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hwp2_04 - Massachusetts Institute of Technology Department...

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�� �� �� �� �� �� �� �� Massachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision Handed out: 2004 Sep. 23 Due on: 2004 Sep. 30 Problem 1: Suppose that the image in a patch D moves with velocity (u, v) . To find the best-fit image velocity we minimize 1 I = (uE x + vE y + E t ) 2 dx dy 2 D by choosing u and v . This is a straight-forward calculus problem—the stationary points of I can be found by solving the pair of equations obtained by setting the derivatives of I with respect to u and v equal to zero. Let the solution so obtained be (u, v) = (u 0 , v 0 ) . There usually is a significant amount of measurement noise in real images, and the estimate of the velocity will be affected by this noise. We need to determine how quickly I changes as we move away from the minimum at (u 0 , v 0 ) in order to determine how certain the location of the minimum is. The solution is dependable if I changes rapidly as we move away from the minimum, since noise will then have less of an effect on the estimation of the location of the minimum. (a) Let I denote the second partial derivative of I in a direction that lies an angle θ clockwise from the u -axis. Show that I (θ) = cos 2 θ E 2 + 2 sin θ cos θ E x E y + sin 2 θ E y 2 , x D D D or I (θ) = (E x cos θ + E y sin θ) 2 dx dy. D Conclude that I 0 for all θ , and that the stationary point consequently must be a minimum (as opposed to a maximum or saddle).
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