Ganglion cells in the peripheral retina have lower density and larger receptive fields than in the fovea. Consequently, the visual signals relayed from the periphery have substantially lower resolution than those relayed by the fovea. The information contained in peripheral ganglion cell responses can be quantified by how well they predict the foveal ganglion cell responses to the same stimulus. We constructed a model of human ganglion cell outputs by combining existing measurements of the optical transfer function with the receptive field properties and sampling densities of midget (P) ganglion cells. We then simulated a spatial population of P-cell responses to image patches sampled from a large collection of luminance-calibrated natural images. Finally, we characterized the population response to each image patch, at each eccentricity, with two parameters of the spatial power spectrum of the responses: the average response contrast (standard deviation of the response patch) and the falloff in power with spatial frequency. The primary finding is that the optimal estimate of response contrast in the fovea is dependent on *both* the response contrast and the steepness of the falloff observed in the periphery. Humans could exploit this information when decoding peripheral signals to estimate contrasts, estimate blur levels, or select the most informative locations for saccadic eye movements.

*α*and

*β*. Here, we define the standard deviation of the windowed responses to be the “response contrast”

*c*. (Note that the response contrast is not restricted to the range 0–1.) Thus, using any two parameters out of the three (

*α*,

*β*,

*c*) serves as a complete description of the power spectrum. Here, we used the response contrast (

*c*) in lieu of

*α*because it is a more intuitive property of the signal and is less correlated with the exponent

*β*.

*c*

_{0},

*β*

_{0}), given each particular observed peripheral pair (

*c*

_{ ɛ },

*β*

_{ ɛ }) at an eccentricity

*ɛ*of interest:

*P*(

*c*

_{0},

*β*

_{0}∣

*c*

_{ ɛ },

*β*

_{ ɛ }). It is important to note that the task is to estimate from the power spectrum of peripheral ganglion cell responses what the power spectrum of the foveal ganglion cell responses would be for the same image patch.

*K*is the constant required for the posterior probabilities to sum to 1.0. First, we binned the foveal power spectra into quantiles containing approximately 1000 spectra, using a kd-tree algorithm (Press, Teukolsky, Vetterling, & Flannery, 2007) by recursively splitting at the median of the subcells' data along alternating dimensions. These quantiles provide an approximation of the prior probability distribution,

*P*(

*c*

_{0},

*β*

_{0}), over the foveal power spectra. Next, we estimated the likelihood distributions,

*P*(

*c*

_{ ɛ },

*β*

_{ ɛ }∣

*c*

_{0},

*β*

_{0}). Note that each of the 1000 foveal power spectra falling within one of the discrete quantile bins maps onto a specific power spectrum (

*c*

_{ ɛ },

*β*

_{ ɛ }) at eccentricity

*ɛ*. We fitted this sample of 1000 contrast and slope pairs at eccentricity

*ɛ*with a Gaussian distribution and used this Gaussian as the estimate of

*P*(

*c*

_{ ɛ },

*β*

_{ ɛ }∣

*c*

_{0},

*β*

_{0}). These estimated prior and likelihood distributions were combined using Equation 1 to obtain the estimated posterior distributions.

*c*and

*β*for each eccentricity. These histograms are plotted in Figure 3 for several eccentricities. The figure shows that the primary change with eccentricity is in the distribution of the slope parameter

*β*indicated by the horizontal shift of the distributions. On the other hand, the distribution of response contrast, indicated by the vertical position, remains relatively constant. This shows that despite the removal of high spatial frequencies due to spatial summation by the peripheral ganglion cells, the simultaneous reduction in the spatial sampling rate of the ganglion cells tends to preserve the average variation of the responses within the image patch (i.e., the response contrast).

*c*and

*β*in the fovea,

*P*(

*c*

_{0},

*β*

_{0}), by quantile binning. The resulting distribution is shown in Figure 4 (note that this is the same data as in Figure 3a replotted with quantile binning). The response contrasts vary from approximately 0.02 to 20, and the spectral falloff parameter varies from approximately 0.2 to 0.45.

