Biomedical Engineering Reference
In-Depth Information
FIGURE 8.4: The optical flow is modeled as smooth flow of a fluid.
two volumes. Therefore the following image constraint equation (also called
brightness consistency constraint) holds true:
I(x;y;z;t) = I(x + x;y + y;z + z;t + t):
(8.2)
Assuming the movement to be small enough, an image constraint can now be
developed from Equation (8.2) at I(x;y;z;t) with Taylor series:
= I(x;y;z;t) +
@I
@x
x +
@I
@y
y +
@I
I(x + x;y + y;z + z;t + t)
@z
z
+
@I
@t
t + H
where H means higher order terms, which should be small enough to be ig-
nored. From these equations we achieve:
@I
@x
x +
@I
@y
y +
@I
@z
z +
@I
@t
t = 0
or
@I
@x
x
t
+
@I
y
t
+
@I
z
t
+
@I
t
t
= 0
@y
@z
@t
which results in
@x
u +
@I
@I
@y
v +
@I
@z
w +
@I
@t
= 0
where u;v;w are the x,y and z components of the velocity V or optical ow
of I(x;y;z;t), and
@I
@x
,
@I
@y
,
@I
@z
and
@
@t
are the derivatives of the image at
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