Multi-Photo Fusion through Projective Geometry

Image Registration Through Projective Transformations

Projective geometry allows us to deal with geometric transformations and offers more powerful methods than conventional Euclidean geometry. For instance, in Euclidean geometry any two distinct points in a plane define a line. On the other hand, any two distinct lines define a point, unless the lines are parallel. Projective geometry overcomes this limitation in a symmetric way: any two distinct points determine a unique line and any two distinct lines intersect at a point. When we consider the image formation process by means of a pinhole camera, it is quite evident that Euclidean geometry is not sufficient. Properties like lengths, angles, areas, and parallelism are not preserved during the imaging process: parallel lines may intersect.

We get to projective geometry by taking Euclidean geometry and adding an extra dimension. A point in an n-dimensional Euclidean space is represented as a point in an (n+1)-dimensional projective space. Suppose we have a point (x, y)^T in the Euclidean space, the homogenous coordinates of a point can be obtained by adding an extra coordinate to the pair as (λx, λy, λ)^T . We say that this 3-vector is the same point in homogeneous coordinates (for any non-zero value λ). An arbitrary homogeneous vector x = (x₁, x₂, x₃)^T represents the point x = (x₁ / x₃, x₂ / x₃)^T in R².