 
Efficient Energies and Algorithms for Parametric Snakes
M. Jacob, T. Blu, M. Unser
IEEE Transactions on Image Processing, vol. 13, no. 9, pp. 12311244, September 2004.
Parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity. However, they have a few drawbacks which limit their performance. In this paper, we identify some of these problems and propose efficient solutions to get around them. The widelyused gradient magnitudebased energy is parameter dependent; its use will negatively affect the parametrization of the curve and, consequently, its stiffness. Hence, we introduce a new edgebased energy that is independent of the parameterization. It is also more robust since it takes into account the gradient direction as well. We express this energy term as a surface integral, thus unifying it naturally with the regionbased schemes. The unified framework enables the user to tune the image energy to the application at hand. We show that parametric snakes can guarantee low curvature curves, but only if they are described in the curvilinear abscissa. Since normal curve evolution do not ensure constant arclength, we propose a new internal energy term that will force this configuration. The curve evolution can sometimes give rise to closed loops in the contour, which will adversely interfere with the optimization algorithm. We propose a curve evolution scheme that prevents this condition.
Errata

Equation (30): the upper bound of the summation should be (N R − 1) instead of (N R).

Equation (35), third line: an increment (dt) should be added at the end of the line.

Equation (36): the upper bound of the summation should be (N R − 1) instead of (N R).

Equation (36): the integer modulus should be taken with respect to (M R) instead of (M).

Equation (37): the lower bound of the summation should be (ℓ = k − N + 1) instead of (ℓ = 0). Meanwhile, the upper bound of the summation should be (k + N − 1) instead of (M − 1).

Equation (45): the lower bound of the inner summation should be (ℓ = k − N + 1) instead of (ℓ = N + 1). Meanwhile, the upper bound of the inner summation should be (k + N − 1) instead of (N − 1).

@ARTICLE(http://bigwww.epfl.ch/publications/jacob0402.html,
AUTHOR="Jacob, M. and Blu, T. and Unser, M.",
TITLE="Efficient Energies and Algorithms for Parametric Snakes",
JOURNAL="{IEEE} Transactions on Image Processing",
YEAR="2004",
volume="13",
number="9",
pages="12311244",
month="September",
note="")
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2004
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