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Unformatted text preview: EE263s Summer 2009-10 Laurent Lessard EE263s homework 5 1. Curve-smoothing. We are given a function F : [0 , 1] → R (whose graph gives a curve in R 2 ). Our goal is to find another function G : [0 , 1] → R , which is a smoothed version of F . We’ll judge the smoothed version G of F in two ways: • Mean-square deviation from F , defined as D = integraldisplay 1 ( F ( t ) − G ( t )) 2 dt. • Mean-square curvature, defined as C = integraldisplay 1 G ′′ ( t ) 2 dt. We want both D and C to be small, so we have a problem with two objectives. In general there will be a trade-off between the two objectives. At one extreme, we can choose G = F , which makes D = 0; at the other extreme, we can choose G to be an affine function ( i.e. , to have G ′′ ( t ) = 0 for all t ∈ [0 , 1]), in which case C = 0. The problem is to identify the optimal trade-off curve between C and D , and explain how to find smoothed functions G on the optimal trade-off curve. To reduce the problem to a finite-dimensional one, we will represent the functions F and G (approximately) by vectors f, g ∈ R n , where f i = F ( i/n ) , g i = G ( i/n ) . You can assume that n is chosen large enough to represent the functions well. Using this representation we will use the following objectives, which approximate the ones defined for the functions above: • Mean-square deviation, defined as d = 1 n n summationdisplay i =1 ( f i − g i ) 2 . • Mean-square curvature, defined as c = 1 n − 2 n − 1 summationdisplay i =2 parenleftbigg g i +1 − 2 g i + g i − 1 1 /n 2 parenrightbigg 2 . In our definition of c , note that g i +1 − 2 g i + g i − 1 1 /n 2 gives a simple approximation of G ′′ ( i/n ). You will only work with this approximate version of the problem, i.e. , the vectors f and g and the objectives c and d . (a) Explain how to find g that minimizes d + μc , where μ ≥ 0 is a parameter that gives the relative weighting of sum-square curvature compared to sum-square deviation. Does your method always work? If there are some assumptions you need to make (say, on rank of some matrix, independence of some vectors, etc.), state them clearly. Explain how to obtain the two extreme cases: μ = 0, which corresponds to minimizing d without regard for c , and also the solution obtained as μ → ∞ ( i.e. , as we put more and more weight on minimizing curvature). (b) Get the file curve smoothing.m from the course web site. This file defines a specific vector f that you will use. Find and plot the optimal trade-off curve between d and c . Be sure to identify any critical points (such as, for example, any intersection of the curve with an axis). Plot the optimal g for the two extreme cases μ = 0 and μ → ∞ , and for three values of μ in between (chosen to show the trade-off nicely). On your plots of g , be sure to include also a plot of f , say with dotted line type, for reference. Submit your Matlab code....
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- Summer '08
- Rank, Invertible matrix, optimal trade-oﬀ curve