Discrete Total Variation: New Definition and Minimization

We propose a new definition for the gradient field of a discrete image, defined on a twice finer grid. The differentiation process from the image to its gradient field is viewed as the inverse operation of linear integration, and the proposed mapping is nonlinear. Then, we define the total variation of an image as the `1 norm of its gradient field amplitude. This new definition of the total variation yields sharp edges and has better isotropy than the classical definition.