Projective reconstruction

Projective basis

Choose 5 points in the first image and their matches in the second (with the matrix ). These points verify the epipolar constraints.

Camera projection matrices

Together with matrix we can compute the camera projection matrices: and , w.r.t the chosen projective basis.

3D reconstruction

based on the classical triangulation principle:

Basic equations:

(Image points (u,v), (u',v') 3D projective point )

Method 1:

eliminating s and s' gives

with , the i-th row vector of . Since the projective coordinates are defined up to a scale factor, we can impose the constraint: . The solution is simply the eigenvector of associated to the smallest eigenvalue.

Method 2:

a physically more meaningful criterion

3D reconstruction in projective space

The projective projection matrices are available HERE for the first image and THERE for the second image. The projective structure of the bi-plane object is available HERE for the first image.

Back projection of the projective reconstruction onto the first image
an orthographic projection of the projective reconstruction (not very meaningful in Euclidean sense)

Author: Zhengyou Zhang
Email: zhang@microsoft.com