Duality of reconstruction and positioning from projective views

Abstract

Given multiple image data from a set of points in 3D, there are two fundamental questions that can be addressed: (1) What is the structure of the set of points in 3D? (2) What are the positions of the cameras relative to the points? In this paper, we show that for projective views and with structure- and position-defined modulo linear transformations, these problems are are dual in the sense that their solution arises from constraint equations where space point and camera positions occur in a reciprocal way. The problem of computing camera positions from m points in n views can be solved with the same algorithm as the problem of directly reconstructing n+4 points in m-4 views. This unifies different approaches for projective reconstruction: methods based on external calibration and direct methods exploiting constraints that exist between space and image invariants.