If it's on your site, you will need to properly configure an SSL certificate in order to use the https protocol. If you have an SSL cert and you need URLs rewritten, then you can change your board's URL to incorporate that, and it should rewrite to https:// for pulling image files and other uploads.
If you are attempting to pull an image from an external site, they will need to have proper SSL setup in order for the security to not get broken by a non-https url.
Once a page attempts to load non-https resources, the security is broken and you no longer have a pure https website.