Tangent screen projection

As explained above, the stimulus display was positioned at a distance of 30 cm from the rat eye. This number was chosen, because the optimal viewing distance for rats has been reported to range between 20 and 30 cm (Wiesenfeld and Branchek, 1976). In addition, earlier behavioral studies from our group have shown that rats are capable of discriminating complex visual objects under a variety of identity-preserving transformations, when viewing the objects at a distance of 30 cm (Zoccolan et al., 2009; Tafazoli et al., 2012; Alemi-Neissi et al., 2013; Rosselli et al., 2015). However, because of such a short viewing distance, objects will appear distorted, when they undergo large translations over the stimulus display. Because of this, position changes will also result in size changes and distortions of the objects’ shape, unless appropriate corrections are applied. To address this issue, we displayed the stimuli under a tangent screen projection. This projection allows presenting the stimuli as they would appear, if they were shown on virtual screens that are tangent to a circle centered on the rat’s eye, with a radius equal to the distance from the eye to the point on the display just in front of it (i.e., 30 cm). Thanks to this projection, the shape, size and aspect ratio of each stimulus were preserved across all the eight azimuth positions tested in our experiment (see previous section).

The tangent projection is explained in Figure 1—figure supplement 2D–F. Panels D and E show, respectively, a top view and a side view of the rat eye and the stimulus display (O₁). R is the distance between the eye and the center of the display. O₂ is an example (virtual) tangent screen, where an object would be shown, if its center was translated of an azimuth angle θ to the left of the center of the stimulus display (while maintaining the default elevation of 0°). The coordinate x₀ indicates the projection of the object’s center over O₁, following this azimuth shift θ. The Cartesian coordinates (x₂, y₂) indicate the position of a pixel of the stimulus image, relative to the object’s center, over the virtual screen O₂, while (x₁, y₁) indicate the projection of this point over the display O₁, i.e., its coordinates, relative to the center of the object (x₀, 0) in O₁. The red line drawn over O₂ shows the distance of this pixel from the object’s center over the tangent screen, while the red line drawn over O₁ is the projection of this distance over the stimulus display.

The projection of any point (x₂, y₂) in the virtual tangent screen O₂ to a point (x₁, y₁) in the stimulus display O₁ can be computed using simple trigonometric relationships. Figure 1—figure supplement 2D shows how x₁ can be expressed as a function of x₂ and θ:

where:

Note that φ is the azimuth position of the point (x₂, y₂), when expressed in spherical coordinates. Similarly, Figure 1—figure supplement 2E shows how y₁ can be expressed as a function of y₂ and θ:

where:

To better illustrate the effect of the tangent screen projection, Figure 1—figure supplement 2D shows how two points at the same distance from (but on opposite sides of) the object’s center in the tangent screen (see, respectively, the red and green lines over O₂) would be projected over the stimulus display (see, respectively, the red and green lines over O₁). As shown by the drawing, these two points would not be equidistant any longer from the object’s center, in the projection over O₁. This can be better appreciated by looking, in Figure 1—figure supplement 2F (top), at the images of an example object (object #8) shown at positions −22.5°, −15°, −7.5° and 0° of visual angle (azimuth) over the stimulus display (these are a subset of the eight different positions tested for each object in our experiment; see above). The distortion applied to the object by the tangent screen projection becomes progressively larger and more asymmetrical (with respect to the vertical axis of the object), the larger is the distance of the object’s center from the center of the display (0° azimuth). Critically, this distortion was designed to exactly compensate the one produced by the perspective projection to the retina, so that, regardless of the magnitude of the azimuth displacement, the resulting projection of an object over the retina will have the same shape, size and aspect ratio as the ones produced by the object shown at center of the display, right in front of the rat’s eye (Figure 1—figure supplement 2F, bottom).