Scale Recognition, Regularization Parameter Selection, and Meyer's G Norm in Total Variation Regularization

We investigate how TV regularization naturally recognizes scale of individual image features and we show how perception of scale depends on the amount of regularization applied to the image We give an automatic method for nding the minimum value of the regularization parameter needed to remove all features below a user chosen threshold We explain the relation of Meyer s G norm to the perception of scale which provides a more intuitive understanding of this norm We consider other applications of this ability to recognize scale including the multiscale e ects of TV regularization and the rate of loss of image features of various scales as a function of increasing amounts of regularization Several numerical results are given Introduction Consider the problem of restoring a noise contaminated or otherwise degraded image inR given a measured image u x nd an approximation u x to the true image utrue x where u Kutrue and where x is the noise or other degradation in the image The work in this paper results from the case in which the blurring operator K is the identity in which case the problem could be considered one of ltering or denoising u utrue Typically our goal is to recover the true image utrue as exactly as possible and or to nd a new image u in which the information of interest is more obvious and or more easily extracted Total variation regularization in image processing Just over a decade ago Rudin Osher and Fatemi proposed to modify the given image by decreasing the total variation