Denoising is a fundamental procedure in image processing and computer vision. The total variation based image denoising model of Rudin, Osher, and Fatemi (ROF) has become one of the standard techniques in the field, and has been applied to a variety of image restoration tasks. In the first part of the talk, I will consider anisotropic versions of the ROF energy and describe properties of their minimizers (joint work with Stanley Osher). In the second part, I will show that certain variants of the ROF model turn out to be convex formulations of non-convex optimization problems encountered in shape denoising and image segmentation applications, thus allowing us to find global minimizers of these non-convex problems via standard convex minimization techniques (joint work with Tony Chan and Mila Nikolova).