Regularization of orthonormal vector sets using coupled PDE´s

We address the problem of restoring, while presenting possible discontinuities, fields of noisy orthonormal vector sets, taking the orthonormal constraints explicity into account. We develop a variational solution for the general case where each image feature may correspond to multiple n-D orthogonal vectors of unit norms. We first formulate the problem in a new variational framework, where discontinuities and orthonormal constraints are preserved by means of constrained minimization and Φ-function regularization, leading to a set of coupled anisotropic diffusion PDE. A geometric interpretation of the resulting equations, coming from the field of solid mechanics, is proposed for the 3D case. Two interesting restrictions of our framework are also tackled: the regularization of 30 rotation matrices and the direction diffusion (the parallel with previous works is made). Finally, we present a number of denoising results and applications