Differential Equations Solutions 51

Differential Equations Solutions 51 - R or D is quite...

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61 Figure 11.2. The images resulting from minimizing R . termination criteria to the scaling of the variables and the function to be minimized; otherwise the function might terminate prematurely (even with negative entries in the cluster centers if, for example, the objective function values are all quite close together) or never terminate (if small changes in the variables lead to large changes in the function). We chose to scale R and D by dividing by 256 and by the number of points in the summation. The solution is very sensitive to the initial guess, since there are many local minimizers. (a) The number of variables is kq . (b) Although the number of variables is quite small (9 for k = 3 and 15 for k = 5), evaluating the function
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Unformatted text preview: R or D is quite expensive, since it involves a mapping each of the 1,000 pixels to a cluster. Therefore, the overhead of the algorithm is insigniFcant and the time is proportional to the number of function evaluations. The functions are not dierentiable, so modeling as a quadratic function is not so eective. This slows the convergence rate, although only 15-25 iterations are used. This was enough to converge when minimizing D , but not enough for R to converge. Actually, a major part of the time in the sample implementation is postpro-cessing: the construction of the resulting image! (c) igures 11.2 and 11.3 show the results with k = 3 , 4 , 5 clusters. The solution...
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