I am looking for projective invariant properties of quadrilaterals or even a group of quadrilaterals. Example:

enter image description here

In Multiple View Geometry in Computer Vision by Hartley and Zisserman I read that

Concurrency, collinearity, order of contact: intersection (1 pt contact); tangency (2 pt contact); inflections (3 pt contact with line); tangent discontinuities and cusps. cross ratio (ratio of ratio of lengths)

are invariant but I don't know how to use these properties as numeric values. I am not a mathematician so it is hard for me to understand how I could use for example concurrency or collinearity. Say I have an image that shows one quadrilateral and a second image that shows the same quadrilateral after a projective transformation. Can I somehow say e.g. the cross ratio of both quadrilaterals is 0.5 and so this is one invariant characteristic of the given quadrilateral?


If I understand correctly, you would like to compute numerical values related to some projectively invariant property (PIP) to test for consistency. I will give some ideas and hope they are helpful to someone.

The four corners of a single quadrilateral are the minimal information to define the homography relating the two images. There is not much to go on in that case.

For more than one such quadrilateral, there may be more information.

Cross-ratios can be computed for four points on a line. Perhaps more importantly, they can also be computed for four lines which intersect in one point. However, to use either of these PIPs the points really must be on a line (respectively, the lines really must intersect in a single point). Another projectively invarant property would be quadrilateral sets https://www-m10.ma.tum.de/foswiki/pub/Lehre/WS0809/GeometrieKalkueleWS0809/ch8.pdf

If you have non additional information, like that your quadrilaterals are in some way regular etc. you might choose four point pairs to compute the homography and consider the relative coordinates of any further points to be PIPs.

Permutation invariants also exist. For example, you pick one side of a specific quadrilateral and intersect the sides of all other quadrilaterals. The order of these intersections should not change (other than reflection). Counting the number of flips may give you a numeric value which expresses consistency.


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