Landmark-Based Shape Encoding and Sparse-Dictionary Learning in the Continuous Domain

We provide a generic framework to learn shape dictionaries of landmark-based curves that are defined in the continuous domain. We first present an unbiased alignment method that involves the construction of a mean shape as well as training sets whose elements are subspaces that contain all affine transformations of the training samples. The alignment relies on orthogonal projection operators that have a closed form. We then present algorithms to learn shape dictionaries according to the structure of the data that needs to be encoded: 1) projection-based functional principal-component analysis for homogeneous data and 2) continuous-domain sparse shape encoding to learn dictionaries that contain imbalanced data, outliers, or different types of shape structures. Through parametric spline curves, we provide a detailed and exact implementation of our method. We demonstrate that it requires fewer parameters than purely discrete methods and that it is computationally more efficient and accurate. We illustrate the use of our framework for dictionary learning of structures in biomedical images as well as for shape analysis in bioimaging.