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Forward projection model of non-central catadioptric cameras with spherical mirrors

Published online by Cambridge University Press:  05 April 2016

Nuno Goncalves Open the ORCID record for Nuno Goncalves [Opens in a new window] ,
Ana Catarina Nogueira  and
Andre Lages Miguel
Nuno Goncalves*
Affiliation:
Institute of Systems and Robotics, University of Coimbra Polo 2, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. E-mails: anacatnog@isr.uc.pt, andrelages@isr.uc.pt
Ana Catarina Nogueira
Affiliation:
Institute of Systems and Robotics, University of Coimbra Polo 2, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. E-mails: anacatnog@isr.uc.pt, andrelages@isr.uc.pt
Andre Lages Miguel
Affiliation:
Institute of Systems and Robotics, University of Coimbra Polo 2, Pinhal de Marrocos, 3030-290 Coimbra, Portugal. E-mails: anacatnog@isr.uc.pt, andrelages@isr.uc.pt
*
*Corresponding author. E-mail: nunogon@isr.uc.pt
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Summary

Non-central catadioptric vision is widely used in robotics and vision but suffers from the lack of an explicit closed-form forward projection model (FPM) that relates a 3D point with its 2D image. The search for the reflection point where the scene ray is projected is extremely slow and unpractical for real-time applications. Almost all methods thus rely on the assumption of a central projection model, even at the cost of an exact projection.

Two recent methods are able to solve this FPM, presenting a quasi-closed form FPM. However, in the special case of spherical mirrors, further enhancements can be made. We compare these two methods for the computation of the FPM and discuss both approaches in terms of practicality and performance. We also derive new expressions for the FPM on spherical mirrors (extremely useful to robotics and graphics) which speed up its computation.

