Differential Equations Solutions 25

Differential Equations Solutions 25 - algorithms reduce...

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35 One common bug in the TSVD section: zeroing out pieces of S A and S B . This does not zero the smallest singular values, and although it is very efficient in time, it gives worse results than doing it correctly. The data was generated by taking an original image F (posted on the website), multiplying by K , and adding random noise using the MATLAB statement G = B * F * A’ + .001 * rand(256,256) . Note that the noise prevents us from completely recovering the initial image. In this case, the best way to choose the regularization parameter is by eye: choose a detailed part of the image and watch its resolution for various choices of the regularization parameter, probably using bisection to Fnd the best pa- rameter. ±igure 6.1 shows data and two reconstructions. Although both algorithms yield images in which the text can be read, noise in the data ap- pears in the background of the reconstructions. Some nonlinear reconstruction
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Unformatted text preview: algorithms reduce this noise. • In many applications, we need to choose the regularization parameter by au-tomatic methods rather than by eye. If the noise-level is known, then the discrepancy principle is the best: choose the parameter to make the residual Kf − g close in norm to the expected norm of the noise. If the noise-level is not known, then generalized cross validation and the L-curve are popular methods. See [1,2] for discussion of such methods. [1] Per Christian Hansen, James M. Nagy, and Dianne P. O’Leary. Deblurring Images: Matrices, Spectra, and Filtering . SIAM Press, Philadelphia, 2006. [2] Bert W. Rust and Dianne P. O’Leary, “Residual Periodograms for Choos-ing Regularization Parameters for Ill-Posed Problems”, Inverse Problems, 24 (2008) 034005....
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