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What is the difference between stereoscopic depth and crossed/uncrossed disparity, and specifically in terms of their use in the methods to manipulate visual stimuli to investigate depth-perception?

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Short answer
Stereoscopic vision is the perception of depth via disparity. Disparity is the mechanism that the visual system uses to create binocular depth perception.

Background
A dictionary definition of Stereoscopic is the following:

[N]oting or pertaining to three-dimensional vision or any of various processes and devices for giving the illusion of depth from two-dimensional images or reproductions, as of a photograph or motion picture.

So stereoscopic depth perception refers to the ability of perceiving depth in a 2D image.

A more visual-scientific description of stereopsis is the following (Kalloniatis & Luu, 2016):

Stereopsis refers to our ability to appreciate depth, that is the ability to distinguish the relative distance of objects with an apparent physical displacement between the objects. [...] Stereopsis is an important binocular cue to depth perception. Stereopsis cannot occur monocularly and is due to binocular retinal disparity [where] two objects stimulate [...] disparate (non-corresponding) retinal points [...].

Monocular cues include occlusion, lights & shade, among others. Binocular cues, i.e., stereoscopic cues are caused by the slightly offset retinal images on our two eyes due to the distance between them. This is called disparity. Disparity depends on the focal distance, and in a nutshell, uncrossed disparity refers to points in space closer than the focus distance and crossed disparity to points more distant than the focal distance (source: Heeger, New York University).

Hence, long story short, stereoscopic depth is a perception, disparity is the mechanism that the visual system uses to create that perception.

Reference
- Kalloniatis & Luu, Perception of Depth. In: The Organization of the Retina and Visual System, Utah University (2016)

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