> Probably the best approach to the problem would be to implement an algorithm which allows to generate pixel positions in the original image along the Line without using the FrontEnd

I think `ImageTransformation` doest just that. Using your definitions:

    r = Norm[id]/2;
    profile = 
      First@ImageData@
        ImageTransformation[
         img, center + #[[1]] AngleVector[α] &, {Round[r], 1}, 
         DataRange -> Full, PlotRange -> Full];

    ListLinePlot[profile, PlotRange -> All]

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


  [1]: https://i.sstatic.net/0Vcdr.png