Simon Woods' cool answer is based on two independent discrete Fourier transforms: one for $x$, one for $y$. This answer differs in that it uses only one, but on complex values:. It's it not a very big difference that is a required step for this kind of things.
img = Import["https://i.sstatic.net/wtJoA.png"]; img = Binarize[img~ColorConvert~"Grayscale"~ImageResize~500~Blur~3]; pts = DeleteDuplicates@Cases[Normal@ListContourPlot[Reverse@ImageData[img], Contours -> {0.5}], _Line, -1][[1, 1]]; z = pts[[All, 1]] + I*pts[[All, 2]]; m = 10; n = Length@z; cn = 1/n*Table[Sum[z[[k]]*Exp[-I*i*k*2 Pi/n], {k, 1, n}], {i, -m, m}]; {f[t_], g[t_]} = {Re@#, Im@#} &@ Sum[cn[[i + m + 1]]*Exp[I*i*t], {i, -m, m}] // ComplexExpand; ParametricPlot[{f[t], g[t]}, {t, 0, 2 Pi}] - $m=10$:
- $m=20$:
To make the difference clear: Simon Woods' code with m = 3 gives
{{4341/8 + 1593/8 Sin[37/11 + (44 t)/7] + 58 Sin[1/3 + (88 t)/7], 3225/11 - 867/8 Sin[6/5 - (44 t)/7] + 142/3 Sin[45/23 + (88 t)/7]}} while the above code gives:
{23.2813 - 38.9784 Cos[t] + 16.5986 Cos[2 t] - 17.5251 Cos[3 t] - 195.161 Sin[t] + 55.6546 Sin[2 t] - 15.5714 Sin[3 t], -101.483 - 101.745 Cos[t] + 44.3024 Cos[2 t] - 34.0713 Cos[3 t] + 37.4785 Sin[t] - 16.4218 Sin[2 t] + 5.10139 Sin[3 t]} The base functions are the same for x and y here.
Note If someone manages to compute my cn using Fourier I'd be happy!