New Geometric Interpretation and Analytic Solution for Quadrilateral Reconstruction

Abstract

A new geometric framework, called generalized coupled line camera (GCLC), is proposed to derive an analytic solution to reconstruct an unknown scene quadrilateral and the relevant projective structure from a single or multiple image quadrilaterals. We extend the previous approach developed for rectangle to handle arbitrary scene quadrilaterals. First, we generalize a single line camera by removing the centering constraint that the principal axis should bisect a scene line. Then, we couple a pair of generalized line cameras to model a frustum with a quadrilateral base. Finally, we show that the scene quadrilateral and the center of projection can be analytically reconstructed from a single view when prior knowledge on the quadrilateral is available. A completely unknown quadrilateral can be reconstructed from four views through non-linear optimization. We also describe a improved method to handle an off-centered case by geometrically inferring a centered proxy quadrilateral, which accelerates a reconstruction process without relying on homography. The proposed method is easy to implement since each step is expressed as a simple analytic equation. We present the experimental results on real and synthetic examples.