Coarea formula

Coarea formula

The coarea formula is a mathematical tool used to compute the integral of the (N −M)-dimensional measure of the level sets of a function by integrating a suitable Jacobian. It is often used in cases where the change of variables formula is applicable and has been established by Federer.From: Carleman estimates for Baouendi–Grushin operators with applications to quantitative uniqueness and strong unique continuation [2022], Nonlinear Analysis [2019]

The Coarea Formula

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Published in Kenneth Kuttler, Modern Analysis, 2017

Kenneth Kuttler

where f : ℝn → ℝm for f a Lipschitz mapping and m ≥ n. It is a version of the change of variables formula for multiple integrals. The coarea formula is a statement about the Hausdorff measure of a set which involves the inverse image of f. It is an extension of Fubini’s theorem. Recall that if n > m and ℝn = ℝm × ℝn − m, we may take a product measurable set, E ⊆ ℝn, and obtain its Lebesgue measure by the formula

Mesh segmentation via geodesic curvature flow

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Published in Computer-Aided Design and Applications, 2018

Zhiyu Sun, Ramy Harik, Stephen Baek

In a level set formulation, the energy functional in Eqn. (3.1) is converted from a line integral to a surface integral on a manifold by the coarea formula [6]: where is a level set formulation of the curve such that the contour of is equal to and being an infinitesimal area defined on the manifold . is the Dirac’s delta function. Consequently, the energy functional in Eqn. (3.2) can further be reduced to the Euler-Lagrange partial differential equation: where is the boundary of and is the outward normal at the boundary. For closed , the boundary condition is ignored automatically.

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The Coarea Formula

Mesh segmentation via geodesic curvature flow