To efficiently estimate preferred spatial frequency across the visual field, we use a novel set of grating stimuli with spatially varying frequency and orientation.
Figure 1 illustrates the logic of the stimulus construction, which is designed for efficient characterization of a system whose preferred spatial frequency falls with eccentricity. Conventional large-field two-dimensional sine gratings will be inefficient for such a system, because the stimulus set will include low-frequency stimuli, which are ineffective for the fovea, and high-frequency stimuli which are ineffective for the periphery. Instead, we construct “scaled” log-polar stimuli, such that local spatial frequency decreases in inverse proportion to eccentricity (
Figure 2B). Specifically, all stimuli are of the form
\begin{eqnarray}
f(r,\theta ) = \cos (\omega _r \ln (r) + \omega _a \theta + \phi ), \quad
\end{eqnarray}
where the coordinates
\((r,\theta )\) specify the eccentricity and polar angle of a retinal position, relative to the fovea. The angular frequency
\(\omega _{a}\) is an integer specifying the number of grating cycles per revolution around the image, and the radial frequency
\(\omega _{r}\) specifies the number of radians per unit increase in
\(\ln (r)\). The parameter
\(\phi\) specifies the phase, in radians. The local spatial frequency is equal to the magnitude of the gradient of the argument of
\(\cos (\cdot )\) with respect to the retinal position (see
Supplement section 1.1):
\begin{eqnarray}
\omega _l(r, \theta ) = \frac{\sqrt{\omega _r^2 + \omega _a^2}}{r}. \quad
\end{eqnarray}
That is, the local frequency is equal to the Euclidean norm of the frequency vector
\((\omega _r, \omega _a)\) divided by the eccentricity (in units of radians per pixel or radians per degree, depending on the units of
\(r\)), which implies that the local spatial period of the stimuli grows linearly with eccentricity. Similarly, the local orientation can be obtained by taking the angle of the gradient of the argument of
\(\cos (\cdot )\) with respect to retinal position (see
Supplement section 1.1):
\begin{eqnarray}
\theta _{l}(r, \theta ) = \theta + \tan ^{-1}\left( \frac{\omega _a}{\omega _r} \right). \quad
\end{eqnarray}
That is, the local grating orientation is the angular position relative to the fovea, plus the angle of the two-dimensional frequency vector
\((\omega _r, \omega _a)\). Note that
\(\theta _{l}\) is in absolute units (e.g.,
\(\theta _{l}=0\) indicates local orientation is vertical, regardless of location). For our stimuli, this depends on the polar angle, but a uniform grating has the same
\(\theta _{l}\) value everywhere in the image (its orientation thus does not depend on polar angle).