The non-tightness of a convex relaxation to rotation recovery

Yuval Alfassi, Daniel Keren, Bruce Reznick

Research output: Contribution to journalArticlepeer-review


We study the Perspective-n-Point (PNP) problem, which is fundamental in 3D vision, for the recovery of camera translation and rotation. A common solution applies polynomial sum-of-squares (SOS) relaxation techniques via semidefinite programming. Our main result is that the polynomials which should be optimized can be non-negative but not SOS, hence the resulting convex relaxation is not tight; specifically, we present an example of real-life configurations for which the convex relaxation in the Lasserre Hierarchy fails, in both the second and third levels. In addition to the theoretical contribution, the conclusion for practitioners is that this commonly-used approach can fail; our experiments suggest that using higher levels of the Lasserre Hierarchy reduces the probability of failure. The methods we use are mostly drawn from the area of polynomial optimization and convex relaxation; we also use some results from real algebraic geometry, as well as Matlab optimization packages for PNP.

Original languageEnglish
Article number7358
Issue number21
StatePublished - 5 Nov 2021

Bibliographical note

Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.


  • Convex relaxation
  • PNP
  • Polynomial optimization
  • Rotation recovery

ASJC Scopus subject areas

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


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