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Medical Image Segmentation: Energy Minimization and Deformable Models
Published in Troy Farncombe, Krzysztof Iniewski, Medical Imaging, 2017
Chris McIntosh, Ghassan Hamarneh
One structure of particular interest in the diagnosis and understanding of many diseases is vasculature (Bullitt et al., 2003). Vessel segmentation remains an interesting application area of energy minimization methods because of its unique challenges. Firstly, the topology is complex, and as such many of the already mentioned topology-adaptive shape models were first demonstrated in their application to vessel segmentation (McInerney and Terzopoulos, 2000). Secondly, the vessels are often of very low contrast motivating advances in image terms (Frangi et al., 1999; Vasilevskiy and Siddiqi, 2002; Wink et al., 2001, 2004). For example, Vasilevskiy and Siddiqi built flux-maximizing geometric flows based on the observation that the gradient vector field near a vessel should be orthogonal to the vessel (Vasilevskiy and Siddiqi, 2002). They define a flux-maximizing geometric flow as one for which the inward normals of the underlying curve align with the direction of the gradient vector field. Near vessels the gradient vector field points inward toward the vessel centerline, and thus maximizing the flux will align the boundary of the segmentation to the boundary of the vessel. The last major challenge in vessel segmentation is that the vessels can be very thin, pushing the boundaries of numerical stability in many techniques and motivating new methods with increased stability to thin structures (Lorigo et al., 2001; Sundaramoorthi et al., 2007). For example, Lorigo et al. modify GAC to deform along the medial axis of a tubular shape, as opposed to its surface (Lorigo et al., 2001). For a complete survey of vessel segmentation techniques, the reader is referred to Lesage et al. (2009).
Mesh segmentation via geodesic curvature flow
Published in Computer-Aided Design and Applications, 2018
Zhiyu Sun, Ramy Harik, Stephen Baek
The geodesic curvature flow (GCF) is a geometric flow, or informally, a continuous evolution of a curve that minimizes the arc length of a curve. Given a closed self-avoiding rectifiable curve lying on a differential -manifold embedded in (), the energy functional of the GCF is defined as follows: where is an infinitesimal segment defined on the curve for the integration. Here, we restrict our curve to be rectifiable in order to make sure that it is integrable. A curve on a manifold is said to be rectifiable if and only if the length of every geodesic polygon formed by vertices , can be bounded from above by the length of the curve for some parameterization , and under the induced metric of . This consequently means that the curve is a function with bounded variations, and thus integrable.