No, because for one, even if that were possible, your computer would have to be running 24/7.
Other than that, the contents of your hard drive is not readily accessible from the outside (for obvious reasons).
You could share a photo on your computer if you
a) install a web server
b) use a service like noip.com to get a domain to point to your computer
c) set up your router to forward port 80 requests to your computer
As you can see, this is a huge hassle compared to simply uploading your photo to imgur.com
can you share a picture using a file path from your computer? not a web url
