Geodesic Thin Plate Splines for Image Segmentation

Thin Plate Splines are often used in image registration to model deformations. Its physical analogy involves a thin lying sheet of metal that is deformed and forced to pass through a set of control points. The Thin Plate Spline equation minimizes that thin plate bending energy. Rather than using Euclidean distances between control points for image deformation, we are using geodesic distances for image segmentation. Control points become seed points and force the thin plate to pass through given heights. Intuitively, the thin plate surface in the vicinity of a seed point within a region should have similar heights. The minimally bended thin plate actually gives a "confidence" map telling what the closest seed point is for every surface point. The Thin Plate Spline has a closed-form solution which is fast to compute and global optimal. This method shows comparable results to the Graph Cuts method.