Fuse, Takashi 
  
2. GRADIENT-BASED APPROACH 
2.1 Gradient Constraint Equation 
An equation that relates the change in image brightness at a point to the motion of the brightness pattern was derived by 
Horn and Schunck (1981). Let the image brightness at the point (x, y) in the image plane at time 7 be denoted by I(x, y, 
D Now consider what happens when the pattern moves. The brightness of a particular point in the pattern is 
constant, so that 
dl 
e 2 1 
di (1) 
The differentiation (1) can be expanded in a Taylor series 
ox dt dy dt ot 
If we let 
NN, and veil, 3) 
dt dt 
and then, 
Iu+JI v+I,=0 (4) 
where we have also introduced the additional abbreviations I, I, and I, for the partial derivatives of image brightness 
with respect to x, y and 1, respectively, that is 
ol ol al 
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This equation expresses a plane, which have normal vector (u,v,1), and a measured point (Z, I, I) is on the plane. Due 
to single linear equation in the two unknowns u and v, the parameter u and v, that is x-component and y-component of 
optical flow respectively, cannot be determined. As a consequence, the optical flow (u, v) cannot be computed locally 
without introducing additional constraints. 
In order to solve this problem, various methods have been proposed. The basic methods of gradient-based approach 
are reviewed in following section. 
2.2 Increase in the Number of Observation Equations 
One of the approaches of solving the gradient constraint equation is increase in the number of observation equations: 
(a) by the assumption that a constant velocity over each spatial neighborhood (spatial local optimization method) 
(Barron, Fleet and Beauchemin, 1994, Kearney, Tompson, and Boley, 1987, Lucas and Kanade, 1981); 
(b) by the constant velocity over temporal neighborhood (temporal local optimization method) (Kearney, Tompson and 
Boley, 1987, Nomura, Miike and Koga, 1991); 
(c) by use of three channels (RGB, HSI) of each pixel (multispectral constraints method) (Markandey and Flinchbaugh, 
1990, Mitiche, Wang and Aggarwal, 1987, Woodham, 1990); 
(d) by use of second order derivatives of each pixel (second order derivative method) (Bainbridge-Smith and Lane, 
1997, Nagel, 1983, Tistarelli and Sandini, 1990, Tretiak and Pastor, 1984, Uras, Girosi et al, 1988) 
(e) by their combination. 
The spatial local optimization method estimates optical flow by solving a group of observation equations obtained from 
a small spatial neighborhood of the image as a system of linear equations. Two observation equations are sufficient to 
arrive at unique solution for (u, v). More than two equations may be included in the system to reduce the effects of 
errors in the observation equations. Let the small spatial neighborhood, Q , be equal to nxn pixels. The observation 
equation at each pixel in small spatial neighborhood can be obtained. We can get n^ observation equations as an 
  
270 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
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