Is there any way to get the compression quality from image (JPEG or JP2000) with ArcObjects or any other software?
Unlikely but not impossible. Depends on what the one who stored t has added to the metadata. If you use gdalinfo.exe on it, you might get something like this:
But as you see: The quality setting in % is not part of this IN THE DATA I have … yours might be different. In general gdalinfo is a good utility to read all header info.
Retrieving data ...