Iso-disparity Surfaces for General Stereo Configurations

This paper discusses the iso-disparity surfaces for general stereo configurations. These are the surfaces that are observed at the same resolution along the epipolar lines in both images of a stereo pair. For stereo algorithms that include smoothness terms either implicitly through area-based correlation or explicitly by using penalty terms for neighboring pixels with dissimilar disparities these surfaces also represent the implicit hypothesis made during stereo matching. Although the shape of these surfaces is well known for the standard stereo case (i.e. fronto-parallel planes), surprisingly enough for two cameras in a general configuration to our knowledge their shape has not been studied. This is, however, very important since it represents the discretisation of stereo sampling in 3D space and represents absolute bounds on performance independent of later resampling. We prove that the intersections of these surfaces with an epipolar plane consists of a family of conics with three fixed points. There is an interesting relation to the human horopter and we show that for stereo the retinas act as if they were flat. Further we discuss the relevance of iso-disparity surfaces to image-pair rectification and active vision. In experiments we show how one can configure an active stereo head to align iso-disparity surfaces to scene structures of interest such as a vertical wall, allowing better and faster stereo results.