Abstract:
The objective of this thesis is to find and analyse practical numerical algorithms for the minimisation and gradient-flows of the Mumford-Shah and Mumford-Shah-Euler functionals for unit vector fields.
The motivation for these questions is twofold: First, these are interesting model-problems combining non-convex functionals with a non-convex constraint, as an extension of existing works on harmonic maps to the sphere.
Second, bot functionals were originally introduced in image processing: The Mumford-Shah functional for segmentation, and the Mumford-Shah-Euler functional for inpainting; and the sphere-constraint can be used to implement the chromaticity and brightness colour model in this context.
In the first part of the thesis, two schemes for the minimisation of the Mumford-Shah functional for unit-vector fields are presented and discretised using first-order finite elements.
The first scheme uses a projection approach to enforce the sphere-constraint. It works well in simulations, but we only have partial convergence results.
The second scheme uses a penalisation approach, which only approximates the sphere-constraint, but allows for a complete proof of convergence.
In the second part of the thesis, two schemes for the gradient-flow of the Mumford-Shah-Euler functional for unit-vector fields are presented and discretised, again using first-order finite elements.
The first scheme is an extension of the penalisation approach from part one, which again allows for a complete proof of convergence.
The second scheme uses a Lagrange multiplier to enforce the sphere constraint, and we can again only present partial convergence results.
Both parts are concluded by simulations comparing the two corresponding algorithms with each other and presenting comparisons between the chromaticity and brightness and the conventional RGB colour model.