By Arnfried Kemnitz
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Additional info for Mathematik Zum Studienbeginn
Example text
M. 1 Pixel values in original (unblurred and un-noisy) image with the text, as reference. 2 The generic form of the L-curve. 1 The L-Curve Perhaps the most convenient graphical tool for setting ¸ is the “L-curve”. When we plot log kd ¡ Af ¸ k versus log kf¸ ¡f 1 k (for a discrete problem) we get the characteristic L-shaped curve with a (often) distinct corner separating vertical and horizontal parts of the curve. The rationale for using the L curve is that regularization is a trade-o¤ between the data mis…t and the solution seminorm.
However, the data mis…t is increasing sharply with more …ltering. Along this portion, our preference for the more highly …ltered solutions increases only slightly at the cost of a rapidly rising data mis…t, and so it is desirable to choose a solution which lies not too far to the right of the corner. The following …gure shows the L-curve that is associated with the deblurring example in chapter 1. The curve is labelled parametrically with the values of the regularization parameter. M. 3 L-curve for the deblurring example of the position of largest upwards-pointing curvature.
The left-hand side is the matrix A ∗ acting on the data vector (1) ,and since A = a∗1 (x) = 1∗ , in this case, then A∗ = a1 (x) = 1 (x) and the function 1 (x) acting on the number 1 simply results in the function 1 (x) which is the constant function with value 1. M. 7 The three right singular vectors 0. 82706 0. 54744 0. 12766 which has the normalized eigenvectors: u1 = 0. 46039 , u2 = −0. 52822 , and u3 = −0. 71375 , 0. 32248 −0. 64909 0. 68868 with eigenvalues 1. 4083, 0.
Mathematik Zum Studienbeginn by Arnfried Kemnitz
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