**How to add a legend showing the amplitudes instead of brightness values:**

With the help of Julien Kluges answer I have the following code, which is also showing the "frequency" along the axes: 

    img = ColorConvert[ExampleData[{"TestImage", "Lena"}], "Grayscale"];
       
    fft = Fourier[ImageData[img]];
    fft = RotateLeft[fft, Floor[Dimensions[fft]/2]];
        
    fftAbsData = Log[Abs[fft]] - Min[Log[Abs[fft]]];
        
    {fftAbsMin, fftAbsMax} = MinMax[fftAbsData];
    
    colorFunc[v_] := Blend[{Blue, Green, Yellow, Orange, Red}, v];
    
    g = Show[Colorize[
        Image[(fftAbsData - fftAbsMin)/(fftAbsMax - fftAbsMin)], 
        ColorFunction -> colorFunc]];
    
    myColorTable = 
      Flatten@{Table[{Blend[{Blue, Green, Yellow, Orange, Red}, x]}, {x, 
          1/256, 1, 1/256}]};
    
    myLegend = 
      BarLegend[{myColorTable, {Exp[fftAbsMin + Min[Log[Abs[fft]]]], 
         Exp[fftAbsMax + Min[Log[Abs[fft]]]]}}, 
       LegendLabel -> "amplitude", LegendMarkerSize -> {40, 300}, 
       LabelStyle -> {FontFamily -> "Calibri", FontSize -> 15}];
    
    Legended[Graphics[Inset[g, Scaled[{.5, .5}], Automatic, Scaled[1]], 
      Frame -> True, 
      FrameLabel -> {{"cycles/pixel", ""}, {"cycles/pixel", ""}}, 
      PlotRange -> {{-0.5, 0.5}, {-0.5, 0.5}}, 
      AspectRatio -> ImageAspectRatio@g], myLegend]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/WLm1L.png