PDEs-Based Image Processing for Matrix Fields

Nonlinear parabolic partial differential equations(PDEs) are an important tool to perform filtering and denoising procedures for scalar images. Prominent examples are PDEs that steer morphological operations or govern isotropic and anisotropic diffusion processes.

Similar image processing tasks arise for matrix-valued data, so-called matrix fields. This data type is, for example, the output of diffusion tensor magnetic resonance imaging (DT-MRI), a modern medical image acquisition technique.

This talk is intended to introduce a generic framework that allows us to find the matrix-valued counterparts of PDEs relevant in the scalar setting. This framework relies on an operator algebraic point of view on symmetric matrices and exploits their rich algebraic structure. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting.

Numerical experiments performed on both synthetic and real world data confirm the effectiveness of this generic matrix-valued framework.