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I have this image

colormaped image

that I want to turn into data using this gradient

gradient

black being 0, white being 1, and all the colors in between accordingly, so I could later use ListPlot3D to make a height map.

I tried using ImageData, to work something out using RGB values of each color point, but in the end the purple was higher than the blue, so that didn't workout.

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Import data, take only one column from the legend, and extract ImageData from the legend:

img = Import["https://i.stack.imgur.com/1JxTr.png"];
leg = ImageTake[Import["https://i.stack.imgur.com/MRqiH.png"], All, {1, 1}];
legdata = ImageData[leg, DataReversed -> True][[All, 1, 1 ;; 3]];

Connect the RGB values to a number in $[0,1]$:

rules = Rule @@@ Transpose[{legdata, Subdivide[Length@legdata - 1]}];

Extract ImageData info from the image:

imgpoints = ImageData[img][[All, All, 1 ;; 3]];

Employ Nearest to find the best approximation of the color in img using colors available from leg:

pts = Table[
    Nearest[legdata, imgpoints[[i, j]]][[1]], {i, 1, 
     Length@imgpoints}, {j, 1, Length@imgpoints[[i]]}] /. rules;

Draw points:

ListPlot3D[pts, ColorFunction -> "Rainbow", BoxRatios -> {1, 1, 1}]

enter image description here

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