Biomedical Engineering Reference
In-Depth Information
where i is the current pixel position.
The updated elements of the flow should be immediately used for the next
step, as required by the Gauss{Seidel method. Thus only one variable array
is needed for the calculation which is constantly updated.
8.7 Preserving discontinuities
Smoothing ensures high density in the optical flow but it has the negative
impact of fading out the flow across the boundaries of moving objects. Phys-
iologically the organs in the human body perform different types of motion
but the tissue of one organ never flows inside another. Thus it is necessary to
accentuate the organ boundaries, which amounts to preserving discontinuities
in the flow.
A whole range of methods has been proposed for preserving these discon-
tinuities. The techniques include using variable values for or selecting only
a part of equations in the case of local methods [4] or to use a function that
tries to preserve discontinuities by weighting the smoothing term [8], [20], [24].
An appropriate weighting function should have the property of reducing
the effect of the smoothing term in areas of discontinuity, i.e., at object bound-
aries, but to retain the effect in areas of homogeneous flow such as inside the
objects. We have adopted a function by weighting the smoothing term in ac-
cordance with the image gradient. Thus places where edges are present will
be smoothed less than areas inside organs which usually have fewer edges and
a lower gradient. This in turn leads to the reformulation of equation (8.13) as
1
0
2i
(S) =
q
:
(8.14)
1 +
jI
x
jjru
i
j
2
+jI
y
jjrv
i
j
2
+jI
z
jjrw
i
j
2
)
2
i
The discontinuity-preserving algorithm uses coupled pairs of equations. The
optical flow is calculated iteratively in a double loop. The outer loop updates
the variables and the inner loop updates the u;v;w optical ow components.
The algorithm in pseudo code can be described as follows:
initializeu,v,w,psi_1,psi_2
forouterloop
forx,y,z
updatepsi_1' asinequation1.12
updatepsi_2' asinequation1.15
end
forinnerloop
forx,y,z
updateu
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