A Variational Shape Optimization Framework for Image Segmentation

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Image segmentation is one of the fundamental problems in image processing. The goal is to partition a given image into regions that are uniform with respect to some image features and possibly to extract the region boundaries. Recently methods based on PDEs have been found to be an effective way to address this problem. These methods in general fall under the category of shape optimization, as the typical approach is to assign an energy to a shape, say a curve in 2d, and to deform the curve in a way that decreases its energy. In the end when the optimization terminates, the curve is not only at a minimum of the energy, but also at a boundary in the image.

In this thesis, we emphasize the shape optimization view of image segmentation and develop appropriate tools to pursue the optimization in 2d and 3d. We first review the classes of shape energies that are used within the context of image segmentation. Then we introduce the analytical results that will help us design energy-decreasing deformations or flows for given shapes.

We describe the gradient flows minimizing the energies, by taking into account the shape derivative information. In particular we emphasize the flexibility to accommodate different velocity spaces, which we later demonstrate to be quite beneficial. We turn the problem into the solution of a system of linear PDEs on the shape. We describe the corresponding space discretization based on the finite element method and an appropriate time discretization scheme as well.

To handle mesh deterioration and possibly singularities due to motion of nodes, we describe time step control, mesh smoothing and angle width control procedures, also a topology surgery procedure that allows curves to undergo topological changes such as merging and splitting. In addition we introduce space adaptivity algorithms that help maintain accuracy of the method and reduce computational cost as well.

Finally we apply our method to two major shape energies used for image segmentation: the minimal surface model (a.k.a geodesic active contours) and the Mumford-Shah model. For both we demonstrate the effectiveness of our method with several examples.