plotting - Count the number of regions made by Lissajous curve

How many disconnected regions does Lissajous curve divide the plane into?

For example: Let

g = ParametricPlot[{Sin[2 t], Sin[t]}, {t, 0, 2 π}]

The number of disconnected regions made by curve g is 3.

Answer

I don't like to think too much :P

Manipulate[
 {#, Composition[
      # - 1 &,

      Length,
      Union,
      Flatten,
      MorphologicalComponents,
      Binarize,
      Rasterize
      ]@#} &@
  ParametricPlot[{Sin[ n t], Sin[m t]}, {t, 0, 2 Pi}, Axes -> False, 
   PlotStyle -> Thick]
 , {n, 2, 10, 1}, {m, 1, 9, 1}]