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Next: Rectification Up: Camera model and epipolar Previous: Camera model

Epipolar geometry

Let us consider a stereo rig composed by two pinhole cameras (Fig. 2). Let tex2html_wrap_inline1165 and tex2html_wrap_inline1167 be the optical centres of the two cameras. A 3D point tex2html_wrap_inline1111 is projected onto both image planes, to points tex2html_wrap_inline1171 and tex2html_wrap_inline1173 , which constitute a conjugate pair. Given a point tex2html_wrap_inline1171 in the left image plane, its conjugate point in the right image is constrained to lie on a line called the epipolar line (of tex2html_wrap_inline1171 ). This line is the projection through tex2html_wrap_inline1167 of the optical ray of tex2html_wrap_inline1171 ; indeed tex2html_wrap_inline1171 may be the projection of an arbitrary point on its optical ray.

Figure 2: Epipolar geometry.

Furthermore one should observe that all the epipolar lines in one image plane pass through a common point called the epipole, which is the projection of the conjugate optical centre:


The parametric equation of the epipolar line of tex2html_wrap_inline1185 writes:


In image coordinates it becomes:


where tex2html_wrap_inline1187 and tex2html_wrap_inline1189 is the projection operator extracting the ith component from a vector.

Figure 3: Top row: a stereo pair (Copyright SYNTIM-INRIA). Bottom row: the rectified pair. The right pictures plot the epipolar lines corresponding to the point marked in the left pictures.

When tex2html_wrap_inline1165 is in the focal plane of the right camera, the right epipole is at infinity, and the epipolar lines form a bundle of parallel lines in the right image. Analytically, the direction of each epipolar line can be obtained by taking the derivative of the parametric equations (13,14) with respect to tex2html_wrap_inline1161 :


Note that the denominator is the same in both components, hence it does not affect the direction of the vector. The epipole is rejected to infinity when tex2html_wrap_inline1197 . In this case, the direction of the epipolar lines in the right image doesn't depend on tex2html_wrap_inline1199 any more and all the epipolar lines becomes parallel to vector tex2html_wrap_inline1201 .

A very special case is when both epipoles are at infinity, that happens when the line containing tex2html_wrap_inline1165 and tex2html_wrap_inline1167 (the baseline) is contained in both focal planes, or the retinal planes are parallel to the baseline. Epipolar lines form a bundle of parallel lines in both images.

Any pair of images can be transformed so that epipolar lines are parallel and horizontal in each image as in Fig. 3. This procedure is called rectification.

next up previous
Next: Rectification Up: Camera model and epipolar Previous: Camera model

Andrea Fusiello
Tue Feb 3 17:18:41 MET 1998