Multiphase Segmentation with Convex Relaxation of Phase Functions

This work presents a multiphase functional to segment and approximate a given gray scale image where the resulting image is smooth on the support of each phase function. Therefore unique existence of the model functions is shown and then existence of a minimum of the functional with respect to functions in a convex relaxation of a set of characteristic functions is shown. This research puts a focus on image denoising and image segmentation simultaneously and uses both approaches in one functional. Moreover since it is unsure if the functional is convex a semi-gradient descent approach is established in a spatially continuous setting and later on in a finite dimensional setting. In order to prove certain features a mollifying operator was introduced. Furthermore a mapping that binds the update of the optimization process to have range in [0 1] was used. The segmentation was performed with up to four characteristic functions whereas segmentation with only two showed the best result. The work presented here has profound implications for future studies of a concurrent algorithm of image segmentation and denoising.