List categories, gallery-style on front page

On the front page, I want to display my categories in a grid with thumbnails per category, like one would do with a main index with each post showing a thumbnail that would link to the full single post. In this instance the thumbnail would link to that category’s category page.

I don’t believe I can set custom variables for categories, but I might be able to name the thumbnails by category id and have them called that way. But I’m at a loss of how to call a “loop” of category links.