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Currents and Finite Elements as Tools for Shape Space

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Abstract

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper, we study a general representation of shapes as currents, which are based on linear spaces and are suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the \(H^{-s}\) norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element-based discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples.

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Notes

  1. This also happens to be Euclidean-invariant, but this is not relevant to the sequel.

  2. For example, when \(S^1\) acts on \({\mathbb C}^n\) by \(z_i \mapsto e^{i\theta } z_i\), the set of invariants \({\bar{z}}_i z_j\), \(1\le i,j\le n\)—\(n^2\) real invariants in all—is complete, and \(n^2\) is much larger than \(\mathrm {dim}({\mathbb C}^n/S^1)=2n-1\). One can find smaller complete sets, limited by the dimension of the smallest Euclidean space into which \({\mathbb C}^n/S^1\) can be embedded. However, in such sets the individual invariants are more complicated [4], so that the total complexity is \(\mathcal {O}(n^2)\).

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Authors and Affiliations

  1. Palmerston North, New Zealand

    James Benn

  2. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Stephen Marsland

  3. Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

    Robert I. McLachlan

  4. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

    Klas Modin

  5. Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway

    Olivier Verdier

  6. Department of Mathematics, KTH, Stockholm, Sweden

    Olivier Verdier

Authors
  1. James Benn
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  2. Stephen Marsland
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  3. Robert I. McLachlan
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  4. Klas Modin
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  5. Olivier Verdier
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Correspondence to Stephen Marsland.

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Benn, J., Marsland, S., McLachlan, R.I. et al. Currents and Finite Elements as Tools for Shape Space. J Math Imaging Vis 61, 1197–1220 (2019). https://doi.org/10.1007/s10851-019-00896-x

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