Consider a practical example where the 2D curve—could be a sequence of points—denotes the measured border of a certain water body. The measurement contains random noise and must be smoothed for certain use (for example, hydrodynamic simulation). We don’t want the area to be affected by the smoothing. The question is—which smoothing method should be used?
Below is an example of low-pass smoothing, the area does appear to be conserved.
I read from this old 1995 paper [https://graphics.stanford.edu/courses/cs468-01-fall/Papers/taubin-smoothing.pdf] which states "the method of Fourier descriptors does not produce shrinkage...", but couldn’t figure out why.
I know that with proper parameterization, the coordinates of a closed curve (in any dimension, including 2D) could be considered as a $2π$-periodic function. For example, we have $(a \cos t,b \sin t )$ for an ellipse. I also know that the area enclosed by a 2D curve $(x(t),y(t))$ could be found via Green's formula $$A={1 \over 2} \oint_0^{2\pi} x y'-x' y \, \mathrm{d}t.$$ But it seems the area formula could not be analyzed in the Fourier space, so I post the question here expecting some Fourier guru to help out. Thanks!