Output png file with exact same pixels as image plot, and which also contains figure axes.

I created a figure using the image function because I needed to access the individual pixels to color them directly. To the image was added axes and axes labels. I now want to output a png file that is true to the original pixels in the image, but contain the addition of the axes. I've tried print (using paper position), imwrite (problem keeping axis), getframe/frame2im/imwrite (lose resolution), and though I can get approximately correct results, I can't get exactly the same image down to the pixel resolution (seen when you zoom into the png file). Any ideas would be appreciated.

FYI, am trying to reproduce this plot using MATLAB

https://upload.wikimedia.org/wikipedia/commons/5/50/Logistic_Bifurcation_map_High_Resolution.png