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Binocular disparity refers to the difference in images of an object seen by left and right eye resulting from the eyes' horizontal separation. The brain uses binocular disparity to extract depth information from the two-dimensional retinal images in stereopsis.

Geometry
In humans left and right eye are horizontally separated by about 50-75 mm (interpupillary distance) depending on each individual. Thus, each eye has a slightly different view of the world, which can be easily seen when alternately closing one eye while looking at a vertical edge.

Focusing on some object in ones sight means to move the eyes such that the image of the object falls on the fovea, the spot of sharp vision, in both eyes. This is called a vergence movement. After the vergence movement the focused object is said to be projected to corresponding points on the two retinae. (see illustration for what corresponding points are) Because of the different views observed by left and right eye, things which are not focused (lying in front or behind the point of fixation) do not fall on corresponding retinal points.

In the visual neurosciences binocular disparity is defined as the difference between the positions of the corresponding point and the actual point of projection in one of the eyes and is usually expressed in degree visual angle (see figure for more exact account).

Tricking neurons with 2D images
Brain cells (neurons) in a part of the brain responsible for processing visual information coming from the retinae (primary visual cortex) can detect the existence of disparity in their input from the eyes. Specifically, these neurons will be active, if an object with "their" special disparity lies within the part of the visual field to which they have access (receptive field).

Researchers investigating precise properties of these neurons with respect to disparity present visual stimuli with different disparities to the cells and look whether they are active or not. One possibility to present stimuli with different disparities is to place objects in varying depth in front of the eyes, but obviously this has many drawbacks and is not precise enough. Instead neuroscientists use a trick as schematised in figure ??. The figure shows that the disparity of an object with different depth than the fixation point can alternatively be produced by presenting an image of the object to one eye and a laterally shifted version of the same image to the other eye. This is what neuroscientists usually do with random dot stimuli to study disparity selectivity of neurons, but this principle has also been applied in magic eye illusions.

capture figure 1: Corresponding points on the two retinae. The small black dot represents the fovea. It is defined as centre. The coloured arrows point to corresponding points. They are characterised by having the same distance to the fovea and lying on the same side of it in both eyes.

capture figure 2: Definition of binocular disparity (far and near). The full black circle is the point of fixation. The blue object lies nearer to the observer. Therefore it has a "near" disparity dn. Objects lying more far away (green) correspondingly have a "far" disparity df. Binocular disparity is the angle between two lines of projection in one eye. One of which is the real projection from the object to the actual point of projection. The other one is the imaginary projection running through the focal point of the lens of the one eye to the point corresponding to the actual point of projection in the other eye. For simplicity reasons here both objects lie on the line of fixation for one eye such that the imaginary projection ends directly on the fovea of the other eye, but in general the fovea acts at most as a reference. Note that far disparities are smaller than near disparities for objects having the same distance from the fixation point.

capture figure 3: Simulation of disparity from depth in the plane. The full black circle is the point of fixation. Objects in varying depths are placed along the line of fixation of the left eye. The same disparity produced from a shift in depth of an object (filled coloured circles) can also be produced by laterally shifting the object in constant depth in the picture one eye sees (black circles with coloured margin). Note that for near disparities the lateral shift has to be larger to correspond to the same depth compared with far disparities (relates to figure 2).