Total Variation - Applications

Applications

Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems

• Numerical analysis of differential equations: it is the science of finding approximate solutions to differential equations. Applications of total variation to this problems are detailed in the article "total variation diminishing"

• Image denoising: in image processing, denoising is a collection of methods used to reduce the noise in an image reconstructed from data obtained by electronic means, for example data transmission or sensing. Total variation denoising is the name for the application of total variation to image noise reduction; further details can be found in the paper (Caselles, Chambolle & Novaga 2007).