Stochastic level-sets and random sets in image processing with partial differential equations
Final Report Abstract
In this project, gray value uncertainty in image processing with partial differential equations (PDEs) has been considered. To model gray value uncertainty, images are replaced by stochastic fields. Using such stochastic images in PDE based models yields stochastic PDEs (SPDEs) which we discretize with contemporary numerical approaches like the generalized polynomial chaos approach. In the reported project phase, the segmentation of stochastic images with a stochastic version of the well known geodesic active contour model has been investigated. Such model is challenging for numerical discretizations as it involves a stochastic velocity, i.e. the advective transport of a level-set function is with a velocity that is a random variable. Key to our numerical treatment is a local diagonalization of the matrix that describes the advective coupling of the stochastic modes. Together with an Strang operator splitting we arrive at a scheme that can be dealt with by classical numerical upwinding techniques or other known approaches to hyperbolic PDEs. We have evaluated the new model with numerical tests, artificial 2D image data, and some real world 2D medical images.
Publications
- Stochastic Partial Differential Equations for Computer Vision with Uncertain Data, Synthesis Lectures on Visual Computing 9(2), 160 pages, Morgan & Claypool Publishers, 2017
T. Preusser, R. M. Kirby, T. Pätz
(See online at https://doi.org/10.1007/978-3-031-02594-5) - Stochastic upwind method for level-set segmentation on images with uncertain data, GAMM 2021
E. Theilen, T. Preusser
