Optical flow-based representation for video action detection - Emerging Trends in Image Processing, Computer Vision, and Pattern Recognition - page 275

(1)
The equation is writen for a point in a video frame. The point is assumed to change its
pixel value over time. ( x , y , t ) is defined as the 3D point composed of the 2D coordinates (pixel
value) of the given point at time t . I represents the intensity function giving the image intens-
ity value of a given pixel value at a given time. The equation is based on the assumption that
enormously small amount of change in the pixel position of the point in enormously small
amount of time period converges to zero change in the intensity value. In other words, the
intensity value of a pixel in a frame is equal to the intensity value of another pixel having
the same point (the point in the former pixel) in the next frame. The point moves enormously
small amount of distance (pixel change) in enormously small amount of time.
LHS of the equation is expanded by using the Taylor Series Expansion [ 26 ] :
(2)
(3)
By some further derivations on Equation (3) , we obtain Equations (7) (8) . The variables rep-
resented by I x , I y , I t are the derivatives of intensity function according to all dimensions.
(4)