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Metrics of Curves in Shape Optimization and Analysis

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Level Set and PDE Based Reconstruction Methods in Imaging

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2090))

Abstract

In these lecture notes we will explore the mathematics of the space of immersed curves, as is nowadays used in applications in computer vision. In this field, the space of curves is employed as a “shape space”; for this reason, we will also define and study the space of geometric curves, which are immersed curves up to reparameterizations. To develop the usages of this space, we will consider the space of curves as an infinite dimensional differentiable manifold; we will then deploy an effective form of calculus and analysis, comprising tools such as a Riemannian metric, so as to be able to perform standard operations such as minimizing a goal functional by gradient descent, or computing the distance between two curves. Along this path of mathematics, we will also present some current literature results (and a few examples of different “shape spaces”, including more than only curves).

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Notes

  1. 1.

    We will provide more detailed definitions and properties of the “actions” in Sect. 3.8.

  2. 2.

    Due to Fréchet, 1948; but also attributed to Karcher [26].

  3. 3.

    The precise definition of what the gradient is is in Sect. 3.7.

  4. 4.

    A different gradient descent flow for curve length will be discussed in Remark 10.32.

  5. 5.

    The derivatives are computed in distributional sense, and must exists as Lebesgue integrable functions.

  6. 6.

    The fattened sets are not drawn filled—otherwise they would cover A.

  7. 7.

    That is, any A, B ∈Ξ c can be connected by a Lipschitz arc γ: [0, 1] →Ξ c .

  8. 8.

    “Generically” is meant in the Baire sense: the set of exceptions is of first category.

  9. 9.

    There is a slight abuse of notation here, since in the definition N = T ⊥ given for planar curves in 1.12, we defined N to be a “vector” and not the “vector space orthogonal to T”.

  10. 10.

    We are considering only reparameterizations in Diff+(S 1).

  11. 11.

    It seems that S is Lipschitz-arc-connected, so d g(x, y) < ∞; but we did not carry on a detailed proof.

  12. 12.

    Indeed, the continuous lifting is unique up to addition of a constant to α(s), which is equivalent to a rotation of ξ; and the constant is decided by Φ 1(α) = 2π 2.

  13. 13.

    It is though possible to define Sobolev metrics for any \(j \in \mathbb{R},j > 0\); see Prop. 3.1 in [57].

  14. 14.

    Though, a scale-invariant Sobolev metric is proposed in [39] in Sect. 4.8 as a sensible generalization.

  15. 15.

    The detailed proof has not yet been written…

  16. 16.

    Note this definition is different from (13) in [56].

  17. 17.

    We use the definition (42) of H 0.

  18. 18.

    Though, an alternate method that does not need the tracing of the curve is described in Sect. 5.3 in [56]—but it was not successively used, since it depends on some difficult-to-tune parameters.

  19. 19.

    In a sense, the focus of research in active contours has been mostly on the numerator in (52)—whereas we now focus on the denominator.

  20. 20.

    We use the definition (42) of H 0; the directional derivative was computed in 4.6.

  21. 21.

    H 0 j was defined in (39). Note that H 0 j is a seminorm on T c M, since its value is zero on constant fields; but H 0 j is a norm on D c M, by Poincaré inequality (43).

  22. 22.

    We use the definition (42) of H 0.

  23. 23.

    A.k.a. as the Picard–Lindelöf theorem.

  24. 24.

    Actually, by tracking the first part of the proof in detail, it possible to prove that all other constants \(a_{1},a_{2},a_{3},a_{4},P,Q\) may be bounded in terms of these two quantities I(t), N(t).

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    Andrea C. G. Mennucci

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Mennucci, A.C.G. (2013). Metrics of Curves in Shape Optimization and Analysis. In: Level Set and PDE Based Reconstruction Methods in Imaging. Lecture Notes in Mathematics(), vol 2090. Springer, Cham. https://doi.org/10.1007/978-3-319-01712-9_4

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