Scale recognition, regularization parameter selection, and Meyer's G norm in total variation regularization

We investigate how TV regularization naturally recognizes the scale of individual features of an image, and we show how this perception of scale depends on the amount of regularization applied to the image. We give an automatic method driven by the geometry of the image for finding 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 effects 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.