Is increasing the DPI of a low resolution image a good idea? Today after a paper of mine got accepted I received this from the publisher (Elsevier):


  We have received the above article for publication, unfortunately
  figures x1, x2 and x3 are of low resolution and of insufficient quality
  to reproduce well.
  
  The minimum resolution requirements are 1200 dpi for black and white
  figures and 300 dpi for colour figures. If you are converting your
  figures to TIFF files, please ensure that the resolution is correct
  before scanning them.


The figures are digital 512×512 gray-scale images in the PNG format. The source of the images is not me. I had converted them to EPS files using Inkscape. The source of some of the images is: https://github.com/cszn/DnCNN/tree/master/testsets/Set12.

How do I resolve the issue? Should I just save the same images using 1200 DPI in EPS format and send it to the publisher?
The comment you received is aimed at obtaining images of higher quality for publication.
Processing the files you sent them will not solve this issue.
If it did, the typesetters would already have done this themselves – this is what “insufficient quality to reproduce well” hints at.

In your specific case, the images seem to be digitalisations of historic photographs.
Thus, the only way to improve the quality is to obtain higher-resolution digitalisations of these photographs.
If you cannot do this, write exactly this to the publisher; they are most likely going to accept this.
Also note that if those images are test cases for some image-processing algorithm or similar, they should be printed exactly as they are, and this is what you should write the publisher.

The entire remark about the resolution probably doesn’t apply to you but was made for illustrations, plots, and similar.
These are almost always best provided in a vector format (which has infinite resolution, if you so wish) and even if using a pixel format, they can be produced at arbitrary resolution.
The resolution requirements are aimed at those researchers who insist on working with pixels in this case.