First your image is added:
img =

Looking at the Dimensions you can see it has 3 color channels:

ImageData[img] // Dimensions

Therefore it is converted to Grayscale.

img2 = ColorConvert[img, "Grayscale"]

The graylevel coding is also inverted to match the use of Arrayplot, the array data are retrieved using ImageData

idata = ImageData[ColorNegate[img2]];

dims = idata // Dimensions

ArrayPlot[idata]

Now the array data is reorganized to fit ListPlot3D. First a matrix is created that holds the 2D indices plus a placeholder for the GrayLevel

lst = Outer[{#1, #2, 0} &, Range[First[dims]], Range[Last[dims]]];

Next the GrayLevel data are reconfigured to macth this format in order to be able to add both together

lst2 = Map[{0, 0, #} &, idata, {2}];
lst3 = lst + lst2;

Finally the data is displayed using ListPlot3D

ListPlot3D[Flatten[lst3, 1]]

Using Image the data can be converted back into a image, and a colornegate is required

ColorNegate[Image[idata]]