Dual Computation of Projective Shape and Camera Positions from Multiple Images

Stefan Carlsson*, Daphna Weinshall

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations


Given multiple image data from a set of points in 3D, there are two fundamental questions that can be addressed: • What is the structure of the set of points in 3D? • What are the positions of the cameras relative to the points? In this paper we show that, for projective views and with structure and position defined projectively, these problems are dual because they can be solved using constraint equations where space points and camera positions occur in a reciprocal way. More specifically, by using canonical projective reference frames for all points in space and images, the imaging of point sets in space by multiple cameras can be captured by constraint relations involving three different kinds of parameters only, coordinates of: (1) space points, (2) camera positions (3) image points. The duality implies that the problem of computing camera positions from p points in q views can be solved with the same algorithm as the problem of directly reconstructing q + 4 points in p - 4 views. This unifies different approaches to projective reconstruction: methods based on external calibration and direct methods exploiting constraints that exist between shape and image invariants.

Original languageAmerican English
Pages (from-to)227-241
Number of pages15
JournalInternational Journal of Computer Vision
Issue number3
StatePublished - 1998

Bibliographical note

Funding Information:
This work was partially performed under the ESPRIT-BRA project VIVA, and with support from the Swedish National Board for Industrial and Technical Development, NUTEK. Vision research at the Hebrew University is supported by the U.S. Office of Naval Research under Grant N00014-93-1-1202, R&T Project Code 4424341—01. Both authors acknowledge the support of the EC-Israel Exploratory Collaboration Activity, EC-IS-003, SAM, Shape and Motion


  • Duality
  • Epipolar geometry
  • Multiple views
  • Positioning
  • Projective shape
  • Reconstruction


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