I(x, y, t). Now consider what happens when the pattern
moves. The brightness of a particular point in the pattern
is constant, so that
"-0.
dt
(1)
The differentiation (1) can be expanded in a Taylor series
If we let
and then,
dl dx dl dy dl
+ — +— = 0.
dx dt dy dt dt
(2)
dx , dy
u = — and V = — ,
dt dt
(3)
I x u +1 V + I t = 0
(4)
Flinchbaugh, 1990, Mitiche, Wang and Aggarwal, 1987,
Woodham, 1990);
(d) 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 2
observation equations:
hx U +
I 2x U +
I ly v + I it = 0
12y V + ^21 = 0
lixU+iiyV + iit =0
(5)
2J impositi 0 "
Another apP rc
equation is 1,n P
Schunck,15
(b) temporal i
method);
(c) their combii
One way to ex[
the square of t
flow velocity:
/fl«
(obi
The total error,
where we have also introduced the additional abbreviations
I x , I y and 7, for the partial derivatives of image brightness
with respect to x, y and t, respectively, that is
a/
dt'
(5)
This equation expresses a plane, which have normal vector
(w,v, 1), and a measured point (7 X , 7 y , 7 t ) 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
is compiled in following section.
HI
Converting to vector notation, overdetermined linear
equation system (5), is given by
Gf=-b (6)
where
Iu
V
'V
G =
lix
h
II
u
, b =
i it
1 2
I 2
V
I 2
n X
n‘y
n t
has the least squares solution
+a 2
i
The minimizati
values for the
calculus of van
However, it wi
simultaneously
Jordan elimina
by iterative me
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
/ =[G T G)~ l G T b
provided that the inverse of G T G exists.
(8)
In the temporal local optimization method, small temporal
local neighborhood is n frames, and then we can obtain n
observation equations. Multispectral constraints method
has three observation equations. These overdetermined
linear equation systems have the least squares solution in
the same way as spatial local optimization method.
Ì № cha,
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