How do I deter woodpigeons from fouling the birdbath?

I love woodpigeons and really enjoy seeing them in the garden.  They, along with most of my garden visitors, make a great deal of use of the birdbath.  The problem is that after they finish drinking or bathing they like to perch on the rim looking outwards contemplating life for five minutes or so, with the inevitable consequence!

My worry is that the other birds have to drink polluted water until I can clean out the mess, which I have to do most days.  I hope it doesn't harm them at all.