In this section, we shall very briefly discuss variational methods in image processing, with an example application in denoising. Image processing tasks can be viewed as function estimation (for example, segmentation can be thought of as finding a smooth closed curve between an object and the background). Calculus of variations can be used for minimization of the appropriately defined energy functionals (with the Euler-Langrange method) for a specific image processing task, and the gradient descent method is used to evolve towards the solution.

The following diagram describes the basic steps in an image processing task, represented as a variational optimization problem. First, we need to create an energy functional E that describes the quality of the input image u. Then, with the Euler-Lagrange equation, we need to calculate the first variation. Next, we need to set up a partial differentail equation (PDE) for the steepest descent minimization and discretize it and evolve towards the minimum: