothesis testing method applied to the eigenvalues of the second-order moment matrix to deal with this issue. To tackle the third problem, we chose to detect the noise space for each subject separately because there are individual differences. Then, the

*R*-squared values for all 16 subjects were averaged.

*y*denote the vector containing the

*t*time samples,

*A*be a gain matrix of the signal space, and Σ be the covariance matrix of the noise with scaling factor,

*σ*

^{2}. Then, the population model for the second-order moment matrix containing all time samples is the

*t*by

*t*matrix

*A*exactly is, only the assumption that

*A*is uncorrelated to the noise is important. From this second-order moment matrix, the noise subspace can be determined by considering the number of multiplicities of eigenvalues. If the noise is uncorrelated, then the eigenvalues equal to the scaling factor

*σ*

^{2}of the noise correspond to the noise subspace. For this to work, we need to make Σ the identity matrix (with ones on the diagonal and zeros elsewhere). If Σ were known, then we would simply use the inverse of Σ and decorrelate

*t*

_{1}and

*t*

_{2}, we can use the first set to estimate Σ. An unbiased estimate of Σ is

*S*

_{ y}

*S*

_{ e}

^{−1}, where

*y*from the second batch of trials. Next, we compute the eigenvalues

*l*

_{1},

*l*

_{2}, …,

*l*

_{ n}of

*S*

_{ y}

*S*

_{ e}

^{−1}to determine the noise space by considering which of these eigenvalues are equal to the noise scaling factor,

*σ*

^{2}. If we use a sufficiently large number

*t*

_{1}of trials for

*S*

_{ e}, then we can treat this matrix as fixed and assume that the distribution of the eigenvalues of

*S*

_{ y}

*S*

_{ e}

^{−1}is the same as the distribution of the eigenvalues of

*S*

_{ y}. We used 65% of the total number of trials for estimation of

*S*

_{ e}and 35% for estimation of

*S*

_{ y}. With 1599 trials in this experiment, this amounts to

*t*

_{1}= 1039 and

*t*

_{2}= 560. We can sequentially test for

*k*eigenvalues of the signal space, or equivalently, we can test for

*q*=

*n*−

*k*eigenvalues equal to the noise scaling factor. We begin with

*q*= 0 (all noise) and continue to test the null hypothesis

*H*

_{ q}:

*l*

_{ k + 1}=

*l*

_{ k + 2}= … =

*l*

_{ q}=

*σ*

^{2}until the test says that there is no difference between the eigenvalues anymore. At this point, we have found that the noise space can be constructed by

*q*eigenvectors. We use the Bartlett corrected test of the likelihood ratio test for equality of likelihood (Muirhead, 1982):

*χ*

_{( q + 2)( q − 1) / 2}

^{2}with (

*q*+ 2)(

*q*− 1) / 2 degrees of freedom. In order to cover the signal space as best as possible (so as not to underestimate the maximum value in

*R*-squared), we used a significance level of 0.05 for each test.