“Modeling and Rendering of Impossible Figures” by Wu, Fu, Yeung, Jia and Tang

  • ©Tai-Pang Wu, Chi-Wing Fu, Sai-Kit Yeung, Jiaya Jia, and Chi-Keung Tang




    Modeling and Rendering of Impossible Figures



    This article introduces an optimization approach for modeling and rendering impossible figures. Our solution is inspired by how modeling artists construct physical 3D models to produce a valid 2D view of an impossible figure. Given a set of 3D locally possible parts of the figure, our algorithm automatically optimizes a view-dependent 3D model, subject to the necessary 3D constraints for rendering the impossible figure at the desired novel viewpoint. A linear and constrained least-squares solution to the optimization problem is derived, thereby allowing an efficient computation and rendering new views of impossible figures at interactive rates. Once the optimized model is available, a variety of compelling rendering effects can be applied to the impossible figure.


    1. Agarwal, S., Ramamoorthi, R., Belongie, S., and Jensen, H. W. 2003. Structured importance sampling of environment maps. ACM Trans. Graph. 22, 3, 605–612. 
    2. Alexeev, V. 2008. Impossible world. http://im-possible.info/english/.
    3. Bookstein, F. 1989. Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Patt. Analy. Mach. Intell. 11, 6, 567–585. 
    4. Elber, G. 2002. Escher for real. http://www.cs.technion.ac.il/~gershon/EscherForReal.
    5. Ernst, B. 1987. Adventures with Impossible Figures. Tarquin, Stradbroke, England.
    6. Heyden, A. 1996. On the consistency of line-drawings, obtained by projections of piecewise planar objects. J. Math. Imaging Vis. 6, 4, 393–412. 
    7. Huffman, D. A. 1971. Impossible objects as nonsense sentences. Mach. Intell. 6, 295–323.
    8. Ju, T., Zhou, Q.-Y., and Hu, S.-M. 2007. Editing the topology of 3d models by sketching. ACM Trans. Graph. 26, 3, 42. 
    9. Khoh, C. W. and Kovesi, P. 1999. Animating impossible objects. www.csse.uwa.edu.au/~pk/Impossible/impossible.html.
    10. Lipson, A. 2002. Andrew lipson’s LEGO page. http://www. andrewlipson.com/lego.htm.
    11. Matusik, W., Pfister, H., Brand, M., and McMillan, L. 2003. A data-driven reflectance model. ACM Trans. Graph. 22, 3, 759–769. 
    12. M.C. Escher Foundation. The official M.C.Escher website. http://www.mcescher.com.
    13. Nealen, A., Igarashi, T., Sorkine, O., and Alexa, M. 2007. Fibermesh: Designing freeform surfaces with 3D curves. ACM Trans. Graph. 26, 3, 41. 
    14. Owada, S. and Fujiki, J. 2008. Dynafusion: A modeling system for interactive impossible objects. In Proceedings of the Conference on Non-Photorealistic Animation and Rendering (NPAR). 65–68. 
    15. Penrose, L. S. and Penrose, R. 1958. Impossible objects: A special type of illusion. Brit. J. Psychol. 49, 31–33.
    16. Rademacher, P. 1999. View-dependent geometry. In Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’99). ACM Press/Addison-Wesley, New York, 439–446. 
    17. Savransky, G., Dimerman, D., and Gotsman, C. 1999. Modeling and rendering Escher-like impossible scenes. Comput. Graph. Forum 18, 2, 173–179.
    18. Schattschneider, D. and Emmer, M., Eds. 2003. M.C.Escher’s Legacy: A Centennial Celebration. Springer.
    19. Simanek, D. E. 1996. The principles of artistic illusions—Adding depth to illusions. http://www.lhup.edu/~dsimanek/3d/illus2.htm.
    20. Sugihara, K. 1982. Classification of impossible objects. Percept. 11, 65–74.
    21. Sugihara, K. 1986. Machine Interpretation of Line Drawings. The MIT Press. 
    22. Sugihara, K. 1997. Three-Dimensional realization of anomalous pictures–An application of picture interpretation theory to toy design. Patt. Recogn. 30, 7, 1061–1067. 
    23. Sun, J., Jia, J., Tang, C.-K., and Shum, H.-Y. 2004. Poisson matting. ACM Trans. Graph. 23, 3, 315–321. 
    24. Tsuruno, S. 1997. The animation of M.C. Escher’s “Belvedere.” In Proceedings of the ACM SIGGRAPH Visual Conference. 237. 
    25. Uribe, D. 2001. A set of impossible tiles. http://im-possible.info/english/articles/tiles/tiles.html.
    26. Wahba, G. 1990. Spline Models for Observational Data. SIAM.
    27. Wu, T.-P., Tang, C.-K., Brown, M. S., and Shum, H.-Y. 2007. Shapepalettes: Interactive normal transfer via sketching. ACM Trans. Graph. 26, 3, 44. 
    28. Yu, J. and McMillan, L. 2004. A framework for multiperspective rendering. In Proceedings of the 15th Eurographics Workshop on Rendering Techniques (EGSR). 61–68. 

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