Abstract

Nonlinear partial differential equations (PDE) are now widely used to regularize images. They allow to eliminate noise and ar- tifacts while preserving large global features, such as ob ject contours. In this context, we propose a geometric framework to design  PDE flows acting on constrained datasets. We focus our interest on flows of matrix- valued functions undergoing orthogonal and spectral constraints. The corresponding evolution PDE’s are found by minimization of cost func- tionals, and depend on the natural metrics of the underlying constrained manifolds (viewed as Lie groups or homogeneous spaces). Suitable nu- merical schemes that fit the constraints are also presented. We illustrate this theoretical framework through a recent and challenging problem in medical imaging: the regularization of di?usion tensor volumes (DT- MRI).