Do image backlinks count as backlinks?

Yes, the ALTernate attribute acts as 'anchor text' for links that contain images.

A recent test found both Bing and Google indexed/ranked ALT attribute values: http://www.webmasterworld.com/google/4265397.htm

Also, you need to consider that Google operates functionally like the lowest common browsing denominator (like a Lynx browser, or as one SEO put it 'a blind five year old'), so if you ran a site through the Lynx browser, those ALT attributes are replacements for anchor text.

More recommendations from Google here: http://support.google.com/webmasters/bin/answer.py?hl=en&answer=114016