Total variation regularization has been used for image denoising for about twenty years now. These regularizations have other uses as well; in particular, they can be used to detect differences in scales in data. I have been studying the geometric and regularity properties of minimizers for the associated variational problems with the goal of better understanding them. In this talk I will describe some new results in this area. Some of these results describe minimizers where total variation is defined using an anisotropic norm for the gradient; the motivation for this work, which is joint with Kevin Vixie, is that some computational schemes for computing minimizers naturally use a polygonal approximation to the standard Euclidean metric to define total variation.