*β*

_{0}do not correspond to large changes in the peripheral likelihood distributions. For example, the cyan and blue bins map to highly overlapping distributions at all eccentricities. This means that little information about

*β*

_{0}is carried by the peripheral power spectrum. On the other hand, differences in the foveal response contrast

*c*

_{0}are preserved in the likelihood distributions, although the separation diminishes with eccentricity. Finally, the distributions tilt as eccentricity increases. This means that knowledge of the falloff parameter of the peripheral power spectrum

*β*

_{ ɛ }is informative about the contrast of the corresponding foveal power spectrum. For example, in Figure 5b, the patches indicated by the arrows have an equal observed response contrast but different falloff values. Clearly, the point indicated by the left arrow is more likely to have come from the cyan bin than from the red bin. Thus, the falloff steepness of the spatial power spectrum of a small patch of peripheral ganglion cell responses is informative about the response contrast that the same small image patch would generate had it been projected to the fovea. Further, this becomes more relevant at greater eccentricities.

_{opt},

_{opt}), given observed peripheral values (

*c*

_{ ɛ },

*β*

_{ ɛ }). Figure 6 illustrates the systematic behavior of the means of the posterior distributions. In each panel, the points on the right represent a regular grid spanning the distribution of (

*c*

_{ ɛ },

*β*

_{ ɛ }) values. For each grid point, the mean of the posterior distribution is indicated by the corresponding point on the left side of the panel (marked with a solid line). There are two obvious trends visible in the plots. First, for any given peripheral response contrast, the optimal estimate of foveal response contrast is, on average, approximately constant. Specifically, the average orientation of the solid lines for any given peripheral response contrast is approximately horizontal. Second, for any given peripheral response contrast, the optimal estimate of foveal response contrast varies systematically with the falloff parameter

*β*

_{ ɛ }, especially at larger eccentricities. Specifically, the optimal estimate of foveal response contrast tends to decrease as the magnitude of

*β*

_{ ɛ }increases (this is related to the tilt of the likelihood distributions in Figure 5). These two trends are illustrated more fully in Figure 7a, which plots the optimal estimate of foveal response contrast for a large set of image patches randomly sampled at 15° eccentricity.

_{opt}) and the peripheral values (

*c*

_{ ɛ },

*β*

_{ ɛ }) are much weaker. This may be related to the observation that the variation in the peripheral falloff is much greater than in the foveal falloff. The weaker trends are illustrated in Figure 7b, which plots the optimal estimate of foveal falloff for image patches randomly sampled at 15° eccentricity. The estimated falloff increases with observed contrast from about 0.3 to 0.35, and there is an even weaker effect of the observed falloff. We conclude that a rational strategy for estimating the foveal power spectrum from a peripherally encoded power spectrum is to take into account both the peripheral response contrast and the falloff in estimating the foveal response contrast but only take into account the peripheral response contrast in estimating the foveal falloff.

*f*is a hyperbolic function of eccentricity,

*g*is a quadratic function of eccentricity, and

*k*is a constant. The full six-parameter function, as well as the fitted parameter values, is given in 1.

^{6}patches of width 2°) presented at various retinal eccentricities.

^{6}pairs of parameter values for each retinal eccentricity, we then asked the following question: Given the observed neural responses to a stimulus presented in the periphery, how accurately could the brain predict what the neural responses would be to that stimulus if it were presented in the fovea? In other words, we asked how accurately one can predict the pair of parameter values in the fovea from the pair of values observed in the periphery. Obviously, the more accurate the prediction, the less uncertainty there is about the peripheral stimulus, and the less the need to fixate the peripheral stimulus (Raj, Geisler, Frazor, & Bovik, 2005).

*r*= 0.99 and

*r*= 0.88, respectively. In other words, it appears that learning to optimally estimate the retinal image in the periphery can be accomplished in large part by learning to optimally estimate what would be the foveal responses to the peripheral retinal image.

*k*, was used to describe the strong relationship between an observed peripheral response contrast and an estimated foveal response contrast. Two parameters were used to describe the changing impact on the estimated foveal contrast of the measured peripheral slope with eccentricity: