hwp4_04

# hwp4_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 Nov 4th Due on: 2004 Nov 12th Problem 1: In the continuous version of the optical ﬂow problem, we ﬁnd the functions u(x, y) and v(x, y) that minimize 2 2 2 (uE x + vE y + E t ) 2 + λ(u 2 + u y + v + v y )dx dy x x D The Euler equations for this problem yield λ±u = (uE x + vE y + E t )E x λ±v = (uE x + vE y + E t )E y where E x , E y , and E t are the derivatives of image brightness. What are the natural boundary conditions? Problem 2: The formulation of the optical ﬂow problem as above uses as a measure of “unsmoothness’’ the sum of squares of ﬁrst partial derivatives of the components of the optical ﬂow. Consider a sphere centered at X = Y = 0 ,of radius R , rotating about an axis parallel to the y -axis with angular velocity ω . Suppose the sphere is far away from the camera (in relation to its radius) and we approximate perspective

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hwp4_04 - Massachusetts Institute of Technology Department...

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