A Fast and Reliable Solution to PnP, Using Polynomial Homogeneity and a Theorem of Hilbert

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One of the most-extensively studied problems in three-dimensional Computer Vision is "Perspective-n-Point" (PnP), which concerns estimating the pose of a calibrated camera, given a set of 3D points in the world and their corresponding 2D projections in an image captured by the camera. One solution method that ranks as very accurate and robust proceeds by reducing PnP to the minimization of a fourth-degree polynomial over the three-dimensional sphere S3. Despite a great deal of effort, there is no known fast method to obtain this goal. A very common approach is solving a convex relaxation of the problem, using "Sum Of Squares" (SOS) techniques. We offer two contributions in this paper: a faster (by a factor of roughly 10) solution with respect to the state-of-the-art, which relies on the polynomial's homogeneity; and a fast, guaranteed, easily parallelizable approximation, which makes use of a famous result of Hilbert.

Original languageEnglish
Article number5585
Issue number12
StatePublished - Jun 2023

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© 2023 by the authors.


  • polynomial optimization
  • the PnP problem

ASJC Scopus subject areas

  • Analytical Chemistry
  • Information Systems
  • Atomic and Molecular Physics, and Optics
  • Biochemistry
  • Instrumentation
  • Electrical and Electronic Engineering


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