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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. [ABSTRACT FROM AUTHOR]

A point of a discrete object is called simple if it can be deleted from this object without altering topology. In this paper, we present new characterizations of simple points, which hold in dimensions 2, 3, and 4 and lead to efficient algorithms for detecting such points. In order to prove these characterizations, we establish two confluence properties of the collapse operation which hold in the neighborhood of a point in spaces of low dimension. We develop this work in the framework of cubical complexes, which provides a sound topological basis for image analysis and retrieves the main notions and results of digital topology, in particular the notion of simple point. [ABSTRACT FROM AUTHOR]