** Next:** Conclusion
** Up:** Multi view geometry
** Previous:** Three view geometry
** Contents**

##

Multi view geometry

Many people have been studying multi view relationships [59,156,34]. Without going into detail we would like to give some intuitive insights to the reader. For a more in depth discussion the reader is referred to [88].

An image point has 2 degrees of freedom. But images of a 3D point do not have degrees of freedom, but only 3. So, there must be independent constraints between them. For lines, which also have 2 degrees of freedom in the image, but 4 in 3D space, images of a line must satisfy constraints.

Some more properties of these constraints are explained here. A line can be back-projected into space linearly (3.9). A point can be seen as the intersection of two lines. To correspond to a real point or line the planes resulting from the backprojection must all intersect in a single point or line. This is easily expressed in terms of determinants, i.e.
for points and that all the subdeterminants of
should be zero for lines. This explains why the constraints are multi linear, since this is a property of columns of a determinant. In addition no constraints combining more than 4 images exist, since with 4-vectors (i.e. the representation of the planes) maximum determinants can be obtained. The twofocal (i.e. the fundamental matrix) and the trifocal tensors have been discussed in the previous paragraphs, recently Hartley [54] proposed an algorithm for the practical computation of the quadrifocal tensor.

** Next:** Conclusion
** Up:** Multi view geometry
** Previous:** Three view geometry
** Contents**
Marc Pollefeys
2002-11-22