Ultimately, Google Analytics does exactly the same thing, however, the image URL is created using Javascript.
If you really, really wanted to use Google Analytics for your site, (albeit with limited functionality) you might take a look at Troubleshooting The Tracking Code to see how the image URL works - here's the bare minimum to push data to Google:
http://www.google-analytics.com/__utm.gif?
utmhn=<hostname>&
utmac=<google acct id>
So if your hostname were "mydomain.com" and your Google account ID were "UA-123ABC-99" then you could use the following image URL on your site:
http://www.google-analytics.com/__utm.gif?utmhn=mydomain.com&utmac=UA-123ABC-99
This limited implementation should allow you to track per-page statistics, unique/returning visitors, and more (you'll be missing any Javascript-enabled features, though - including event tracking).