Affine Bases for Affine Spaces

Gabriel Dogadov, Marc Alexa; Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Findings, 2026, pp. 213-222

Recent work suggests quadratic distance fields (QDFs) as a representation of affine subspaces of Euclidean space (often called flats), as they enable an efficient computation of weighted means: compute the weighted mean of the QDFs and then project back onto the closest QDF representing a flat. Considering this operation as an affine combination of flats, we 1) analyze the prerequisites for flats forming a basis of the space of flats and 2) provide an algorithm for projecting flats into a such a basis. Intriguingly, the projection requires nothing more than the solution of a linear system. This allows computing with flats similar to how we commonly compute with points in Euclidean spaces, with comparable cost despite the curved nature of the manifold of flats. We demonstrate the versatility of this idea by triangulating local spaces of affine lines, to be used, e.g., in image based rendering.