Horopter

In studies of binocular vision the horopter is a volume centred on the fixation point that contains all points in space that yield single vision. This volume can be defined theoretically as the points in space which are imaged on corresponding points in the two retinas, that is, on anatomically identical points. More usually it is defined empirically using some criterion.

History of the term

This section is based on Glanville (1933).Glanville, A. D. (1933). The psychological significance of the horopter. The American Journal of Psychology, 45, 592-627.] The term "horopter" was introduced by Franciscus Aguilonius in the second of his six books in optics in 1613. In 1818, G. U. A. Vieth argued from geometry that the horopter must be a circle passing through the fixation-point and the centers of the lenses of the two eyes. A few years later J. Müller made a similar conclusion for the horizontal plane containing the fixation point, although he did expect the horopter to be a surface in space. The theoretical/geometrical horopter in the horizontal plane has come to be known as the Vieth-Müller circle. In 1838, Charles Wheatstone invented the stereoscope, allowing him to explore the empirical horopter.Wheatstone, C. (1838). Contributions to the physiology of vision.—Part the First. On some remarkable, and hitherto unobserved, phænomena of binocular vision. Philosophical Transactions of the Royal Society of London, 128, 371-394.] He found that there were many points in space that yielded single vision; this is very different from the theoretical horopter.

Theoretical horopter

Two theoretical horopters can be distinguished from geometry. One is in the horizontal plane containing the eyes and the fixation point. It is the Vieth-Müller circle.

The other is in the vertical plane containing the fixation point and passing between the eyes. It is a straight line passing through the fixation point and tilted away from the eyes above the fixation point and towards the eyes below the fixation point.

These theoretical horopters depend on the there being no rotation of the eyes about their primary axes ("cyclovergence") and there being no turn of both eyes to the left or to the right ("version"). In the general case of cyclovergence and version, the geometrical horopter is a spiral.Schreiber, K. M., Tweed, D. B., & Schor, C. M. (2006). The extended horopter: Quantifying retinal correspondence across changes of 3D eye position. Journal of Vision, 6(1):6, 64–74, http://journalofvision.org/6/1/6/, doi:10.1167/6.1.6.]

Empirical horopter

As Wheatstone (1838) observer, the empirical horopter, defined by singleness of vision, is much larger than the theoretical horopter. This was studied by P. L. Panum in 1858. He proposed that any point in one retina might yield singleness of vision with a circular region centred around the corresponding point in the other retina. This has become known as "Panum's fusional area", although recently that has been taken to mean the area in the horizontal plane, around the Vieth-Müller circle, where any point appears single.

These empirical investigations used the criterion of singleness of vision, or absence of diplopia to determine the horopter. Other criteria used over the years include the "drop-test horopter", the "plumb-line horopter", and "identical-visual-directions horopter", and the "equidistance horopter". Most of this work has been confined to the horiontal plane or to the vertical plane.

The most comprehensive investigation of the three-dimensional volume of the empirical horopter used the criterion of identical visual directions.Schreiber, K. M., Hillis, J. M., Filippini, H. R., Schor, C. M., & Banks, M. S. (2008). The surface of the empirical horopter. Journal of Vision, 8(3, 7), 1-20. http://journalofvision.org/8/3/7/article.aspx] Schreiber et al. (2008) found that the empirical horopter is a thin volume slanted back above the fixation point for medium to far fixation distances and surrounding the Vieth-Müller circle in the horizontal plane.

Horopter in computer vision

In computer vision, the horopter is defined as the curve of points in 3D space having identical coordinates projections with respect to two cameras with the same intrinsic parameters. It is given generally by a twisted cubic, i.e., a curve of the form "x" = "x"(θ), "y" = "y"(θ), "z" = "z"(θ) where "x"(θ), "y"(θ), "z"(θ) are three independent third-degree polynomials. In some degenerate configurations, the horopter reduces to a line plus a circle.

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