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.

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