Count the number of regions made by Lissajous curve

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}]

Graphics`Mesh`MeshInit[];
eps = 1/100000000;
Manipulate[Labeled[plt = ParametricPlot[{Sin[n t], Sin[(n - 1) t]}, {t, eps, 2 Pi},
    Axes -> False, PlotStyle -> Thick]; plt /. Line -> Polygon, 
  Grid[{{"n", "p"}, {n, 2 + Length@Graphics`Mesh`FindIntersections[plt]}}, 
       ItemStyle -> Directive[16, "Panel"]], Top], {n, 2, 10, 1}]