hwp1_04 - Massachusetts Institute of Technology Department...

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Unformatted text preview: Massachusetts Institute of Technology Department of Computer Science and Electrical Engineering 6.801/6.866 Machine Vision Handed out: 2004 Sep. 9th Due on: 2004 Sep. 16th Problem 1: In class we solved the least squares problem of estimating the speed at which an image moves in a one dimensional camera, based on the constant brightness assumption. The constant brightness assumption (in one dimension) can be expressed in the form uE x (x, t) + E t (x, t) = 0, where E x and E t are the x and t derivatives of image brightness E(x, t) , while u = dx/dt is the speed of the image motion. Now suppose that the illumination is not constant, but changing in such a way that uE x + E t = k (rather than being equal to zero). Clearly this is equivalent to using the incorrect value E = E t − k for the time derivative of the brightness t in the original equation. Suppose we do not know the value of k ahead of time. Let us first blindly apply the least squares method for estimating u . Suppose that the “image’’ runs from x 1 (with brightness E 1 ) to x 2 (with brightness E 2 ). (a) What is the least squares solution for u we would get based on the “cor- rupted’’ time derivative ( E t ) of brightness? (b) Is there any error in the estimated speed u if the brightness at the right end of the image matches that at the left end? Now let us try to take into account the change of overall illumination with time. We do not know the magnitude of k . We will try and estimate both u and k using a least squares method....
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hwp1_04 - Massachusetts Institute of Technology Department...

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