Variational principles in image processing and the regularization of orientation fields

Variational principles and partial differential equations have proved to be fundamental elements in the mathematical modeling of extended systems in physics and engineering. Of particular interest are the equations that arise from a free energy functional. Recently variational principles have begun to be used in Image Processing to perform basic tasks such as denoising, debluring, etc. Great improvements can be achieved by selecting the most appropriate form for the functional.
In this article we show how these ideas can be applied not just to scalar fields (i.e. grayscale images) but also to curved manifolds such as the space of orientations. This work is motivated by the denoising of images acquired with Magnetic Resonance scanners using diffusion-sensitized magnetic gradients.