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Precise invariants with respect to projective transformation of space coordinates
u = (a0 + a1x + a2y)/(c0 + c1x + c2y)
v = (b0 + b1x + b2y)/(c0 + c1x + c2y)
computed from moments does not exist ([9] is probably maximum), that is why projective invariants computed from some other measurements of objects on images are searched. A possible approach is point-based projective invariants. If we can find some significant points in all projectively deformed versions of an object, we can calculate the point-based projective invariant from their coordinates.
If we use some simple version of a cross-product, we need to know sorting of the points. When the correspondences between the points are not known, but the points are vertices of a polygon, then combined invariants to the projective transform and cyclic shift of the vertices is satisfactory. We can utilize also invariants to the projective transform and to the complete permutation of the points. Another approach is computing convex layers of the points. An example of the images registered by this method:
A cut of a Landsat Thematic Mapper image of north-east Bohemia (surroundings of the town Trutnov) from 29 august 1990 (256 x 256 - first image) was registered with an aerial image from 1984 (180 x 256 - second image) with relatively strong projective distortion. 16 points was selected in the aerial image, 18 points was selected in the satellite one, 10 of them have counterparts in the other image (labeled x, the points without counterparts are labeled +). The aerial image was registered by means of projective and permutation invariants (third image) and a color composition of the registered images is shown (fourth image).
Relevant literature:
10. R. Lenz, P. Meer: Point configuration invariants under simultaneous projective and permutation transformations. Pattern Recognition, 27 (1994), 1523-1532.
Contact person: Tomáš Suk
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