hwp5_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 Nov 18th Due on: 2003 Nov 30th Problem 1: The motion field is particularly simple when a camera is moving without rotation in a fixed environment. All parts of the image then appear to be streaming away from the “focus of expansion’’—the projection of the transla- tional motion vector t into the image plane. (a) Show that the focus of expansion is the vanishing point of the family of parallel lines ( r = r + k t ) in the direction of translational motion. (b) Show that, the location of the focus of expansion (x , y ) is given by 1 1 (x , y ) = (U, V ) f W if the instantaneous translational velocity is t = (U, V , W ) T . (c) Show that (x − x )E x + (y − y )E y = at “critical’’ points, that is where E t = . (d) Estimate the position of the FOE by minimizing n (x i − x )E x i + (y i − y )E y i 2 , i = 1 where (x i , y i ) are the positions of n “critical’’ points in the image, while (E x i , E y i ) are the brightness gradients at these points....
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hwp5_04 - Massachusetts Institute of Technology Department...

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