Keywords

non central catadioptric camerasforward projection modelquadric mirrorsreflection point
Type
Articles
Information
Robotica , Volume 35 , Issue 6 , June 2017 , pp. 1378 - 1396
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Agrawal, A., Taguchi, Y. and Ramalingam, S., “Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras,” European Conference on Computer Vision (ECCV), Springer Berlin Heidelberg, Heraklion, Greece (Sep. 2010), pp. 129–143.Google Scholar
2. Agrawal, A., Taguchi, Y. and Ramalingam, S., “Beyond Alhazen's Problem: Analytical Projection Model for Non-Central Catadioptric Cameras with Quadric Mirrors,” IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Colorado Springs, US (2011), pp. 2993–3000.Google Scholar
3. Baker, M., “Alhazen's problem,” Am. J. Math. 4 (1), 327–331 (1881).CrossRefGoogle Scholar
4. Barreto, J. and Araujo, H., “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27 (8) (2005) 1327–1333.Google Scholar
5. Bjorke, K., “Finite-Radius Sphere Environment Mapping,” In: GPU Gems – Programming Techniques, Tips, and Tricks for Real-Time Graphics (Addison-Wesley, 2004) ISBN: 0-321-22832-4.Google Scholar
6. Blinn, J. F. and Newell, M. E., “Texture and reflection in computer generated images,” Commun. ACM 19 (10), 542–547 (1976).CrossRefGoogle Scholar
7. Born, M. and Wolf, E., Principles of Optics (Pergamon Press, Oxford, UK, 1965).Google Scholar
8. Chen, M. and Arvo, J., “Perturbation methods for interactive specular reflections,” IEEE Trans. Vis. Comput. Graph. 6, 253–264 (2000).Google Scholar
9. Chen, M. and Arvo, J., “Theory and application of specular path perturbation,” ACM Trans. Graph. 19 (4), 246–278 (2000).Google Scholar
10. Dias, T., Miraldo, P., Goncalves, N. and Lima, P., “Augmented Reality on Robot Navigation using Non-Central Catadioptric Cameras,” IEEE/RSJ International Conference on Intelligent Robots and Systems - IROS, Hamburg, Germany (Sep. 2015), pp. 4999–5004.Google Scholar
11. Ding, Y., Yu, J. and Sturm, P. F., “Multiperspective Stereo Matching and Volumetric Reconstruction,” In: Proceedings of the International Conference on Computer Vision (ICCV) Kyoto, Japan (Sep. 2009), pp. 1827–1834.Google Scholar
12. Dupont, L., Lazard, D., Lazard, S. and Petitjean, S., “Near-optimal parameterization of the intersection of quadrics: I. The generic algorithm,” J. Symb. Comput. 43 (3), 168–191 (2008).Google Scholar
13. Estalella, P., Martin, I., Drettakis, G. and Tost, D., “A gpu-Driven Algorithm for Accurate Interactive Specular Reflections on Curved Objects,” Proceedings of the 2006 Eurographics Symposium on Rendering (2006).Google Scholar
14. Estalella, P., Martin, I., Drettakis, G., Tost, D., Devillers, O. and Cazals, F., “Accurate Interactive Specular Reflections on Curved Objects,” Vision Modeling and Visualization (VMV) (2005) pp. 8.Google Scholar
15. Gasparini, S., Sturm, P. and Barreto, J., “Plane-Based Calibration of Central Catadiotpric Cameras,” IEEE International Conference on Computer Vision (2009), pp. 1195–1202.Google Scholar
16. Glaeser, G., “Reflections on spheres and cylinders of revolution,” J. Geom. Graph. 3 (2), 121–139 (1999).Google Scholar
17. Goncalves, N., “On the reflection point where light reflects to a known destination in quadric surfaces,” Opt. Lett. 35 (2), 100–102 (Jan. 2010).CrossRefGoogle Scholar
18. Goncalves, N. and Nogueira, A. C., “Projection through quadric mirrors made faster,” ICCVW: 9th Workshop on Omnidirectional Vision, Camera Networks and Non-Classical Cameras, Kyoto, Japan (Oct. 2009), pp. 2141–2148.Google Scholar
19. Hecht, E., Optics (Addison-Wesley, Massachusetts, USA, 1987).Google Scholar
20. Levin, J., “A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces,” Commun. ACM 19 (10), 555–563 (1976).Google Scholar
21. Levin, J., “Mathematical models for determining the intersection of quadric surfaces,” Comput. Graph. Image Process. 11 (1), 73–87 (1979).Google Scholar
22. Lhuillier, M., “Automatic scene structure and camera motion using a catadioptric system,” Comput. Vis. Image Underst. 109 (2), 186–203 (2008).CrossRefGoogle Scholar
23. Martin, A. and Popescu, V., “Reflection Morphing,” ACM SIGGRAPH 2004 Sketches, SIGGRAPH '04, ACM (Los Angeles, California, USA, 2004), pp. 152–156.Google Scholar
24. Mei, C. and Rives, P., “Single View-Point Omnidirectional Camera Calibration from Planar Grids,” In: IEEE International Conference on Robotics and Automation, Rome, Italy (Apr. 2007), pp. 3945–3950.Google Scholar
25. Micusik, B. and Pajdla, T., “Autocalibration & 3d Reconstruction with Non-Central Catadioptric Cameras,” IEEE Conference on Computer Vision and Pattern Recognition (Washington, DC, USA, 2004) pp. 58–65.Google Scholar
26. Mitchell, D. and Hanrahan, P., “Illumination from curved reflectors,” SIGGRAPH Comput. Graph. 26 (2) (1992) 283–291.Google Scholar
27. Ofek, E. and Rappoport, A., “Interactive Reflections on Curved Objects,” SIGGRAPH '98, New York, NY, USA, ACM (1998) pp. 333–342.Google Scholar
28. Press, W., Teukolsky, S., Vetterling, W. and Flannery, B., Numerical Recipes - The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, UK, 2007).Google Scholar
29. Roger, D. and Holzschuch, N., “Accurate specular reflections in real-time,” Comput. Graph. Forum 25 (3), 293–302 (2006).Google Scholar
30. Roth, S. and Black, M. J., “Specular Flow and the Recovery of Surface Structure,” In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR), New York, USA (Jun. 2006), pp. 1869–1876.Google Scholar
31. Scaramuzza, D., Martinelli, A. and Siegwart, R., “A Toolbox for Easily Calibrating Omnidirectional Cameras,” In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, China (Oct. 2006), pp. 5695–5701.Google Scholar
32. Sturm, P. and Barreto, J., “General Imaging Geometry for Central Catadioptric Cameras,” In: IEEE European Conference on Computer Vision, Springer Berlin Heidelberg, Marseille, France (Oct. 2008), pp. 609–622.Google Scholar
33. Swaminathan, R., Grossberg, M. and Nayar, S., “Non-single viewpoint catadioptric cameras: Geometry and analysis,” Int. J. Comput. Vis. 66 (3), 211–229 (2006).Google Scholar
34. Unger, J., Wenger, A., Hawkins, T., Gardner, A. and Debevec, P. E., “Capturing and Rendering with Incident Light Fields,” Proceedings of the 14th Eurographics Workshop on Rendering Techniques, Leuven, Belgium (Jun. 25–27, 2003), pp. 141–149.Google Scholar
35. Whitted, T., “An improved illumination model for shaded display,” Commun. ACM 23 (6), 343–349 (Jun. 1980).Google Scholar
36. Ying, X. and Hu, Z., “Catadioptric camera calibration using geometric invariants,” IEEE Trans. Pattern Anal. Mach. Intell. 26 (10), 1260–1271 (2004).Google Scholar
37. Yu, J., Yang, J. and McMillan, L., “Real-Time Reflection Mapping with Parallax,” Proceedings of the 2005 Symposium on Interactive 3D Graphics and Games, I3D '05, ACM (Washington, DC, USA, 2005) pp. 133–138.Google Scholar