Identifying a pizza with a slice taken out

The title might be slightly misleading, because we will not be talking about pizza slices. But let's say you are in a situation where you have an image containing different types of pizzas with different shapes. Now, somebody has taken a slice out of one of those pizzas. How would we automatically identify this?

We cannot take the approach we took earlier because we don't know what the shape looks like, so we don't have any template. We are not even sure what shape we are looking for, so we cannot build a template based on any prior information. All we know is the fact that a slice has been taken from one of the pizzas. Let's consider the following image:

It's not exactly a real image, but you get the idea. You know what shape we are talking about. Since we don't know what we are looking for, we need to use some of the properties of these shapes to identify the sliced pizza. If you notice, all the other shapes are nicely closed; that is, you can take...