I understand the initial steps to perform HE of tiles in the image. But can't figure out how to perform the interpolation in the last step to remove the artificial boundaries between the tiles? Can someone explain this to me? It would be really helpful if someone explains it with a figure.
In step 8 of the reference you provide, you obtain 4 possible values for a given pixel (based on the values obtained from its 4 neighbours, which I assume are the usual North-East-South_West neighbours). Then, to avoid introducing tile artifacts, you can use any sufficiently regular interpolation method to obtain the value in the center of these 4 pixels.
For simplicity and computational issues, I suggest that you try bilinear interpolation. It will not introduce ringing or ripple artifacts and will not overflow the dynamic range of the input pixels.