Fitting Flats to Flats

Affine subspaces of Euclidean spaces are also referred to as flats. A standard task in computer vision, or more generally in engineering and applied sciences, is fitting a flat to a set of points, which is commonly solved using the PCA. We generalize this technique to enable fitting a flat to a set of other flats, possibly of varying dimensions, based on representing the flats as squared distance fields. Compared to previous approaches such as Riemannian centers of mass in the manifold of affine Grassmannians, our approach is conceptually much simpler and computationally more efficient, yet offers desirable properties such as respecting symmetries and being equivariant to rigid transformations, leading to more intuitive and useful results in practice. We demonstrate these claims in a number of synthetic experiments and a multi-view reconstruction task of line-like objects.