By Stiefel E.
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Example text
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. 2 The Discrepency Principle This is an alternative criterion advocated by those who believe that “the data come …rst”. The value of ¸ is chosen so as to place the solution on the edge of the feasible set as de…ned by C (f ) · C0 : Since C (f ) is the sum of squares of noise terms, its expectation value can be calculated if one knows the amount of noise that is present.
As we shall see later, this is a result of fundamental importance which is called Bayes’ theorem.
As n becomes large, the problem of model fitting merges into that of indirect imaging. In the regime of indirect imaging, the number of points in image space is chosen to be so large that it can adequately represent the object of interest. Since there are now many images which map to the same data, it becomes necessary to choose from among them using an additional criterion of optimality. In terms of the singular value decomposition, the rank r of the forward map A is less than n when the parameters of the model are not independent.