Suppose that in Example 10.28 motion along the x-axis is set to zero. The object now moves...

Suppose that in Example 10.28 motion along the x-axis is set to zero. The object now moves only along the y-axis at 1 pixel per frame for 32 frames and then (instantaneously) reverses direction and moves in exactly the opposite direction for another 32 frames. What would Figs. 10.63 and 10.64 look like under these conditions?

EXAMPLE 10.28: Detection of a small moving object via the frequency domain.

Figures 10.61 through 10.64 illustrate the effectiveness of the approach just derived. Figure 10.61 shows one of a 32-frame sequence of LANDSAT images generated by adding white noise to a reference image. The sequence contains a superimposed target moving at 0.5 pixel per frame in the x-direction and 1 pixel per frame in the y-direction. The target, shown circled in Fig. 10.62, has a Gaussian intensity distribution spread over a small (9-pixel) area and is not easily discernible by eye. Figures 10.63 and 10.64 show the results of computing Eqs. (10.6-8) and (10.6-9) with a1 = 6 and a2 = 4, respectively. The peak at u1 = 3 in Fig. 10.63 yields V1 = 0.5 from Eq. (10.6-10). Similarly, the peak at u2= 4 in Fig. 10.64 yields V2= 1.0 from Eq. (10.6-11).

(10.6-8)

(10.6-9)

(10.6-10)

(10.6-11)

FIGURE 10.61 LANDSAT frame. (Cowart, Snyder, and Ruedger.)

FIGURE 10.62 Intensity plot of the image in Fig. 10.61, with the target circled. (Rajala, Riddle, and Snyder.)

FIGURE 10.63 Spectrum of Eq. (10.6-8) showing a peak at u1 = 3. (Rajala, Riddle, and Snyder.)

Guidelines for the selection of a1and a2can be explained with the aid of Figs. 10.63 and 10.64. For instance, suppose that we had used a2= 15 instead of a2= 4. In that case the peaks in Fig. 10.64 would now be at u2= 15 and 17 because V2 = 1.0, which would be a seriously aliased result. As discussed in Section 4.5.4, aliasing is caused by undersampling (too few frames in the present discussion, as the range of u is determined by K). Because u = aV, one possibility is to select

FIGURE 10.64 Spectrum of Eq. (10.6-9) showing a peak at u2 = 4.(Rajala, Riddle, and Snyder.)