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Epipolar Geometry
The epipolar geometry is the intrinsic projective geometry between two views. It is independent of scene structure, and only depends on the camera’s internal parameters and relative pose. The fundamental matrix encapsulates this intrinsic geometry. It is a
matrix of rank 2. If a point
in space is mapped to one image as
, and to the second image as
, then the image points satisfy the relation
.
can be computed if both camera parameters are known. More about this in Section 2.3.2.
The relation between a projected point in one’s view and its corresponding point
in the other view can be described geometrically. In Figure 2.7, we can see that for every point
a corresponding line, on which
can be found, exists. These lines are called epipolar lines. The intersection between the baseline
and the image planes is called epipole. All epipolar lines intersect at the epipole.
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If a point in space is known, the corresponding epipolar lines can be found by computing the intersection between the two image planes, and the plane that is formed by the baseline
and the line from the optical center
to the point
. In Figure 2.7 the epipolar lines are denoted by
and
. Supposing now that we know only
, we may ask how the corresponding point
can be found. The plane
is determined by the baseline and the ray defined by point
. From above we know that the point
lies inside the line
which is the intersection of plane
with the second image plane. In terms of a stereo correspondence algorithm the benefit is that we do not have to search the entire image for the corresponding point
but only along the line
. The fundamental matrix
is the algebraic representation of the epipolar geometry and is discussed in the next section.
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