concave

i'm trying to get around the rule of only being able to form convex shapes in the SFML c++ library. To do this I'm planning on testing given vertices, and if concave, splitting the vertices into groups, testing each groups' concaveness, and repeating until a full set of concave shapes results that look just like the original shape when put together What I would like to know is... What the equation for testing a shapes concaveness is: What is it and how does it work? How would i split up the vertices of the concave shape so in the end the shape is formed out of as few convex shapes as possible?

I have a set of discrete points shown in an image, like the following I want to reconstruct or up sampling (I'm not sure what's the correct way to describe it) the image, so that the result image would be like the following . It doesn't need to be exactly the same as the example image, but the main idea is to fill up the original one. I have an initial idea about how to do it. But I don't know how to do it after the first step. My idea is to first separate image using kmeans and find out the different objects. And I have successfully done it. The resulting images after kmeans are: . After