I guess I am confused.
If you were making backups, shouldn't your original files be right where they were before the crash?
As I understand it, your backup was interrupted by the crash and some of the originals were lost before they could be backed up. Correct?
You're probably looking at the Lightroom previews and asking if you can salvage the images that were lost from the previews. I'm not a Lightroom expert, so perhaps someone else can chime in here. Just wanted to clarify what was being asked.
(You should be able to extract the previews somehow, but I imagine they're low resolution versions of the originals).
In the future, be sure to never remove the images from your memory card before ensuring that they're in two locations elsewhere (e.g. on your computer and backed up externally). Good luck with recovery!
The images are not in LR. The images resided on your hard drive. If the Hard Drive crashed and is no longer accessible you may have to utilize specialized software or services to recover your data. If your computer crashed and a reboot restored everything to normal LR may have lost the association between its database and the files in the folders. You might try reestablishing this link. You can also create a link to the files that were backed up. A little more information on your crash would be helpful to determine the next steps.
For the future what I would suggest is to have LR create backups to an external drive during import. It takes a little longer but IMO it is well worth it. Here is what I do once a week or more often if the files are critical. I copy all the files on my external backup to another external backup drive and take it to work with me and store it in a cabinet. This way if there is lightning of some other calamity I have an off site copy of all my data.