The Ovuscule

P. Thévenaz, R. Delgado-Gonzalo, M. Unser

IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 2, pp. 382–393, February 2011.

Download PDFDownload PSDownload Global BibTeX

We propose an active contour (a.k.a. snake) that takes the shape of an ellipse. Its evolution is driven by surface terms made of two contributions: the integral of the data over an inner ellipse, counterbalanced by the integral of the data over an outer elliptical shell. We iteratively adapt the active contour to maximize the contrast between the two domains, which results in a snake that seeks elliptical bright blobs. We provide analytic expressions for the gradient of the snake with respect to its defining parameters, which allows for the use of efficient optimizers. An important contribution here is the parameterization of the ellipse which we define in such a way that all parameters have equal importance; this creates a favorable landscape for the proceedings of the optimizer. We validate our construct with synthetic data and illustrate its use on real data as well.

The associated software is available here.

Erratum

• p. 386, first column, Equation (12), some bounds of the integrals are incorrect. The corrected equation should read J_D = (ƒ₀ ∕ (π R²)) ∫_(−π)^(π) (∫₍₀₎^{(R ∕ √2 − 1 ∕ √2)} (−1) r dr + ∫_{(R ∕ √2 − 1 ∕ √2)}^{(R ∕ √2 + 1 ∕ √2)} √2 (r − R ∕ √2) r dr + ∫_{(R ∕ √2 + 1 ∕ √2)}^{(R − 1)} (1) r dr + ∫_{(R − 1)}^{(R + 1)} (1 ∕ 2) (1 − r + R) r dr) dϑ = 0.

• p. 392, Appendix D, The vector a is incorrect. Its corrected components should be a₁ = −sgn(e₁₂) (a² e₂₂ − α²)^½ and a₂ = (a² e₁₁ − α²)^½, where a is defined in Appendix E.

@ARTICLE(http://bigwww.epfl.ch/publications/thevenaz1101.html,
AUTHOR="Th{\'{e}}venaz, P. and Delgado-Gonzalo, R. and Unser, M.",
TITLE="The Ovuscule",
JOURNAL="{IEEE} Transactions on Pattern Analysis and Machine
	Intelligence",
YEAR="2010",
volume="33",
number="2",
pages="382--393",
month="February",
note="")

DOI: 10.1109/TPAMI.2010.112