Legend showing amplitudes of the FFT of an image

The following code performs the FFT manually, rescales it to $[0,1]$ and displays it with the BarLegend you wanted with about 0.2 seconds kernel time and 1s additional display-frontend time needed.

{w, h} = ImageDimensions[img];
fftData = Abs[Fourier[ImageData[ColorConvert[img, "Grayscale"]]*
     Table[(-1)^(i + j), {i, 1, h}, {j, 1, w}]]];
{fftMin, fftMax} = MinMax[fftData];

colorFunc[v_] := Blend[{Black, Green, Yellow, Orange, Red, White}, v];

Legended[Show[{Colorize[
    Image[(fftData - fftMin)/(fftMax - fftMin)*255], 
    ColorFunction -> colorFunc]}, Frame -> True], 
 BarLegend[{colorFunc[#] &, {0, 1}}, 
  LegendLabel -> Style["normalized\namplitude", 14]]]

(Save the graph with the Exportcommand otherwise the legend is not included.)

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 = Abs[fft];

{fftAbsMin, fftAbsMax} = MinMax[fftAbsData];

myColorTable = 
  Flatten@{Table[{Blend[{Blue, Green, Yellow, Orange, Red}, x]}, {x, 1/256, 1, 1/256}]};

g = Colorize[Image[fftAbsData], ColorFunction -> (Blend[myColorTable, #] &)];

myLegend = 
  BarLegend[
    {myColorTable, {fftAbsMin,fftAbsMax}}, 
    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
]

With a modified color table:

myColorTable = 
  Flatten@{Table[{Blend[{Blue, Green, Yellow, Orange}, x]}, {x, 1/50, 
      1, 1/50}], 
    Table[{Blend[{Orange, Red, White}, x]}, {x, 1/206, 1, 1/206}]};

one gets:

The same way the FFT for the phases is:

img = ColorConvert[ExampleData[{"TestImage", "Lena"}], "Grayscale"];

fft = Fourier[ImageData[img]];
fft = RotateLeft[fft, Floor[Dimensions[fft]/2]];

fftPhasesData = Arg@fft/Pi;

{fftPhasesMin, fftPhasesMax} = MinMax[fftPhasesData];

myColorTable = 
  Flatten@{Table[{Blend[{Blue, Green, Yellow, Orange, Red}, x]}, {x, 1/256, 1, 1/256}]};

g = Colorize[Image[(fftPhasesData - fftPhasesMin)/(fftPhasesMax - 
        fftPhasesMin)], ColorFunction -> (Blend[myColorTable, #] &)];

myLegend = 
  BarLegend[
    {myColorTable, {fftPhasesMin, fftPhasesMax}}, 
    LegendLabel -> "phase (\[Pi])", 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
]