Semi-metric Space: A New Approach to Treat Orthogonality and Parallelism

Abstract

We propose a new method to recover 3D structures of artificial objects from scene pictures using orthogonality and parallelism. A new transformation group, “semi-metric space,” is defined to describe the scenes of artificial objects consisting of orthogonal and parallel line features. A metric invariant called conic dual to the circular points has a simple diagonal form in the semi-metric space. Furthermore, under some assumptions, the metric reconstruction is possible using some affine properties. The algorithms are verified with real images captured with a camera in a commercial mobile phone.