the physical stimuli that represent the model input variables are : physical velocity of monitor in space m ; velocity of the image ( grating ) on the monitor r ; and eye velocity in space e. the psychophysical constants of the model are : the weber fractions for the difference between retinal and reference signals w , and for the difference between two retinal velocity signals w ; and a stimulus - dependent gain factor g for the visual perception of velocities
. let 
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 $ $ \theta { \left ( { x , t } \right ) } $ $ \end{document } be a function defined by : 
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 $ $ \theta { \left ( { x , t } \right ) } = 0{\text { if } } { \left| x \right| } < t\quad { \text{and}}\quad \theta { \left ( { x , t } \right ) } = x - t $ $ \end{document } otherwise .
the perceived velocity of the monitor in space is related to the difference between the retinal velocity of the monitor vmon / ret  =  e    m , and the reference signal vref which is equal to e for a still observer and to e  +  vhead / space for a moving observer : \documentclass[12pt]{minimal } 
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 $ $ \hat{v}_{{{\text{mon / space } } } } = \theta { \left ( { v_{{{\text{ref } } } } - v_{{{\text{mon / ret } } } } { \text { , jnd}}_{1 } } \right ) } , $ $ \end{document}where \documentclass[12pt]{minimal } 
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 $ $ { \text{jnd}}_{1 } = - { \text{sign}}(v_{{{\text{mon / ret } } } } ) \,w{\kern 1pt } { \sqrt { v_{{{\text{ref } } } } ^{2 } + v_{{{\text{mon / ret } } } } ^{2 } } } .
$ $ \end{document}the perceived velocity of the grating in space is related to the difference between the retinal velocity of the grating vgrat / ret e    ( m  +  r ) , and the reference signal vref : \documentclass[12pt]{minimal } 
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 $ $ \hat{v}_{{{\text{grat / space } } } } = \theta { \left ( { v_{{{\text{ref } } } } - v_{{{\text{grat / ret } } } } { \text { , jnd}}_{2 } } \right ) } , $ $ \end{document}where \documentclass[12pt]{minimal } 
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 $ $ { \text{jnd}}_{2 } = - { \text{sign}}(v_{{{\text{grat / ret } } } } ) \,w{\kern 1pt } { \sqrt { v_{{{\text{ref } } } } ^{2 } + v_{{{\text{grat / ret } } } } ^{{\text{2 } } } } } . $ $ \end{document}the perceived velocity of the grating relative to the monitor is related to the difference between the two perceived velocities in space , using a jnd based on the difference between the two retinal signals involved .
these are assumed to be : the retinal velocity of the monitor vmon / ret  =  e    m ; and the velocity difference between the image of the monitor and the grating within it vdelta  =  (m    r ) .
thus : \documentclass[12pt]{minimal } 
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 $ $ \hat{v}_{{{\text{grat / mon } } } } = \theta { \left ( { \hat{v}_{{{\text{grat / space } } } } - \hat{v}_{{{\text{mon / space } } } } { \text { , jnd}}_{3 } } \right ) } , $ $ \end{document}where \documentclass[12pt]{minimal } 
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 $ $ { \text{jnd}}_{3 } = - { \text{sign}}(\hat{v}_{{{\text{mon / space } } } } { \text{)}}\,w_{\delta } \,{\sqrt { v_{{{\text{mon / ret } } } } ^{2 } + v_{{{\text{delta } } } } ^{2 } } } .
$ $ \end{document}this value is then to be factored by a stimulus - dependent gain g to model the experimentally measured perceived velocity of the grating on the monitor .
note that the model is deterministic and treats jnd s as signed values , the sign of which is defined as opposite to that of the perceived velocity of the monitor or grating in space . however , since we think of a jnd as intrinsic noise , it should be an interesting challenge to translate the model in more probabilistic terms .
 when a slowly moving pattern is presented on a monitor which itself is moved , the pattern appears to freeze on the screen ( mesland and wertheim in vis res 36(20):33253328 , 1996 ) even if we move our head with the monitor , as with a head mounted display ( pavard and berthoz in perception 6:529540 , 1977 ) . 
 we present a simple model of these phenomena , which states that the perceived relative velocity between two stimuli ( the pattern and the moving monitor ) is proportional to the difference between the perceived velocities of these stimuli in space , minus a noise factor . 
 the latter reflects the intrinsic noise in the neural signals that encode retinal image velocities . with noise levels derived from the literature 
 the model fits empirical data well and also predicts strong distortions of visually perceived motion during vestibular stimulation , thus explaining both illusions as resulting from the same mechanism . 
Mathematical formulation of the model