Trifocal tensor

The Trifokalgeometrie is the extension of epipolar geometry to three images. If the position of an object point in two images is known, is its position in the third image of the intersection of the two epipolar lines. Thus, in contrast to the image pair, there is a clear result, if the point is not in the Trifokalebene (the plane, which is formed from the three centers of projection ) is or three projection centers lie on a line. The arrangement in which the 3-D point lies on the Trifokalebene is referred to as a singular event.

The Trifokaltensor

The Trifokaltensor t is a tensor, which includes the geometric relationship between the three cameras. It consists of three homogeneous 3 × 3 matrix, and has 18 degrees of freedom.

Calculation of Trifokaltensor

To calculate the Trifokaltensors the three projection matrices P of the cameras must be known. Are these with P1 = [I | 0], P2 = [ aij ] and P3 = [ bij ] ( I is the identity matrix and thereby 0 is the zero vector ) denotes, then the Trifokaltensor calculated t with

Extensions to more than three images

It is possible to extend the geometric relationships on more than three images. This is common practice in only four views. Here exists the so-called quadrifokale tensor, which describes the relationship of image points and lines between these views. However, no mathematical relationships have been studied for more than four views.