We present SurfCut, an algorithm for extracting a smooth, simple surface with an unknown 3D curve boundary from a noisy 3D image and a seed point. Our method is built on the novel observation that ridge curves of the Euclidean length of minimal paths ending on a level set of the solution of the eikonal equation lie on the surface. Our method extracts these ridges and cuts them to form the surface boundary. Our surface extraction algorithm is built on the novel observation that the surface lies in a valley of the eikonal equation solution. The resulting surface is a collection of minimal paths. Using the framework of cubical complexes and Morse theory, we design algorithms to extract ridges and valleys robustly. Experiments on three 3D datasets show the robustness of our method, and that it achieves higher accuracy with lower computational cost than state-of-the-art.

We propose new dense descriptors for texture segmentation. Given a region of arbitrary shape in an image, these descriptors are formed from shape-dependent scale spaces of oriented gradients. These scale spaces are defined by Poisson-like partial differential equations. A key property of our new descriptors is that they do not aggregate image data across the boundary of the region, in contrast to existing descriptors based on aggregation of oriented gradients. As an example, we show how the descriptor can be incorporated in a Mumford-Shah energy for texture segmentation. We test our method on several challenging datasets for texture segmentation and textured object tracking. Experiments indicate that our descriptors lead to more accurate segmentation than non-shape dependent descriptors and the state-of-the-art in texture segmentation.