2d to 3d image point projection, epipolar geometry and epiline: A question

• Oct 10th 2011, 09:43 PM
x3bnm
2d to 3d image point projection, epipolar geometry and epipolar line: A question
Definitions:
Epipolar geometry:

Epipolar geometry is the geometry of stereo vision. When two cameras view a 3D scene from two distinct positions, there are a number of geometric relations between the 3D points and their projections onto the 2D images that lead to constraints between the image points. These relations are derived based on the assumption that the cameras can be approximated by the pinhole camera model.

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Epipole or epipolar point:

\$\displaystyle X\$ is the 3d point in both left and right image and \$\displaystyle O_L\$ and \$\displaystyle O_R\$ are focal points of left and right camera.

Since the two focal points of the cameras are distinct, each focal point projects onto a distinct point into the other camera's image plane. These two image points are denoted by \$\displaystyle e_L\$ and \$\displaystyle e_R\$ and are called epipoles or epipolar points. Both epipoles \$\displaystyle e_L\$ and \$\displaystyle e_R\$ in their respective image planes and both focal points \$\displaystyle O_L\$ and \$\displaystyle O_R\$ lie on a single 3D line.

Epipolar line:

The line \$\displaystyle O_{L}–X\$ is seen by the left camera as a point because it is directly in line with that camera's focal point. However, the right camera sees this line as a line in its image plane. That line \$\displaystyle (e_{R}–x_{R})\$ in the right camera is called an epipolar line. Symmetrically, the line \$\displaystyle O_{R}–X\$ seen by the right camera as a point is seen as epipolar line \$\displaystyle e_{L}–x_{L}\$by the left camera.

An epipolar line is a function of the 3D point \$\displaystyle X\$, i.e. there is a set of epipolar lines in both images if we allow \$\displaystyle X\$ to vary over all 3D points. Since the 3D line \$\displaystyle O_{L}–X\$ passes through camera focal point \$\displaystyle O_L\$, the corresponding epipolar line in the right image must pass through the epipole \$\displaystyle e_R\$ (and correspondingly for epipolar lines in the left image). This means that all epipolar lines in one image must intersect the epipolar point of that image. In fact, any line which intersects with the epipolar point is an epipolar line since it can be derived from some 3D point \$\displaystyle X\$.

My question:
I want to calculate 3d coordinate from 2d coordinate of a point using epipolar geometry and capturing 2 frames of same image point with 2 cameras. I know how to reconstruct 3d point from 2d points.

The cameras are located in a circular region radius of 7 feet and the cameras are mounted exactly at opposite sides of the circle's diameter. If the camera1 is mounted at 0 degree of the circular area then camera2 is located at 180 degree of the same area.

If I take a picture of a 2d point which coincides with epipoles in image frames of camera1 and camera2 then can epipolar geometry be used when cameras are located at two opposite sides of a region? Sorry if I'm wrong but doesn't in this case epipolar line will be a point?

Can anyone kindly tell me if epipolar geometry be used to find 3d coordinate in this special case? If not is there any other method that I can use to calculate 3d coordinate of this 2d point?