Horopter

Schematic representation of the theoretical (T) and the empirical (E) horopter.

In studies of binocular vision the horopter is the locus of points in space have the same disparity as fixation. This can be defined theoretically as the points in space which project on corresponding points in the two retinas, that is, on anatomically identical points. The horopter can be measured empirically in which it is defined using some criterion.

History of the term

The horopter as a special set of points was first mentioned in the eleventh century by Ibn al-Haytham, known to the west as "Alhazen".[1] He built on the binocular vision work of Ptolemy[2] and discovered that objects lying on a horizontal line passing through the fixation point resulted in single images, while objects a reasonable distance from this line resulted in double images. Thus Alhazen noticed the importance of some points in the visual field but did not work out the exact shape of the horopter and used singleness of vision as a criterion.

The term horopter was introduced by Franciscus Aguilonius in the second of his six books in optics in 1613.[3] In 1818, Gerhard Vieth argued from geometry that the horopter must be a circle passing through the fixation-point and the nodal point of the two eyes. A few years later Johannes 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 (i.e., not restricted to the horizontal plane). The theoretical/geometrical horopter in the horizontal plane became known as the Vieth-Müller circle.

In 1838, Charles Wheatstone invented the stereoscope, allowing him to explore the empirical horopter.[4][5] He found that there were many points in space that yielded single vision; this is very different from the theoretical horopter, and subsequent authors have similarly found that the empirical horopter deviates from the form expected on the basis of simple geometry.

Theoretical horopter

Later Hermann von Helmholtz and Ewald Hering worked out the exact shape of the horopter almost at the same time. Their descriptions identified two components for the horopter. The first is in the plane which contains the fixation point (wherever it is) and the two nodal points of the eye. The locus of horopteric points in this plane takes the form of the arc of a circle (the Vieth-Müller circle) going from one nodal point to the other in space, passing through the fixation point. The second component is a line (the Prévost–Burckhardt line[6]) which is perpendicular to this arc, cutting it at the point midway between the two eyes (which may, or may not, be the fixation point). That the horopter is composed of a circle in the fixation plane and a perpendicular line remains true as long as the eyes are in primary or secondary position (resp. straight ahead or on the horizontal or vertical meridians). When the eyes are in tertiary position, cyclorotation of the eyes have to be taken into account and was calculated by Helmholtz. In this case the horopter becomes a spiral converging toward the vertical horopter at the top and bottom extremities and passing through the nodal point of the two eyes.[7][8] Finally, when fixation is at infinity the horopter takes on the special form of a plane perpendicular to the fixation line(s).

This description depends on the horopter being defined as the locus of points which make the same angle at the eyes (i.e. that have zero relative disparity with fixation) - which was the original definition used by Aguilonius. If one considers a slightly different definition, based on the projections into space of corresponding retinal points, then Schreiber and colleagues have shown that a different theoretical form emerges.[9] As Helmholtz predicted, and Solomons subsequently confirmed,[10][11] in the general case which includes cyclorotation of the eyes, the theoretical horopter takes the form of a twisted cubic.[9]

Empirical horopter

As Wheatstone (1838) observed, 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 on 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 early empirical investigations used the criterion of singleness of vision, or absence of diplopia to determine the horopter. Today the horopter is usually defined by the criterion of identical visual directions (similar in principle to the apparent motion horopter, according that identical visual directions cause no apparent motion). Other criteria used over the years include the apparent fronto-parallel plane horopter, the equi-distance horopter, the drop-test horopter or the plumb-line horopter. Although these various horopters are measured using different techniques and have different theoretical motivations, the shape of the horopter remains identical regardless of the criterion used for its determination.

Consistently, the shape of the empirical horopter have been found to deviate from the geometrical horopter. For the horizontal horop[ter this is called the Hering-Hillebrand deviation. The empirical horopter is flattefd than predicted from geometry at short fixation distances and becomes convex for farther fixation distances. Moreover the vertical horopter have been consistently found to have a backward tilt of about 2 degrees relative to its predicted orientation (perpendicular to the fixation 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.

References

  1. ↑ Smith, A. Mark (2001). Alhazen's theory of visual perception. Vol. 2 English translation. American Philosophical Society.
  2. ↑ Smith, A. Mark (1996). Ptolemy's Theory of Visual Perception. American Philosophical Society.
  3. ↑ Aguilonius, Franciscus. Opticorum libri sex.
  4. ↑ Glanville AD (1993). "The Psychological Significance of the Horopter". The American Journal of Psychology 45 (4): 592–627. doi:10.2307/1416191. JSTOR 1416191.
  5. ↑ Wheatstone C (1838). "Contributions to the Physiology of Vision. Part the First. On Some Remarkable, and Hitherto Unobserved, Phenomena of Binocular Vision". Philosophical Transactions of the Royal Society of London 128: 371–94. Bibcode:1838RSPT..128..371W. doi:10.1098/rstl.1838.0019. JSTOR 108203.
  6. ↑ Howarth PA (2011). "The geometric horopter". Vision Research 51 (4): 397–9. doi:10.1016/j.visres.2010.12.018. PMID 21256858.
  7. ↑ Tyler, Christopher W (1991). The horopter and binocular fusion. In Vision and visual dysfunction 9. p. 19-37.
  8. ↑ Howard, Ian P; Rogers, Brian J (2002). Seeing in depth, volume 2: Depth perception. Ontario, Canada: I. Porteous.
  9. 1 2 Schreiber KM, Tweed DB, Schor CM (2006). "The extended horopter: Quantifying retinal correspondence across changes of 3D eye position". Journal of Vision 6 (1): 64–74. doi:10.1167/6.1.6. PMID 16489859.
  10. ↑ Solomons H (1975). "Derivation of the space horopter". The British journal of physiological optics 30 (2–4): 56–80. PMID 1236460.
  11. ↑ Solomons H (1975). "Properties of the space horopter". The British journal of physiological optics 30 (2–4): 81–100. PMID 1236461.
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