Easiest way to do this - as I have done in the past - is to simply name the images according to the category id - then they will automatically be called and you don't have to worry about adding a path for every single one. You'd just use this in place of the category title:
<img src="{$ImagesDir}/cat-images/{$current_category.categoryid}.jpg" alt="{$current_category.category|escape}" />
Then just create a folder called cat-images in the common_files/images folder called 'cat-images' and upload your jpg images (or png if you prefer) there.
Note that this will not be as good from an SEO perspective as real text with an h1 tag around it though.