[names.gif] [titlesmall.gif]

Projective Geometry

Before starting any registration method, we need to define a search space. That is, we have choose a model for the coordinate tranformation between images, or, equivalently, choose a model for the camera and its motion between shots.

General Projective Transforms


We made some usual assumptions like considering that the camera optics had no non-linear effects and that parallax effects between shots were negligible. Based on those two assumptions, it can be shown that the mapping between two snapshots is reduced to a 8-parameter projective transform. If we use homogeneous coordinates [homCoords.GIF],those transforms have a convenient 3x3 matrix representation:


A 8-parameter projective transform is determined by a set of 4 corresponding points in both images. It is possible to recover the homogeneous form of the transform by solving a simple linear equation system, as mentioned in the part devoted to feature-based registration.

Lastly the transform accepts an equivalent representation that makes its different parameters a lot more intuitive than a to h. We basically write the homography matrix as a composition of elementary transforms matrices that include image rotation R, skew angle L, isotropic scale factor S, aspect ratio A, translation T, and perspective or keystone factors P:


We coded and extensively used conversions between those three representations for the registration and blending phases.

Single View-Point Model


However we restricted ourselves to the case of a single point-of-view model. Indeed considering that all shots are taken from the same place and that the zooming factor is kept constant, the eight parameters are reduced to 3 angles [angles.GIF] and a constant focal length f. The angles are actually describing the camera orientation in a spherical parametrization as shown on the following figure:


In that model, the coordinate transforms between images can be rewritten as 3D rotations after a projection from the image planar space to the unit sphere:


Previous (Proposal) · Up (Main) · Next (Registration)

© 2000 Laurent Meunier and Moritz Borgmann