Heike Benninghoff

Heike Benninghoff, Harald Garcke

In this paper, we introduce a novel parametric method for segmentation of three-dimensional images. We consider a piecewise constant version of the Mumford-Shah and the Chan-Vese functionals and perform a region-based segmentation of 3D image data. An evolution law is derived from energy minimization problems which push the surfaces to the boundaries of 3D objects in the image. We propose a parametric scheme which describes the evolution of parametric surfaces. An efficient finite element scheme is proposed for a numerical approximation of the evolution equations. Since standard parametric methods cannot handle topology changes automatically, an efficient method is presented to detect, identify and perform changes in the topology of the surfaces. One main focus of this paper are the algorithmic details to handle topology changes like splitting and merging of surfaces and change of the genus of a surface. Different artificial images are studied to demonstrate the ability to detect the different types of topology changes. Finally, the parametric method is applied to segmentation of medical 3D images.

Heike Benninghoff, Harald Garcke

In this article, a new method for segmentation and restoration of images on two-dimensional surfaces is given. Active contour models for image segmentation are extended to images on surfaces. The evolving curves on the surfaces are mathematically described using a parametric approach. For image restoration, a diffusion equation with Neumann boundary conditions is solved in a postprocessing step in the individual regions. Numerical schemes are presented which allow to efficiently compute segmentations and denoised versions of images on surfaces. Also topology changes of the evolving curves are detected and performed using a fast sub-routine. Finally, several experiments are presented where the developed methods are applied on different artificial and real images defined on different surfaces.

Heike Benninghoff, Harald Garcke

In this paper, we introduce a novel approach for active contours with free endpoints. A scheme is presented for image segmentation and restoration based on a discrete version of the Mumford-Shah functional where the contours can be both closed and open curves. Additional to a flow of the curves in normal direction, evolution laws for the tangential flow of the endpoints are derived. Using a parametric approach to describe the evolving contours together with an edge-preserving denoising, we obtain a fast method for image segmentation and restoration. The analytical and numerical schemes are presented followed by numerical experiments with artificial test images and with a real medical image.

Heike Benninghoff, Harald Garcke

Curve evolution schemes for image segmentation based on a region based contour model allowing for junctions, vector-valued images and topology changes are introduced. Together with an a posteriori denoising in the segmented homogeneous regions this leads to a fast and efficient method for image segmentation and restoration. An uneven spread of mesh points is avoided by using the tangential degrees of freedom. Several numerical simulations on artificial test problems and on real images illustrate the performance of the method.