Editing Visual acuity (section)

== Motion acuity ==

The eye has acuity limits for detecting motion.<ref name="Lappin-2009">{{Cite journal |last1=Lappin |first1=Joseph S. |last2=Tadin |first2=Duje |last3=Nyquist |first3=Jeffrey B. |last4=Corn |first4=Anne L. |name-list-style=vanc |date=January 2009 |title=Spatial and temporal limits of motion perception across variations in speed, eccentricity, and low vision. |journal=Journal of Vision |volume=9 |pages=30.1–14 |doi=10.1167/9.1.30 |pmid=19271900 |quote=Displacement thresholds for peripheral motion were affected by acuity limits for speeds below 0.5 degrees/s. [0.0087 radians/s] |doi-access=free |number=30}}</ref> Forward motion is limited by the ''subtended angular velocity detection threshold'' (SAVT), and horizontal and vertical motion acuity are limited by lateral motion thresholds. The lateral motion limit is generally below the looming motion limit, and for an object of a given size, lateral motion becomes the more insightful of the two, once the observer moves sufficiently far away from the path of travel. Below these thresholds [[subjective constancy]] is experienced in accordance with the [[Stevens' power law]] and [[Weber–Fechner law]].

=== Subtended angular velocity detection threshold (SAVT) ===
There is a specific acuity limit in detecting an approaching object's looming motion.<ref name="Weinberger-1971">{{Cite journal |last=Weinberger |first=Hershel |name-list-style=vanc |date=19 February 1971 |title=Conjecture on the Visual Estimation of Relative Radial Motion |journal=[[Nature (journal)|Nature]] |volume=229 |issue=5286 |page=562 |bibcode=1971Natur.229..562W |doi=10.1038/229562a0 |pmid=4925353 |s2cid=4290244 |doi-access=free}}</ref><ref name="Schrater-2001">{{Cite journal |last1=Schrater |first1=Paul R. |last2=Knill |first2=David C. |last3=Simoncelli |first3=Eero P. |name-list-style=vanc |date=12 April 2001 |title=Perceiving visual expansion without optic flow |journal=[[Nature (journal)|Nature]] |volume=410 |issue=6830 |pages=816–819 |bibcode=2001Natur.410..816S |doi=10.1038/35071075 |pmid=11298449 |s2cid=4406675 |quote=When an observer moves forward in the environment, the image on his or her retina expands. The rate of this expansion conveys information about the observer's speed and the time to collision... this rate might also be estimated from changes in the size (or scale) of image features... we show, ... observers can estimate expansion rates from scale-change information alone, and that pure scale changes can produce motion after-effects. These two findings suggest that the visual system contains mechanisms that are explicitly sensitive to changes in scale.}}</ref> This is regarded as the [[subtended]] angular velocity detection threshold (SAVT) limit of visual acuity.<ref>{{Cite journal |last1=Hoffmann |first1=Errol R. |last2=Mortimer |first2=Rudolf G. |name-list-style=vanc |date=July 1996 |title=Scaling of relative velocity between vehicles |journal=Accident Analysis & Prevention |volume=28 |issue=4 |pages=415–421 |doi=10.1016/0001-4575(96)00005-X |issn=0001-4575 |pmid=8870768 |quote=Only when the subtended angular velocity of the lead vehicle exceeded about 0.003 rad/s were the subjects able to scale the relative velocity}}</ref> It has a practical value of 0.0275 rad/s.<ref>{{Cite journal |last1=Maddox |first1=Michael E. |last2=Kiefer |first2=Aaron |name-list-style=vanc |date=September 2012 |title=Looming Threshold Limits and Their Use in Forensic Practice |journal=Proceedings of the Human Factors and Ergonomics Society Annual Meeting |volume=50 |pages=700–704 |doi=10.1177/1071181312561146 |s2cid=109898296 |quote=A number of laboratory researchers have reported values of the looming threshold to be in the range of 0.003 radian/sec. Forensic practitioners routinely use elevated values of the looming threshold, e.g., 0.005–0.008, to account for the complexity of real-world driving tasks. However, only one source has used data from actual vehicle accidents to arrive at a looming threshold – and that value, 0.0275 rad/sec, is an order of magnitude larger than that derived from laboratory studies. In this study, we examine a much broader range of real-world accident data to obtain an estimate of the reasonable upper end of the looming threshold. The results show a range of 0.0397 to 0.0117 rad/sec... |number=1}}</ref> For a person with SAVT limit of <math>\dot\theta_t</math>, the looming motion of a directly approaching object of size {{mvar|S}}, moving at velocity {{mvar|v}}, is not detectable until its distance {{mvar|D}} is<ref name="Weinberger-1971" />

