Flat refractive geometry

Tali Treibitz, Yoav Y. Schechner, Clayton Kunz, Hanumant Singh

Research output: Contribution to journalArticlepeer-review


While the study of geometry has mainly concentrated on single viewpoint (SVP) cameras, there is growing attention to more general non-SVP systems. Here, we study an important class of systems that inherently have a non-SVP: a perspective camera imaging through an interface into a medium. Such systems are ubiquitous: They are common when looking into water-based environments. The paper analyzes the common flat-interface class of systems. It characterizes the locus of the viewpoints (caustic) of this class and proves that the SVP model is invalid in it. This may explain geometrical errors encountered in prior studies. Our physics-based model is parameterized by the distance of the lens from the medium interface, besides the focal length. The physical parameters are calibrated by a simple approach that can be based on a single frame. This directly determines the system geometry. The calibration is then used to compensate for modeled system distortion. Based on this model, geometrical measurements of objects are significantly more accurate than if based on an SVP model. This is demonstrated in real-world experiments. In addition, we examine by simulation the errors expected by using the SVP model. We show that when working at a constant range, the SVP model can be a good approximation.

Original languageEnglish
Article number5770266
Pages (from-to)51-65
Number of pages15
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number1
StatePublished - 2012
Externally publishedYes

Bibliographical note

Funding Information:
The authors thank the reviewers for their insightful comments and Hank Chezar of the USGS and Boaz Zion of Volcani Center for letting them use their images in Fig. 1. They thank Ben Herzberg and Gal Gur-Arye for help in the experimental dives. Yoav Y. Schechner is a Landau Fellow —supported by the Taub Foundation. This work was supported by the US-Israel Binational Science Foundation (BSF grant 2006384) and the Israeli Ministry of Science, Culture and Sport (Grant 3-3426). This reasearch was supported by the Ollendorff Minerva Center for Vision and Image Science. Minerva is funded through the BMBF. Yoav Y. Schechner was partially supported by US Department of the Navy Grant N62909-10-1-4056 issued by the US Office of Naval Research (ONR) Global and the United States has a royalty-free license throughout the world in all copyrightable materials contained herein. Funding was also provided in part by the CenSSIS ERC of the US National Science Foundation (NSF) under Grant EEC-9986821. Tali Treibitz was partailly supported under grants from NSF ATM-0941760 and ONR N00014-08-1-0638. Tali Treibitz is an Awardee of the Weizmann Institute of Science—National Postdoctoral Award Program for Advancing Women in Science.


  • 3D/stereo scene analysis
  • Computer vision
  • camera calibration
  • imaging geometry
  • vision and scene understanding

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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