Shape Projectors for Landmark-Based Spline Curves

We present a generic method to construct orthogonal projectors for two-dimensional landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, and scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves, which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology.