Title
Discrete total variation: new definition and perspectives
Abstract
In 1992, Rudin, Osher, and Fatemi introduced the total variation (TV) for imaging problems. Since then, a variety of papers has demonstrated the effectiveness of TV minimization to recover sharp images, by preserving strong discontinuities, while removing noise and other artifacts. For continuously defined two-dimensional functions, the TV is simply the L1 norm of the gradient amplitude. But for discrete images, it is a nontrivial task to properly define the gradient using finite differences, as is well known in the community of computer graphics and visualization. We demonstrate how a new definition for discrete total variation, first proposed by Condat in 2017 can affect modern TV minimization-based algorithms. This new definition of the total variation corrects some drawbacks of the classical definition, yields sharper edges and structures, and has potential applications going far beyond imaging algorithms. We investigate what TV-based algorithms and schemes can benefit from the use of a new approach to TV discretization, for instance, one can consider edge detection based on the gradient amplitude and stable schemes for computing PDE flows based on the gradient orientation.