Learning Epipolar Geometry

Drag the mouse above the applet to move around the world ...

And click any key to toggle the state of the applet and drag the point M around ...

One View

The model for a single camera is pretty simple. For a pinhole camera centered at C, looking at a point M, the projection m is the intersection of the line (C,M) with the image plane.

When you move M around, it moves within the plane z=2.

Two Views

Things become interesting with a second viewpoint. If we know m the projection of a point M onto a camera, its corresponding point in the other camera is constrained to lie on a line. This line is called the epipolar line, and the correspondence between m and this line is described by the Fundamental matrix.

The epipolar lines are the trace of the plane (C1,C2,M) in the image planes.

Notice that the points e-c1c2 and e-c2c1 don't move as you drag M arround. They are called the epipoles. It is where one camera is seen from the other camera.

Three Views

Lets consider a third camera. In each camera, the epipolar lines related to the 2 others cameras are now crossing at a point. If a point M is registered within two images, its projection in the third one is uniquely defined, through the trifocal tensor. Notice how the two epipolar lines merge into one when the point M approches the plane (C1,C2,C3) called the trifocal plane.

When M is not in the trifocal plane, the position of the projection in the third camera can be predicted by intersecting the epipolar lines issued from the two other cameras

Some specific cases

There are specific configurations that give non generic results.

Parallel image planes

The good one appears when you only translate your camera without rotating it at all. The image planes become parallel and the epipolar lines as well.

Moreover, when you translate your camera on the ground plane parallel to the first image plane, the epipolar lines become parallel to the X axis of the image planes. This special configuration (sometimes called standard geometry, or rectified images) is very helpful during the matching process.

Aligned optical centers

The bad one appears when a third image is taken from a position aligned with the previous ones: The plane (C1,C3,M) is equal to the plane (C2,C3,M) (C1, C2, C3, M are coplanar). Thus, in the third image, the epipolar lines become equal and no prediction can be done by intersecting the epipolar lines from two cameras in the third image! Unfortenately this configuration appears quite often in real situations. Note: you can still do prediction using the trifocal tensor.

For any comments or suggestions, mail to Sylvain Bougnoux <bougnoux at imra-europe.com>.

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