Abstract
We analyze the rate in which image details are suppressed as a function of the regularization parameter, using first order Tikhonov regularization, Linear Gaussian Scale Space and Total Variation image decomposition. The squared L 2-norm of the regularized solution and the residual are studied as a function of the regularization parameter. For first order Tikhonov regularization it is shown that the norm of the regularized solution is a convex function, while the norm of the residual is not a concave function. The same result holds for Gaussian Scale Space when the parameter is the variance of the Gaussian, but may fail when the parameter is the standard deviation. Essentially this imply that the norm of regularized solution can not be used for global scale selection because it does not contain enough information. An empirical study based on synthetic images as well as a database of natural images confirms that the squared residual norms contain important scale information.
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Gustavsson, D., Pedersen, K.S., Lauze, F., Nielsen, M. (2009). On the Rate of Structural Change in Scale Spaces. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_69
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DOI: https://doi.org/10.1007/978-3-642-02256-2_69
Publisher Name: Springer, Berlin, Heidelberg
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