Say we have a continuous function $u(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$.
I have seen several textbooks that make the following assertion:
The length/area element of the zero level set of $u$ is given by $\lvert\nabla H\left(u\right)\rvert = \delta(u)\lvert\lvert\nabla u\rvert\rvert$, where $H\left(u\right)$ is the Heaviside step function, $\delta(u)= \partial H(u) / \partial u $ is the Dirac delta function.
We can measure the length of the zero level set as $ \int\int \lvert \nabla H(x,y)\rvert dx dy = \int\int \delta (u(x,y))\lvert \nabla u (x,y)\rvert dx dy$
I fail to see this. Why is the length (or/and area) element of the zero level set of $u$ given by $\lvert\nabla H\left(u\right)\rvert $ or $\delta(u)\lvert\lvert\nabla u\rvert\rvert$?
Some papers and textbooks that make this assertion:
- Zhao, H.K. et al., 1996. A Variational Level Set Approach to Multiphase Motion (available online). Journal of Computational Physics, 127(1), p.179-195 (page 2, Equations 2.2a and 2.2b)
Geometric Partial Differential Equations and Image Analysis. Guillermo Sapiro. Cambridge University Press 2001 (e.g. page 92).
Image Processing and Analysis. Variational, PDE, Wavelet, and stochastic methods. Tony Chan and Jianhong Shen. SIAM 2005 (e.g. page 46)