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Apparent Contours for Projections of Smooth Surfaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2165)

Abstract

Our goal as described in the overview is to provide a complete analysis of the views of natural scenes involving geometric features, shade/shadow, and apparent contours resulting from viewer movement. Our approach to this will involve progressively adding more detailed structure to simpler situations. The starting point for this is the case where we have a single object whose boundary is a smooth surface \(M \subset \mathbb{R}^{3}\) without geometric features. Hence, for the remainder of this chapter we always assume M is a compact smooth surface without boundary.

Keywords

  • Generic Transition
  • Natural Scene
  • View Direction
  • View Projection
  • Infinitesimal Deformation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 3.1
Fig. 3.2

Notes

  1. 1.

    In general a smooth function f whose 4-jet at (0, 0) is f 4(x, y) = L 2 + B + C, where L, B, C are linear, cubic and quartic forms in x, y respectively, has type at least A 3 at x = y = 0 if and only if B = LQ for a quadratic form Q and exactly A 3 provided in addition L does not divide Q 2 − 4C.

  2. 2.

    We are indebted to Jan Koenderink for this example and for the insight that with some care the general case can be deduced from this very special one.

References

  1. V.I. Arnol’d, Indices of singular 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces. Russ. Math. Surv. 34, 1–42 (1979)

    CrossRef  MATH  Google Scholar 

  2. J.W. Bruce, P.J. Giblin, Curves and Singularities, 2nd edn. (Cambridge University Press, Cambridge, 1992)

    CrossRef  MATH  Google Scholar 

  3. R. Cipolla, P. Giblin, Visual Motion of Curves and Surfaces (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  4. T. Gaffney, The structure of \(T\mathcal{A}(f)\), classification and an application to differential geometry. Part I. Proc. Symp. Pure Math. 40, 409–427 (1983)

    MathSciNet  Google Scholar 

  5. P.J. Giblin, Apparent contours: an outline. Proc. R. Soc. Lond. A 356, 1087–1102 (1998)

    MathSciNet  MATH  Google Scholar 

  6. J.J. Koenderink, Solid Shape (MIT Press, Cambridge, 1990)

    Google Scholar 

  7. J.J. Koenderink, A.J. van Doorn, The singularities of the visual mapping. Biol. Cybern. 24, 51–59 (1976)

    CrossRef  MATH  Google Scholar 

  8. H.I. Levine, Singularities of differentiable mappings, in Notes on Bonn Lectures by Rene Thom, Proceedings of Liverpool Singularities Symposium, ed. by C.T.C. Wall. Springer Lecture Notes, vol. 192 (Springer, Berlin, 1970), pp. 1–89

    Google Scholar 

  9. J. Martinet, Deploiements versels des applications differéntiables et classification des applications stables, in Singularités d’Applications Differéntiables. Plans-Sur-Bex, Springer Lecture Notes, vol. 535 (Springer, Berlin, 1975), pp. 1–44

    Google Scholar 

  10. J.N. Mather, Stability of C mappings I: the division theorem. Ann. Math. 87 (1), 89–104 (1968)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. J.N. Mather, Stability of C mappings II: infinitesimal stability implies stability. Ann. Math. 89 (2), 254–291 (1969)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. J.N. Mather, Stability of C mappings III: finitely determined map-germs. Publ. Math. IHES 35, 127–156 (1969)

    CrossRef  MATH  Google Scholar 

  13. J.N. Mather, Stability of C mappings VI: The Nice Dimensions, in Proc. Liverpool Singularities Symposium. Springer Lecture Notes, vol. 192 (1970), pp. 207–253

    Google Scholar 

  14. J.N. Mather, Generic projections. Ann. Math. 98, 226–245 (1973)

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. I.R. Porteous, Geometric Differentiation: For the Intelligence of Curves and Surfaces, 2nd edn. (Cambridge University Press, Cambridge, 2001)

    MATH  Google Scholar 

  16. H. Whitney, On singularities of mappings of Euclidean spaces: I, mappings of the plane into the plane. Ann. Math. 62, 374–410 (1955)

    CrossRef  MathSciNet  MATH  Google Scholar 

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Damon, J., Giblin, P., Haslinger, G. (2016). Apparent Contours for Projections of Smooth Surfaces. In: Local Features in Natural Images via Singularity Theory. Lecture Notes in Mathematics, vol 2165. Springer, Cham. https://doi.org/10.1007/978-3-319-41471-3_3

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