The level-set (LS) method uses a signed-distance function to capture the interface in two-phase flows. Geometrical properties can be easily obtained, and merging and splitting of the interface are handled automatically by the LS method. However, it is not inherently volume conserving. Several methods found in literature that aim to solve this problem are discussed in this report, including the interface-correction level-set (ICLS) method. The ICLS method uses an additional advection step with a correction-velocity field to restore global volume loss/gain. We present the volume-of-fluid-based local interface-correction level-set (VOF-LICLS) method. This is an extension to the ICLS method that aims to restore volume locally. This is achieved by coupling the ICLS method with the VoF method. In each time step we evolve both the level-set function and the volume fraction function. The VoF advection is performed using a Lagrangian-Eulerian method on a dual mesh. From the advected LS field volume fractions are constructed and compared to the advected VoF field. This allows us to use local volume fluxes for the velocity field, instead of a global volume flux. The construction of the correction-velocity is performed with the use of an analytic equation and the local volume fluxes. The novel volume correction procedure is then executed iteratively. The level-set equation is discretized in space with a finite element approach. Instabilities occur for the standard Galerkin approach for pure advection equation, so the SUPG method is used in order to obtain stable solutions. The performance of the developed method in this thesis is compared with LS and ICLS for three different test cases. We report a significant improvement for local volume conservation, and we observe that VOF-LICLS yields more accurate interface positions.

Deformable energy-minimizing contours are used in image analysis, particularly to detect boundaries from objects. From the Mumford-Shah model we derive an equation to move such contours towards the boundary, which minimizes the functional. The external force is computed using the mean value of the inside and outside area of the contour. Various image filtering methods are proposed to enhance this model. We introduce an improved Mumford-Shah model to improve the computational time without losing accuracy. An artificial force is added at the beginning of the simulation to quickly yield an approximation of the contour. Finally, we consider a simplified version of the model, which we solve analytically. The solution is used to find suitable parameters for the model.