Mathematical models of obtaining stereo images from two-mirror catadioptric systems with regard to lens distortion
Stepanov D.N.


Program Systems’ Institute named after A.K. Ailamazyan of Russian Academy of Sciences (PSI RAS),  Pereslavl-Zalessky, Russia


In this paper, we develop and research mathematical models that describe image acquisition from two-mirror catadioptric systems, which allow one to generate stereo images using a single camera and two flat mirrors. The main difference between the proposed models and existing solutions in this area is that real camera distortion is introduced into the images. An overview of the main methods of creating 3D models using optical technologies is presented, as well as an overview of types of catadioptric systems depending on the number of mirrors and their shape (curved or flat). The methodology of the research consists in the mathematical modeling of a two-mirror catadioptric system, as well as in a computer experiment using real images that were obtained using a stereo nozzle with two flat mirrors and synthetic images which were generated based on the ray tracing algorithm. Results of the experiments on the camera calibration with a stereo nozzle, as well as results of image rectification using the calibration data and developed mathematical models are presented. The results of the experiments allow us to verify the adequacy of the developed models. The proposed models expand the theory of computer vision and can be used in the creation and research of computer vision systems for robotic complexes.

computer vision, optical devices, calibration, math modeling, ray tracing, stereovision, optical characteristics, stereo nozzle.

Stepanov DN. Mathematical models of obtaining stereo images from two-mirror catadioptric systems with regard to lens distortion. Computer Optics 2019; 43(1): 105-114. DOI: 10.18287/2412-6179-2019-43-1-105-114.


  1. Popov SB. The use of structured lighting in computer vision systems [In Russian]. Computer Optics 2013; 37(2): 233-238.
  2. Hartley R, Zisserman A. Multiple view geometry in computer vision. 2nd ed. New York, NY: Cambridge University Press; 2003. ISBN: 978-0-521-54051-3.
  3. Lin S-S, Bajcsy R. Single cone mirror omni-directional stereo. Source: < >.
  4. Burbridge C, Nehmzow U, Condell JV. Omnidirectional projections with a cone mirror and single mirror stereo. Proceedings of OMNIVIS 2008.
  5. Genovese K, Casaletto L, Lee Y-U, Humphrey JD. Panoramic stereo DIC-based strain measurement on submerged objects. In Book: Proulx T, ed. Optical measurements, modeling, and metrology, Vol 5. New York, Dordrecht, Heidelberg, London: Springer; 2011: 257-263. DOI: 10.1007/978-1-4614-0228-2_31.
  6. Nene SA, Nayar SK. Stereo with mirrors. Sixth International Conference on Computer Vision 1998: 1087-1094. DOI: 10.1109/ICCV.1998.710852.
  7. Cabral ELL, de Souza JC, Hunold MC. Omnidirectional stereo vision with a hyperbolic double lobed mirror. Proc ICPR 2004; 1: 1-9. DOI: 10.1109/ICPR.2004.1333989.
  8. Li W, Li YF. Single-camera panoramic stereo imaging system with a fisheye lens and a convex mirror. Opt Express 2011; 19(7): 5855-5867. DOI: 10.1364/OE.19.005855.
  9. Jiang W, Shimizu M, Okutomi M. Single-camera multi-baseline stereo using fish-eye lens and mirrors. In Book: Zha H, Taniguchi R-i, Maybank S, eds. Computer Vision – ACCV 2009. Pt II. Berlin, Heidelberg: Springer-Verlag; 2009: 347-358. DOI: 10.1007/978-3-642-12304-7_33.
  10. Gluckman J, Nayar SK. Rectified catadioptric stereo sensors. IEEE Transactions on Pattern Analysis and Machine Intelligence 2002; 24(2): 224-236. DOI: 10.1109/34.982902.
  11. Clark AF, Chan SW. Single-camera computational stereo using a rotating mirror. Proc British Machine Vision Conference 1994; 2: 761-770. DOI: 10.5244/C.8.75.
  12. Wang R, Li X, Zhang Y. Analysis and optimization of the stereo-system with a four-mirror adapter. Journal of the European Optical Society Rapid Publications 2008; 3: 08033. DOI: 10.2971/jeos.2008.08033.
  13. Pachidis T, Lygouras J. A pseudo stereo vision system as a sensor for real time path control of a robot. Proc 19th IEEE IMTC/2002 2002; 2: 1589-1594. DOI: 10.1109/IMTC.2002.1007197.
  14. Chai X, Zhou F, Chen X. Epipolar constraint of single-camera mirror binocular stereo vision systems. Opt Eng 2017; 56(8): 084103. DOI: 10.1117/1.OE.56.8.084103.
  15. Vernon D. An optical device for computation of binocular stereo disparity with a single static camera. Proc SPIE 2002; 4877: 38-46. DOI: 10.1117/12.463762.
  16. Aggarwal R, Vohra A, Namboodiri AM. Panoramic stereo videos with a single camera. Proc CVPR 2016: 3755-3763. DOI: 10.1109/CVPR.2016.408.
  17. Gorevoy AV, Machikhin AS. Optimal calibration of a prism-based videoendoscopic system for precise 3D measurements. Computer Optics 2017; 41(4): 535-544. DOI: 10.18287/2412-6179-2017-41-4-535-544.
  18. Goshtasby A, Gruver WA. Design of a single-lens stereo camera system. Pattern Recognition 1993; 26(6): 923-937. DOI: 10.1016/0031-3203(93)90058-5.
  19. Gluckman J, Nayar SK. Planar catadioptric stereo: geometry and calibration. Proc IEEE Computer Society Conference on Computer Vision and Pattern Recognition 1999; 1: 22-28. DOI: 10.1109/CVPR.1999.786912.
  20. Mariottini GL, Scheggi S, Morbidi F, Prattichizzo D. Catadioptric stereo with planar mirrors: multiple-view geometry and camera localization. In Book: Chesi G, Hashimoto K, eds. Visual Servoing via Advanced Numerical Methods. Berlin, Heidelberg: Springer-Verlag, 2010: 3-21. DOI: 10.1007/978-1-84996-089-2_1.
  21. Bradski G, Kaehler A. Learning OpenCV. Sebastopol, CA: O'Reilly Media Inc; 2008. ISBN: 978-0-596-51613-0.
  22. Bouguet J-Y. Camera calibration toolbox for matlab. Source: < >.
  23. 3Dberry – A kit for studying computer vision from your Raspberry Pi! [In Russian]. Source: < >.
  24. Sampson PD. Fitting conic sections to ‘very scattered’ data: An iterative refinement of the Bookstein algorithm. Computer Graphics and Image Processing 1982; 18(1): 97-108. DOI: 10.1016/0146-664X(82)90101-0.

© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846)332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20