data1  =  
      Table[Sign[-0.8 + Sin[20*2*Pi*i/128]*Sin[20*2*Pi*j/128]], 
         {i,  0,  128},  {j,  0,  128}];

ListDensityPlot[data1,  Mesh  ->  False];

data2 = Abs[Fourier[data1]];

data3 = 
      Transpose[
        Join[data2[[Range[65, 129],  Range[1, 129]]], 
          data2[[Range[1, 64],  Range[1, 129]]]]];

data4 = 
      Transpose[
        Join[data3[[Range[65, 129],  Range[1, 129]]], 
          data3[[Range[1, 64],  Range[1, 129]]]]];

ListDensityPlot[data4,  PlotRange  ->  {0, 4},  
      Mesh  ->  False];