Parallax
In a stereoscopic image every point appears twice: once for the left eye and once for the right eye. The horizontal distance between the two images is parallax. The brain turns this difference into depth.
Screen parallax
On the screen, the position of the left-eye image is subtracted from the position of the right-eye image:
where: x_right and x_left are the horizontal positions of the point in the right-eye and left-eye images (on the screen, mm)
- P = 0: the point appears on the screen plane.
- P > 0 (positive, uncrossed): the point appears behind the screen.
- P < 0 (negative, crossed): the point appears in front of the screen, toward the viewer.
- P = e: the lines of sight are parallel; the point appears at infinity.
- P > e: the rays never cross; the eyes have to turn outward. This is divergence, and it has no counterpart in nature.
Perceived distance
The viewer's eyes see a point where the two rays through the two images on the screen cross. From similar triangles:
where: Zp perceived distance (from the viewer), V viewing distance, e interocular distance, P screen parallax
This relation is the viewing side of the model by Woods, Docherty and Koch (1993). If the parallax is as negative as the interocular distance (P = −e), the point appears at half the viewing distance, that is, exactly midway between the screen and the viewer.
Percent, millimeters, pixels
Parallax is usually given as a ratio of image width, because this ratio is independent of resolution and screen. Its value on the screen is found by multiplying by the screen width, and its value in pixels by multiplying by the horizontal resolution:
where: d parallax as a ratio of image width, Ws screen width, N_px horizontal pixel count
Although the ratio stays the same, its value in millimeters grows with the screen. In a 1920-pixel image, 1% parallax is 19 pixels:
| Screen | Width | 1% parallax |
|---|---|---|
| Phone 6.1" | 0.14 m | 1.4 mm |
| TV 55" | 1.22 m | 12.2 mm |
| Cinema screen | 10 m | 100 mm: larger than the interocular distance, divergence |
This is why a film is prepared for a single screen; if it is to be shown on another screen, its parallax is readjusted. See Mastering.
Angular parallax
What the eye actually perceives is the difference between the vergence angle to the two images and the vergence angle to the screen (angular disparity):
where: δ angular parallax. 1° = 60 arcminutes (′)
A commonly used comfort limit is 1° (Lambooij et al.). On a TV viewed from 2.5 m, this corresponds to about 44 mm of parallax.
Disparity and parallax
The equivalent on the camera sensor is called disparity. When the image is enlarged to the screen, disparity becomes screen parallax: P = (Ws / Wc) · d_sensor (Wc sensor width). The difference on the retina of the eye is called retinal disparity.