Parametric Manifold of an Object under Different Viewing Directions

  • Xiaozheng Zhang
  • Yongsheng Gao
  • Terry Caelli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7576)


The appearance of a 3D object depends on both the viewing directions and illumination conditions. It is proven that all n-pixel images of a convex object with Lambertian surface under variable lighting from infinity form a convex polyhedral cone (called illumination cone) in n-dimensional space. This paper tries to answer the other half of the question: What is the set of images of an object under all viewing directions? A novel image representation is proposed, which transforms any n-pixel image of a 3D object to a vector in a 2n-dimensional pose space. In such a pose space, we prove that the transformed images of a 3D object under all viewing directions form a parametric manifold in a 6-dimensional linear subspace. With in-depth rotations along a single axis in particular, this manifold is an ellipse. Furthermore, we show that this parametric pose manifold of a convex object can be estimated from a few images in different poses and used to predict object’s appearances under unseen viewing directions. These results immediately suggest a number of approaches to object recognition, scene detection, and 3D modelling. Experiments on both synthetic data and real images were reported, which demonstrates the validity of the proposed representation.


pose manifold 3D object in-depth rotations viewing directions appearance prediction object rendering 


  1. 1.
    Basri, R., Jacobs, D.W.: Lambertian reflectance and linear subspaces. IEEE Trans. Pattern Anal. Mach. Intell. 25(2), 218–233 (2003)CrossRefGoogle Scholar
  2. 2.
    Belhumeur, P.N., Kriegman, D.J.: What is the set of images of an object under all possible illumination conditions. Int. J. Comput. Vis. 28(3), 245–260 (1998)CrossRefGoogle Scholar
  3. 3.
    Blanz, V., Vetter, T.: Face recognition based on fitting a 3D morphable model. IEEE Trans. Pattern Anal. Mach. Intell. 25(9), 1063–1074 (2003)CrossRefGoogle Scholar
  4. 4.
    Blanz, V., Grother, P., Phillips, P.J., Vetter, T.: Face recognition based on frontal views generated from non-frontal images. In: Proc. IEEE Conf. CVPR (2005)Google Scholar
  5. 5.
    Bookstein, F.L.: Principal warps: thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 11(6), 567–585 (1989)zbMATHCrossRefGoogle Scholar
  6. 6.
    Caglioti, V.: On the space requirements of indexing 3D models from 2D perspective images. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 718–723 (2000)Google Scholar
  7. 7.
    Cootes, T.F., Edwards, G.J., Taylor, C.J.: Active appearance models. IEEE Trans. Pattern Anal. Mach. Intell. 23(6), 681–685 (2001)CrossRefGoogle Scholar
  8. 8.
    Fermuller, C., Aloimonos, Y.: On the Geometry of Visual Correspondence. Int. J. Comput. Vis. 21(3), 223–247 (1997)CrossRefGoogle Scholar
  9. 9.
    Gross, R., Matthews, I., Baker, S.: Appearance-based face recognition and light-fields. IEEE Trans. Pattern Anal. Mach. Intell. 26(4), 449–465 (2004)CrossRefGoogle Scholar
  10. 10.
    Jacobs, D.: The space requirements of indexing under perspective projections. IEEE Trans. Pattern Anal. Mach. Intell. 18(3) (1996)Google Scholar
  11. 11.
    Prince, S.J.D., Warrell, J., Elder, J.H., Felisberti, F.M.: Tied Factor Analysis for Face Recognition across Large Pose Differences. IEEE Trans. Pattern Anal. Mach. Intell. 30(6), 970–984 (2008)CrossRefGoogle Scholar
  12. 12.
    Shashua, A.: Geometry and Photometry in 3D Visual Recognition. Ph.D. MIT (1992)Google Scholar
  13. 13.
    Shashua, A.: On photometric issues in 3D visual recognition from a single 2D image. Int. J. Comput. Vis. 21(1-2), 99–122 (1997)CrossRefGoogle Scholar
  14. 14.
    Sim, T., Baker, S., Bsat, M.: The CMU pose, illumination, and expression database. IEEE Trans. Pattern Anal. Mach. Intell. 25(12), 1615–1618 (2003)CrossRefGoogle Scholar
  15. 15.
    Ullman, S., Basri, R.: Recognition by Linear Combinations of Models. IEEE Trans. Pattern Anal. Mach. Intell. 13(10), 992–1006 (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xiaozheng Zhang
    • 1
    • 2
  • Yongsheng Gao
    • 1
    • 2
  • Terry Caelli
    • 3
  1. 1.Biosecurity Group, Queensland Research LaboratoryNational ICT AustraliaAustralia
  2. 2.Computer Vision and Image Processing LabGriffith UniversityBrisbaneAustralia
  3. 3.Victoria Research LaboratoryNational ICT AustraliaAustralia

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