<math display="block">D \lessapprox \sqrt{\frac{S \cdot v}{\dot{\theta_{t}}}-\frac{S^2}{4}},</math>

where the {{math|S<sup>2</sup>/4}} term is omitted for small objects relative to great distances by [[small-angle approximation]].

To exceed the SAVT, an object of size {{mvar|S}} moving as velocity {{mvar|v}} must be closer than {{mvar|D}}; beyond that distance, [[subjective constancy]] is experienced. The SAVT <math>\dot\theta_t</math> can be measured from the distance at which a looming object is first detected:

<math display="block"> \dot\theta_t \approx \frac{4S \cdot v}{S^2 + 4D^2}, </math>

where the {{math|S<sup>2</sup>}} term is omitted for small objects relative to great distances by [[small-angle approximation]].

The SAVT has the same kind of importance to driving safety and sports as the static limit. The formula is derived from taking the [[derivative]] of the [[visual angle]] with respect to distance, and then multiplying by velocity to obtain the time rate of visual expansion ({{math|d<var>θ</var>/d<var>t</var> {{=}} d<var>θ</var>/d<var>x</var> · d<var>x</var>/d<var>t</var>}}).

=== Lateral motion ===
There are acuity limits (<math>\dot\theta_t</math>) of horizontal and vertical motion as well.<ref name="Lappin-2009" /> They can be measured and defined by the threshold detection of movement of an object traveling at distance {{mvar|D}} and velocity {{mvar|v}} orthogonal to the direction of view, from a set-back distance {{mvar|B}} with the formula
<math display="block"> \dot\theta_t \approx \frac{B \cdot v}{B^2 + D^2}. </math>
Because the [[Tangent (trigonometry)|tangent]] of the [[subtended angle]] is the ratio of the orthogonal distance to the set-back distance, the angular time rate ([[radians|rad]]/[[second|s]]) of lateral motion is simply the [[derivative]] of the [[inverse tangent]] multiplied by the velocity ({{math|d<var>θ</var>/d<var>t</var> {{=}} d<var>θ</var>/d<var>x</var> · d<var>x</var>/d<var>t</var>}}). In application this means that an orthogonally traveling object will not be discernible as moving until it has reached the distance

<math display="block"> D \lessapprox \sqrt{\frac{B \cdot v}{\dot\theta_t} - B^2}, </math>

where <math>\dot\theta_t</math> for lateral motion is generally ≥ 0.0087 rad/s with probable dependence on deviation from the [[Fovea centralis|fovia]] and movement orientation,<ref name="Lappin-2009" /> velocity is in terms of the distance units, and zero distance is straight ahead. Far object distances, close set-backs, and low velocities generally lower the salience of lateral motion. Detection with close or null set-back can be accomplished through the pure scale changes of looming motion.<ref name="Schrater-2001" />

=== Radial motion ===
The motion acuity limit affects [[Rotational speed|radial motion]] in accordance to its definition, hence the ratio of the velocity {{mvar|v}} to the radius {{mvar|R}} must exceed <math>\dot\theta_t</math>:
<math display="block">\dot\theta_t \lessapprox \frac{v}{R}.</math>

Radial motion is encountered in clinical and research environments, in [[Fulldome|dome theaters]], and in [[virtual-reality]] headsets.