Projective Area-invariants as an extension of the Cross-Ratio

Lars Nielsen, Gunnar Sparr

Research output: Contribution to journalArticlepeer-review


Projective invariants provide a framework for computer vision where the image of an object is described by its intrinsic properties, independently of the particular view. It is advantageous if these intrinsic properties are defined in terms of computationally simple features. An area-measurement provides a good candidate that is easy to reliably compute from a particular image of the object. The main contributions of this paper are the definition and justification of area-invariants in projective geometry and the indication of its relevance in image analysis. A framework that covers one-dimensional intervals and two-dimensional figures has been developed. In the linear case, the invariants are linear only in two cases. The first case is the well known cross-ratio, and the second case is called the polar case. The generalization to the plane can be done in different directions. One can use either points (on the line or in the plane) or the geometric figures (intervals, triangles, circles) as the basic entities involved. The first view was adopted already by Möbius, who generalized the cross-ratio in various directions. The second view used here leads to another generalization of the cross-ratio, where the invariants are relations between the areas of a class of geometric figures, related to each other in a certain manner. Remarkably enough, these invariants turn out to be linear if the figures involved are related in a pole/ polar configuration
Original languageEnglish
Pages (from-to)145-159
JournalGraphical Models
Issue number1
Publication statusPublished - 1991

Subject classification (UKÄ)

  • Mathematics
  • Computer Vision and Robotics (Autonomous Systems)


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