Cone Fields and Topological Sampling in Manifolds with Bounded Curvature

Abstract

A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μ∈(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold.

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Fig. 3

Notes

  1. 1.

    θ∈[0,π/2].

  2. 2.

    In particular, for any compact set K and any bound on Hausdorff distance δ>0 there is a compact set L with zero μ-reach such that d H (K,L)<δ.

  3. 3.

    ∥∇ x f∥ is the non-negative number \(\max\{0,\limsup_{y\to x} \frac {f(y)-f(x)}{d(y,x)}\}\). That this is ∥∇ K (x)∥ follows from our geometric construction of ∇ K (x) and from the cosine rule.

Abbreviations

\(\mathcal {M}\) :

is a smooth Riemannian manifold which forms the ambient space.

A :

is a compact subset of \(\mathcal {M}\) which we desire to reconstruct.

S :

is a noisy point cloud sample of A.

δ :

is a bound on the Hausdorff distance between two compacts sets.

\(\mathit{UT}\mathcal {M}\) :

is the unit tangent bundle of \(\mathcal {M}\).

\(T_{x}\mathcal {M}\) :

is the tangent plane to \(\mathcal {M}\) at the point x.

γ :

is a geodesic on \(\mathcal {M}\) (usually unit speed and always constant speed).

x,y,z :

are points in \(\mathcal {M}\).

exp x :

is the exponential map from the tangent plane at x to \(\mathcal {M}\).

Γ(γ):

is the isometry between tangent planes induced by parallel transport along γ.

w,v :

are unit tangent vectors.

β,θ :

are angles. We mainly care about acute angles.

C(w,β):

is a cone. It is a ball in the unit tangent sphere at a point in \(\mathcal {M}\) with center w and radius β.

W :

is a cone field. Also denoted by {(x,C(w x ,β x ))}.

W′:

denotes the complementary cone field to W when W is an acute cone field. For W above it is {(x,C(w x ,π/2−β x ))}.

X :

is a vector field.

K,L :

are compact subsets of \(\mathcal {M}\).

d K :

is the distance function from K.

K a :

is the a-offset of K. That is, \(\{ x\in \mathcal {M}: d_{K}(x)\leq a\}\).

K [a,b] :

is the [a,b] annulus of K. That is, {xM:ad K (x)≤b}.

K :

is the gradient vector field for d K .

References

  1. 1.

    N. Amenta, S. Choi, T.K. Dey, N. Leekha, A simple algorithm for homeomorphic surface reconstruction, in Proceedings of the Sixteenth Annual Symposium on Computational Geometry (ACM, New York, 2000), pp. 213–222.

    Google Scholar 

  2. 2.

    D. Attali, A. Lieutier, D. Salinas, Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes, Comput. Geom. (2012).

  3. 3.

    A. Bissacco, P. Saisan, S. Soatto, Gait recognition using dynamic affine invariants, in Proc. Int’l Symp. Math. Theory of Networks and Systems, Citeseer (2004).

    Google Scholar 

  4. 4.

    G. Carlsson, Topology and data, Bull. Am. Math. Soc. 46(2), 255–308 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    J.M. Chang, M. Kirby, H. Kley, C. Peterson, B. Draper, J. Beveridge, Recognition of digital images of the human face at ultra low resolution via illumination spaces, in Computer Vision—ACCV 2007, (2007), pp. 733–743.

    Google Scholar 

  6. 6.

    F. Chazal, S.Y. Oudot, Towards persistence-based reconstruction in Euclidean spaces, in Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry (ACM, New York, 2008), pp. 232–241.

    Google Scholar 

  7. 7.

    F. Chazal, D. Cohen-Steiner, A. Lieutier, A sampling theory for compact sets in Euclidean space, Discrete Comput. Geom. 41(3), 461–479 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    J. Cheeger, D.G. Ebin, American Mathematical Society, Comparison Theorems in Riemannian Geometry (1975). AMS Chelsea Publishing.

    MATH  Google Scholar 

  9. 9.

    T.K. Dey, K. Li, E.A. Ramos, R. Wenger, Isotopic reconstruction of surfaces with boundaries, in Computer Graphics Forum. Wiley Online Library, vol. 28 (2009), pp. 1371–1382.

    Google Scholar 

  10. 10.

    G. Doretto, A. Chiuso, Y.N. Wu, S. Soatto, Dynamic textures, Int. J. Comput. Vis. 51(2), 91–109 (2003).

    Article  MATH  Google Scholar 

  11. 11.

    H. Edelsbrunner, E.P. Mücke, Three-dimensional alpha shapes, in Proceedings of the 1992 Workshop on Volume Visualization (ACM, New York, 1992), pp. 75–82.

    Google Scholar 

  12. 12.

    J.H.G. Fu, Tubular neighborhoods in Euclidean spaces, Duke Math. J. 52(4), 1025–1046 (1985).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    R. Ghrist, Barcodes: the persistent topology of data, Bull. Am. Math. Soc. 45(1), 61–75 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    K. Grove, Critical point theory for distance functions. in Proc. of Symposia in Pure Mathematics, vol. 54 (1993).

    Google Scholar 

  15. 15.

    M.C. Irwin, Smooth Dynamical Systems. Pure and Applied Mathematics (Academic Press, San Diego, 1980).

    MATH  Google Scholar 

  16. 16.

    J.M. Lee, Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics (Springer, Berlin, 1997).

    MATH  Google Scholar 

  17. 17.

    A. Lieutier, Any open bounded subset of \(\mathbb{R}^{n}\) has the same homotopy type than its medial axis, in Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications (ACM, New York, 2003), pp. 65–75.

    Google Scholar 

  18. 18.

    A. Lytchak, Open map theorem for metric spaces, Algebra Anal. 17(3), 139–159 (2005).

    MathSciNet  Google Scholar 

  19. 19.

    P. Niyogi, S. Smale, S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39(1–3), 419–441 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    V. Patrangenaru, K.V. Mardia, Affine shape analysis and image analysis, in Stochastic Geometry, Biological Structure and Images, Dept. of Statistics, University of Leeds (2003), pp. 56–62.

    Google Scholar 

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Correspondence to Katharine Turner.

Additional information

Communicated by Herbert Edelsbrunner.

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Turner, K. Cone Fields and Topological Sampling in Manifolds with Bounded Curvature. Found Comput Math 13, 913–933 (2013). https://doi.org/10.1007/s10208-013-9176-6

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Keywords

  • Distance function
  • Surface and manifold reconstruction
  • Deformation retraction
  • Fibre bundle

Mathematics Subject Classification

  • 54-C-15
  • 55-S-40
  • 53-B-20