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U. Glaeser

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FIGURE 28.7 A two-component, 2-D signal in translational motion.<br />

Extending the analysis to the 3-D case ( f(x, y, t)), let the velocity r = (rx, ry) . Then<br />

© 2002 by CRC Press LLC<br />

(28.19)<br />

Each temporal frequency is shifted by the dot product of the spatial frequency vector s = (ξx, ξy) and the<br />

image velocity vector r = (rx, ry). If the image was originally static, then<br />

(28.20)<br />

Geometrically, the image motion changes the static image transform (which lies in the (ξ x, ξ y) plane)<br />

into a spectrum in a plane with slope −r y in the (ξ y, ξ t) plane and −r x in the (ξ x, ξ t) plane. As in the 2-D<br />

case, the shifted points lie on a line through the origin. Note that this represents a relatively sparse<br />

occupation of the frequency domain (of interest for compression applications). A 3-D volume of data<br />

has been “compressed” into a plane. This compactness is not observed in the spatiotemporal domain.<br />

In summary, the spectrum of a stationary image lies in the (ξ x, ξ y) plane. When the image undergoes<br />

translational motion, the spectrum occupies an oblique plane which passes through the origin. The<br />

orientation of the plane indicates the speed and direction of the motion. It is, therefore, possible to<br />

associate energy in particular regions of the frequency domain with particular image velocity components.<br />

By filtering specific regions in the frequency domain, these image velocity components can be detected.<br />

As will be seen shortly, other effects (such as the visual impact of temporal aliasing) can also be understood<br />

in the frequency domain.<br />

3-D Sampling<br />

In its simplest form (regular sampling on a rectangular grid, the method used here), 3-D sampling is a<br />

straightforward extension of 2-D (or 1-D) sampling (Fig. 28.8). Given a bandlimited sequence<br />

with<br />

F ξ x, ξ y, ξ t<br />

(- ξ ξ x0, - y0,<br />

r ξ +r ξ x0 y0)<br />

x y<br />

(- ξ ξ x0,- y0,0)<br />

ξ t<br />

the continuous sequence can be reconstructed from a discrete set of samples whenever<br />

ξ y<br />

( ξ ξ x0, y0,-<br />

r ξx - r ξ x )<br />

0 y y0<br />

( ξx , ξ<br />

0 y0,0)<br />

f( x– rxt, y – ryt, t)<br />

F<br />

→ F ξ ( x, ξy, ξt + rxξ x + ryξ y)<br />

ξt = – r ⋅ s = – ( rxξ x + ryξ y)<br />

f( x, y, t)<br />

F<br />

→ F ξ ( x, ξy, ξt) ( ) = 0 whenever ξx > ξx0 , ξy > ξy0 , or ξt > ξt0 ξxs ><br />

2ξ x0 , ξ ys<br />

><br />

2ξ y0 , and ξ ts<br />

ξ x<br />

> 2ξt0 (28.21)<br />

(28.22)<br />

(28.23)

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