How to make a plot over some external figure?

Question

Consider the following figure:

I would like to plot a test function

func[ma_]=1000*ma^5

over it. Is it possible somehow to do?

Answer 1

You can use Prolog and Inset to position the image behind the plot, but some calibration is needed to get correct coordinates.

First, calibrate the image by determining the location of the bottom-left and top-right corners of the plot.

img = Import["https://i.stack.imgur.com/rUBAY.png", "Image", 
 ImageResolution -> 72];

imageData = ImageData[img, DataReversed -> True];
Manipulate[
 Graphics[{Raster[imageData, {{0, 0}, ImageDimensions[img]}]}, 
  ImageSize -> 500], {{bottomLeft, {0, 0}}}, {{topRight, {300, 
    300}}}, {{bottomLeft, {0, 0}}, Locator}, {{topRight, {300, 300}}, 
  Locator}]

Then assign the corresponding $x$ and $y$ range*, and use this data to calculate the correct aspect ratio and image scaling.

bottomLeft = {182, 151};
topRight = {1687, 966};

{xRange, yRange} = {{0, 3}, {-3, 7}};

aspectRatio = 1/Divide @@ (topRight - bottomLeft);
scaling = Abs[Subtract @@ xRange]*
   First@ImageDimensions[img]/First@(topRight - bottomLeft);

Finally, use Plot* to draw your function. Make sure the axes are properly aligned, then use Frame -> False and remove SetAlphaChannel to get the final image.

func[ma_] := 1000 * ma^5;

Plot[Log10[func[ma]], {ma, xRange[[1]], 
  xRange[[2]]}, {PlotRange -> {xRange, yRange}, 
  PlotStyle -> {Thick, Red}, 
  Prolog -> {Inset[
     SetAlphaChannel[img, .3], {xRange[[1]], yRange[[1]]}, bottomLeft,
      scaling]}, AxesOrigin -> {0, -3}, AspectRatio -> aspectRatio, 
  Axes -> False, ImageSize -> 500, Axes -> True, ImagePadding -> 50, 
  PlotRangeClipping -> False, Frame -> True}]

---

* Due to plot coordinate system, it is more convenient to use Plot[Log10[ ...]] instead of LogPlot[].