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U. Glaeser

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FIGURE 28.21 An instance of the aperture problem.<br />

To view this limitation in another way, the constraint equation can be rewritten in vector form, as<br />

so that the component of optical flow in the direction of the intensity gradient ( f x, f y) T is<br />

© 2002 by CRC Press LLC<br />

(28.47)<br />

(28.48)<br />

However, the component of the optical flow perpendicular to the gradient (along isointensity contours)<br />

cannot be determined. This is a manifestation of the aperture problem. If the motion of an oriented<br />

element is detected by a unit that is small compared with the size of the moving element, the only<br />

information that can be extracted is the component of motion perpendicular to the local orientation of<br />

the element. For example, looking at a moving edge through a small aperture (Fig. 28.21), it is impossible<br />

to tell whether the actual motion is in the direction of a or of b.<br />

One way to work around this limitation is to impose an explicit smoothness constraint. Motion was<br />

implicitly assumed smooth earlier, when a Taylor’s expansion was used and when the higher order terms<br />

were ignored. Following this approach, an iterative scheme for finding the optical flow for the image<br />

sequence can be formulated:<br />

and<br />

rx( k, l)<br />

n+1<br />

=<br />

=<br />

ry( k, l)<br />

n+1<br />

Edge<br />

b<br />

Aperture<br />

( fx, fy) ⋅ ( rx, ry) = – ft ft 2 2<br />

fx + fy -------------------<br />

rx( k, l)<br />

n λf 2<br />

x rx( k, l)<br />

n<br />

λ fx fyr y( k, l)<br />

n<br />

+ + λ fx ft – -----------------------------------------------------------------------------------<br />

2 2<br />

1 + λ( fx + fy )<br />

rx( k, l)<br />

n f<br />

λ f<br />

xrx( k, l)<br />

x<br />

n<br />

fyr y( k, l)<br />

n<br />

+ + ft – ------------------------------------------------------------<br />

2 2<br />

1 + λ( fx + fy )<br />

=<br />

ry( k, l)<br />

n<br />

f<br />

λ f<br />

xrx( k, l)<br />

y<br />

n<br />

fyr y( k, l)<br />

n<br />

+ + ft – ------------------------------------------------------------<br />

2 2<br />

1 + λ( fx + fy )<br />

a<br />

(28.49)<br />

(28.50)

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