I have a function like $f(s,t,x,y)$ I want to numerically find the region defined by: $$\lbrace(x,y)\in D\ |f(s,t,x,y)>f_0 ,\, \forall (s,t)\in D\rbrace$$
I can check for all $(x,y)$ whether for all pairs $(s,t)$ the relation $f(s,t,x,y)>f_0$ is satisfied and if so add the pair $(x,y)$ to the region. But this approach requires checking all of the possible combinations. I need a faster way that gives me an estimate of the boundary of such a region or the biggest convex hull that fits inside this region.
Here is an example of the regions that I'm dealing with:
Here $D$ is a square region. The above shape is achieved by an exhaustive search on the pixels of the square. I think this is like using exhaustive search for continuous optimization while a more efficient method, i.e. gradient descent, exists. Is there any iterative algorithm available for boundary (locus) estimation?